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MIT_5111_Principles_of_Chemical_Science_Fall_2014

15_Thermodynamics_Bond_and_Reaction_Enthalpies.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at osw.mit.edu. CATHERINE DRENNAN: Next hand out-- thermodynamics. Yes. Yes. I love thermodynamics. All right. So what is thermodynamics? So thermodynamics and kinetics I feel go together, but for kind of weird reasons we do thermodynamics now and we do kinetics at the very last unit of the semester. Part of the reason for this is that kinetics is often a unit that students can pick up really fast, and so I like doing it at the end when everything-- your world is sort of crazy and you have something that you can get a grasp on pretty easily for the last unit. So anyway, I'll tell you a little bit about kinetics now because we won't get to a lot of it until later. So thermodynamics deals with energy change and spontaneity of reactions. And thermodynamics brings you three of my favorite things in chemistry, which are [MUSIC PLAYING] delta H, [CYMBAL CRASH] Delta S, and [DRUM ROLL] delta G. I love these. I live my life around these things. I believe entropy should always be increasing. And I love nothing more than free energy. That's great stuff. So today we're going to talk about delta H. Next week we have delta S and delta G. What about kinetics? What does kinetics bring us? Well, kinetics brings us the rate or speed of a reaction. It can bring us fast reactions and it can bring us slow reactions. I like kinetics too. I kind of like the fast to be honest with you. All right. So thermodynamics and kinetics. One. thermodynamics tells us whether something is going to happen spontaneously or not, but kinetics tells us the rate at which it happens. So let's just think of an example for a minute. You may have heard-- this commercial happens a lot around Valentine's Day. Diamonds are forever. So actually, thermodynamically, graphite is favorable to diamonds. It's more stable. So graphite or coal-- thermodynamically, this is the stuff. Diamonds-- diamond is not forever. That's really a kinetic statement. It's there for a very long time, but you know it isn't inert. So here's an important question. What is the best ring for one geek to give another geek? The thermodynamically stable one or one that is more kinetically slow to react, more stable, more inert. Inert is reaction. What do you-- I should have a clicker question on this, I know. I don't know what you think. But actually, the answer is, in my opinion, neither of these. Come on. There's only really one ring that any geek really wants. Green Lantern's ring has the power of chemistry at your fingertips. Who cares if something's inert? If you're the Green Lantern you can do whatever. So that's the ring. Anyway-- AUDIENCE: What about the one ring? CATHERINE DRENNAN: Oh, one ring. Yeah. I like the Green Lantern ring, but I guess next-- maybe we should open a blog on this. It's a really important question. So let's think about bonding for a minute. So thermodynamics is telling us about energy change. It's telling us about spontaneity. And so we need to think about energy that's going in. So kinetics is telling us about how fast. Thermodynamics is really telling us about stability. How stable is something? How much does it cost to break it apart? So we're back to bond association energies. We have the change in the dissociation energy E to the little d. It's the energy to break a bond. We've seen this plot before. Now we have methane. We're breaking off a hydrogen from it, which is actually very hard to do. Scientists would love to break apart methane and make methanol, but it's a hard thing to do. Up here we have unfavorable reactions. When the atoms are too close, then you have a sweet spot where the positions of the atoms are just right to form a bond. That's when you get methane. And then, if you put in energy, you can pull this off, is they go far apart with the radius this way. Your bond will dissociate. You always have to put energy in to get it to dissociate. So now we can think about it-- we thought about this in terms of bond dissociation. We've seen this before, but now we can think about it in terms of a new term, which is delta H B, or bond enthalpy. So bond enthalpy is the change in heat accompanying the dissociation of a bond and that's measured at a constant pressure. In fact, if we relate delta H to Delta E-- delta H equals delta E plus whatever change in pressure or volume. And often, this term is pretty small, so people often really think about reactions in terms of these being pretty similar to each other. So for gases, the difference really is 1% to 2%. And if you're talking about a liquid or a solid it's really a negligible difference. So we often really kind of think about these things in the same way. We think about the energy going into a system to break the bond, or we think about the bond enthalpies. And bond enthalpy, delta H, is often easier to measure, so it's very convenient. So again, bond enthalpies You always have to put energy in if you're going to break a bond. So it's always going to be positive. It always takes heat. It always takes something to break a bond. And so breaking a bond is endothermic. Heat must be added. Whereas bond formation is exothermic. Heat is being released. And so we can think about this-- again, when you form a bond those-- we saw with MO theory-- there's more electrons in lower energy in the bonding orbitals than in the antibonding orbitals. They're happy. This is a lower energy state. So if you're going to break that bond, breaking up is hard to do. And there's a song that verifies that statement. Breaking up is hard to do. You always have to have heat. But when you go from that stable stay out-- when you form those bonds, it's like kind of the married couple, often get a little boring. They're in a low energy state. It's hard to get them out of the house and so they release all of their energy and they form this nice, little, happy, stable couple. So when you do bond formation that's an exothermic process. So we can talk about standard bond enthalpies. When you see this little circle up there that means it's a value that's at standard conditions, where your reactants and products are in their standard states. And so we can think about what are some delta H or some bond enthalpies for different kinds of carbon hydrogen bonds. So here's our friend methane. If you're going to pull off a hydrogen you have to put energy in to do that, and the bond enthalpy for that is 438. Now we can think about some other kinds of carbon bonds. We talked about this one. We have it here in the classroom. So if we pull off a hydrogen from that, it's plus 410. So similar but not the same. If you now substitute three flourines for three of the hydrogens that changes your value a little bit, but not much. If you substitute chlorine it changes it a lot more. Same with bromine. So it depends. The bond enthalpy depends on what else is around that atom that you're pulling around. So it's not always the same value. It's different. It depends on what else is there. So often you'll have a table that will report mean bond enthalpies. And they take all the bonds enthalpies and they get the mean value, and they're usually within 8% of each other. And for a carbon hydrogen bond it's around 412 is the mean bond enthalpy. But when you're using mean bond enthalpies to calculate something, you have to realize that there can be some pretty big differences depending on what's around that bond that you're going to break. So those are some bond enthalpies. Why are they important? Well, they're important because the difference in bond enthalpies between a product and a reactant can tell you about the enthalpy of that reaction. The enthalpy of the reaction of breaking of bonds and forming new bonds, or the enthalpy of reaction, which is delta H R-- sub R. And that is in the standard state in that case. So let's talk about enthalpies of reactions. So we have this symbol again. Standard bond enthalpy for a reaction. So if it's a little B it's a bond enthalpy. If it's a little R it's a reaction enthalpy. If it's negative value it means it's an exothermic reaction, and if it's a positive value it means that it's an endothermic reaction. So we'll use these terms a lot and you'll get very familiar with them. So let's look at some examples of reactions. And here is one of my favorites, and it is yes. [MUSIC PLAYING] It has it's own song. - (SINGING) Photosynthesis Aah. Photosynthesis. Aah. Photosynthesis. Aah. Photosynthesis does not involve a camera or a synthesizer, although that's interesting too. Photosynthesis is how the plants take in light from the sun and turn it into energy. It's actually a thing on which most life depends here on the planet Earth. Photosynthesis. Aah. CATHERINE DRENNAN: OK. That gives you an idea. Unfortunately, every time you will hear the word photosynthesis you'll go Ah. It happens. I'm sorry about that. So photosynthesis. Amazing reaction. People right now are trying to duplicate it in industry to solve the energy problem. Good luck with that. But I know. I wish them good luck. That would be awesome. We use the opposite of that reaction for our energy. So we take sugar and use oxygen to break it down, which is an awesome thing because this has a really negative enthalpy of reaction minus 2816 kilojoules per mole. It's huge. And we store this in something called ATP. So since this is-- and I'm going to need the help of the TAs for a minute because we're going to do a very quick demo at the end of today's class. This reaction is exothermic big time. It's a big negative, which raises the question, if it's that exothermic-- really big value. We have sugar in air. Why we should feel heat? Heat should be released. So I think we should do this demo now and see whether that's true. So I have a bag of sugar and it is sealed under nitrogen so there's no oxygen in there. And I forgot my safety glasses but I'll try to-- sorry about the front room. I should have had a stellar announcement that you might want to sit back. But I'm going to cut this open and let O2 in. So there should be a lot of heat coming out. AUDIENCE: [INAUDIBLE] the things inside it individually wrapped? CATHERINE DRENNAN: Oh, you know what? They are individually wrapped. All right, so this is not going to work. So I need the TAs to come down here, please, and you've got to help me unwrap them. Has anyone done the experiment yet? Do you feel heat coming out? AUDIENCE: Yeah. CATHERINE DRENNAN: You do? [LAUGHTER] All right. I better try it up here. Let's see. I'm going or unwrap mine. It's not working very well. So it turns out that heat should be released but this is very slow. So we don't feel the heat when we unwrap our Hershey's Kisses. I encourage everyone to try this experiment at least once. But the way that we harness this energy in our bodies is that we have catalysts, which are enzymes, that speed up the reaction. And that's how we get the full force of this reaction out. So that is actually our introduction to thermodynamics. And next time we're going to talk about how we're going to calculate these delta Hr, these heats of reaction. So we were talking about delta H, and so we want to pull out the handouts from last time. And we were at the bottom of page two with three different ways to calculate delta H. So our delta H of reaction, delta Hr, the reaction enthalpy. So I introduced you to bond enthalpies, and today we're going to look at how you use bond enthalpies to calculate reaction enthalpies. And remember, bond enthalpies-- sometimes it has nothing. Just delta H. Sometimes it's delta H sub B. Capital B for bond. And we're going to look at that. Then we're going to look at how you can calculate delta H for reaction from the standard enthalpies of formation, and I'll introduce you to what that means. And then, also tell you about Hess's law where you can combine known reactions that have known delta H's to get a new equation and calculate a new delta H for that reaction. So three different ways. So we're going to start with way one, which is bond enthalpies. So here is the equation for calculating bond enthalpies. So we have the delta H0 of the reaction equals the sum of all of the reactants bond enthalpies minus the sum of all the product bond enthalpies. And so this is bonds broken minus bonds formed. And so let's think about this for a minute and think about what would be true. If you had stronger bonds in the products than in the reactants, what would be true? And this is a clicker question. All right. Let's just take 10 more seconds. OK Yep. So now let's think about why this is true. So it's good news that most of you know that negative means exothermic and positive means endothermic. And let's look at why this is true. So let's look at both of these. So if we have bonds stronger in the products-- you can just think about the equation. So if you are bond stronger in the products this is a bigger number and that's the smaller number, which is going to give you a negative answer. And a negative value is exothermic. And you can think about the equation stronger bonds here. A bigger number minus the smaller number. Positive or endothermic. But let's think for a minute about why this is the case and rationalize it because on an exam this is one of the equations that you're not given, so let's help you remember why this would be true. We can think about this-- if you're going to break bonds-- and this isn't in your notes but people get confused by this, so I'm just going to write a little bit on the board. So if you're going to break bonds you need to put energy into the system to break bonds. And we talked about this before. And since we have exam two coming up we'll just do a little review of some of the things that might be on the exam. So you don't have this in the handout we're doing right now, but you had this in the lecture nine handout. And something like this might be on the exam so we should be thinking about it. So remember, if there was no energy that you needed to put in to break a bond-- if breaking the bond required no energy there would be no bond. So when the energy is zero there's no bonds. These two are not-- these things are not bonded together. And when you do form a bond, you go down an energy here so it's at a lower state. It's more stable. That's why it forms a bond. If it was less stable it wouldn't be forming a bond. But if it's more stable, lower in energy, a larger negative number, then a bond forms. So to break this bond you have to put energy into the system. So breaking bonds always involves energy in. But forming bonds-- so if we're forming bonds then we're going to have energy out. So we're at a lower place here. So if we want to break bonds we have to put energy in, but if we're forming bonds then we're going to have energy that these guys had that is going to be released somewhere. So energy goes out of the system. And the farther down we have the stronger bonds, the more energy you have to put in to break. But also, the more energy that comes out when the bonds form. So energy in to break a bond, but when a bond is forming it goes to a lower state and that energy is released. So now we can think about what happens if you have a reactant with weak bonds. So if the reactant then has weak bonds, how much energy do you have to put in if it has weak bonds to break them? Not a lot. So we have just sort of a little bit of energy in. Little energy in to break those bonds. Now in the products, if we have strong bonds, how much energy goes out if we're forming strong bonds? A lot. So energy out. We have lots of energy out. So that was the first case that we had. So we had something where the bonds were stronger in the product and we said that this was negative. So net here we have heat or energy out is released, and so that's an exothermic system. Oh, the boards work today. And if we have the other-- if we have, say, strong bonds in the reactants, then we have to put a lot of energy in. Big energy in. And if we have weak bonds that are being formed, we're not getting much energy back so the net here is that you have heat in or heat absorbed. And it's an endothermic reaction. So this is just one way to think about it. Remember, whenever you are going to break bond you always have to put energy in to break the bond. And when a bond is formed that energy is released. So we are thinking about the net of the processes, and that's why this equation works for us. So keep this in mind. This is one of the points that people get confused on the exams. And sometimes like they say, oh, thermodynamics. I just don't understand it, and they're not keeping calm and sciencing on. They're getting all stressed by thermodynamics, and it's only this confusion. That's it. So if you work this out then thermodynamics will be your friend and you will love thermodynamics like I do forever. Just kind of keep this in mind and those diagrams in mind and you'll be all good. All right. So let's do an example now. So we can use these bond enthalpies in this equation where we're summing up all our reactants. And sometimes you see some of a little i here for i reactants minus j products. So the sum of all of the products. And it really is a lot here because we're talking about breaking every bond. We're talking about forming every bond. So this is not a huge molecule, but let's think about how many bonds we're actually going to be breaking here. So these are all the bonds that are broken. They're not quite as many being formed here. So bonds broken. We have carbon hydrogen bonds, and we have seven of those. So 1, 2, 3, 4, 5-- let's see if I can count them-- 1, 2, 3, 4, 5, 6, 7. There it is. I need my glasses. OH bonds. We have these guys up here. One, two, three, four, five. We have CO bond over a one double bond over here. So we also are going to have these ones here. One, two, three, four, five. We have the double bond over there. We have five carbon bonds. The one single bond here. And the carbon bond is one, two, three, four, five. And we have OO bonds. We have six of those. Thank goodness. I didn't have to count anymore. It's already labeled. And then the bonds formed. So we're going to have these. So it's six of those, so we have 12 altogether and we have also 12 over here. So first you have to count. And counting is not one of my strengths, so I don't like doing it this way and I'm going to show you two other ways to calculate the same thing. But we can take this and sum these all up. We can look up the mean bond enthalpies for every single one of these types of bonds, multiply them by the appropriate coefficients, and come up with a sum for all the bonds for i number of bonds that you have in the reactants. And you can do the same in the products for j number of bonds that you have in the products and come up with these numbers. So if you were told that you have to do it this way-- use bond enthalpies and you know how to do it-- or if it's an easier problem and you're only, say, breaking two things and forming two things this isn't a bad way to do it. For big molecules this is definitely a nuisance. And if we sum all of this up together-- and so for the total number we have reactants minus products. And so if we subtract this we get minus 2,740. And the actual value is minus 2,816. So it's not even the best agreement when you do it this way. And the reason was, if you remember last time, we were talking about the bonds. Mean bond enthalpy is about 8% difference. So if you had, say, CH in a system that has all the rest of the atoms on carbon or H, that's a somewhat different value than if all of those other H atoms were substituted with bromine or if all those other H atoms were substituted with carbon. Then the bond enthalpy for that CH-- it depends on what else is bonded to the C. And so there's about 8% difference usually in the values. And so overall, you're not going to get much better. You certainly are not going to get better than eight. So agreement of 3% is pretty good, but it's not all that precise because we're using these mean bond enthalpies, which don't depend on the actual value in that particular system. So we can do better than this, and it can be also easier. And it'll be easier if we use standard heats of formation. So this is delta H sub f, f for formation. So the delta H sub f not for standard value is equal to the reaction, delta H, if you're talking about a reaction that involves one mole of compound being derived from its pure elements in their most stable state and in their standards state. So this is standard state 1 bar and room temperature. So let's calculate for the same reaction glucose plus oxygen going to CO2 plus water and see if we can get a little bit more accurate value that way. So let's think about what's happening in this reaction. So every time we oxidize glucose we're forming water. And so we can think about the heat of formation for liquid water. So again, this would be one mole coming from pure elements in their most standard state. So we have to think about where the hydrogen is coming from and where the oxygen is coming from. So hydrogen in its most stable form is H2 gas, and oxygen in its most stable form is O2 gas. So that's then the equation balanced for one mole of H2O liquid being formed. And we can look up the delta H for this-- that delta H of formation-- for this reaction as written is the delta H of formation, and it's minus 285.83 kilojoules per mole. So now let's consider what else that we're forming, water. And we're also forming CO2. So CO2 is derived from carbon in its most stable state, which is graphite as we discussed before, and also O2, oxygen. And O2 oxygen gas is the most stable state there. So for this reaction as written that is the delta H of formation of CO2 gas and it's minus 393.5 kilojoules per mole. So those are our products. We also have two reactants. One of our reactants is O2. So it's what's doing the oxidation. And we're going from O2 gas to O2 gas. This is the most stable state. So what do you think the value is here? AUDIENCE: 0. CATHERINE DRENNAN: 0. Yes. So if you have an element already in its most stable state, its heat of formation is going to be 0. Because it's already the most stable state, so the heat of formation is 0. And every year, I think, on an exam someone's trying to see if they can calculate a delta H of a reaction and they're looking and they're like, oh, I want to use heats of formation because I know that's a lot easier but a value is missing from my table. And they're like, the value is missing from the table. And the TA doesn't know how much information or whatever to give. And if you think you should have a value on an exam and you don't think about, is that element already in its most standard state? Perhaps it's zero and that's why it's not listed on the table. So keep this in mind. This can be very useful to remember. All right. One more thing is involved in the equation. We have glucose. So we can think about the reaction that forms glucose from elements in its most stable state, and we've actually talked about all these already. We have O2. That's in its most stable state. Carbon graphite. H2 gas. And so this reaction as written-- it has a heat of formation of minus 1,260 kilojoules per mole. So now we can calculate the delta H for the oxidation of glucose. The delta H of the reaction from these delta H's of formation. And here's the equation. Delta H of the reaction is equal to the sum of all of the delta H's of formation of the products minus the sum of the delta H's of formation of the reactants. So this now is one of the sources of confusion because if you're using bond enthalpies it's reactants minus products. If you're using delta H of formation it's products minus reactants. So that's why I spend a little time over here thinking about what's going on with the bond enthalpies, so hopefully no one will fall into this delta H pitfall over here and you'll keep the reactions-- the equations straight. So now we can plug it in. If you remember the equations this is pretty easy. So we have our delta H of reaction. We have 6 times the heat of formation of our products over here. CO2. 6 times the first product and then 6 times the second product, which is water, minus the first reactant, which is our glucose, and we have one of those, and we have 6 oxygens. So products minus reactants. Pay attention to the stoichiometry. You need to multiply the heats of formation by the number of molecules so then we can put in the values that we just saw. C02 minus 392 for our water minus 285 minus-- and here we have a minus 1,260 for glucose. And again, 6 times 0 because the oxygen is already in its most stable state. And if we do the math correctly, you get minus 2,816 kilojoules per mole, and that is exactly the experimental value. And it's because the heats of formation are also experimental, so this is a very precise number. When you use the heats of formation you're going to get a much closer value to experimental. And this was a bit easier than thinking about every bond that would be broken and every bond that would be formed. One more way that you can do this. And this takes advantage of something known as Hess's Law and the fact that enthalpy is a state function, which means that it's independent of path. So if you were climbing a mountain and you wanted to go from point A to point B, you could climb all the way up to the top and go back down or you could just go right from A to B and it wouldn't matter. Your delta H would be the same in both cases because it's independent of path. So it only matters what the values are for your reactants and your final products. It doesn't matter how you get from the reactants to the products. Delta H is going to be the same. And because of this, you can take different routes if their equations for different parts of your reaction that are already known with values of delta H-- you can add those equations together and then add together the delta H's to get a new value. So Hess's Law-- if there are two or more equations that are added to give another chemical equations then you can add up the delta H for the reactions of each of the individual equations to get the sum for your new equation. So let's do this now, again, for glucose and oxygen. So if we have these three equations here-- this one is showing glucose plus oxygen being broken down to the elements that are in the most stable state. So graphite, H2 and O2 for glucose. And then our 6 O2s are there on both sides because it's already in the most stable state. We're going to be forming CO2 from the elements in the most stable state and also water. So we can add these together, paying attention to the stoichiometry. So we need to multiply this equation by 6 and this equation by 6, and then we should be able to do some canceling and make sure that we're getting our equation of interest. So we can cancel these 6 O2s with these, we can cancel these O2s with these, and we can cancel this H2 with this. And that leaves us with glucose plus 6 oxygens going to 6 CO2s and 6 waters. So this is going to work now. And now, since we added this together to get this, we can add our delta H of reactions together to get a new delta H of reaction. Oh, sorry. I forgot to cancel my graphites. There we go. Now we're good. Didn't notice them there. So our delta H for reaction. We saw before that the formation of CO2 from elements in its most stable state was minus, so now we've just changed the sign because now we're going the opposite direction. So we have a positive value for that delta H of reaction. Now we have 6 times the heat of formation of CO2 and 6 times the heat of formation of water because that's what those equations are. Those are the heat of formation reactions. And if we add this all together then we get the number that we saw before. So it doesn't matter what path we take. We're going to get to that same answer. And this one, since we're using information that all has to do with heat of formation, it's not really very different from the one we did before. But you can use Hess's Law for delta H of reactions that are not heats of formation. If equations are available that can be added or summed to get your net reaction, then you can add or subtract these values to get a new delta H. Don't forget kilojoules per mole. So we have our three different ways-- bond enthalpies, heat of formation, and Hess's Law.

MIT_5111_Principles_of_Chemical_Science_Fall_2014

32_Kinetics_Reaction_Mechanisms.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. CATHERINE DRENNAN: All right. So 10 more seconds. OK. Let's quiet down. So that is the-- you got the 70% with the right answer. If people can just yell out what was wrong with number one. AUDIENCE: Sig figs. CATHERINE DRENNAN: Sig figs. And what about numbers three and four? What equation was that using? AUDIENCE: Second order. CATHERINE DRENNAN: Second order. That's right. So this is a good clicker question for a problem that's going to be coming up on exam four and also on the final exam to a larger degree. On the final exam we have equation sheets that have all the equations from the whole semester, and so you need to figure out and remember which equation goes with which problem. It doesn't say, oh, here is the expression for first order. Here's the equation for second order. You need to look and remember which equation goes with which thing. But in terms of first order, what's a way that you can remember what a first order equation would be? What's missing from first order? Yeah, the concentration in the material. So for first order it's independent of the original concentration in the material, which is why we can use first order equations for nuclear chemistry. Because the rate of decay of radioactive nuclei are independent of the nuclei around them, so it's a first order process. So when you think a little bit about these equations, you should be able to identify which equation goes with which problem. And so you can identify the type of first order problems also. And for the second order it almost always says, for this second order process, which gives you a nice hint that that's a second order process. And then you just have to identify the equation. OK. So today is more kinetics. So we're in the kinetics unit. We'll be in the kinetics unit for the rest of the semester. It's our last unit. So today I think is one of the most important lectures in terms of the kinetics unit because we're talking about reaction mechanisms, and that is really an important part of kinetics. So investigating reaction mechanisms. So if you're going to describe how a reaction takes place-- often reactions don't occur in one step. It's really uncommon for reactions to occur in one step. So you want to describe the different steps. And you describe those steps, which are also called elementary reactions. So you break down a complex reaction into a series of steps, and then you try to figure out if that mechanism, if those series of steps, are consistent with the data that you've collected on this particular reaction. And over here I just show you some steps in the natural biosynthesis of a vitamin biotin. Biotin is important vitamin for us. It's also used a lot in feedstock. And so people buy a lot of biotin to put in feed stock, and so there's a huge market of biotin. Now right now it's made by what is estimated to be a 13 step organic synthesis which produces-- that's a lot of steps. It's really expensive, and there's a huge amount of organic waste associated with making biotin. And we're talking about in the tons level of waste. So researchers have been trying to figure out the mechanism by which nature makes biotin because that would be a lot more environmentally friendly. So when you're thinking about this, what you want to say is, OK, if we write out a mechanism is it consistent with the experimental data and are there fast and slow steps? Because if there's a step that's really slow maybe you can do something about it. Maybe if you're talking about an enzyme you could re-engineer the enzyme so it would be a better enzyme. Maybe natural selection didn't particularly work. Maybe the cell doesn't really need as much biotin as we do commercially, so there was no need to make it fast. But now we have a need to make it fast, so maybe we can do some evolution of this enzyme and design something to be better. So when you're talking about reaction mechanisms you want to know what's fast, what's slow. And if you want to use that product for something, you want to figure out how you can change things to have a better mechanism-- maybe avoid some really slow steps so that you can do better. Today is also World AIDS Day. And understanding the mechanism of HIV protease was really essential in designing inhibitors against that enzyme. And if you inhibit the enzyme you stop the development of the disease. So this was a very important thing, and we have some pretty good molecules to treat AIDS right now. And some of the challenges have moved on to other things, like less good health care in parts of the world where this has affected. So I feel like AIDS has kind of taken a backseat to hearing about Ebola recently, but AIDS is still a very important problem and one that smart people like you could address. So again, understanding reaction mechanisms is very, very important. All right. So let's go to a simpler problem and a simpler reaction mechanism. We'll go to our friend over here where we have two molecules of NO reacting with one molecule of O2 going to two molecules of NO2. So someone measured some rates for this and came up with the following rate law where you have a rate constant, which is just called k obs for observed. And we'll talk about that more in a little bit. And they discovered it is second order with respect to NO and first order with respect to O2. So there's a couple of things right away that we can ask about this. One is, what is the overall order then of this reaction from this experimental data? What would that be? AUDIENCE: Three. CATHERINE DRENNAN: Three. Again, if some of you missed it. 2 plus 1 is 3. This is not where you want to lose your points on the exam. There's going to be some tricky significant figures in the kinetics units. Save your points to lose there. Count 2 plus 1 and get 3. So is one step likely to have three different things come together at the same time? So how likely is it that all three things are going to merge in one step? No, it's not very likely, so it's no. But if it did work that way, what would it be called? What molecular reaction? Do you remember? AUDIENCE: [INAUDIBLE]. CATHERINE DRENNAN: So if you have three things it's a termolecular. So they're rare. So that's not how this works. All right. So let's look at some rate laws and try to write a rate law for a reaction. We're going to take our overall reaction, we're going to divide it to two steps and write a rate law for that and see if that's consistent with the experiment. So we finally got this up here. Sorry the handwriting isn't perfect. I got here early but then-- the first time I'm going to use the boards a lot today. Squeegee was gone. There's always something. So we're going to break this reaction down into two steps. So in the first step we have our two molecules of NO coming together to form an intermediate N2O2. And this is a reversible step. So in the second step of the reaction we have our O2 molecule coming in, reacting with our intermediate and forming two molecules of NO2. And so this is really pretty common that when you have a multistep reaction that you form an intermediate and your intermediate goes away. So now we can think about how we would write the rate law for this particular mechanism. So starting up here. This is being written as a series of steps, which were also called elementary reactions. And for an elementary reaction it occurs exactly as written. That's its definition of an elementary reaction or a step. So now we can write the rate law for the forward direction of this exactly as it is written. So that would be writing-- and these are my little K's for rate constants. So we have K1 times the concentration of NO to the 2. So we're writing it exactly as written. I said that for an overall reaction you can't just look at the stoichiometry. You have to think about experiment. But for an elementary reaction you can write it just from the stoichiometry, and so this is how we would write it just from the stoichiometry. So what would be the order of that reaction? Just the forward reaction? 2. And that makes it a what kind of molecular reaction? AUDIENCE: Bimolecular. CATHERINE DRENNAN: Bimolecular. Bimolecular. OK. So let's write out the rate law for the reverse reaction now. And again, exactly as written. So we're going to have K minus 1 times the concentration of our intermediate N2O2. And that would be the rate law for the reverse reaction. So what would be the order for this reaction now? 1. Right. We only have one thing in here. And what do you call an x molecular when there's one thing? AUDIENCE: Uni. CATHERINE DRENNAN: Uni. Unimolecular. For step two now you're good at this. Let's do a clicker question. All right. 10 more seconds. Yup. So we're just going to write this exactly as written. It doesn't say it's a reversible reaction. So it's K2-- and I'll put this down here. K2-- our rate constant for the second step-- and then times the reactants. So we have O2 and our intermediate N2O2. So what would be the order of this reaction or this step? Two? We have two things in there. And again, we call that bimolecular. So we have uni-, bi-, and ter- molecular reactions for order of 1, 2, and 3. All right. So we've written this out. Now what we want to do is we want to write out an overall rate law for this. So we've written out the rate laws for all the individual steps. And we'll put that-- Yeah, I guess I can put it-- maybe I'll try to write it down here. I was going to have this all organized but then the boards weren't cooperating today. So I'm going to write the overall rate. Let me try it here. And this is for formation of N2O2. And we're going to put a 2 in there, and I'll explain that in a minute. Then our K2-- so we're just writing this now from the last step-- times O2 and our intermediate N2O2. So the overall rate of forming NO2 is the 2 because 2 moles are formed. So as these guys disappear-- as O2 and N2O2, our intermediate, disappear-- and O's going to form twice as fast because there are two of them being formed. So we put a 2 in there. And then we just have a rate order or rate law for the last step, K2 O2 intermediate. So you can always write the rate for the overall reaction from that last step. Although sometimes we'll see if they're fast and slow steps you can also write it from the rate determining step. But we're not done here because we have an intermediate in this expression. And in a rate law you cannot have an intermediate in there. You need to solve for the rate in terms of your rate constants, your reactant concentration, and your product concentration. So we need to get rid of this. We need to solve for this. So how are we going to do that? How are we going to solve for this? So we want to think about now, what is the net rate for this formation? How is that intermediate being formed, and how is it being consumed given those steps up here? So it's only being formed in the forward direction of the first step. And let me just grab this. So the forward direction of the first step, which is K1 times NO to the 2. And so this is where our intermediate is formed. Our intermediate is going away in two different steps. So it's going away in the reverse direction of step one. So it's decomposing in that step. And so it's decomposing at the rate of K minus 1 times its concentration. The intermediate is also being consumed in the second step. It's reacting with oxygen and being consumed. So here it decays and here it is consumed. And it's being consumed by the reaction K2 times the concentration of O2 times its concentration. So that's the second step. So now we just need to take that equation and solve for the intermediate N2O2. But we don't know net rate over there. We have too many variables right now. So at this point we have to use what's known as a steady state approximation. So steady state approximation and pretty much everything-- I'm talking about reaction mechanisms-- we're going to use the steady state approximation. And that is that the rate of formation of intermediates equals the rate at which they go away. So the net rate is 0. So we can set this whole thing now equal to 0. So steady state approximation net rate is 0, or the rate at which an intermediate forms equals the rate at which that intermediate decays. So I can rearrange this equation now, and I'm going to bring the terms that have the intermediate to one side. So I'm going to put them over here. So we have the term at which it decays-- K minus 1 times our concentration of our intermediate. And it had a negative but I brought it over here to this side with a 0, so now it's positive. And I'm going to bring the same for the rate law at which it's consumed over. So that's K2 times our intermediate concentration and our oxygen concentration. And now on the other side we'll have the rate at which the intermediate is formed, which is K1 No to the 2. So this is another way to express the steady state approximation. The rate at which the intermediate goes away equals the rate at which it's formed. That's the steady state. There's no sort of flux in the intermediate. It's being formed and going away at equal rates. So now we can use this to solve for the intermediate. Now we're set. Now we can solve for the intermediate. And so let's do that over here. So I'm going to pull out-- I had a straight line here. This one's a little crooked. I used to write a lot on the board and used to evaluate professors by their good handwriting and my ratings were always-- my overall rating was limited by my handwriting to a large degree and so I stopped writing on the board. But now they've got rid of that as a criteria, so I can write on the board again. So I'm going to pull out the concentration of the intermediate-- our N2O2. and I'll pull it out of the expression leaving K minus 1 and K2 times O2. So I just pulled out the concentration of the intermediate over here. And then we leave the other side the same. Our K1 times our NO squared. So now I can solve for the concentration of the intermediate. N2O2 equals K1 times NO squared over K minus 1 plus K2 times O2. Great. So we've solved for the concentration of the intermediate now. Now we can take this and bring it back over here and put that whole term into this. And then we'll have a rate law that is expressed only in terms of our rate constants and our reactants and/or products. So let's do more on the PowerPoint here. So this is what we just came up with. So the concentration of our intermediate K1 times NO-- one of our reactants second order-- over K minus 1 plus K2 times the concentration of oxygen. Now I'm going to plug that into this expression, which we just got from writing the rate law for the last step using a 2 because we have two molecules of product being formed. So we're going to plug it in there, and that's going to give us this. So we have our 2, we have a K1 from here, a K2 from here, we have our oxygen concentration, we have our NO overconcentration second order over K minus 1 plus K2 times concentration of oxygen. So that might be the complete answer to some problems but here we were given an experimental rate law, which was second order in NO and first order in O2. That does not match this. Term has an O2 at the bottom. This one doesn't have an O2 at the bottom. These are not the same. So that must mean that the mechanism has fast and slow steps and that we need to reconsider this expression in terms of fast and slow steps. So let's go back now and think about our mechanism. And so if we come over here-- let's say that the first step-- let's bring this down again. Let's say the first step is fast. And we already said that it was reversible, but we'll put that down too. Is fast and reversible. And step two is slow. So I'm just going to propose that these are true, and then we will recalculate what the rate law would be if you have a fast reversible step followed by a slow step and see if that agrees with experiment. So to do that we have to consider what it means if we have a fast step followed by a slow step. So let's introduce a term-- very important term-- which is the rate determining step. Also known as the rate limiting step. So the slow step of a reaction, if it's truly a very slow step, is going to govern the overall rate of the reaction. So let's think about this a minute. So I told you that the extra problems for exam four are long. They're very long. Sorry about that, but there were a lot of problems and I wanted to get you ready for exam four. You also have problem set nine due tomorrow. So you've got a lot of problems to do. So after class today I feel that many of you are going to be really inspired to get started on those problems. And you may run out of here. You might leap over the chair in front of you and race out the door. You may clear the table on the way out because you're in a real hurry to start those extra problems. You will run to the library to look for a table, but all the tables will already be taken. How did your classmates get out of class so fast to get all of the tables in the library? And they're already finishing problem set nine and starting on the extra problems. So you race back to your dorm, but all the tables in the downstairs of the dorm are also filled with 5.111 students. So finally, on the fifth floor of one of the dorms you find an empty table, and then you're really fast. You got the problems out, you got your pencils out, you got your calculator out, you've got old equation sheets out, and ready to go. It's like two seconds. So it took you 40 minutes to find a free table at MIT that didn't already have a 5.111 student sitting and doing those extra problems and problem set nine. So it was 40 minutes plus 2 seconds to actually start doing the problems. So the rate determining or rate limiting step was finding your table. 40 minutes plus 2 seconds is pretty much 40 minutes, and that's what happens in these reaction mechanisms. If you have a really slow step that governs the overall rate of the reaction. Now a lot of you know also about rate determining steps because some of you may be the rate determining step in your group of friends. The rate at which you get to dinner and eat is determined by you being ready to go. Some are in the room like me, yeah, that's me. You know who you are. So I'm saying that what you gotta do to not let yourself be that person-- that rate determining person, the RDS in your group of friends-- you need to get sleep and you need to eat well and you need to make sure you got your ATP. And that means, of course, to get enough sleep you gotta start problems sets early, especially the extra problems because they're long. Rate determining steps. Very important. So let's get back to our example. And we made a proposal. We made a proposal that step two was going to be rate determining. That was slow. We made the proposal that step one was fast and reversible. Step two was slow. So what that's going to mean then is that our rate law for the first step-- that's fast. That's a big number. The rate for the second step is slow. Rate determining. So that will mean then that K1 is going to be greater than K2 times O2, so we can drop out our concentrations here to think about them. They're going to be the same. So what's left? That means the rate constant for that reverse step is very fast. It's a big number. That's going to be big compared to K2 times O2. So now we can go back and look at our expression for the intermediate. And we note that K minus 1 is in the bottom as well as K2 times O2 is in the bottom of the expression. So if K minus 1 now is really big compared to K2 times O2-- again, that's the fast step; that's the slow step-- then this term pretty much doesn't matter and it can drop out. Because this is really small compared to that. And if we drop out this term we can rewrite the expression like this. The concentration of our intermediate rate constant K1 NO squared over K minus 1. And now we can rearrange this equation, bringing the concentrations to one side and our rate constants to the other side. So we have our intermediate N2O2 over NO squared equals rate constant K1 over K minus 1. What does rate constant K1 over rate constant K minus 1 equal? AUDIENCE: [INAUDIBLE]. CATHERINE DRENNAN: It equals the equilibrium constant K1 for the first step. So if you have a fast reversible step followed by a slow step, the first step is basically in equilibrium. We can make that approximation that it is in equilibrium and solve it by thinking about equilibrium expressions. Fantastic. We can be back to equilibrium expressions. So let's think about this a little bit more. Here I have a pretty picture for you. So here we have our reactants forming our intermediates. The intermediates are also going back and forming our reactants. Reactants are forming the intermediates and then back again. Fast reversible. Every once in a while an intermediate gets siphoned off to products, but if this is a really, really slow step it doesn't happen very often. It doesn't really affect this very much. And so basically, this is in equilibrium. It's like this part doesn't even really matter. It doesn't play in. So when we have a fast reversible step followed by a slow step, we can assume the first step is in equilibrium and we can solve for our intermediate using equilibrium expressions. So let's do that. Let's take our equilibrium expression for the first step and now plug it in to our rate law. So we can substitute this step now or we can have rate constants or we can have an equilibrium constant. You can write either. These are equivalent. And we can put those back into this, which was our original overall rate that we wrote. We weren't done though because we had an intermediate. So we can plug this now in for our concentration of the intermediate. And so now we get 2 times K1 K2 O2 NO squared over K minus 1. Or we could just put that with the big equilibrium constant and get rid of our little K1 over K minus 1. Both of those are equivalent. And now we can take all of our K terms and call them K obs. So K obs is just the experimental rate constant. It's the collection of rate constants that are measured. And we often-- when we measure things, we can't distinguish K1 from K2. We sometimes try to do that, and that's a little more complicated. But in this case, all that was given was an overall K observed. And this was our experimental rate law K observed times O2 first order NO second order. And now we see this expression agrees with this rate. So the fact that we have a good agreement means that a mechanism with this fast reversible step followed by a slow step gives rise to a rate law that's consistent with experiment. It doesn't prove that's the right mechanism. It's very hard to prove mechanisms are right, but at least it's consistent. So we can say this is a good guess, a good proposal, for our mechanism. So let's look at another example. So in this example we have NO again. We have two molecules of NO, and now we have Br2 going to two molecules of NOBr. And we're told that the experimental rate is K obs times NO first order, Br2 first order and asked, for this proposed mechanism, which would be the slow step to give rise to that experimental data? So the first thing that we would want to do with all of these is to write the rate laws for each individual step. So for the forward step we have one molecule of NO reacting with Br2 with rate constant K2. So we get rate constant K1 times the concentration of NO times the concentration of Br2. Again, this is a step or an elementary reaction, so we write the rate law just based on the stoichiometry here. Now we can do the same thing for the reverse rate. So we have K minus 1 times the concentration of our intermediate. So in step two our intermediate, which is formed in step one, is reacting with the second molecule NO, forming our product. And we can write the rate for this as well. K2 times the concentration of our intermediate, NOBr2, times the concentration of NO. So again, these are steps. They're elementary reactions so we can write the rate law based on the stoichiometry in that proposed step. So now we can write the overall rate law for the formation of NOBr, and we can just write it from the second step like we did before. Again, this is an example. We're forming two molecules of product so there's a two in there. We have K2. It's basically just this. K2 times our intermediate, NOBr2, times the concentration of NO. But once again, we're not done because there is an intermediate in the expression and you can't have an intermediate in your rate law. You need to solve for the rate law in terms of rate constants, reactants, and products. So we need to now solve for our intermediate in terms of things that are allowed in the overall rate law. And so we want to think again about what is the change in concentration of our intermediate So we can do the same thing that we did before? So the intermediate is being formed in the first step. So we have the rate law for the first step. K1 times NO times the concentration of Br2. The intermediate is also decaying in the reverse part of the first step. So that's minus the reverse rate minus K minus 1 times the intermediate here. And then it's being consumed in the second step. So it's going away by the rate K2 times the concentration of the intermediate times the concentration of NO. So again, this is exactly what we did with the first example. We think about the change, how it's being formed, and the two different ways that it's being consumed. We can again use a steady state approximation and set all of that equal to 0. So let's do that and we'll solve again for the intermediate. So on this next slide now-- I just put those things up there. This is what you were just copying down. If you didn't finish it's still here. And here is the steady state approximation. So again, the steady straight approximation is the net rate of formation of your intermediate equals the net rate of it's going away. Net rate is 0. So rearranging then we can bring the two terms that involve the decay or the consumption of our intermediate on one side and then set them equal to the rate at which that intermediate is formed. And then, we can pull out our terms for our intermediate. So we pull out NoBr2, leaving K minus 1, leaving K2 and NO, and set it equal to the rate law for the first step in the forward direction. We can solve for the intermediate. Take this, divide it by this term. So we have K1 times NO times Br2 over K minus 1 plus K2 times NO. Now we can take this. We are done solving for our intermediate. We have no more intermediates in there, so we can now plug this back in to the expression we had before. So we can take this, plug it in here, and that gives us this formation. So we have 2 times K1, NO times NO-- NO squared-- Br2 on top, K minus 1 on the bottom plus K2 times NO. Now we were asked, what are the fast and what are the slow steps? So now we want to take this and think about, if there's different fast and slow steps, is it consistent then with the experiment? So first, let's consider if the first step was slow and the second step was fast-- or i.e., if we have K2 NO greater than K minus 1. And this is a clicker question, so why don't you tell me how this then, using this, changes? OK. 10 seconds. OK. Let's now think about why that's true. This involved doing a couple of steps in your head. So if we have a first step that's slow and a second step that's fast, the second step involves the K2 times the concentration of NO. That's the second step. That's fast. That's going to be a big number compared to K minus 1. So if we look, both are on the bottom here. And if this term, K2 NO, is much, much bigger than K minus 1, then we can say K minus 1 goes way. If we get rid of K minus 1, we can simplify the expression even more. We can get rid of our K2s and we can get rid of one of our NOs, which gives us this. So saying that the first step is slow and the second step is fast gives us a very different equation. A lot of things cancel out. So we can also write that expression as K obs times NO times Br2. And the overall order of that reaction would be what? Yell it out. AUDIENCE: Two. CATHERINE DRENNAN: Yes. So this is what we would get for a first step that's slow; second step that's fast. Now let's consider if the first step is fast and the second step is slow. So if the first step is fast that means K minus 1, the rate constant for the reverse step, is going to be a lot bigger than K2 times NO. And so now we can look up at this expression and say, OK, if this is much bigger than this then that cancels out. And then we're left with this expression, which I can put down here. So that leaves us-- we can't cancel any more at this point. So that leaves us with 2 times K1 times K2-- these-- NO squared Br2 over K minus 1. So assuming different things about how fast and slow the steps are gives you very different rate laws. We can also write that to make it look a little simpler as K obs, but you'll note that the overall order is very different. So what's the overall order here? Three. Right. So let's remind ourselves what the experimental rate law was. And it was NO first order Br2 first order. So that means that this one would be consistent. So the mechanism is likely to involve a slow first step and a fast second step. And so that's how you do a lot of these problems. You think about what is going to change when you have different fast and slow steps. One of them will be more consistent with the experimental data and one of them will not be consistent. OK Let's do one more fast example. Here we have rate law for two molecules of ozone O3 going to three molecules of O2. And ozone has been in the news a lot recently. So we want to keep our ozone layer. We don't want it to go away. So we have O3 going to O2 plus O, and you're forming an intermediate O. That intermediate is reacting with O3, forming our two molecules of O2. So let's just write out what our rate is for the forward reaction. So we have K1 times the concentration of O3. For the reverse we have K minus 1 times concentration of O2 times the concentration of our intermediate O. For the next one we have K2 times our intermediate O times the concentration of O3. So now we're told that there's a fast reversible step and a slow step. So the rate will be determined by the slow step. So we can write out the rate of formation of O2 based on the slow step, which happens to be the second step, which is what we've done all along. So there's not really a huge change right now. So the formation-- again, two molecules of O2 were formed so we have a 2. We have K2 times the concentration of our intermediate O times the concentration of O3. But again, O is an intermediate so we need to solve for it in terms of our products reactants of rate constants. But now we're told something about fast and slow steps right up front. And so if we have a fast reversible step followed by a slow step, how can we solve for our concentration of our intermediate in a simpler way than we've been doing? What do we use? Or what can we use? AUDIENCE: [INAUDIBLE]. CATHERINE DRENNAN: We can use the equilibrium expression. So we can put that in. We can say our equilibrium expression products over reactants equals little K1 over K minus 1, or equilibrium constant K1. Solve for O and get either big equilibrium constant K1 or our little rate constant K1 over K minus 1, and we have O3 over O2 here. So this was a lot simpler than doing all of that. So again, if you have a fast reversible step followed by a slow step you can solve for the concentration of your intermediate using an equilibrium expression which you all know how to write. So that makes your life easier. Then we can substitute that back in and we are able to put this back. And we'll solve it for O and we'll plug in our K1 over K minus 1. O3. We had an O3. So that's squared over O2. Or we could write that in terms of K obs O3 concentration to the 2 over O2. So let's end with some fun, thinking about what we would observe here. First, the order and then what would happen if we double things. So what is the order with respect to O3? You can yell that out. AUDIENCE: [INAUDIBLE]. CATHERINE DRENNAN: Yup. What is the order for O2? AUDIENCE: [INAUDIBLE]. CATHERINE DRENNAN: Oh, sorry. I had something here. Double this what happens? AUDIENCE: [INAUDIBLE]. CATHERINE DRENNAN: The rate. Well, four times. Order here. Some people yelled it out. Oh, no. Oh, man. It's a clicker question. I forgot about that. Don't listen. But luckily, everyone yelled out different answers. All right. 10 more seconds. OK. Yup. So it's minus 1. And so if you double it, it will half. And then finally, the overall order would be 1 because again, the overall order is the sum. So 2 minus 1 is 1. And last clicker question. All right. 10 more seconds. I know you're really in a hurry to do those extra problems for exam four. I'm the rate determining step. I am with my clicker question. I admit it. AUDIENCE: Yay. CATHERINE DRENNAN: All right. All right. See you Wednesday. Remember, final clicker competition of the year before the finals.

MIT_5111_Principles_of_Chemical_Science_Fall_2014

31_Nuclear_Chemistry_and_Chemical_Kinetics.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. CATHERINE DRENNAN: So radioactive decay is kind of a classic example of a first-order process. So we are doing one little tiny section of the chapter on nuclear chemistry, and we're doing that all today. And so all we're really covering is problems associated with first-order processes. So this is just a small introduction to this idea. So radioactive decay has a lot of applications. There are medical applications, including imaging organs and bones, including the heart. And so there is a compound that you already saw called Cardiolite. And so we talked about this in transition metals because you have a transition metal. And what is the geometry of this compound? Octahedral, and we have cyanide ligands, which what kind of field strength? Strong. Yeah. So this compound was designed in part by an MIT professor, Alan Davison. You could go talk to him about this incredible discovery and invention. It's used about seven million times a year or to image various organs and has been for a very long time. It's off patent now. But this patent made Allen Davison, MIT, and MIT Chemistry Department an enormous amount of money. And so you could go talk to him about it, except he's happily retired living in one of his homes. [LAUGHTER] So you can't really do that. But anyway, so this uses an isotope of technetium, which is metastable isotope. And so it's 99. It's an isotope of the normal 98 atomic mass. And so the next challenge, you're always looking for the next great thing, the next great imaging agent. So this is still a very active area of research, and there's actually a talk just this week on campus about work in this area. So this is transition metals combined with radioactivity. So it's two topics here in the class. So another, of course, important use is the potential of nuclear energy and the current use of nuclear energy. This has many challenges, and I don't want to go on record of what I think about nuclear energy. I think it's a complicated problem. There are a lot of challenges. But one I'd like to bring up, because I think it's particularly interesting to me, is what to do with the waste. And so one story that I heard about actually there was a documentary made about this. Finland had this idea to create this three-mile long tunnel, and they wanted to store 12,000 metric tons of nuclear waste. And they wanted the containers to store it for 100,000 years. And this documentary asked a number of questions about this idea, such as, what kind of container do you use, and how do you know the material you design your container is going to last 100,000 years? As experimental scientists, we like to test how long things last. But you can't really do this experiment. Also, it kind of brought up the idea, do you guard this facility for 100,000 years? Because you can make bombs out of a lot of this radioactive waste. So you kind of need to protect it. But maybe you should just bury it and then no one knows it's there so you don't have to guard it so they can't find it and use it. But then what if someone stumbles upon it and releases all of this radioactivity? So that would be bad. So do you put warning signs for people who will be around 100,000 years from now, saying, hey, don't go in here. It looks like a pretty tunnel. But, hey, the half-life of the thing stored here are 100,000 years. So this is pretty radioactive still. Don't go inside. And if you write this sign, what language do you put it in? So the documentary pointed out that Neanderthals existed like 40,000 years ago. So 100,000 years from now, what's going to be going on? How do you write a sign to people that long in the future? Anyway, I just think that these are sort of interesting ideas and brings up the point that as scientists and engineers, we need to think not only about the science and engineering of what we're doing, but the ramifications to society and the sociology as well as politics involved in some of this science. So this is an interesting area for that intersection of the social sciences and the natural sciences and engineering. So radioactive decay-- definitely a useful thing, dangerous and useful all at the same time. Oh, look at that. You know the clicker question's coming up at the bottom of the page. We're not there yet. It's OK. We're not there yet. I just added that at the end and apparently didn't animate it well. So the decay of a nucleus is independent of how many nuclei are around it. That's what makes it a first-order process. So because it's a first-order process, we can apply those first-order integrated rate laws that we just derived. So we had our rate log of the concentration of something A equals its original concentration to the e to minus k, which is our rate constant times time and also our half-life equation that we just used. So instead of concentration of A, though, we're going to have a different thing to express what we're interested in here, which is N, the number of nuclei. So we can just write that same expression down. But instead of concentration of A, we're just going to use capital N. So N, the number of nuclei at some particular time, equals how many nuclei were present originally times e to the minus k. And here it is a rate constant still, but it's a decay constant in that the rate you're measuring is radioactive decay. So it kind of has a special name. Although, if you use rate constant for that, that is what it is, so that's OK. t is still time. And yes, N to the o is the original number of nuclei. So we're just going to do a clicker question about how one goes about calculating the number of nuclei. 10 more seconds. So someone want to tell me for one of the Green Lantern T-shirts, what is wrong with the other answers? I think I saw your hand up first. Sorry, folks. AUDIENCE: Let's see. So answers one and two, they have the wrong-- what was it-- the molar mass of technetium. Is that technetium? Yeah And answer four does not multiply by Avogadro's number. So that's going to give you the number of moles of the particle. CATHERINE DRENNAN: Right. Great job. It's Thanksgiving. I thought we needed a good prize today. So right. So one thing you also want to remember make sure that your units are good. And it's really important in doing this-- you can take this back-- to remember to use the number that is here, this atomic mass number, not the one from the periodic table in calculating the problem. Actually, the periodic table disappeared from that. Oh, well. So if you use the periodic table, it's a close answer but sometimes. Sometimes it won't be so close. But remember, when it tells you about the isotope, it always has the atomic mass that you should be using in the problem as part of the questions. So keep that in mind. And yeah, you definitely want to remember Avogadro's number. And the answers are such that it's hard to tell that you messed up. With the wavelength, it's really easy to tell you messed up if you didn't use Avogadro's number because it doesn't make any sense. With these, it's a little harder. So remember to use the isotope's atomic mass and also remember to use Avogadro's number when doing this. And then you should be fine. So this is really similar. It's really similar, depending on whether you're talking about chemical kinetics or nuclear kinetics in doing these problems in terms of the equations. But in chemical kinetics, you're measuring the concentration. Whereas with nuclear kinetics, you're measuring decay events. And so usually how do you measure decay events? And the most common way is here, our Geiger counter. So I just want to-- it's always important every once in a while at MIT to double check that the rooms that you're teaching in have not been contaminated by some wonderful experiments. So, so far, we're good. So here, this is working. You can hear the chips, I think. This is pretty good. You don't have to be concerned about this. There's always some background level. It's fine. So there are gases in here that will get ionized by radiation, which gives off. Then that's translated into that clicking noise. So that's what is happening. So it's measuring, with our thing, whether there are any radioactive events going on. And this is called a Geiger counter. And we use X-rays in my lab. So I went and stole this from our X-ray facility before I came here. Luckily, it's almost Thanksgiving, so no one was collecting any data. So no one will get in trouble for taking this right now. And Hans Geiger is the person who came up with this idea and this device. Does anyone remember where we heard that name before? Think back class two. So he did that amazing gold foil experiment. And so our ping pong balls that we were throwing were duplicating the experiment that he did as a graduate student. And luckily, I think he was smart enough to realize that when you're working with things-- he was working with a lot of radioactivity at that point-- that once you know exactly how much radioactivity you're working with-- and so these were very early day experiments. And he came up with this device that helped him know how safe he was. And this is still sort of the standard thing to have these around and double check that there is no radiation leaks in places like that. So the Geiger counter-- all right. So also a couple of more terminology things. Decay rate is also called activity or specific activity. So you're talking about how active your substance is. That's really how radioactive is it. And activity also has the letter A. So we were talking about the concentration of A. Now we have A again. There's a lot of A's in this unit. So that's the change now in the number of nuclei over time- that's the rate expression, or the rate law-- k, the decay constant, times the number of nuclei. And because activity is proportional to the number of nuclei, we can also take this expression that we had before that had the N's in it and rewrite it with A. So now it's really just like that first-order expression we had but without the concentration term. So we have the activity at some time equals the original activity of the material times e to the minus kt. And all of these equations are going to be on your equation sheet. But if you mess up and use the wrong equation for this, it doesn't matter, as long as you're using the first-order equation. Whether it's concentration or activity, it's the same idea that you can determine, if you know the rate constant or the decay constant, how much material is left, how much activity is left, how many nuclei are left after a given amount of time. So I know what you're all thinking now. You're thinking, this is fantastic, but what about the units? We musts here about the units. So the SI units for activity are the becquerel, BQ. And one becquerel is one radioactive disintegration per second. And this is the newer unit. The older unit was called a curie, and sometimes you will still see this in the literature. And a curie, one curie, was 3.7 times 10 to the 10th disintegrations per second. So it was a much larger number than the current SI unit. So this was what one gram of a radium specific activity was. So they used this big number. But it was not really practical because you want to tell people like how much radiation would be safe for them to have in a year or something like that. And you didn't want to use this giant number for that. So we've moved to here. So does anyone know or want to guess who the older unit was named for of radioactivity? One might think Marie Curie. But a lot of the evidence suggests it was actually her husband, Pierre Curie, who it was named after. It's a bit controversial. But they both worked together, and they worked with Henri Becquerel. And they all won the 1983 Nobel Prize for discovering radioactivity. Three years later, Pierre Curie was killed crossing the street. He slipped when it was raining, and a horse and wagon, I guess, ran over him and killed him. So this is, I think, an example of someone who's so brilliant, but you say, they're so brilliant, but do they look both ways before they cross the street? So you are all very brilliant. And I encourage you look both ways before you cross the street. Anyway, so he died. And some of the stories are that they named the unit after him as a tribute. Others say, well, it's really for both of them. But in any case, now it's named after the third person, Henri Becquerel. So radioactivity, I'm going to tell you a little bit about radioactivity. This chart is not in your handout because you're not responsible for knowing all this information, so I just didn't put it in there. But you can look this up. There's a couple of points I did make in your handout, which is there are different types of nuclear radiation. We have alpha particles, alpha decay, beta decay, gamma decay. Some of those involve a mass change. So like an alpha particle is the same as a helium-4 nucleus-- two protons, two neutrons. Beta decay involves an electron. Gamma is a photon. So there are definitely different types. Some mass change, some not. There are also really dramatic differences in half-life. So again, half-life depends on the material in question. It depends on that decay constant, that rate constant. And if we look at this table, we can see things from milli seconds. And if we look at some of these, d is for a day. a is for year. y is also for year. So sometimes you'll see y for year. Sometimes you'll see a. I think most people guess that y is for a year. That a is for year, I don't really know. But anyway, in this table it's a. So if you see that, don't be confused. And Ga, that's giga years, so 10 to the ninth. That's where the Finland 100,000 years comes from, that we need to keep the stuff safe for a very, very, very, very, very, very long amount of time for giga years. So in some decay processes, such as uranium-238, you have more than one type of nuclear radiation going on. And it can involve a very long and complicated series of different events. So here at MIT, we spend most of our time talking about science and engineering. But I feel like every once in a while we should throw in some poetry into our science classes. So once a year I like to read a chemistry poem to enrich our lives. And today is that day in 2014. And the poem I'm going to read to you is called "The Days of Our Half-Lives," and it is by Professor Mala Radhakrishnan. She got her PhD here at MIT in the Chemistry department. And she wrote this book, which she wanted me to point out is available on Amazon if you're looking for a Christmas present for a very, very geeky friend of yours. And it's illustrated by another MIT chemistry PhD, Mary O'Reilly, who actually did the illustrations for the videos that I've been showing you in class. So MIT chemists, just really multi-talented individuals. So I will read you this poem now. And as I read it to you, I will point out what is happening in this decay process because all of Mala's poetry is scientifically correct. So "Days of Our Half-lives." "My dearest love, I am writing you to tell you all that I've been through. I've changed my whole identity. But loved, I can't pretend to be. When I was uranium-238, you were on my case to start losing weight. For 5 billion years I'd hoped and I'd prayed, and finally I had an alpha decay. Two protons and two neutrons went right out the door. And now I was thorium-234. But my nucleus was still unfit for your eyes, not positive enough for its large size. But this time my half-life was not very long, because my will to change was really quite strong. It took just a month, not even a millennium, to beta decay into protactinium. But still, rejected me right off the bat-- protactinium, who's heard of that? So beta decay I did once more to become uranium-234. Myself again but a new isotope, you still weren't satisfied. But I still had hope. Three alpha decays 'twas hard, but I stayed on through thorium then radium and then radon. I thought that I would finally please you. My mass was a healthy 222. But you said, although I like your mass, I don't want to be with a noble gas. They dress so well. You had a point though. I wasn't reactive. So in order to please you, I stayed proactive. A few days later, I found you and said, two more alpha decays, and now I am lead. But you shook your head. You were not too keen on my mass number of 214. I had a bad experience with that mass before, and an unstable astatine walked right out the door. So in order to change, I went away. But all I could do was just beta decay. My hopes and my dreams started to go under, because beta decay does not change a mass number. To bismuth and polonium, I hoped and I beckoned. My half-life was 1 6 4 microseconds. And then finally, I alpha decayed. And then I was lead with a prize worthy mass of 210. Got to admit, I was getting quite tired, and my patience with you had nearly expired. You were more demanding than any I'd dated. And much of my energy had been liberated. But you still weren't happy, but you had a fix. I really like the number 206. So I waited for years until the day which began with another beta decay and then one more. And finally, in the end, I alpha-ed to lead 206, my friend. To change any further. I wouldn't be able-- not longer active, but happily stable. It took me a million years to do, but look how I've changed, and all just for you. Wait, what did you say? I've gotten so old that you'd rather be with a young lass of gold? Well, I give up. We're through, my pumpkin. Shouldn't all my effort be counting for something? Well, you won't be able to rule me anymore, because I'm leaving you not for one atom, but four. That's right. While you were away diffusing, I found some chlorines that I found quite amusing. And we're going to form lead, Cl4, and you won't be hearing from me anymore. See, over the years, I've grown quite wise. I've learned that love's about compromise. You still have half of your half-lives to live, so go out there. It's your turn to give." Thank you. [APPLAUSE] There's a whole book of them on Amazon. So that is first order. First order is pretty exciting because it has nuclear decay. Second order-- not quite as exciting. But we should talk about it anyway. So second order integrated rate loss-- we're not going to go through a derivation. It's in your book. But here is the equation, if you do the derivation. So now we have 1 over the concentration of A at time t equals rate constant k times t plus 1 over the original concentration of A. And we could plot this one over concentration of t versus time. And if we did that, you would have the opportunity for another clicker question. 10 more seconds. 90s, yeah-- it's kind of hard to come up with clicker questions in this unit, so. But it's fun, for Thanksgiving, we'll have lots of 90s. So we can just look, and this is actually an expression for a straight line again. So we're plotting on the y-axis one over a concentration of A at all the various different times versus time over here. And so our intercept is going to be 1 over the initial concentration of A. And our slope is going to be what? k, right. So again, you can measure your concentration as it changes with time, how the concentration changes, plot it, and just determine your rate constant for that particular material. So second order half-life-- we can do another derivation. But in this case, I will just give you the equation. So half-life equals 1 over k times your original concentration of A. And so this is different from first order. There is a concentration term in the equation. So for second order half-life, it does depend on the starting concentration. So that's really the big difference. In first order, it doesn't depend on the starting concentration. It just depends on the rate constant or the decay constant, which depends on the material in question. With second order, you do need to know how much you had originally. So again, how do you know if it's a first or a second order process? And here, you really have to determine it experimentally. So one thing you could do is measure how A changes over time and then plot your data using the equation for first order. And you may see that, yeah, that does not form a straight line when you're plotting with natural log of a concentration of A. But then if you try plotting it 1 over the concentration of A, you get a beautiful straight line with your data. And so you'd say, that's a second-order process. So again, you're determining these things experimentally, collecting data, plotting the data, determining rate constants, determining the order of the reaction. Now, this is very exciting. What we're going to talk about is the relationship between the rate constants and equilibrium constants. So I love this. I love when we come back to stuff that we've talked about before and see it in a slightly different way. So at equilibrium, we talked about how it's a dynamic process. And you have the rate of the forward reaction equal the rate of the reverse reaction. Reactions are still going. They haven't stopped. But the rates are equal in both directions. So we've talked about how to write an equilibrium constant for reaction. So if we have a reaction of A plus B going to C plus D, we can write our equilibrium constant k and its products over reactants. Unless one of our products or reactants is a solid or a very dilute solution. It's the solvent. And I heard from your TAs that in the last problem set, some people had forgotten what goes into q or k expressions. So it's good to review that for this next unit and exam four and the final-- so products over reactants. Now suppose we tell you that it's a second-order process and the rate of the forward reaction here, A plus B, we can write the rate law for that forward reaction being second order, first order in A and first order in B. So the rate constant for the forward direction is k 1 and then times the concentration of A times the concentration of B. For the reverse reaction, the rate constant is k minus 1. And this is generally true in all the problems. If it's a first step, you have plus 1 k1 on the top, k minus 1 on the bottom. So we have k minus 1 times the concentration of C and D So that's the backward direction. So at equilibrium, these rates are equal. We just talked about that. We've seen that before. The rate of the forward reactions, so k1 times A times B is equal to k minus 1 times C times D when you're at equilibrium. So we can rearrange this equation now and say C and D over here over divide by A and B. It's going to be equal to k1 over k minus 1. And we also just saw that C times D over A times B was equal to k. So therefore, our equilibrium constant k equals k 1, the rate constant for direction, over k minus 1, the rate constant for the reversed direction. So here we're relating equilibrium constants and rate constants. So we thought a lot about what's true if you have a big equilibrium constant. If you have a big equilibrium constant, if you have an equilibrium constant much greater than 1, what's the ratio of products and reactants at equilibrium? Is there more or less products at equilibrium and reactants? More. So we thought about that, and now we can think about the relationship of the rate constants. So if k is greater than 1, is k 1 greater or less than k minus 1? Greater. And so that would then be the case where you have more products than reactants at equilibrium. If k is less than 1, a case where there's more reactants than products at equilibrium, then you have k 1 is less than k minus 1. So again, we can think about this in terms of thermodynamics. We can also now think about it in terms of rates. So one more thing that we need to cover before we end today, and that is about elementary steps and molecularity, which I just love saying that word. So on Monday, we're going to talk about mechanism of reactions. Most reactions do not occur in one step, and we need to think about mechanisms. I said it was Wednesday. But it's actually Monday. So it's coming up. It's very exciting. And we're going to talk a lot about elementary steps when we talk about mechanisms. So an elementary step is one of the steps in the reaction. So reactions usually don't occur in one step. They have many steps, and each step is called an elementary reaction. So we talked about last time that for the overall order of the reaction, you can't just look at the stoichiometry and say what the order of the reaction is. So you can't predict it from stoichiometry for an overall reaction. But if it's an elementary reaction, if it's a step, that elementary reaction is written exactly as it occurs. So in that case, the order and the rate law can be predicted. So this is you're breaking it down into sort of the smallest unit, the smallest step, this elementary reaction, so you can just look at the stoichiometry for a single step, for an elementary reaction, and project to the order and the rate law. So elementary reactions occur exactly as written. So that's what we're going to do on Monday. We're going to break down our mechanisms into elementary steps, write out the rate laws, and then figure out what kind of mechanism we might have. So finally, molecularity-- so molecularity is just the number of things that come together to form a product. And here we have three names-- unimolecular, bimolecular, and termolecular. Unimolecular process, what do you guess? How many reactants are coming together to form product? One. Bimolecular, what do you guess? Two. Termolecular is a little harder, but just give it a whirl. Three, yes. So bimolecular is very common. And termolecular is not. So I have three molecules to come together. And if you try to think about how you get three things to come together all at the same time, that's kind of rare. Usually, when there are three things reacting, there are multiple steps involved. But two is good. Now finally, we'll end with a clicker question. Think about which of these would be examples of unimolecular processes. 10 seconds. So it actually is one and two. So most people got the two. Yes, that's radioactive decay. But the other, you can have a decomposition. So here we have decomposition into its elements is also a first-order process. Happy Thanksgiving, everybody. See you next Monday.

MIT_5111_Principles_of_Chemical_Science_Fall_2014

17_Thermodynamics_Now_What_Happens_When_You_Heat_It_Up.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. INSTRUCTOR: So let's consider a reaction and consider the effect of temperature on this reaction on the thermodynamics of the reaction. So this is the decomposition of sodium bicarbonate. Does anyone know what another name for bicarbonate is? If you were going to send someone to the grocery store to buy some, what would you tell them to get? AUDIENCE: Baking soda. INSTRUCTOR: So you would probably say, look for baking soda. Does anyone know what this is used for or how it works? AUDIENCE: Baking. INSTRUCTOR: Baking, yes. So what is baking soda doing? What step of baking are you using it for? AUDIENCE: [INAUDIBLE] INSTRUCTOR: Yeah, bread rising. So in this reaction, you're forming gas, and that gas helps bread to rise. So this process happening while it's baking allows for bread rising and things like that. So baking soda-- you always want to make sure you add your baking soda. So let's consider the thermodynamics of this reaction. delta H0 is a positive value, 135.6 kilojoules. And why don't you tell me which of these you think is going to be the delta S of these options? AUDIENCE: [CHATTER] INSTRUCTOR: Just 10 more seconds. AUDIENCE: [CHATTER] INSTRUCTOR: So the positive value-- that is a very good guess and that is correct. Because we're going from solids to gases, so you would predict that entropy would be increasing. Things are not moving very much with a solid. With a gas, they can. You have more disorder, more freedom. So that is, in fact, the correct value for this delta S0 is plus 0.334 kilojoules. I already put it in kilojoules for you. And now we can calculate the delta G for this reaction. And let's first do it at room temperature. So we have delta H, positive value and endothermic reaction. We're doing it at room temperature first, in Kelvin so we can cancel our units. And we put in our delta S value here. And then we can calculate it out. And delta G0 is plus 36.1 kilojoules per mole That is not a spontaneous reaction. So our bread is not going to rise, and our baking will be failed, except that we're probably not going to be baking it in the oven at room temperature. So this would be non-spontaneous but when we bake, we're going to heat the oven, and so usually 350 or something, which is 450 Kelvin. So now we can do the equation again. We plug-in our delta H, our new temperature, and our delta S, and we get a negative value. Delta G0 is minus 14.7 kilojoules per mole at 450 degrees Kelvin. So this would be spontaneous. So when you're baking, you want to remember to turn your oven on, heat it up, and put in your baking soda. So let's think about now this type of reaction that has a positive delta H and a positive delta S. So these both have the same sign. And if both delta H and delta S have the same sign, temperature can be used to control spontaneity. It will be non-spontaneous at one temperature but spontaneous at another temperature. And if we assume delta H0 and delta S0 are independent of temperature, which is fine to do-- that's a good assumption-- then delta G0, which is definitely dependent on temperature, is linear-- is a linear function with temperature. So let's plot now those two values of delta G that we just calculated. So here is our plot. We have delta G0 kilojoules per mole on the y-axis and temperature on the x-axis. And let's put in the numbers that we had. So we had calculated at room temperature, around 300 Kelvin, a value of about 26-- positive 26 kilojoules per mole. And we calculated then at about 450 Kelvin a value about minus 14.7, almost minus 15 kilojoules per mole. And when you have two points, you can draw a beautiful straight line. When you have more, sometimes it's more complicated. But it is linear. Delta G0 is linear with temperature. So now let's think about this is a straight line, and we can think about an equation for a straight line. So the equation that we know and love, delta G0 equals delta H minus T delta S, can be now rearranged. We have delta G on the y-axis here. We have temperature is our x-axis. So now we've just rearranged. We've pulled minus delta S0 over here, and delta H0 are over here. So delta H is our y-intercept. And for some reason, your notes just say "y dash i." I don't know what happened to "intercept" part. But anyway, that's the y-intercept is delta H0. And what is the slope? AUDIENCE: [INAUDIBLE] INSTRUCTOR: Yeah. So the slope is negative delta S. So if you plotted delta G's versus temperature, you could get out your delta H, or you could get out from the slope your delta S. And let's think about these different parts of the plot. So over here delta G is greater than 0. It's of positive value. And what does that mean about the spontaneity of the reaction, if its delta G is positive? AUDIENCE: Non-spontaneous. INSTRUCTOR: Right, so that's non-spontaneous. Down here, we have delta G minus value less than zero, so there it's spontaneous. So at some temperatures, the reaction is non-spontaneous, and in other temperatures, the reaction is spontaneous. And there is a certain value of T, T star, that is the temperature at which it switches from non-spontaneous to spontaneous, or if you're decreasing temperature, spontaneous to non-spontaneous. And for a particular reaction, you can calculate what that temperature value is, what T star is. So let's do that. So we can calculate T star. And so again, this is the temperature at delta G equals 0 where you have that switch point between spontaneous and non-spontaneous. So we can set delta G0 equal to 0. And then we can solve for T star. So T star would just be delta H0 over delta S0, because this is set to 0. And we can plug these values in. So delta H, again, was plus 136 kilojoules per mole. And delta S0 was plus 0.334 kilojoules per Kelvin per mole. And we can calculate out a temperature of 406 Kelvin. So this is the temperature at which you have this switch from a spontaneous to a non-spontaneous process. So if you're cooking your bread below 406, you get something that looks like this. The first time-- my husband is the chef in our family. And the first time I was tasked with making cupcakes for my daughter's school, I forgot to put the baking soda in. But if you put enough frosting on it and the kids are four years old, it totally doesn't matter. Anyway, if I had put in the baking soda and cooked it at temperatures above 406 Kelvin, I would have had something that looked a whole lot better, and I wouldn't have even had to put frosting on. But, no, I would have. Four years old, you have to put frosting-- never mind. It all worked out. So there is this temperature at which you have a switch if the sign of delta H and the sign of delta S are the same. So now let's think about another case where you have delta H0 and delta S are both negative values. What would this plot look like? And here are some options for you. AUDIENCE: [CHATTER] INSTRUCTOR: Let's just do 10 more seconds. AUDIENCE: [CHATTER] INSTRUCTOR: 90%, excellent. So that is correct. So we can draw that now in your handout. So here, if they're both negative, down here we have delta S is a negative value, spontaneous. You'll have a negative delta G. And here you have a negative delta H minus T times a negative delta S. So this will be a positive term and this will be a negative term. And to be spontaneous, you want a negative overall. So at low temperatures, the unfavorable delta S is down-weighted. So at low temperatures, you're spontaneous. But as the temperature increases, you'll get to a magic temperature, a T star, in which this delta S term now becomes greater than the delta H term. This will be a big positive value compared to a smaller one, and you'll switch to a positive delta G and a non-spontaneous process. Let's consider all the options now. One more. 10 more seconds. AUDIENCE: [CHATTER] INSTRUCTOR: Great. Yeah, I knew those other clicker questions were going to be deciding the winners, I feel like. So yes, this is always going to be spontaneous. So if we remember the equation-- and if you don't have it memorized yet, you will soon. Even though it will be on an equation sheet, most people don't need it. You use it so much. So when delta H0 is negative and this is positive, delta G is always going to be negative. So it'll always be spontaneous, and so when delta G0 will always be negative at every single temperature. So now let's think about this one. Positive delta H, negative delta S-- what will this be? You can just yell it out. AUDIENCE: Never. AUDIENCE: Never spontaneous, right. So positive, and then here with another positive, a minus, a minus, another positive-- it'll never be spontaneous. How sad for it. So delta G will always be positive at all temperatures. So here are cases where delta H and delta S have different signs. But now we have cases where they're both positive or they're both negative. So for the both positive case, this will be sometimes spontaneous. It will depend on T. So when will delta G be negative and, therefore, the reaction be spontaneous, when T is greater or smaller than T star? AUDIENCE: Greater. INSTRUCTOR: Greater, right. So here, we have a positive delta H endothermic reaction. We have a positive delta S. And so when T is big here, this term will dominate. And you can still get a negative delta G0, so when temperatures are above that magic temperature. And then our last case, when we have a negative delta H, an exothermic reaction, and a negative delta S0-- and again, that will depend on temperature. So when they have the same signs, temperature makes a difference. And here, you will have a negative delta G when you have a smaller temperature, because here you want the delta H term. That's a negative term. And you want that to be the bigger term, and so you want to have smaller temperatures that will down-weight the negative delta S0 over here. So these are our four possibilities-- always spontaneous; never spontaneous; and then sometimes spontaneous, and it depends on the temperature compared to T star. And T star, again, is the temperature at which you switch between spontaneous and non-spontaneous. So temperature is important. Temperature is important to thermodynamics, and temperature is also important to kinetics. Most of the time, you can speed up a reaction, or at least an elementary reaction, when you increase the temperature. So now, let's think about thermodynamics in biological systems, and think about a very important interaction in biological systems which is hydrogen bonding. So this is also great review for the exam on Monday. So hydrogen bonding are interactions between hydrogen bond donors. So what is a hydrogen bond donor? A hydrogen bond donor is a hydrogen in a polar bond. So this is why it's good review for exam 2 because for exam 2 you should be able to identify polar and non-polar bonds. So a hydrogen bond donor is a hydrogen in a polar bond, and a hydrogen bond acceptor is an electronegative atom with a lone pair. And electronegativity is also a topic on exam 2, so might as well learn it for Monday and then continue to learn it, because you'll also need it for thermodynamics and later on as well. So here we have a bond between something X and a hydrogen, and here we have y that's an electronegative atom with a lone pair. And this little, squiggly plus thing here indicates that it has a partial positive charge. And here you have a negative charge, and that's going to make for a nice favorable interaction, this hydrogen bond. So X is something that will lead to a polar bond, such as nitrogen, oxygen or fluorine, nitrogen and oxygen being the most relevant to biological systems. So if you have N or O here, that's a polar bond. And so that will then form a hydrogen bond with an electronegative atom that has a lone pair. And examples of that are also nitrogen, oxygen, fluorine, and nitrogen and oxygen, again, being the most relevant for biological systems. So we have this interaction because these sort of partial charges here that form this nice hydrogen bonding interaction. So if we look at one of the most important molecules that hydrogen bonds which is water, we see that water has polar bonds between the oxygen and the hydrogen, because those have electronegativity difference of greater than 0.4, and it has two lone pairs. So water is capable of being a hydrogen bond donor with its hydrogen and also a hydrogen bond acceptor. And so if we draw the hydrogen bonds as dotted lines here, we see this OH polar bond is a hydrogen bond donor, and this oxygen here with its lone pair is the hydrogen bond acceptor. So here we have this network of hydrogen bonding interactions, both on the board and on my t-shirt. I also have water-hydrogen bonding on my t-shirt today. So these hydrogen bonds in water are really important for life. This is a very, very important property. Now just for exam review, what is the shape of that molecule? AUDIENCE: Bent. INSTRUCTOR: What angle do we expect between the hydrogen, oxygen, and the nitrogen? AUDIENCE: [INAUDIBLE] INSTRUCTOR: Less than 109.5. That's right. What would be the SN number? AUDIENCE: 4. INSTRUCTOR: 4, Yes. Good! If those answers seem like "I have no idea what everyone's talking about," you know what your weekend is going to involve. Those are things that are going to be on the exam. VSEPR, yes. So let's compare now hydrogen bonds with covalent bonds. So here we have a hydrogen bond donor and acceptor versus a covalent interaction. So a covalent interaction where you have a bond, where you have bonding electrons are being shared between the two. So covalent bonds are stronger than hydrogen bonds, for sure. And let's look at some examples. So if we consider this X as an oxygen here, in a oxygen hydrogen polar bond interacting with another oxygen, the value for that hydrogen bond would be about 20 kilojoules per mole as opposed to the covalent bond, here just the hydrogen and oxygen, has 463 kilojoules per mole. So the hydrogen bond is considerably weaker than a covalent bond, but it's still-- weak bonds turn out to be really important in biological systems. Now let's consider a case where we have nitrogen as the acceptor. So we're going to have a nitrogen acceptor here. So if this is OH donating to N, then we have 29 kilojoules per mole. NH interacting with N14, compare to a covalent bond between H and N of 388. So when you compare to covalent bonds, hydrogen bonds are much less, but they are still super important. So for bonds that are made up between molecules here-- intermolecular, between molecules-- hydrogen bonds are the strongest kind of interactions that are between molecules here. So hydrogen bonds can be between molecules. They can also be made within a molecule. So this just shows hydrogen bonding in a protein structure between these are called beta strands here. And hydrogen bonding is incredibly important in forming protein structure-- really, really important. So hydrogen bonds are responsible for protein folding-- very important in proteins. Hydrogen bonds are also really important for DNA. So all of the main in RNA, all of our macromolecular molecules of life, hydrogen bonding is really important. So let's look at a GC base pair that forms in DNA. And we can think about the hydrogen bonds here. So here are three of them. So we have this polar bond between N and H. And it's the hydrogen bond donor, and we have the lone pair on oxygen accepting that hydrogen bond. Here we have a polar bond between nitrogen and hydrogen donating over here to a nitrogen lone pair. And here we have an NH polar bond as a hydrogen bond donor to the lone pair on this oxygen. And this hydrogen bonding pattern is what allows DNA to have its beautiful double helix, and is very, very important. So why don't you give it a try and tell me how many hydrogen bonds you would get between these guys here. And I'll leave this structure up so you can see what it looks like. 10 seconds. AUDIENCE: [CHATTER] INSTRUCTOR: The answer is 2. Let's take a look at that over here. So you have a nitrogen polar bond over here making an interaction with this lone pair. You have a polar NH pond here, hydrogen donor to this lone pair. Why is that not a hydrogen bond? Because it is carbon hydrogen. So carbon hydrogen, the electronegativity difference is not greater than 0.4, so carbon hydrogen is not a polar bond. So you need to have both a polar bond, and you need to have an electronegative atom that has lone pairs to be the hydrogen bond acceptor. So here you have a hydrogen bond acceptor, but you don't have a hydrogen bond donor. And so this actually turns out to be important because you want to specifically recognize one base with another base, and so the hydrogen bonding pattern is essential for that working out. So here, these are hydrogen bonds. I said they're weaker than covalent bonds, but they're strong enough to help stabilize the structure of DNA. But they're not so strong that DNA cannot unzip. And you need to unzip DNA to read it, and by reading, that's essential to make another copy, to have cell division, or to translate your genetic code. So that's, I think, why hydrogen bonds are so important in biology, because you don't want a lot of super strong bonds. You want weaker interactions because in biology things are moving around. So let's talk about the importance of hydrogen bonding. And for this we have another In Their Own Words segment. [VIDEO PLAYBACK] - My name is Lourdes Aleman, and my research is on RNA interference or RNAi. RNA interference is simply a silencing mechanism the cells use to turn down the expression of a gene. Double-stranded RNA pieces have some sort of complementarity to a sequence within the genome. That double-stranded RNA piece binds to a big large protein complex, it unwinds the double stranded piece, takes one of those strands only, and finds the gene that it complements with. And once it does, it binds to that particular RNA from that gene and it destroys it and not allowing protein to be made from that RNA. DNA and RNA both can form base pairs by hydrogen bonding. The short piece of RNA that is found in this protein complex guides that RNA and basically finds its match by hydrogen bonding. So if it forms a hydrogen bond along the whole entire sequence, in knows it has found a match somewhere in the sequence. And that's how it recognizes the gene that its targeting. Macular degeneration is a disease of the retina. There are too many blood vessels in the retina, and they can bleed and scar over time, and eventually these patients can become blind. Some patients with this disease, one of the reasons they have an outgrowth in the retina is because there is a gene called the VEGF that there's too much of. And VEGF tells cells, "Make blood vessels." RNA interference is being used to silence the expression of this gene so that in patients with macular degeneration, you don't get further growth of more blood vessels and more bleeding and scarring as a consequence. My dream for RNAi would be that as a patient you will go into the doctor if you were diagnosed with some sort of disease. The doctor would go into the computer, order you a some double-stranded RNA for the particular gene that has been mutated or is malfunctioning in your disease, and you would then come back, and they would put that double-stranded RNA into you, and you will get better. That would be my dream, that it could be applicable to pretty much any disease or viral infection that you can think of. How many of you have heard of like RNAi or treatments and things and sort of-- a number of people-- like customized medicine. I mean, I think that that would be such an incredible thing if this works out, to really be able to treat every individual based on their DNA. So we're not there yet, but there's a lot of people who are working on this. And there's a lot being done at MIT research in RNA, and that particular work was in Phil Sharp laboratory. Phil Sharp is one of our Nobel laureates here at MIT. So that's why hydrogen bonding is important. So let's just do one more example of thermodynamics in biological systems. And we're back to thinking about ATP, which we talked about already, and the hydrolysis of ATP. So we saw that this is a spontaneous reaction before, and this can be what's called "coupled" to a spontaneous process to drive that non-spontaneous reaction. So the total change in free energy of a coupled reaction is the sum of the individual delta G's. So if you have one that is unfavorable and one that's favorable, one that's positive and one that's negative, you sum that up and if it's overall negative, then it will become a spontaneous coupled reaction. So let's look at this example again. So we have delta G 0 for ATP at 310 Kelvin. Why do you think I'm using 310? What do you think that temperature is? AUDIENCE: Body temperature. INSTRUCTOR: Body temperature. So we have ATP, so you have triphosphates-- that's the TP-- and hydrolysis. You're losing one of the phosphates. You're going to a diphosphate. And hydrolysis means a cleavage reaction that involves water. So we saw before that the delta H0 for this is negative. It's an exothermic reaction minus 24 kilojoules per mole. Delta S0 is plus 22 joules per Kelvin per mole. And so if we plug this into our delta G equation, we have delta H minus T delta S. At body temperature, this is a negative value, negative 31 kilojoules per mole. So this is spontaneous. The hydrolysis of ATP is spontaneous. Now we want to couple this to something that's non-spontaneous. And the reaction we're going to couple to that's non-spontaneous is the addition of a phosphate group to glucose. And this keeps the glucose in the cell, because things that have charge can't come and leave the cell as readily. So nature often does this, its way of holding the things that it wants inside the cell inside the cell. But this is a non-spontaneous process. The delta G0 is plus 17 kilojoules per mole, but if we couple that to the hydrolysis of ATP, which is minus 31 kilojoules per mole, and we can add those together, so 17 minus 31, gives us minus 14 kilojoules per mole. And now this non-spontaneous reaction is driven forward. So we've taken something that wasn't favorable and made it favorable by coupling it to ATP hydrolysis. So if ATP hydrolysis is so favorable and it's spontaneous, why isn't it happening all the time, which would be really bad for us because we store our energy in the form of ATP, so we want to keep it in its good form? And the answer again is kinetics. It's a slow process. So ATP is inert enough that we can use it as an energy storage.

Political_Sociology_Lectures

Political_Sociology_Week_2_Lecture_Neoliberalism_and_Crisis_Part_1.txt

good morning students and welcome to week two of political sociology the lecture today is as i indicated in an email i sent out this morning a very very critical lecture because it lays the groundwork for much of what we're going to be talking about throughout the rest of the semester and it introduces the single most important concept to understand the issues we face in the united states the political dynamic the level of inequality in wealth and income and many other aspects of american society and that is neoliberalism a term that is widely used some of you may have already been exposed to the concept and it's also widely misunderstood so it's important for us to delve into that concept and the origins of neoliberalism within the united states and that's something i want to do in this lecture uh this lecture will also be beginning with what i would call some basic political economic concepts informed by marxist theory which i think will also enhance your ability to understand what's happening in the american capitalist political economic system and it's vital uh you don't need to have a background in economics i don't need to have taken micro or macro in fact that might have done more damage than good so i want you to appreciate the fact that there are some economic ideas and concepts that you can merge with politics that brings life to the study of what is conventionally called economics but which was originally called political economics before they removed the term political we could talk more about the reasons why they did that but use your imagination i think you can come up with some reasons why they might want to eliminate the word political in the kind of work they do um in any case uh please post any questions again we're using voicethread here uh so any thoughts comments reactions questions objections grievances uh post them uh there have been a few postings in voicethread so far this semester the first week uh but not many uh so i am using this particular program for that purpose and if people don't take advantage of that there are probably other ways i can be recording these lectures that would involve fewer steps and would be somewhat less cumbersome so uh let's get started and before we do anything else uh somebody had uh asked me about um on this this flag here looks like the american flag and they were wondering what it was or is it you know is it the american flag you probably can't see it clearly um it's not the american flag as you know it uh i don't uh post the flag i don't wave the flag it's not outside my house i don't have a flag lapel for philosophical reasons i do not support the concept of patriotism which is often associated with flag waving or nationalism there's a lot more we could say about those two concepts uh and the value of them and also the dysfunctional aspects of what i call patriotism and nationalism but these are not um things or sentiments which i express i didn't say the pledge to the flag when i was in high school for a variety of reasons and i pretty much have sustained my view about pledging allegiance to a flag i have issues with that that's my uh problem not your problem uh there's a lot we could say about it and not everyone agrees with me on that that's perfectly fine so what is this that i have behind me then if it isn't the american flag if i'm not trying to as politicians always do wrap themselves in the american flag i don't know if you've noticed this but at the last two at the convention um and anytime there's kind of a political statement uh you will see a politician a standing in front of like 20 or 30 flags behind them the democrats do this because they're always a little defensive about the fact that maybe they're not patriotic enough so it's kind of a defense mechanism i find it to be kind of a shameless form of pandering uh to try to communicate that you are indeed a good american patriot it's unnecessary as far as i'm concerned but it is a part of what we might call in sociology civil uh religion so what is this flag let me just show you briefly what it is uh that's what it is um now if you've ever heard of adbusters you can go to their website it's a organization they've been around for a while uh and it's a very critical analysis that they engage in of american patterns of materialism uh consumerism uh also critical of american foreign policy and of the domination of our entire system and our way of thinking and our psyche by large corporations and so this is obviously not a celebration of of the united states it's actually a criticism of the united states where you see these stars representing states uh you see corporations the messages that these corporations are essentially controlling what happens uh in the united states and largely dictating what we do what we think how we spend our time and how we certainly spend our money so check out the ad busters website i um contribute to the organization i have a subscription to their lively lively colorful image filled and provocative very edgy very subversive publications i like what they do um and some of you might find it uh find it interesting so that's the flag that's up there uh and somebody asked me about that okay let's get to the uh lecture i want to begin by emphasizing that you can't really and i think i mentioned this at the beginning of the semester we are integrating politics and economics in this course in a very big way what i call political economy political economy is a whole area and it's one of the areas that i specialize in as a sociologist and you can't understand much about the political system unless you understand something about the way the economic system works and that economic system is capitalism that's what we call it we don't call it the free market we don't call free enterprise it's a capitalist system so if you think about the capitalist economy there are two necessary conditions for capitalist economic growth capitalist economic expansion what marxist you sometimes might hear this term capital accumulation it's a very marxian term but basically we're talking about growth expansion of a capitalist economy in the most simplistic terms so there are two necessary conditions for the capitalist economy to expand and grow and have capital accumulation the first is you must have a capacity to produce and notice i put in parentheses that are supply okay capacity to produce what does that mean it means that you the economy will not in any way expand or grow there will be no capital accumulation if capitalists do not invest their money that's what i mean by the capacity to produce there must be a capacity to encourage capitalists to take the money the capital the financial resources they have and invest it in production are only going to do that if they perceive that that investment will translate into profit that is a return on investment roi as they say over in business school okay so you have to have conditions you can just think of this as like the business climate you have to have conditions that are ripe for capital investment and that means conditions that encourage the capitalists to invest their money in anticipation of making a profit if they do not believe they're going to make a sufficient profit they will not invest their money if they don't invest their money we're all screwed because the capitalist economy depends on the private investment decisions of capitalists this is one of the major criticisms one can make of capitalism so you have to have a capacity to produce secondly you have to have a capacity to consume and what that means is that the goods and the services that are being generated and produced by the capital investment must be purchased must be consumed must be bought by people who have money this is the demand side you understand so there's the supply the production side capacity to produce and there's a capacity to consume the demand side if capital capitalists produce enormous amounts of goods but they can't sell them they don't make it profit right so you have to have it on one hand capacity to produce conditions that encourage investment and a capacity to consume that is sufficient amount of money in the hands of people to purchase what is being produced demand okay now one of the wonderful things about marx's theory is it identifies all of the many inherent contradictions that exist in capitalist economies that's one of the beautiful aspects of marxist analysis i know that you read a little bit about marx in the first reading by janowitz that doesn't even scratch the surface of the value of marxian analysis which is the inner workings of the capitalist economy so let's consider one of the most significant contradictions and i think this is something everybody can grasp and it's fundamental there's a whole series of contradictions this is just one but to me it's a very central contradiction of capitalism on the one hand if you want to encourage capitalists to invest their money the capacity to produce you have to have low costs and you have to have low wages capitalists aren't going to invest their money if they have to pay workers so much that they're not going to make the level of profit they demand or anticipate or desire so the capacity to produce encouraging investment keep wages low on the other hand and i think you're beginning to see where i'm going here the capacity to consume that is the ability for the capitalists after they produce all these goods cheaply to sell them and realize the profit depends on higher wages that is that people have to have sufficient amounts of income to buy the goods fundamental contradiction in capitalist economies okay so that's the first point i want to make this is simply one way to think about the way the capitalist economy works and ultimately if you understand these two concepts you can understand how certain kinds of economic crises emerge in a capitalist economy and i'm going to give examples from american uh economic history all right so let's consider two different kinds of uh crises what is a demand side crisis okay now we call it a demand side crisis this is one capital capitalist the capitalist class the bourgeoisie they're very strong they impose their conditions on the society on the economy on the working class and when they do this if they're powerful enough they keep wages very very low okay because they want to maximize profit the lower the wages the higher the profit all things being equal okay so when the capitals class is very strong the capacity to consume tends to be weak because the capitalists are exercising their political power to keep wages low this creates what's called a wage squeeze which produces a demand side crisis now it's a demand side crisis because wages are low capitalists can produce cheaply but they can't sell the goods that they're producing because wages are insufficient okay that's a demand side crisis and i have this little mathematical equation ctp the capacity to produce is more powerful greater than the capacity to consume so when capitalists talk about different kinds of crisis they talk about a wage squeeze crisis a crisis where wages are too low for the mass of citizens the mass of the population to consume what is being produced this is a problem okay now what about a supply-side crisis think about the other side when labor is relatively strong i'm going to say relatively strong because labor is never in the driver's seat of the capitalist economy because they're in a dependent position vis-a-vis the capitalist class but let's say labor develop some organizational strength we'll talk about that okay this could be through labor unions etc working working workplace power working-class power when labor is strong and they're able to bring up their wages and their salaries and demand certain kinds of working conditions of the capitalist class this increases costs and as the cost rise remember the capacity to produce is weakened because now capitals look at the situation they say well you know costs are rising labor is too strong wages are too high we can't make a sufficient profit i'm not going to invest you understand now i have that little mathematical formula the capacity to consume is greater than the capacity to produce that's a profit squeeze you understand one is a wage squeeze producing a demand side crisis profit squeeze supply side crisis and that crisis originates when capitalists decide not to invest the economy slows down it goes into recession you have unemployment etc so again this contradiction this relationship between the capacity to produce and the capacity to consume okay so let's go back to the great depression uh by the way there have been books thousands of books and articles written on the great depression and what caused it and what factors contributed to the economic crisis of the 1930s we're going to spend you know like five minutes okay so this is a simplified view and my analysis of the great depression is in the context of that contradiction between the capacity to produce and capacity to consume so the crisis of the 1930s the great depression was a demand side crisis and what i mean by that is at that time 1920s you had very very favorable capacity to produce conditions you had a large influx of immigrant labor so you had surplus labor uh labor unions did not really exist to any extent so workers were not well organized and you had the beginning of a mass production industrial system that could produce many goods rapidly and productively uh the beginnings of the assembly line so labor surplus lots of immigrant labor coming to the united states poorly organized not in labor unions new technologies all of this contributed to a very favorable very favorable investment climate the capacity to produce was very favorable and there was massive amounts of investment and speculation that's the key point speculation produce lots of goods because conditions are ripe for investment anticipating a high level of profit so that's what happened lots of investment lots of speculation about an expanding economy high profits etc but what was missing yes we had a strong capacity to produce what didn't we have during that time a good capacity to consume wages were very very low and so you had a crisis of what's sometimes called under consumption all right under consumption insufficient ability of the population to consume all of the goods that the capitalists are investing in if the capitals can't invest and make a profit by selling the goods you're going to have a crisis all right so that it's sometimes called a realization crisis in marxist theory what that means is that capitalists can produce cheaply but they can't realize they can't realize the profit so just imagine you're producing lots of goods very cheaply because wages are low workers are disorganized there's a labor surplus you can keep wages as low as possible pay people the minimum minimum minimum produce lots of goods very cheaply you put them in a warehouse you still haven't realized the profit there's a lot of profit embedded potential profit embedded in each one of those little items little commodities that you've produced but you don't realize the profit until that product is exchanged in a market for money and if there isn't sufficient money in the hands of capital of the population workers you don't realize profit and you have a crisis all right so how did we get out of this mess well there's lots of explanations and theories about this uh but we do know that one of the things that happened and that came out of this crisis was a reorganization of the political economic system of the united states under fdr and some of you are familiar franklin delano roosevelt some of you are familiar with his history and uh he was not a socialist uh he was not a communist uh he was actually from a very wealthy family he was a capitalist himself or at least he came from capitalist roots and he realized that the political economic system ultimately to get the united states out of this crisis had to be reorganized and this brings us to what's called the new deal and a new deal was essentially designed to have massive amounts of government spending and investment in the economy in order to stimulate demand you understand put money in the hands of workers you had jobs programs you had infrastructure programs you had all kinds of conservation core programs you could it's an amazing moment in american history where you see the state the government stepping in and ultimately really i would say saving capitalism from itself this is the great irony is that what people call socialism some people call the new deal from socialism um actually save capitalism because capitalism is a self-destructive system and it has to be bailed out you know constantly and this is the case historically in the united states so under the new deal you have the introduction of social security social insurance programs to make sure that people have money to spend when they're old when they're retired you have progressive taxation and redistribution which means you tax the rich the wealthiest tax rates went up and that money was spent on government programs on redistribution putting money ultimately in the hands of the population that could spend it you had the national labor relations act or what was called the wagner act that allowed unions to be formed in um in organizations and workplaces in factories and so you had a sharp rise in the percentage of workers who were unionized and who could negotiate for a higher wage and you had this expansion of the social welfare state all of this is part of the new deal you understand so in order to resolve the demand side you have to put money into the hands of people and the private sector isn't going to do that the state steps in this is what's called keynesian economic policy okay john maynard keynes maybe you've heard of john maynard keynes the economist british economist and his theory was that capitalist economies inevitably will suffer from these demand side shortages demand side crises and the only way to get out of them is for the government to spend money in a variety of ways and another way they could spend money is to expand the military and so i have a term here called military keynesianism it's a pretty neat little concept and the idea there is that spending on war actually stimulates the economy sadly clearly i think we would all want to stimulate economic growth and expansion and increase wages and create jobs in ways that aren't related to human suffering and destruction but anyway there is during it and remember we had world war ii okay and world war ii during world war ii it was you know wasn't like well you know we we have lots of needs that have to be met during the war to make sure that we can you know provide the truth we didn't leave that to the market by the way we had essentially a command economy uh during wartime directed by the government right and massive amounts of money being pumped in okay that got us out of the great depression and the economic hardship of that period okay so then we move into this what's called the post-war period when i say post-war i mean post-world war ii it's important to point that out because we live in a period now where we have permanent war right we're told the war on terror is permanent and so for your generation the idea of post-war may not make much sense but if you read historical pieces when people use the term post-war they mean post world war ii so world war ii ends you have in place a kind of keynesian political economic system you have this new deal you have social legislation you have a social welfare state and you have the um you know beginning of the expansion of various agencies within the federal government uh designed to ensure uh the safety of workers the safety of food um etc okay we'll talk more about some of those agencies throughout this semester and what they do or what they aren't doing um but a very important part of this period this post-world war ii period and when i say this this um let's just say the period from about 1945 to 1975 okay those were really the glory years of american capitalism right uh and during this period there was something called a capital labor accord and you can think of this as a kind of agreement between the capitalist class and working class which actually ensured uh that workers were well rewarded and compensated for an extended period of time during this period and the capital labor accord essentially worked this way uh labor unions and labor unions are very widespread particularly in the manufacturing sector of the economy at this time labor unions essentially uh came to an agreement with the capitalist class okay that's why capital labor accord an agreement and it went something like this the workers said look now if they're really socialists they say you know we want to take over the workplace you know workers control of the means of production making all decisions they decided there was a compromise they said to the owners we'll let the managers that you've hired decide how work is organized all right you turn that over to managers workers no longer have control over the way the work place is organized you turn that over to managers we'll let the managers figure that out they can assign people to different positions they can organize the assembly line the way they want etc etc we will concede that on the condition that as productivity rises wages and salaries rise proportionally you understand there is a relationship between increasing productivity and increasing wages this was the capital labor accord so i have this term here the social structure of accumulation some people have talked about phases of capital expansion and they call it a social structure of accumulation remember i use this term accumulation before social structure so there are phases of capitalist expansion that involve having in place particular arrangements and institutions like the capital labor accord that agreement some people have called the capital labor accord the social structure of accumulation during this post-war period okay uh some people call the keynesian fortist model now notice i have something called uh in the parentheses there non-zero sum logic this is important the non-zero sum logic right in a zero-sum situation any gain for one side is a loss to the other that's a zero-sum gain and that produces high levels of conflict and antagonism okay because any gain for workers would be regarded as a direct cost to capitalists and vice versa so in this arrangement the capitalist class to some extent okay accepted this accord and basically they agreed to this non-zero sum logic which was yes paying workers a higher wage obviously it's not what we want to do we want to keep wages low as possible we want to maximize profit but we realize that if we're going to sell all of the products in this mass production economy to the population they need to have higher wages you understand so henry ford was you know the person often the example every four or five dollars a day you know how are you going to sell the automobiles if you don't pay your workers enough to buy them right that's basically the idea so is it near a non-zero sum logic yes it's a cost to us to pay workers more but there's also a benefit to that because those workers then turn around and spend the money so this social structure of accumulation you can call it the new deal arrangement you can call it the capital labor accord uh the fortis model there's lots of terms that people have used this fueled the most prolific expansion period of capitalist growth in our history period these were the glory years i have 1950 to 1973 okay i said 45 to 75 roughly that period okay almost everything we talk about today in regards to the amazing aspects of american capitalism really pertain to that period so you might ask yourself um was that an anomaly maybe this isn't really the character of american capitalism maybe that was just a anomalous period that was unusual for a variety of reasons having to do with the great depression and then world war ii and then you know american global dominate whatever right we'll talk about what happens later who's that i can't hear you okay well you're not here uh that's tricky dick uh that's what we called him back in the day dick nixon richard nixon criminal left office in disgrace um he was my nemesis i have to say when i was in when i was in high school i was very political in high school the one thing that energized me more than anything else and probably shaped my entire political outlook if you want to talk about political socialization i was the vietnam war a horrible disgusting exercise in american imperialism a criminal uh enterprise that had horrible horrible impact certainly on the vietnamese people the cambodian people the laotian people and of course it had enormous impact on american politics and lives of americans you should do a little research on the vietnam war because it's one of those things maybe some people have forgotten in any case i could go on and on but he was my nemesis and um he energized my of vitriol and uh hatred in some ways um but he's a republican and remember he's a republican he said we are all keynesians now or at least he would people attribute that quote to him there's some dispute about these quotes but it's true to some extent that even though he was a republican and by the way if nixon were alive today and he was pursuing the policies he pursued then he would be drummed out of the republican party because he would be too um liberal in their view and he would be too liberal because one of the things that um nixon did was he did pretty much subscribe to the idea that government did have some role to play even though he's a republican and you know he had certain kinds of conservative principles um during his um administration uh you had the creation of the occupational a safety and health administration osha that's to protect workers uh today much of the republican party would like to eliminate osha or at least roll back what they do uh environmental protection agency what yes a republican environmental protection agency today republicans would probably like to eliminate that because it interferes with the free market and the ability of corporations to make more profit um clean water clean air bills so there are domestically you know um there's lots of issues i have with some of his domestic policies politics but domestically he was to some extent subscribing to a kind of keynesian framework and so we had this this funny quote coming from of course his foreign policy is what you know ultimately produced the protests and the hatred toward nixon who when he came in office that he was going to end the vietnam war and he actually expanded and okay i could go on and on all right um i hope you can see this let me see if i can move this thing can i move this oh where am i going what the hell is this all right i can't i can't move it um so i'm just gonna leave a dick for now because i'll screw things up um so something happened in the 1970s there was a desire by the capitals class at a certain point the capitalist class were getting a little um restive if you want to call it that and there was a presentation by lewis powell who became a supreme court justice uh to the chamber of commerce and it was a call to arms and um basically uh what powell was saying and i'm not gonna read this you can read it it's very interesting you can read the whole memo if you go online um it's widely cited as kind of a critical point in american political economic history uh and that is that powell essentially said look uh things have gotten out of hand the new deal has gone too far uh workers have too much power uh there's an attack on the free enterprise system uh we as capitalists we as conservatives we who value the free enterprise system need to you know mount a counter offensive uh against this expansion i would call an expansion of democracy um and that's one aspect of neoliberals and we'll talk about but this power memo is a call to arms enough of the new deal enough of the expansion of government this is intruding on the property rights of the capitalist class all right so let me be a little more specific about the four things that the capitalist class begin to complain about and here's where we begin to move into this was the next major economic crisis we had after the great depression was the crisis of the 1970s and into the early 1980s this was not a demand side crisis remember the demand side crisis of the great depression was based on the fact that workers did not have enough money in their hands to buy the goods that were being produced this was a supply-side crisis that emerged because the capitalist class didn't like the conditions that had emerged over the post-war period and over time they decided we can't make enough profit conditions aren't favorable we don't like the business climate we are going to stop investing and when you stop investing if you're a capitalist and the capitalist class does this generally we all get screwed because the economy slows down we go into recession we have unemployment not a good situation all right so what was it what were the complaints and i have them here taxes are too high regulations are too stringent unions are too strong and the welfare state is too generous i want you to remember those three things taxes regulations unions and the welfare state those are the four items that the capitalist class took aim at they wanted taxes lower they wanted deregulation they wanted unions to be weakened or eliminated and they wanted the welfare state to be rolled back there were these jokes free the fortune 500 because you know they were complaining that they were being uh you know enslaved by the new deal political economic system because taxes were too high there were too many regulations unions are too strong the welfare state was too generous it would give people money even though they weren't working it's the last thing a capitalist wants ever is to have people able to survive without selling their labor power for a wage so you have this investment under investment you have recession we go into this supply side uh crisis okay so as i said the supply side crisis the capital capacity to consume is stronger than the capacity to produce remember workers have wrote some relative strength over this period that produces the supply-side crisis this is a profit squeeze right not a wage squeeze wages were pretty high at this time okay and by the way the mid 70s was the point in which the level of economic inequality was the lowest in the united states think about that so the success of programs produces a backlash by the capitals class so you have the profit squeeze sometimes it's sometimes called a capital strike right in other words workers strike by withdrawing their labor power they leave the workplace or you could have a capital strike that is where the capital say you know what we're not investing and when the capitalists engage in a capital strike capital strike the economy gets screwed if it's a capitalist economy where the entire economy and its ability to grow and expand and employ people depends on the private investment decisions of capitalists the achilles heel of capitalism so what's the solution neoliberalism now at the time this wasn't the term that was used so you have the 1970s um you have nixon leaving office you have jimmy carter being elected uh you have gerald i'm sorry gerald ford filled in for uh nixon when he resigned uh ford lost the election to uh jimmy carter and jimmy carter loses the election in a pretty severe a beat down to ronald reagan ronald reagan is a republican and ronald reagan uh to a large extent ushers in uh this neoliberal model and in the early stages some people called it reaganomics some people called it supply side economic supply side because the idea is to put in place conditions that encourage investment which makes some economic sense market fundamentalism the market is the fundamental driver and we should eliminate anything that interferes with the market and let the market do its magic okay um voodoo economics we'll talk more about that later because that has some specific reference all right so neoliberalism becomes the way to describe what essentially is the political economic arrangements that have existed in the united states since 1980 this is a very very 40 years of a neo-liberal political economic model there's an ideological dimension there's a policy dimension the ideological dimension is markets are good governments are bad private is better than public if the government has any role it's to create conditions that encourage investment you understand the role of government is to create conditions that encourage capital investment not to do anything that might discourage capital investment okay and the role of government is also to protect the market from democratic demands we'll talk more about this conflict between capitalism and democracy some people assume they go hand in hand in fact they're fundamentally antagonistic so you have this ideology this sort of anti-government a privatization anti-public sector anti-public goods let the market decide philosophy market fundamentalism okay and then you've got the policies right if taxes are too high you need to lower them if uh regulations are too stringent you need deregulation if unions are too strong you need to beat up on unions and if the welfare state is too generous you need to reduce the welfare state by the way at that time welfare was not unpopular if you asked most people there were surveys done you know about you know do you think people when they're suffering from economic hard times uh should have access to welfare relief questions like that you know there was a majority who said yes of course you know people suffer from economic hard times so one thing that reagan did and this has enormous political consequences over the long haul one thing that reagan did was that he essentially began to describe people who received welfare uh in racist terms he talked about the welfare cadillac okay people receiving welfare drive cadillacs welfare queens okay now these are dog whistles okay these communicate to people that those who are getting welfare actually don't deserve it it's a certain segment of the population what they're saying is it's largely black and brown people who aren't working who don't deserve it we get this term that emerges the deserving and the undeserving poor and you begin to turn people against welfare because you give them the impression that a it's not deserved that people could be working but instead they're collecting they're on the dole and it's the other that is engaged in this behavior uh so he uses that um method that technique that has run through american politics and emerges to this very day when trump talks about the end of the suburbs people are coming to destroy the suburbs we know who he's talking about he doesn't really use the dog whistle right they say he uses the bullhorn so this is not the first time but certainly a way to get people to support the cutbacks in welfare because again there was massive amounts of support now the thing is everyone knew that these policies okay this reaganomic policy set the supply side economic policies what became known as kneeler everyone knew that in the short run over time this was going to put more and more wealth and income in the hands of the capitals class there was no question about it every aspect of what they were asking for would have essentially a redistribution effect when people talk about redistribution they often think about redistribution from the capitalist class to working no this is redistribution from the bottom in the middle up to the top right how did they justify this they justified it by indicating that yes there will be a period where there will be rising inequality concentration of income and wealth but that wealth and income will be concentrated in the hands of the doers of the makers and the capitalist class and they will take all that money that they have and they'll see how wonderful the economic environment is for investment and they will reinvest all that money and we will have a massive renaissance of economic expansion and a expansion of employment and jobs and industry and so yes the income will go up at the beginning but eventually it will all trickle down that was the promise that was one of the ways in which they got away with this here's a few quotes i just like to show these because it gives you sort of sense of the flavor of the ideological message i'm going to talk more about these because they have a direct impact on the current ability of us you know political economic system to manage the greatest crisis we face since the great depression the pandemic government is not the solution to our problem government is the problem okay you understand this is part of the neoliberal ideology ronald reagan when he was running for office and by the way people now they look back at ronald reagan and you know they said oh you know reagan republicans i'm a reagan republican let me tell you something first of all before reagan became president uh he was considered a buffoon and a kind of a joke and there were people just like before trump was elected who said if reagan wins not leaving the country okay so there's that aspect of history that you may not know about when people glorify the reagan years the other is the reagan years weren't so wonderful especially because he instituted these kinds of policies and this kind of ideology the 10 most dangerous words in english language are high i'm from the government and i'm here to help so the idea was the government had no significant positive role to play at all and this became a deep-seated ideology that is communicated to americans to you as you're growing up you you know you ask the typical person you know you want a private solution or a public well private solutions got to be bad market or government market i believe in the market so ideology and then you have margaret thatcher who is the british prime minister uh who also at the same time was promoting neoliberalism in britain okay so how did they resolve so to speak the economic crisis of the 1970s well they did exactly what you would expect they responded to the demands of the capitals class they cut taxes particularly on the wealthy and on business uh they eliminated government regulations deregulation deregulation deregula this has been going on for 30 years this deregulation movement and most americans again are brainwashed into thinking you know regulations are bad deregulations are good okay weakening labor unions all kinds of efforts to make it more difficult for workers to join unions and policies which directly attack labor unions now these were promoted and by the way people thought this is good because you know labor unions raise costs and if costs go up capitals aren't going to invest so we need to get rid of labor unions because they're screwing up the economy and of course attacking and gutting the welfare state and the reason you want to do that is you want to force more and more workers into the labor market you never want workers to be able to receive income without working right and you remove restrictions on the mobility of capital this is another aspect of it okay during this period because capital during this period i talked more about this in my social change in international development course because we talk about globalization in depth uh and that is the freedom of capital to move across borders to other countries with very favorable tax uh consequences uh so any restrictions on capital mobility all that kind of stuff all of this is um a part of the package okay the neoliberal package um okay so some of this i've already i've already mentioned you know i try to continue to get people to just think about the various ways in which uh you can conceptualize this idea of neoliberalism because it's so important but uh one source of confusion can you hear that okay i got these neighbors with these like i don't know garden tools and it just pisses me off how often they use them all right in any case so one point i want to make is it's a source of confusion is this idea of neoliberalism people are sometimes confused by this because they think liberalism liberalism is democratic party liberalism isn't that sort of left-leaning or isn't that you know pro-worker or you know there's some confusion there the liberalism we're talking about here is not american style democratic party liberalism what we're talking about here is classical philosophical liberalism classical liberal theory is very much pro-market anti-government intervention okay there's much more i could say about classical liberal theory but we're talking about liberalism the way people describe liberalism in europe and in britain it doesn't mean sort of welfare state policies democratic party policies it means market-based policies so neo new liberalism okay taking the philosophy of liberalism classical liberalism and introducing it neo in the contemporary setting right that's what we mean here okay so don't confuse this with liberalism okay it's an entirely different thing okay so these are some additional points that i think i've already uh covered all right so what are the consequences of neoliberalism that we've witnessed um over the last 30 plus years uh one is de-industrialization i told you that the limits on capital mobility have been knocked down so you have factories closing moving shifting resources shifting investment from the northeast in the midwest you have you know cities and communities that are entirely devastated because the entire economy revolved around a particular company factory the production of something in that region in that community those factories close those communities are devastated they are economically depressed um the prototypical case is detroit detroit at one time motown motor town was the center of manufacturing productivity expansion high wages comfortable blue collar working jobs look at detroit today okay and you know if you grew up in florida you don't realize this but if you drive through the midwest and northeast through these old factory towns uh you see what happened uh so you have de-industrialization and then the first phase is moving a lot of the a lot of the factories and the investment from the midwest northeast to the sun belt right the sun belt the south which is much more accommodating to the interest of capitalists weak unions um not much of a history of labor organization and uh a very pro pro capitalist pro free market orientation and then eventually of course the capitalists they want to make even higher profits let's just shift everything uh offshore uh to china and um to mexico etc right financialization the financial sector of the economy continues to expand in relationship to the goods producing economy we'll talk more about that we have fiscal crises that is you have this massive pressure to reduce taxes and that means that you have less revenue coming in but you have often the same level of um requirements for states to meet basic needs and so you have fiscal crises we have the great financial recession great financial crisis and you have stagnation you have stagnation of wages over this period and you have largely a very stagnating economy the levels of economic growth during the neoliberal period are much much weaker and smaller if you want to focus on economic growth as one measure than prior periods in economic history so when people say that the neoliberal model you know unleash the market and produce this massive uh expansion of the economy it's simply not true we've had relatively modest rates of growth on the other hand you've had massive massive levels of profit concentrated in the hands of fewer and fewer people so um david harvey if you haven't read anything by david harvey and i think um uh i did have some readings of harvey in the past i know i do in my social change and international development any case david harvey is a marxist geographer anthropologist you know social science person whatever you want to call him very interdisciplinary he's written wonderful accounts of neoliberalism so if you're interested in learning more about the meaning of neoliberalism the intricacies of it i would strongly encourage you to check out david harvey he's got some great youtube presentations uh where he sort of lays out on a board a whiteboard all kinds of different features of the way capitalism operates generally and neoliberalism in particular and his point is neoliberalism was from the very beginning a project to achieve the restoration of class power you have to understand these periods of expansion and growth and transition as a function of the balance of economic power and remember it was a capitalist class it was very upset during the 1970s the expansion of the new deal expansion of state expansion of the government they wanted to restore their power class power so a political project to reestablish the conditions of capital accumulation restored the power of economic elites and they did that they were very successful and the united states stands out today as the most extreme neoliberal industrialized economy in the world and that explains a lot of why we are where we are today okay i think this is gonna end uh at 60 minutes so i'm going to say a couple quick things and then i'm going to have to um end part one okay so under neoliberalism you have de-industrialization i mentioned that before there's this wonderful book written and i think this is 1982 this book had a huge huge impact on the way i was thinking about my own work my own research i wrote a few papers on the migration of capital uh from one geographic area to another of the de-industrialization of america um you have as again i mentioned that you have this attack on labor and you have this steady steady steady steady decline in the percent of the workforce that is a member of a union okay this has enormous enormous political and economic consequences for the united states and most students with all due respect are utterly utterly ignorant about the role of labor unions the history of unions in the united states the critical role that unions play politically and that's why i devote an entire week to labor and politics because you can't understand many things about what's happened over the last 30 to 40 years if you don't understand what has happened to labor right and notice that most of the uh workforce that is now unionized or in public sector public sector unions not private sector unions which have been essentially devastated demolished eliminated okay i'm going to stop here because i want to say more about this chart because it's very important and it will take more than uh two minutes because i'm at 58 minutes uh so i will stop here that is the end of part one i hope you are appreciating the importance of this idea of neoliberalism again it's absolutely vital for your understanding of what the hell is happening and not just domestically there's also global neoliberalism so if you can begin to get an understanding of this it will explain a lot of what's happened over the last 40 years and the political dynamics that currently exist in the united states okay i will see you in part two you

Political_Sociology_Lectures

Week_6_Lecture_Who_Rules_Part_1.txt

okay welcome back to political sociology and this is week six we're moving quickly through the semester i hope everyone is doing well and you're staying tuned in to the political activities and various events that have been emerging in the united states you're doing a great job posting various media articles that you come across and the explanation for your selection so thank you very much for that as you can see i'm still in a different location than i was at the beginning of the semester outside of asheville very cold out in the 40s this morning but it's going to be a beautiful fall day here in western north carolina and i'm hoping that all of you are doing well i don't believe i have seen one person from this class during office hours and that's quite all right it's optional but do not hesitate to check in and as i've said before i'm curious about how things are going with your campus courses since i'm doing exclusively online so this week we actually have a lot to cover and it's a significant substantive area within political sociology some people might extend these topics over several weeks uh in some way we've touched on them in different ways but we're going to attack them directly this week and one of the big questions is um and this is the way we used to pose it back in the day uh when i was in political science um and doing political sociology in sociology was who rules is there a ruling class who runs the joint who makes the decisions um the larger question of course and the two largest conceptual uh ideas that drive this course our power power and democracy who's got the power and to what extent is that power used in ways uh that may or may not enhance democratic procedure democratic institutions democratic uh processes and decision-making so we get to one of the big debates historically in political sociology and that is the question of who rules and where the power lies so let's get started and i want to talk a little bit about this concept of pluralism because much of the debate that has occurred within the context of political sociology in terms of who rules is this debate between pluralists on the one hand and elitists they aren't elitist themselves that is they focus on the role of elites rather than what the pluralists focus on which is a broader distribution of power uh so when you talk about elites you tend to talk about some level of concentration of power when you talk about pluralism you're talking more about a dispersion of power uh so i just wanted to touch on what we mean by pluralism and we use this term a lot in um social science and there's really uh two ways at least to think about pluralism at least two ways one is social pluralism and when we talk about social pluralism we're really talking about the whole range of statuses positions uh institutional affiliations organizations voluntary associations identities that individuals are connected to um the ways in which that those shape the way we approach politics approach life interact with people uh and the idea is that in a democratic polity uh you have civil society and in a civil society you have lots of opportunities for different forms of affiliation organizational associational and these can shape your political outlook so very early in the semester we talked about mutually reinforcing versus cross-cutting cleavages and the fact that uh you know your affiliation may be in a labor union um and your affiliation at the church those are two forms of social affiliation part of the larger context of social pluralism they might not reinforce each other they might be cross-cutting on the one hand the labor union pushes you left and the church affiliation might push you in a more conservative direction and the fact is those are cross-cutting what we would call cross-cutting social pluralist affiliations uh voluntary associations as i said there was an enormous literature back in the 60s and 70s on the importance of voluntary associations in terms of engaging people in public life and one person who spent a lot of time looking at this not so long ago and i think he made some important contributions and actually he just came out with another book on the current disastrous state of american society i think we can all agree uh that the country is in um shape uh on almost any uh significant dimension uh that you can think of and you're all mature uh intellectuals so this isn't obviously going to come as any surprise to you for me to say that uh so robert putnam wrote this book i make reference to this book all the time for a variety of reasons and i think it's kind of interesting that he titled the book bowling alone which is a metaphor um for the decline of associational connections and affiliations in american society what we often call and what has come to be known as social capital so if there was a lot of talk about social pluralism back in the 60s and 70s this idea of social capital has become very very significant uh in the social sciences and people use it to describe a slightly different phenomenon uh but for uh putnam the idea was that these different forms of associational affiliation voluntary associations that we were members of different groups that we interact with in a variety of different social settings are fundamental to the vitality of democracy he's a political scientist and he focuses on that feature of society which generates what we might call uh tolerance uh openness um i i have norms of reciprocity there's a great article written by alvin gouldner one of my favorite sociologists on the norm of reciprocity and these diverse social networks that we're connected to constitute our social capital um now some people when they talk about social capital they're looking at it as a form of capital as an asset as connections which ultimately can benefit us personally materially in terms of our career but what putnam is interested in is the way in which because we have these diverse connections associational organizational they said they subject us and expose us to a variety of different kinds of people with different kinds of perspectives um outlooks political orientations uh etc and that is a valuable way to maybe moderate but also to develop and cultivate an appreciation for a wide range of perspectives we talk about polarization today so that pushes us in an entirely entirely different direction from what putnam is talking about here now he has this bowling alone and i often ask students you know what does he mean by this well actually the first thing i ask students is how many of you bowl very few people bold uh i like to say that bowling is a a kind of working class sport uh if you can call it a sport or recreation or hobby uh it has tended to be associated with uh the working class uh there is a joke that uh the larger the ball of the lower the class the smaller the ball is being used in the athletic activity the higher the class uh now you might say okay well bowling is working class and golf it's more upper class but that's just uh a way one person tried to with a little humor talk about differences in social class and different social class activities but what putnam is getting at here is people used to bowl in leagues okay that's the key they bowled in leagues and they gathered together once a week in a bowling league with people who came from different walks of life different backgrounds different experiences different occupations and while they were bowling they talked and discussed various issues personal political social whatever and that constituted for putnam a vital vital element and ingredient of democratic vitality that you had these experiences these these associational connections and you participated in different organizations or leagues which exposed you to other people and you developed a broader more open um tolerant worldview now people who have studied this idea of social capital say you know we have to make a distinction between bonding social capital and bridging social capital bonding is where you have affiliations but they're all reinforcing your preconceived political position you're interacting with people that are very much like you you don't have any kind of diverse range of contacts or associations or acquaintances that's bonded what putnam is talking about is bridging social capital that is bridging meaning that there are connections being made across different experiences different groups different backgrounds ethnic racial political generational so what somebody might say is today we do have certain forms of social capital but it tends to be bonding and that is it simply reinforces our preconceived ideas we're associating with people like us and it's an echo chamber and that essentially reinforces polarization there's lots of interesting ways we can talk about the idea of social capital all right now i haven't really gotten to the issue entirely of what i mean by pluralism versus elitism so let's keep moving so back in probably the 60s maybe even into the 70s depends what discipline you were in what courses you took as a student of pluralism was to a large extent the dominant political science approach to understanding uh political power the nature of democracy the nature of legislation and decision making in democratic institutions and the idea is that just as i mentioned with social pluralism where you have a whole range of affiliations and associations and organizations that you're connected with the idea is that in society there are different interests uh that are expressed by different groups occupational groups organizational groups business groups religious groups you you name it and so you have social pluralism and it is typically uh expressed organizationally and those organizations have different interests on particular uh policy issues and so there are lots of interest groups and interest groups are a key part of the pluralist model and the interest groups sort of compete with each other uh to shape legislation policy and various kinds of social political economic programs but the point is there's a wide range of interests there's a wide range of interest groups they organize depending on the issue that's being discussed within the legislature on some issues they have no interest on other issues they do they organize they lobby they express their views and the bottom line here is that you have a whole range of interests no single interest no single group no single social class dominates sometimes they win in one venue in one area in one issue arena sometimes they lose okay so the idea is there's a lot of compromise there's a lot of bargaining when interest groups are competing with each other sometimes interest groups may not even have an interest to participate to try to influence policy in one particular area but they mobilize in another and so you've got this panoply of groups organizations interests they attempt to shape policy outcomes by the state at different points in time and so again the key point here is that no single group has a position of dominance and political overriding influence in the political system right uh doll is associated with the term polyarchy ruled by many okay paulie multiple sources centers of power reflected in the range of interest groups ruled by large numbers depending again on what the issue is etc so this was the idea of pluralism and it was a very agreeable way to think about american democracy in other words people thought well this is fair this is fine uh if you're concerned with some policy area then you simply organize um and you attempt to shape that policy you express your interest through the political system through the conventional institutional means and it's fair and sometimes there has to be compromise sometimes i have to be bargaining if there are two groups that have very different views about the way legislation should go and that's the nature of politics so this was a very dominant image of the american political system in largely political science so i want to take a quote from c wright mills because we're going to move to him in a moment almost all of the sociology students have heard about c wright mills and he says not wishing to be disturbed over moral issues of political economy and i've emphasized political economy because when you start talking about political economy you begin to realize that there is in fact a concentrated source of political power and influence americans cling to the notion he says that the government is a sort of automatic machine regulated by the balancing of competing interests a machine that sort of accepts and takes the expressions of interest of different groups and processes those demands processes those organizational efforts and spits out a policy based on the particular compromise bargaining resolution of competition between interest groups now what c wright mills is basically uh inferring and suggesting here if that ain't the way it works and what does he think actually happens so excuse me for my clearing of the throat this is my morning congestion um elite theories i'm going to talk so you have the pluralist model and then you have these elite theories and the elite theories as i said focus on concentrated power among particular groups among particular elites um mills is associated with the power elite we'll talk about that in a moment you know we'll talk about uh domhoff who's also an elite theory so the point here is elite theories are essentially an alternative and within them an embedded critique of pluralism they reject the pluralist model and they advance a different view of understanding american democracy okay so you have pluralist approaches and you have elite approaches i have to tell you that over um the many years since robert dahl and a pluralist begin to present their model um different elite theories have largely supplanted a pluralism that doesn't mean that there aren't lots of different organizations that attempt to influence the political system uh the question is how successful are they and does that level of competition if you like uh actually produce the kinds of varied outcomes that one would associate with pluralism next week we're going to look at a very a well-known study that tried to look at exactly whether in fact empirically there was any support for a pluralist model so we're going to hold off on that but for the moment elite theories are presented as an alternative to pluralism uh okay so let's keep going so there he is the late hasn't been around for a long time late great sea wright mills if you're a sociology major you've heard of c wright mills uh he is held in uh sort of a a kind of folk hero within sociology a level of esteem one thing about sea wright mills which made him a sociologist that many wanted to emulate many respected he was a maverick um he was a social critic uh he never fell into the trap uh of many what i would call post-war post-world war ii uh social scientists who essentially became spokespersons for the superiority of american democracy uh american a foreign policy he was always he always had this critical edge even during a period when it was very unpopular um and in some ways during the 1950s certainly when you had mccarthyism and all of the red scare it was not a necessarily a wise career decision to be questioning whether in fact the united states was the speaking of democracy or whether american foreign policy actually promoted uh democracy uh and the social welfare of people around the world so c right mills was largely a maverick in that regard because at the time when at the very time when 50s and 60s when american society was regarded as the single greatest country in the world democracy in the world c right mills comes out and says well it's not really a democracy the way people think it is actually uh there are elites uh there are ruling elites and they ultimately control the society they make the decisions so he called into question at a very early point in the critical tendencies of uh social science and particularly sociology this notion of american democracy and he did this in a number of publications but the most significant the one that many students are exposed to even an intro to sociology is the power uh elite and i would put the power elite into the uh category of uh institutional elite theory and what i mean by that is uh c wright mills although he was a radical and some people might associate him with marxism and he probably had some sympathy with marxist analysis given the way he wrote about social class his identification of the elite was largely focusing on the individuals who headed up particular uh institutions within society okay so the representatives from particular institutions so it's an institutional analysis you identify which institutions are the most powerful and the people who tend to be in the highest level positions within those institutions and that becomes the way in which he thinks about the power elite so the power elite are drawn from military business and government those are the three institutions that he focuses on he views those as a dominant institutions in society and he notices that there is some significant movement across those institutions by individuals so they um head up those institutions they share a certain particular outlook and they will move from one institution to another so you might have people for the military moving into the government you might have people from the corporate world moving into uh government from government into the corporate world so there's a lot of movement around these three major institutions now i'm also identifying here in this slide a few other ways to think about these multi-centers multi multiple centers of power when we talk about the military it is very important to recognize something many people bring this up because we have totally ignored the warning by president eisenhower when he left office let's remember eisenhower was a republican uh he was a general uh that was the basis for his popularity and he was an extremely popular president he won two elections uh handily over adelaide stevenson and eisenhower warned when he left office about the military industrial complex and his concern was that the military was becoming too powerful it was connected closely with corporate business interest defense contractors and that this complex what he called it the military combined with corporate interests was essentially making significant decisions about foreign policy military policy and domestic policy and he warned that they had become far too powerful this was a republican military man making this warning as he left office we obviously did not heed the warning today the military and these corporate um appendages uh the defense industry are as powerful and more powerful than they've ever been right so we've totally ignored that um but he was identifying those sources of power now if you read the article by don hoff he makes reference to michael mann networks of power this is important to think about the sources of power in the society ideological power right who disseminates the ideas that are put in our head that shape our political opinions that shape whether or not we're going to act or not whether we're going to be passive the ideological apparatus what gramsci called ideological hegemony obviously economic power corporate power class power military power as eisenhower had warned us um and political power people who occupy positions within uh political institutions so those are the networks of power and for uh mills he wanted to look at the relationship between these all right so he looked at the economic the political and the military so in some ways people have said um you know mills is more of a vaberian than a marxist because there's sort of this multi-dimensional approach to understanding uh power rather than focusing on what is being dominant in relationship to the others all right so the big debate the big battle i say the battle lines have been drawn were between robert dahl who i just mentioned before and probably his best known book that advances the pluralist model titled who governs and there's a photograph of a robert dahl if you're a radical political sociologist you might view doll negatively over the many years i have come to respect uh his work uh later in life he realized that there were um significant sources of concentrated power uh that totally challenged uh his view of pluralism i believe he passed away in like 2014. anyway he's a giant in political science and he put out the book who governs and then we have the sociologists maybe a psychologist sociologist william donov there's a picture of donov he's still around he still writes he still attends the american sociological association meetings i've seen him there i've met him a few times a charming individual uh but don hoff takes on doll to some extent you could say that domhoff's life work and he actually has two strands of intellectual interest i'll mention the other in a moment he he his life work is essentially showing in every possible way that there is a ruling class in the united states and he wants to identify that ruling class and he wants to show how the ruling class rules and he does this by publishing book after book after book amazing amazing career record prolific writer engaging speaker and a very pleasant human being uh i might add um his other area of intellectual interest and he uh i think has published uh several books on this topic are about dreams that's why i said he might have a background in psychology but we claim him as a sociologist um how did he make this connection between you know studying dreams and what dreams mean how you interpret people's dreams and um the ruling class well he was studying the dreams of the very very wealthy he was actually studying the dreams of what we might call the capitals class so there was that connection so let me show you the next slide gives you a little flavor i don't think this covers all of the books that don hoff has written with regard to this question of who rules um but any time the pluralists and doll was not the only pluralist there you know political science was dominated by pluralism so there were all kinds of pluralists a guy named nelson polsby uh published pieces uh showing no no you're wrong dom huff you're wrong in fact we do live in a pluralist democracy and here's why and don hoff would say okay i'm going to show you why you're wrong and he write another book so let me just give one example um they said he said well you know the corporate elite dominate the capitalist class dominates the pluralists come out and say look we have a party system we have two parties just two we know the problems with that but basically and there's a party that represents labor in the working class so there's pluralism even in this narrow two-party system tom hoff says i don't think so so what does he do he writes a book titled i love the title fat cats and democrats you can see it up there in the top do you understand what i'm saying so anytime the pluralist came back with like yeah yeah you know you say there's a ruling class but he says no there is a ruling class and he writes a book and he makes an argument and he lays out a theoretical conceptual and empirical strategy to identify the ruling class this is one of his major contributions the reading you have of donov is part of a larger website he has which is designed to assist people who are interested in doing what we call uh power uh power elite analysis investigations power structure i should say power structure uh analysis you can do a power structure of the city of jacksonville uh who rules jacksonville i actually put together a small group of students um some time ago after i arrived at the university of north florida me and professor christa paulson put together this little seminar group and the students collected information about various individuals and sources of power in jacksonville who rules jacksonville so power structure research is what he has made the most significant contribution to in terms of other scholars taking his model taking his approach taking his strategy for locating the centers of power in any kind of a social universe it could be state level it could be city community nation etc so i wrote all these other books who rules america now which was an update uh based on additional information that had been gathered over several different administrations and the last book uh that's listed here the myth of liberal ascendancy i'm gonna talk a little more about that um later so i'm really um you know trumpeting to some extent uh the wonderful contributions of william tomhoff and again he tries to help us locate or answer the question whether in fact there is a ruling class and one way he does that and then you're reading you saw this you should ask three questions who benefits the most from policies and decisions when there's a contest in competition who tends to win consistently and who actually governs who is making the decisions and on what basis so in his empirical strategy he tries to answer these questions and he tries to show that if we ask these questions in the context of american society we can see that in fact there is a ruling class and again the idea of a ruling class is not something that most americans are comfortable with even though if you ask in surveys do you think you know political decisions tend to be made by the people or by a small group of you know people tend to indicate that they understand that in fact decisions tend to be made by a small number of people they may not call it a ruling class but they do understand that in fact uh decision making tends to be concentrated in certain hands a kind of oligarchy the nature of the oligarchy the source of the power lots of debates about that but when domhoff started using this term ruling class you know to rub people the wrong way and certainly it rubbed political scientists the wrong way those who were the defenders of the notion that american democracy was the highest form of democracy in the world and a model for all other countries now for don hoff the key point is social class okay it's not institutional position it's social class the power lies in the economic domain and the source of power is the ownership of property okay for dom hof that's the main network and in that sense he is certainly more consistent with although i'm not sure he necessarily describes himself as a marxist but more consistently in line with a marxist analysis which would focus on class power then uh for example uh c right mills uh who is more maybe vaibarian in looking at the multiple institutional sources of power now how do they rule again you know you ask these three questions you see whether there is in fact this group and then the question is how do they do it how do they translate corporate power economic power capitalist class power into political power and he focuses on and he's written enormous amounts on this and again if you go to that website there's an enormous elaboration on each one of these the special interest process and he talks about the particular groups that tend to have the greatest amount of power to lobby congress to lobby the legislative branch corporations he talks about the policy making process where policies are developed where policy ideas get shaped and he looks at these various what we call think tanks institutions that are dominated and funded by corporations which put together white papers policy papers proposals on political and economic policy and then he talks about the candidate selection process and the role of money and the fact that the people who ultimately end up before you 9 out of 10 times when you have a choice between candidates which presumably is all that's required to label something in democracy that choice is between two people that have already been screened by the corporate elite because if you do any research on campaign contributions and the role of money in being a viable candidate in almost every race candidates must raise money to run a campaign that money must come from the sources of finance which is corporate power they have the money they have the dollars they're not going to give it to somebody who represents political economic interests that are opposed to theirs they fund those campaigns and so on to some extent by the time you're voting the candidates have already been screened by the capitals class they wouldn't have gotten there otherwise because they wouldn't have had the resources and the money to fund their campaigns in the first place so campaign finance campaign contributions and of course this has gotten even more grotesque uh with the citizens united and mccutcheon decisions of the supreme court that supreme court we're going to hear much more about the supreme court in the next few weeks all right something else that is very important is analysis is um this idea that there is a ruling class and there is a cohesion uh that they communicate with each other uh that they have a common understanding again this was one of the challenges of the pluralists they said you know you talk about this ruling class you know there are lots of people that have lots of money but that doesn't mean that they they conspire to uh you know make policy you know that there's somehow this cabal uh that gets together uh and you know secretly decides on things that should be done tom huff writes a book called the bohemian grove check that out it's a very very fascinating study uh of just one probably unique extreme case of where the most powerful corporate political individuals gather and engage in all kinds of bizarre rituals a place called the bohemian grove outside of san francisco somehow he got in there somehow he had people who provided him with information about what goes on there because it's a very private secretive kind of um retreat for the rich and the wealthy but his whole point is that there is social class cohesion so this is another part of his analysis there is cohesion there is corporate culture these people do talk to each other there are interlocking directorates meaning you have one person who sits on multiple boards of corporations so if you have lots of people sitting on these boards they see each other they talk to each other they socialize with one another they are members of the same social clubs when we did our analysis in jacksonville we actually were looking at what country clubs are the most exclusive and who is the member of those clubs right this is where people get together they play golf together they sit down together they have lunch they have drinks this is where many decisions are made i had the debutante there because when i was um a graduate student in political science in st louis washington university in st louis i worked with the sociologists as i was shifting over to sociology and he was looking at the power elite essentially of st louis and you looked at the debutant ball these daughters of the wealthy who are they come to have this coming out it's called that has that is a slightly different meaning today but then the coming out was they would appear at these debutante balls and so this person that i worked for his name was richard ratcliffe he would go through the newspaper and pull out the names of these women young girls becoming women who are participating in the debutante ball as a way to locate because only the rich and the wealthy engaged in this kind of activity it's very kind of southern sort of thing as well but not exclusively southern but it is associated more with the south in any case again this reflected a certain like style of life gatherings of people who have enormous amounts of money and of course they're passing on this level of privilege to their children he looks at the merging of ownership and management interests within corporations and i mentioned the bohemian grove study so my point is to have a ruling class you have to have a class that has some level of cohesion right they can't be totally uh dispersed they can't be totally unorganized they need to in some sense communicate with each other now i'm going to introduce a book that was written recently that talks about the fragmentation of the capitalist class and there's some debate about that but it's an interesting argument about the more recent developments within the capitalist class and by recent i mean maybe the last 25 to 30 years this work like the bohemian grove is probably done in the 1970s or 1980s okay so again class cohesion very important the policy planning network i mentioned before there were these there are these think tanks they're called think tanks uh but they're major foundations major institutions where policy is developed by the most powerful sources in the country corporate sources of power so most of these institutes if you go to their web page and you look at the board of the uh institution of the foundation or you look at the affiliations uh it's all big money big corporate power driving all of these one of the most significant today is the heritage foundation i have links to them here but just you know if you're curious check them out go to the heritage foundation and just see what issues they're pushing currently in the context of what's happening in american politics today the debates over policy today so the big ones of the heritage foundation the american enterprise institute these are conservative i would call them somewhat libertarian sometimes right wing financed by corporations that basically hammer out policy statements white papers reports that are passed on to legislators the american legislative exchange council we talked about that one before they tend to focus on state legislatures uh the hoover institute council on foreign relations very very powerful organization and i have a um image of a book that was written about the council council on foreign relations called wall street's think tank that gives you some idea of the approach that that book takes linking the development of our foreign military policy to the interest of the corporate lead to wall street and the cato institute uh those are five of probably the most significant policy planning foundations that influence policy making in the united states and since supreme court issues are coming up you can check out the federalist society since that is the source for candidates for positions on the supreme court um i like to throw out these quotes occasionally this one again from c right mills it is important not to confuse freedom uh with social power and this is something that americans often do well we live in a free society and because it's a free society it's a democratic society and we all have the right to do this and the right to do that and therefore we all exercise power he wants to make a distinction between this notion of free to do what you want by the way later we're going to have a reading where the question of whether we actually have as much freedom as we think is called into question but for the moment let's focus on c right mills quote here there is a distinction between social power and individual freedom when it comes to social power most americans have very little if and again this goes back to his argument they are not part of the power elite they do not hold positions within the major institutions of the military corporations and government um i put this uh study in here because i think it indicates how association connections that we have shape politics a couple weeks ago we talked about the important role of labor unions if you're a member of a labor union you have an educational function and it's a way to develop a level of class consciousness and norms of equity and it mobilizes people to participate in politics in a particular way well as it turns out and we might say that this is probably just a reflection of the neo-liberal dominance of our political economic system today and that is that it's the employers that seem to be shaping the political orientation of their workers think about that okay so i highlighted read a few of the main points of this study very interesting i would urge you to check it out so the question is you know how does this affiliation that is affiliation with our employer shape our politics right and in what ways do employers deliberately deliberately try to direct our politics in a certain direction so american employers are increasingly engaging their workers in the political process channeling their employees into politics in ways intended to support corporate interests they do lots of research to show the different ways they say what's most effective is when employers use warnings of job loss to motivate participation and when employers could monitor the behavior of their employees suggesting that employers are indeed acting as a type of political machine so when we think about all of the ways in which corporate power translates into political power just as we talk about the ways in which labor union members as a member of an organization might have their political orientation shaped what we now see is that it's employers and corporations that have a much more powerful influence and because of the total almost universal absence of labor unions there's no countervailing power important concept countervailing power okay i don't know if i'm going to get through all this in 60 minutes let's see oops okay all right so we're talking about these different sources of power and i just want to bring up some uh terms concepts that people have used to try to understand the sources of power the um duo backrack and barrett's i read their stuff back in the probably 70s i loved it because it was very critical of the pluralist model and they said there's two faces of power i know i have three faces here because i'm going to add a third by the sociologist luke's maybe a political scientist political sociologist let's call him that in any case uh bachrach and barrett's said there's the open face and the secret of face the open face is the ability and we can see this of particular groups to make binding decisions to make decisions that impact and influence all of us right we may not like it but we can see it but they say there's something equally important that we don't see that people don't pay enough attention to it's not the open face of power that is the raw exercise of power by powerful groups it's the secretive face and what they mean by that is that there are another source of power significant source of power is determining what in fact is put on the table for discussion what policy options are actually open there's this concept called the overton window right meaning that if the window can be widened there's a wider range of possible ways to think about different kinds of policies but if you narrow that you basically say okay we have a policy decision to make here are the options who decides what those options are right this is what they mean by the secretive face who sets the agenda what is on the agenda why is one thing on the agenda and not something else why is it that universal health care for years until just recently was not on the agenda at all when obama developed obamacare affordable care act universal health care medicare for all was taken off the table it wasn't on the agenda okay so what they're saying is one of the significant sources of power is who determines what's on the agenda and of course if we believe that it's class and corporate power there's obviously some things which are acceptable to them and there's other things that are unacceptable the third face which luke's talks about is the deceptive face and that is the ability of the corporate elite uh the powerful uh the ruling class if you like to use dom hof's term the ability of the ruling class to shape the way we think so that's the ideological dimension right so we had the two faces of power luke said let's add a third face which is equally important and that is ideological manipulation ideological hegemony in the language of uh antonio gramsci what i sometimes call cognitive capture i'm not sure if i've made reference to this term before because we talk about regulatory capture and cognitive capture and cognitive capture is where you basically have penetrated the thinking and the understanding of uh policymakers for example or of the larger population where they almost automatically think that oh in order to see the economy expand we need to simply reduce taxes and cut regulations that becomes almost an automatic reflex of almost all political figures democratic party and republican party the neo-liberal cognitive capture right so ideological manipulation ideological hegemony shaping the way we understand how the world works and what is feasible and what is possible right so think about all of these different faces of power okay i'm getting close to 60 minutes i forget where i am um i think we talked about this before restraining myths restraining myths are a source of cognitive capture a source of ideological uh hegemony of ideological manipulation because we have these ideas in our head and once we believe them we are less likely to call into question to act politically to be active as political participants the american dream anybody can make it it's a fair equal opportunity society there's freedom there's mobility or tina there is no alternative this is the best possible system we can have so we talk about those myths right that have penetrated american consciousness and ultimately shape our politics i also mentioned limiting constraints it's another way for people to think about deception imagination constraints that people simply automatically oh we can't do that it's too radical oh we can't do that that would never pass um that's not feasible in our society often when you propose radical policies maybe i shouldn't call them radical policies policies that could substantially improve the quality of human life people say we can't do that or what they often say and we're going to address this in more detail in a few weeks there's no way we could afford that that's too expensive we don't have the resource who's going to pay for that these ideas have been shoved into the brains of americans to the point where they have no political imagination to think about an alternative world an alternative way that we can organize this and by the way the elite want you to think this way they will thank you for thinking this way it saves them a lot of trouble because you're not even going to demand a significant change in policy because you don't believe it's policy pie in the sky utopian too expensive remember imagining alternative possibilities is what politics is about that's what energizes people these individuals do not want you to be energized i'm hitting 60 minutes i'm going to get cut off so i will see you on the other end you

Political_Sociology_Lectures

Week_5_lecture_Class_and_Politics_plus.txt

okay students welcome back to political sociology and as you can see i am in a different location an undisclosed location for my own safety since i know there are forces that are interested in dragging me away for my political views just kidding in any case it's tuesday morning it's a little chilly out i'm outside of asheville north carolina and i'm happy to be back talking about political sociology and this week the topic is class and politics as i was thinking about this i thought it might be a little redundant since the entire uh area field of political sociology is essentially founded on the question of the relationship between class and politics and we've certainly touched on this significant ways already throughout the semester and we will continue to do that when we discuss questions about who rules the ruling class and the relationship between capitalism and democracy so by virtue of living in a capitalist society it's inevitable that class becomes one of the most significant and for some people the most significant there's arguments about this the most significant factor in understanding what happens politically so let's get started and i can't use with voicethread my remote mouse so i'm going to have to move it here and advance the slides by leaning forward not what i like to do but that's voicethread now if it turns out nobody ever uses voicethread and a few of you have thank you very much but if it turns out nobody's using voicethread for the purpose it's intended which is to as you're viewing the slides assuming you are viewing the slides you would pose a question or a comment or reaction there is another method i'm using in my intro class that's probably simpler and it does allow me to use my remote mouse which is important to me okay what can we say about class and politics well let's at least acknowledge that the most significant social theorist in my view of course is uh karl marx and we could spend an entire semester really talking about the relationship between marx's theory his understanding of capitalism class and politics uh so i'm just introducing a couple concepts that you may have been you may be familiar with you may have been exposed to in other classes maybe social theory um so as you may recall one thing we uh discussed earlier in the semester uh was the view of marx in terms of the rise of capitalism and the proletarianization that means that the capitalism can only survive can only operate if you have uh people who have nothing to sell but their labor power they have no access to property no access to independent means of subsistence they have to work for somebody else if you don't have a large percent of your population uh that falls into that category uh capitalists have nobody to exploit and you don't have capitalism uh so one of the you know when we talk about like social change and international development we talk about the rise of capitalism the transition from feudalism to capitalism uh as uh marxists like to talk about it and um it's a process of proletarianization creating the proletary so in any case um when marx talks about a class in itself he's basically saying that there is objectively a huge huge percent of the population in all capitalist societies that are objectively working class that are objectively being exploited because exploitation is simply a necessary condition or for capitalism uh so that's a class in itself right and we could we could spend a lot of time talking about this concept okay so i'm skimming over something which um perhaps i've spent too much time thinking about and therefore i'm qualifying all these things right um so a class in itself you have you know a certain percentage clearly the vast vast majority of people in the united states sell their labor power for a wage even these people in the middle strata the middle class this contradictory class locations those things we talked about now the point is just because you're objectively working does not mean that you have class consciousness that you understand that your class position exists in antagonism in tension with in opposition in conflict with other classes particularly if you're working class the capitals class of bourgeoisie so the key to what we would regard as class politics where you're mobilizing people around the issue of class would be moving from a class in itself that is you have a class structure to a class for itself where that class structure translates into political struggle so when marx talked about class consciousness it's class consciousness that's required to translate a class in itself to a class for itself some of you made reference to the term hegemony or hegemony however you want to pronounce it a cultural ideological hegemony and of course this is a way to penetrate the consciousness of people in capital societies and if you have a dominant ruling class ideological hegemony what we call ideology right uh this will thwart and is intentionally designed to thwart the extent to which workers develop class consciousness make reference to restraining myths beliefs about the way the society works which makes it less likely that people are going to engage in radical political activity which might both question and ultimately overturn the existing arrangements and that ideology is extremely powerful we talked about neoliberalism as one aspect of that okay so i just wanted to say something about this idea of class in itself class four itself how do you translate one into the other this is the big big big question uh for those who want to mobilize the working class around class politics so a few little items i wanted to mention with regard to this some of these we sort of touched on before but in the context of having a discussion about class in politics um as you know most people don't like to talk about class in the united states there was a time when well we really don't have social classes you know it's a middle class society middle majority uh we talked about that you read about that in the first week some of you found that topic uh that concept that middle majority concept very interesting and the idea is well you know we live in a middle class society and if you listen to the rhetoric in the language of most people who are talking politically they rarely rarely talk about the working class they talk about the middle class we want everyone to have an opportunity to be live a middle class lifestyle and uh biden is currently putting out an ad and i had some criticism of this that i posted on facebook um where he's promoting middle class values biden represents middle class values and there is a racial element to this notion of middle class values i won't go into it now but the point is all of the focus tends to be on this middle class this amorphous difficult to define often more associated with lifestyle than with one's actual position in the realm of the economic system the social relations of production that is at least from a marxist perspective how you define class you don't define class by how much money you make you don't define class by where you live if you live in a suburb you don't define class by uh the commodities you own from a marxist perspective you define it by whether or not you sell your labor power for a wage or you own property that provides you with independent income not requiring you to go out and sell your labor powerful wage where do we see class and politics right because we don't really see it in the political system we've talked about this we have this two-party system the two-party system is dominated largely by corporate interest both parties are dominated by corporate interests when we talked about labor last week we talked about the fact that you know why is there more labor activity union membership in canada versus the united states same kind of thing we ended up talking about organization and political party one place where you do see some sharp class differences between voters and non-voters and again you all read an article about this if non-voters voted if they were mobilized if they had an incentive to vote okay i'm not going to shame them uh for not voting then you would probably have a much different kind of political policy landscape because the people who don't work i'm sorry the people who don't vote tend to be working class less educated would have an economic interest in progressive dare i say radical political social democratic policies and voters tend to be more highly educated uh wealthier and therefore you have some notion of class politics but it's between voters non-voters by the way just to make this point clear we have a two-party system versus the european countries you can even include canada here okay because they have the new democratic party so they have something of a multiple party system certainly they have a viable third party where you have multiple parties unlike the united states you tend to see not only higher levels of voting and political participation but you do not see the sharp class distinction between voters and non-voters so think about the role of these institutions party institutions right and that brings me to the sec the next point no working class or labor party we have no working class or labor party so some people say class produces party some people actually say it's party organization that produces class what do you mean by producers class consciousness or as we made reference to last week the class idea that exists in canada the other aspect of american politics which tends to minimize the effect of class and the way we like to think about in terms of economic terms is that there's a great deal of focus between the two parties not so much on economic issues and the organization of the economy remember they're both neo-liberal parties but there's an emphasis on the cultural issues and that tends to confound the relationship between social class and the political party one identifies with so you see working-class people identifying with the republican party not because they agree with the republican party necessarily on economic issues but because the economic debates and questions about how to organize the economy are off the table and all we're left with are cultural issues the social issues on which the working class tends to be more conservative so if the democrats actually wanted to be really successful they would be focusing much more on economic issues there are probably reasons why they don't do that we'll talk about that we also have in the united states you can't rule this out this is a whole another conversation and later we will have some readings on race and politics but in the united states race has divided the working class this is a deliberate policy uh by the capitalist class by the ruling class to try to create divisions within the working class which makes it harder for the working class to mobilize and so this gets us into the whole issue of identity politics universal policies which would be applicable to all rather than means tested i mean we can we'll talk more about that as we get into the semester but unified working class political party doesn't exist at all and i've just touched on a few of the aspects of that [Music] okay so this is what we would expect i think we've talked about this already and the right party would uh essentially mobilize and draw support from the wealthy the rich uh the middle you know whatever the middle is okay and i'm gonna use that term right now just for the sake of this this graphic illustration you know some may go right some may go left but the working class goes left right now this is if you have a left party and a right party right now what does that mean in the united states we sometimes say well the democrats are left and the republicans are right but hopefully you understand how narrow the differences are in many ways between the two parties and we only have two right we don't have a working-class party we don't have a labor party so there really isn't a left party there's a party where presumably the left has no choice but to go there because they're not going to go to the right point party well we know what happened there because working class people in fact do support the right party in the united states and that is the republicans all right so i'm just laying this out conceptually graphically so we've already talked about this quite a bit um but we have no left party we have two-party convergence on neo-liberalism this has essentially been the case since the 80s and so under these conditions where you know voters perceive working-class voters perceive that you know one party comes one party goes i don't really see any significant difference in the way they're pursuing economic policies it is impacting me as a working-class person then of course the socio-cultural issues become more salient that becomes the major dividing point between the two parties and as i said before working class people while they're economically left they tend to be on average okay socially culturally conservative and if they don't perceive a difference on the economic dimension then the republicans have shrewdly cleverly lure them on the basis of what do we say god's guns and gays the social cultural issues and there's many more right immigration all these other kinds of things um so you know the democratic party has essentially seated the working class okay to the republican party because of the focus on cultural issues and then what you get are the working class and i'm not saying all working-class people move to the republican party but in this dynamic working class people may be moving to the right in ways you wouldn't normally expect theoretically and you've got people in the middle again moving both ways but you have the um highly educated professional class professional managerial class pmc as it's called um moving to the democratic party largely because they're more liberal on social and cultural issues based on their education all right i think that's one of the most significant things you need to understand about the current state of american politics okay well where is there uh some class of differences uh what i would say is and i actually put this together during the uh 2020 campaign a primary for the democratic party and you can see that i have here um two wings of the party so there actually is um class voting fairly sharp class division within the democratic party today um much of this is a result of the consequences of the obama administration uh the frustration that people had uh during that period after the great financial crisis uh the rise of occupy wall street is a social movement and ultimately bernie sanders within a democratic party organizing people around a social democratic left-leaning political program that attracted great great support from the working class right this is not the constituency that the democratic party is necessarily interested in mobilizing we could say more about that but sanders was on the other side you've got the progressive neoliberals we've we've talked that we use that term and nancy fraser term we're going to go back to that as well uh buddha judge i sort of put him in that category i had some friends who thought buddha judge was a great candidate i am you know no not my kind of candidate at all i have no regard for the guy uh in any way but i can see how he would be uh attractive to uh the professional highly educated class uh given his pedigree given his biography given the way he speaks and talks so we do see some sharp class divisions with the democratic party i don't know how all this is going to shake out last time i said i thought there would be a realignment and i think it's going to be the progressive neo-liberal wing of the democratic party will essentially be the democratic party it has largely been the democratic party before but what i'm saying is they will continue to dominate and it looks like they're attracting people who are alienated from what has become an extreme right-wing republican party they'll join together and i think the progressive social democrats will be out in the cold and therefore they held a third-party people's party uh convention recently to talk about what it would mean to have a third party you should know now if you read uh lee rudman's a piece on the two-party doom loom doom loop that the you know you're not going to have a successful third party in the united states or viable third party until you change the electoral system away from the single member district winner take all first pass to post electoral model but people are talking about and that's that's that's a good thing uh because we need more parties uh we certainly need a third a fourth a fifth party i would support all of that where people can actually identify a party that represents their interests rather than year after year election national election after a national election lesser of two evils lesser two evils this is not you know promoting democratic vitality okay i get carried away with these things let's see if i can move this thing somewhere there we go i don't know if you can let's put let's put it over there that's good all right let's put it over there um okay so uh this was a graph thomas ferguson is a political scientist somebody who i have enormous respect for he developed the investment theory of politics and um he's been doing a lot of analysis on um the debate between whether obama not obama i'm sorry whether trump's support uh was a function of racial resentment or whether it had to do with economic uh insecurity and hardship and he tends to focus more on the social class uh economic aspect but here you can see uh the point we're trying to make where the democratic party uh is uh divided uh by social class uh where we're looking at the relationship uh between the um let's make sure that we have this median household income in these various towns okay and the vote for sanders i'm going to talk a little bit about this level of analysis data okay because these little dots here are not people okay they're towns and this is data aggregated to that town level okay so we'll talk more about that um but this is simply some of the analysis that's being done by people like ferguson showing that there are some significant class cleavages if you like class differences class divisions within the democratic party so i like his work and i wanted to share that with you and you can go to the source i have the link there um if you read uh richard florida's article uh richard florida has kind of a long history of developing some interesting ideas some of which turn out to be somewhat problematic uh one was uh there was um an analysis he was doing you know maybe 15 years ago where he was looking 10 to 15 years ago gray was looking at the relationship between the characteristics of urban areas and the extent to which they were attracting jobs in high paying high technology sectors he came across a very interesting finding and i'll just bring this up because i think it's interesting and it um also poses some methodological questions and that is he found that one of the strongest predictors of whether a city an urban area a region attracted um high-tech industry jobs uh was the percent of the population that was gay now it's kind of an intriguing finding right um so what was going on there why did he find this strong correlation between the percentage of the population gay in the city the urban area and the attraction of jobs high-tech sector good well-paying jobs um now in my uh data analysis class we sort of talked about this as a spurious relationship it's not that you know having uh a population of gay citizens in your uh city uh automatically produces an increase in the number of good jobs rather these two things are correlated with something else right and what he found was it had to do with the larger climate and environment that existed in these cities in these cities where there were lots of cultural amenities there was good restaurants there were museums uh there typically was often an institution of higher education located in proximity and so the key was that if you wanted to attract industry high-tech industry at that time that's what every city was trying to recruit because they bring good jobs high-paying jobs lots of tax revenue if you wanted to attract them you had to understand that they would only locate facilities they would only bring their workers to some place they could only recruit their workers to some place that had a certain political and social climate right and that political social climate is reflected in lots of cultural social amenities as well as and i have the three t's there so i just want to point he's he said the key to urban economic development is talent technology and tolerance tolerance okay that is these highly educated workers are very liberal and they do not want to nor would the company that locates there be able to attract them to a location uh that was reactionary and conservative now when i talk about this i often mention oh by the way when i first moved to jacksonville somebody told me that the most powerful institution in jacksonville was the first baptist church case closed okay you understand if that is the most powerful organization institution in jacksonville we ain't gonna be recruiting these industries and these workers this is not the kind of place they're going to want to be now of course jacksonville you know has changed over time and you know hopefully there's a little more tolerance um but that that has something to do with the fact that this is not a high-tech mecca at all we've never been able to attract those kinds of jobs so he was he was interested in this and he got a lot of um prominence and he was invited by all of these or urban planners and mayors to to come and tell them you know how could we you know resuscitate our city and he pointed to his his model of the creative class right he called these the talented people he called him the creative class mental highly educated mental labor highly educated professionals you want to bring them to your city now what ultimately happened i'll just tell you how the story ended because it's kind of interesting and richer florida is a decent human being he's a good guy he does but what happened was a lot of the the cities and the urban areas that followed his development prescription ultimately went down the road of gentrification so city after city that tried to institute what uh richard florida was recommending ultimately ended up gentrifying neighborhoods pushing out working class um citizens in that community um displacing them and creating this kind of highly gentrified high cost of living high cost of housing communities and there was kind of a backlash against florida richard florida and he actually came out and did sort of a mia culpa and said you know what i i realized that what i was proposing has could use the great sociological concept unintended consequences or unanticipated consequences he didn't think that would place so when you read his article he's looking at the relationship between uh the occupation the dominant occupations in different states and the political leaning of that state right so occupation what occupations dominate and is the state democratic republican he uses the term red and blue and of course he talks about this class party inversion issue that was discussed uh by lane kenworthy in his article and that is that states that um seem to have the highest per capita income and of course that's related to certain kinds of occupations oh you tend to see that they go to the democratic party and working class states tend to go lower income states tend to go to the republican party right and so the key dynamic there as many of you identified in your reading and writing assignment is education the liberalizing effect of education now i just want to make one methodological caution here and this is something i always talk about in my data analysis class i'm going to bring it up here because it's it's important i just mentioned the spurious relationship between the percentage gay and the increase in the percentage of jobs and technology in an urban area you have to be that's a spurious problem we have a different uh problem here it's called the ecological fallacy and what this means is that when you are looking at the relationship between variables at a level of analysis above the individual right so we have survey data right that's individual level data and then we do collect data for cities counties states right and we find correlations like what percent of is there a correlation between the percent of the population in professional managerial occupations and a percentage of the vote democratic for the states is there a correlation there and of course florida's whole analysis is based on that the ecological fallacy is that even though you see a correlation between the percentage of professional managerial occupations workers in professional managerial occupations in the state and the percent voting democratic you cannot infer from that i guess you could infer but you cannot conclude that it was professional managerial class workers who were voting democratic that's all you can say is in a state that has a high level of professional managerial occupations and workers you have a high level of voting or higher level voting for the democratic party because this is not individual level data right so we can't infer to the individual level that's the ecological fallacy and people make this mistake all the time like you'll see a correlation between the percentage of poverty in a state and the crime rate that doesn't mean that poor people are the ones committing the crime you cannot infer that because the data that you have based that correlation on is not individual level data it's state level data okay so that's something that i think florida needs to be very careful with when he does his analyses and sometimes i think he's a little sloppy there all right okay you have read a chapter or two from piston's book this is a book i came across last year he's a political scientist and i'm going to give more credit to political scientists because they're beginning to do what i think is sociological work what took him so long um so piston is trying to show that in fact class is a very very salient factor in american politics now he's looking at it a little differently he's not looking at it the way maybe marx would look at it that the working class has certain kinds of attitudes right and they develop class consciousness and that translates in a radical political activity that's really not how he's uh entering this question of class and politics in the united states right what he's looking at is the extent to which people have attitudes positively or negatively toward the rich and the poor okay and what the consequences of that are and i actually was surprised to find um in his analysis and he utilizes a wide wide range of survey data to establish his points and hopefully you read this or you will read it and you'll appreciate it um the extent to which i was surprised the extent to which uh there are these kinds of difference particularly um sympathy for the poor okay so sympathy for the poor resentment of the rich okay so people think about the social world in these kinds of class terms based on his survey data right so i'm going to spend a lot of time in these charts but i'll let you look at them [Music] um there's lots of different methods that we use in surveys and there is a one survey that he spends a lot of time drawing from and it's a survey that has open-ended questions that's not common in your typical survey usually it's you know you make a statement like uh rich people are uh rich people deserve to be taxed more highly just a statement right and then people can agree or disagree with that and of course he looks at that kind of data too um but he also dug into the what we would call more qualitative data open-ended question that asks people talking about poor you know when people start talking about the poor it produces a dislike for republicans and they like for democrats okay you understand so when people start thinking about the situation of the poor poverty sympathy for them this translates okay this kind of class talk if you want to call it that rich versus poor that's the kind of class talk he's referring to translates into uh advantages for one party or the other right and talking about the rich produces the same kind of pattern so that's the basis of the book now he goes into all kinds of detail of various explanations for why you know there isn't more class political activity in the united states but this sort of gets at the heart and so what i'm putting in the slides are these tables which i i don't remember if each one of these tables was in the chapters you read so that's i'm just trying to provide you with a little more information about what he what he did okay now this is the more common um way to go about putting together this survey with the closed-ended question uh where you say do the poor uh have more or less money than they deserve and then you give people um choices right it's not an open-ended question uh they have to choose a lot more somewhat more slightly more the right amount slightly less somewhat less a lot less notice the construction of this survey item is designed to produce as much variation which is very important you're going to do statistical enough as much variation as possible so that you have quantitative differences a lot more the high end a lot less the low end but you have all of these intermediate levels in between okay so take a look at this um if these were not included in the chapters you read uh they ask the same question do the rich have more or less money than they deserve okay very interesting uh survey results there okay so i'm going to let you spend some time looking at these uh making sense of them thinking about it uh some of you may be interested in going to uh graduate school maybe in political science maybe in sociologists to get a flavor for some of the things you can you can do and remember these surveys he didn't create these surveys these are surveys that are done nationally everyone has access to them in my class we use the general social survey and then you can access these surveys and answer interesting questions like these okay so i hope that some of you will um think about that or maybe when you're taking the methods class or the data analysis class in sociology or the the one in political science uh that you might explore some question with the existing surveys that you're using in that class which is what we do in data analysis um all right so again um piston is actually making a kind of political argument in the sense that or at least one infers from his work that if the democrats want to be successful they need to start talking about rich and poor they need to stop talking about class all right i want you to think about that and that brings me to an article that came out by eric levitz uh he writes great stuff on the political landscape in the united states uh always insightful and basically levitz uh he read the piston book and you can find this article online and based on that he said democrats must reach out to moderates in 2020 by waging a vicious a vicious class war right now remember only in american politics if somebody says you know what the rich need to pay a higher tax will somebody say oh you're waging a class war right i mean this is the unsophisticated level of american politics right class war i was joking when i created this slide uh back in um i guess last year led by billionaire michael bloomberg right i was being facetious bloomberg uh remember he ran in the democratic primary and just this week he has indicated that he is going to invest i don't know is it a million a billion i don't remember massive massive amounts of money on ads in florida against donald trump so um if you want trump out of office and you know you want biden to win uh you might view this as something positive that he's gonna invest all this money but what does it really say about the nature of american politics when the billionaire has to come forward with massive amounts of his own money to somehow shape the results of the election we talk about russian meddling we never talk about the meddling that takes place routinely legally in our political system money my friends okay you know if you're going to make a pitch to voters on the basis of class wealth privilege concentration of power this is called populism and unfortunately and this has trump has a lot to do with this uh although there's a historical way in which particularly in the american context uh populism has been discussed there's a distortion of what populism means and i would strongly recommend that you um and you did read uh a couple articles by thomas frank from listen liberal uh thomas frank just wrote uh another book uh i'm in the process of reading it it's called the people know and it's a history of populism in the united states but it's also largely an indictment of the democratic party's failure to embrace populism because the democratic party associates populism with as i say right-wing reactionary demagogues like trump um racism anti-semitism xenophobia they all associate this with populism they assume that the population the people uh harbor these kinds of attitudes uh and therefore uh we should essentially minimize the level of participation to some extent of these what did hillary call them deplorables right uh so what he points out is that if you look at american history if you look at the history of populism in the united states uh it involves the progressive mobilization of people that's really what populism is the mobilization you know when you think about democracy it is about the mobilization of the people the deimos the people against inequality against the concentration of wealth and corporate power and institutionalized systems of exploitation why wouldn't the democratic party take advantage of the fact that and as revealed in the research by piston that you can mobilize people when you make reference to rich poor and play on okay you could say that the fact that people do have some resentment about the concentration of of wealth and power in the united states was any of that mentioned during the democratic convention they were too busy inviting republicans in to support a bible so you know this is this is a question um in terms of political strategy and the viability of the political parties to even engage in this kind of thing obviously we have right wing a certain kind of right-wing populism let me just mention some of these terms somebody came up with this wonderful term pluto populism which describes this person was trying to describe a trumpian uh right-wing populism and basically uh plutocrat right that's the rich the wealthy um the owner class the capitals class you know plutocracy whatever you want to you know call it a rule by the rich ruled by the wealthy uh he he this author i can't remember his name but he threw out this term pluto populism basically what that means is that you mobilize people around issues like immigration or you tell them you're going to cut taxes and that kind of stuff because that's good for them but ultimately the populism you're generating support for yourself but ultimately you're instituting policies that almost exclusively benefit the wealthy so the tax reform act of 2017 under trump would be a classic example of pluto populism you phrase it in the context of you know less government cutting taxes you know giving people back the money they deserve and then you put in place a tax reform bill which disproportionately disproportionately benefits the top 10 population right um going back to nancy frazier you read one of her articles she talked about reactionary populism yes there are forms of reactionary populism but there's progressive populism democratic party is afraid of populist afraid of the people the people know um so as i said democrats reject populism uh populism populist appeals uh would bring many working-class people back to the democratic party that have moved over to the republican party because the republican party does appeal in ways to the working class in ways even though they obviously have no interest in promoting the interest of the work they can appeal to them i don't know the democratic party thinks they're above all that kind of thing so why don't they do it well you could say it's this irrational fear of populism right this association of populism with reactionary right-wing demagogues or you could say well even though they might win some elections that way maybe the 2020 election the constituency that supports the democratic party financially has no interest in promoting those kinds of appeals or policies because it repre it threatens their political economic security calling out the financial sector i mean that would be a wonderful kind of populist appeal that would have massive massive levels of support among the population you don't hear a word about it well the democratic party realized heavily heavily on donors in the financial sector what we call the financial oligarchy right and you saw this earlier presentation i made when kamala harris was selected wall street sigh of relief thank goodness you know i mean we know biden's on our side we weren't so sure who he's gonna pick as a but okay it's come on harris everything's okay we will continue to provide the democratic party with the money and bloomberg come on bloomberg represents wall street okay i don't know what to say who dare to say this okay this is something you don't hear in american politics this person said if there is going to be class warfare in this country it's about time the working class won the war because remember we've had neoliberalism since the 1970s early 1980s it's been a class war my friends and the working class has gotten their ass kicked on every possible measure of working class quality of life somebody said it's about time the working class one this is class struggle language this is class conflict language i'm gonna let you guess who you think that was who said this want me to tell you some of you may know it's bernie sanders remember we said that that division a class division within the democratic party quite significant you will not hear this language from biden you will not hear this language from harris you will not hear this language from pelosi or schumer or any of those members of the democratic party establishment and i think there's going to be a price to be paid for this fear of populism failure to address the hardship that many americans have been suffering and continue to suffer and of course during this period of code 19 you would think this would be the perfect opportunity to propose the most aggressive bold agenda to restructure economic production the organization of the political economy massive investment of the new deal sort it's the only thing that's going to really get us out of this economic depression over the long run we don't hear much about that all right i think i'm going to stop here because uh voicethread does cut me off at 60 minutes and i'm at 52. so i want to add a second section to this lecture which shifts a little bit away from the class and politics uh and talks about uh this idea of generational effects and lane kenworthy and some of you mentioned this in your reading and writing assignment made reference to post materialist values and i want to talk a little bit about the origin of that idea uh that comes out of the work of ronald ingleheart and some of the ways that perhaps that concept doesn't apply the way he thought it would and the way it might actually be applied today in some kind of unexpected ways so i'll do that in the second half for now i will say goodbye and thank you for listening you

Political_Sociology_Lectures

Week_3_Lecture_Elections_Parties_Voters_Part_1.txt

good morning students welcome to week three of political sociology and this is the week we will be discussing i guess we can call it electoral sociology there are some connections between what we're going to be doing today and what we have been doing um at least last week in terms of the articles by frank and my piece on the double backlash boomerang and also nancy frazier's piece so to some extent this is a continuation but it also provides you with some additional insights concepts theories on how we understand political dynamics largely in the united states so important for you to grasp these things given what's happening in the world at this moment in the united states and there's a lot to say about that but i'm talking about in the context of the political races that are taking place so before we get started i wanted to touch on a few items that either i didn't discuss in earlier lectures or which i need to introduce in order for you to expand your intellectual horizons all right so let's see if i can move these slides from over here all right this was a slide that actually was included in one of the powerpoint presentations i think week one uh and the reason i'm presenting this is because there's an enormous amount of attention in the media by the democratic party constantly pointing out after trump makes some idiotic ridiculous a statement or he has a press briefing and he uh makes claims that are simply uh absurd and they say if he's lying he's lying again he's a liar he has told this many lies during his entire administration well i understand how this can upset people and i understand why they harp on this uh but my point is uh that it's essentially counterproductive it's a waste of time to point out that donald trump is lying that's the first point i want to make right because everyone knows that and you're not going to convince anybody now based on a lie he tells that somehow that is going to sway them one way or another with regard to donald trump but there is a very important distinction to be made and this was made by a philosopher his name is harry frankfort and he wrote something uh quite some time ago so a short piece he's a he's an internationally world-renowned philosopher and he wrote a book some people call it a pamphlet and the title is on now you might say jaffe this guy is a internationally known highly respected intellectual academic philosopher what kind of title is that on well this just goes to show you that just as in sociology philosophers study just about anything and they bring insights to ways we understand the world that we had not had prior to their contributions now frankfurt makes a distinction between lying and bullshitting and the point i would like to make and other people have done this that is they've read frankfurt's work they've observed donald trump and they have written pieces that trump isn't lying he's bullshitting okay uh and i have the title of one of the uh media pieces on this at the bottom of the slide and the basic point is that liars uh know they're deceiving you and they understand that there is truth and that there is value in truth but they don't want to tell you the truth so they tell you something that isn't factual but they know that they're lying they know they're being deceptive and that's a certain level of acknowledgement and consciousness that liars have trump does not operate that way when trump says things it doesn't he has no concern with the truthfulness the factualness of what he's saying he says the things for the effect they have for how it impacts the people who are listening it's a rhetorical device he has no interest no concern with facts with the truth that's bullshitting and remember where does trump come from he comes into real estate sector right he's a builder he builds you know high-rises he builds hotels he sells real estate he tries to get people to invest that takes a lot of okay lots of people say what we need is a businessman well you know there's a lot of bullshitting that goes on in the business world and he's a master bullshitter so you know it doesn't do us much good to constantly say i can't believe what trump did it's such a lie it isn't it isn't factual it isn't truthful what are we gonna don't waste your time okay he doesn't care his followers don't care they're responding to the message and the way it's presented and the effect it has on them emotionally that's bullshitting okay so i think that's a an interesting way to take something that was written before trump got in office and became a prominent political figure in the united states and president uh by frankfurt and here we are taking the concept of and bullshitting okay and applying it to political speech share that with your friends and parents okay let's go to the next slide all right here's another piece i came across i just want to share this with you briefly and that is that um let me see if i can move my screen i'm going to try something here okay this is not gonna work oh yeah there it is okay i don't know if that if you can see that but it was blocking some of the text okay there were some arguments you know among the democratic party uh during the primary and you know some people were supporting sanders uh who was not a mainstream candidate uh he was not a centrist candidate he was on the left wing of the democratic party um that's largely why he was so popular because he wasn't a democratic centrist and people had enough of that democratic centrism and you've read about some of that uh hopefully you've actually read read that is opened it up looked at it on the screen and read the entire article if you did uh you will have learn something and you will be smarter and you'll be more intelligent if you're not doing the reading uh you will remain ignorant okay that's a little editorial comment so if you read that uh you do know that um uh the the democratic party has to you know emphasize the fact that you know you gotta have these moderates you gotta have these centrists voters will not vote for uh people who are outliers or more extreme candidates on the left to the right uh so there was an interesting study done by uh this political scientist and i have to say sometimes i criticize political science uh because i think sociology and the way we study politics is richer uh and uh more interdisciplinary but i i have to say there's been a lot of good political science research has been done uh recently and i'm gonna actually cite some of that uh today uh because i think that they're making some good contributions so anyway the title of this i do love the title of this right i mean the key to getting something published is to come up with a flashy little type man bites blue dog i like that okay a blue dog democrat you know what that is look it up okay i can't i can't tell you everything you have to autodidactism you need to look up some stuff yourself man bites blue dog are moderates really more electable than ideologues well that's the assumption in american politics that's the assumption in political parties and so people say well you know that candidate he's a little too far out there he's not going to appeal to the center of the political spectrum where presumably most of the voters are and this shows that historically it's been the case that moderates tend to do better but over the past several uh cycles it turns out that um candidates that are more on the um far end of the spectrum left or right within the party you know and it's pretty narrow to begin with uh they've done pretty well and they've done as well as moderates so the point is uh you can win elections by taking a more uh extreme if you want to call it that um perspective or one that deviates from uh the middle of the road the moderate the centrist position i think that's very important okay all right let's keep going here albert hershman was a political economist no longer alive wrote wonderful stuff about economic development and political dynamics i will touch on something else that he uh worked on and wrote about that pertains to different kinds of uh ways to argue about policies but he came up with this beautiful wonderful uh what i'll call a conceptual a triad and one of the things i mentioned and something i sent out to you today was you know what is a con do you know what a concept is because when i ask for two concepts you need to know what a concept is in order to extract and identify concepts that you find in the readings right exit voice and loyalty so he came up with this little conceptual triad and basically his point is that you know when you're facing a kind of unsatisfactory situation you're dissatisfied um with the existing state of affairs okay this could be the political party that you're in this could be your workplace uh you can apply this to many different situations okay and that's the key that's why i emphasize concepts because they're durable because you can apply them as a framework to make sense of things that are out there happening so exit voice and loyalty what the hell does he mean by this let's suppose you have a job and you're not happy you're dissatisfied you're poorly treated you're poorly paid working conditions suck he says when you find yourself in these kinds of situations and that's just one example you have three options exit leave find another job right that's sort of the market response right or if you don't like uh the customer service at a a business that you frequent uh i'm an exit i'm going to take my business elsewhere i'm going to take my money elsewhere right um if you have a job you don't like i'll just quit okay that's one option right uh the other option uh is voice now this is very different this isn't sort of operating in this kind of market mentality that as an individual i will simply move and go somewhere else as if you have the unrestricted freedom to do that voice is the democratic response that is conditions are unsatisfactory you speak up you try to do something to exercising your ability to express your dissatisfaction maybe voice would involve organizing with other people to change the conditions you don't like albert hershman wanted people to think more about exercising voice rather than exit because he placed a high value on democracy and the ability of people to engage and express themselves and to voice their concerns and to ultimately because he believed in progress that was the avenue to making things better and loyalty is simply well these people hired me you know i'm talking about you know the job you have that you don't like they hired me they're paying me i'm just going to be loyal to the company even though i'm not that happy i'll just go along with this so that's one example here's another example and this came up uh actually in 2016 where you had the people who were supporting bernie sanders and i guess you could say it's happened again in 2020 um what do they do about the democratic party they're not happy about the democratic party they're not happy that bernie sanders uh they believe was treated poorly uh that perhaps there was a scheme uh to undermine his candidacy uh during the primary and so what do people who maybe are uh registered as democrats i do even though they're very dissatisfied with the democratic party well there's exit that's it i'm done i'm going to go vote for another party i'll vote for the green party i'll vote for some kind of socialist party whatever or i just won't vote at all i'm going to have nothing to do with the democratic party that's sort of the outside strategy there's an inside strategy because people talk about outside inside you know you got to do both what's the inside strategy you organize within the democratic party with other people who share your views about the way you want the party to move and you change the party you understand and then loyalty i remember the democratic party i'm going to support the party no matter what even though i don't like the candidate even though i don't like the way the primary went even though i thought that maybe one of my candidates that i preferred i was treated unfairly those are the choices okay now often what happens when people are engaged in certain kinds of protest activities for example people will say love it or leave it i don't know if you've ever heard that slogan i heard this a lot when i was growing up and i would protest the vietnam war and other aspects of american society and foreign policy and imperialism and i was critical of the united states and i didn't stand up for the pledge of allegiance in high school people said love it or leave it jaffee now what are they saying here they're saying you have two options loyalty or exit not voice right they cross out voice this is a very undemocratic response and you hear this all the time from people right largely conservatives love it or leave it love america or leave america right exit or exhibit unconditional loyalty but don't voice your complaints don't criticize don't protest okay so you can see how this conceptual scheme applies to a lot of different situations trying to provide you with some tools concepts or tools you can use them and once you are able to use them and apply them your brain will be working at a much much higher level that's one thing we would probably want students to do for their brain to work at a higher level all right i cannot use my remote mouse in voicethreat robert merton some of you have heard of robert martin if you're a sociology major hopefully he was one of the major preeminent uh social theorists in sociology during the post-war post-world war ii period writing in the 40s 50s 60s 70s and um i've always liked some of the concepts concepts that he presented one was the law of unintended consequences some of you are familiar with this i talk about in almost every one of my classes it's a vital vital conceptual tool you've probably heard this or maybe something like the law of unanticipated consequences and uh there are sociologists one sociologist in particular alejandro portez who argued that the law of unintended consequences is the single most significant contribution sociology makes to sort of understanding and making sense of what happens in the world that's a pretty pretty powerful statement so what is the law of unintended consequences what robert merton basically said was that there are lots of instances where you have um agencies organizations institutions putting in place certain kinds of programs and policies and these programs and policies are intended to achieve a goal so you put in place certain conditions as the means to achieve ends to achieve some goal however what merton points out is almost inevitably any policy or program or strategy will have unintended consequences that is there will be consequences that you had not intended or that you did not anticipate this gets at this concept that i also emphasize in my class paradox okay it's paradoxical law of unintended consequences uh the war on terrorism that produces more terrorists that's a classic case of the law and unintended consequence obviously the war on terrorism is designed to uh extinguish terrorism uh in fact uh it had the opposite effect when we launched the war on terrorism everything's a war remember that so just think about all the wars that we you know the war on poverty the war on drugs etc etc unintended consequences so you know you go out and you uh take a military approach uh to a problem uh which is terrorism well you begin to invade and occupy other countries and expand your military might across the globe you piss people off you piss more people off and they get pissed off and they become quote unquote because that's just a label you put on people you want to kill you know justifiably kill terrorists right so you can think of lots of instances and i i asked my students in my intro class to do this you know think of an example of some policy that was put in place in some a workplace that you were in or maybe in some school or something and how it had unintended consequences and often quite quite quite often what happens is the consequences the unintended consequences totally nullify the very ultimate purpose of the policy the program the strategy to begin with okay law of unintended consequences this is something everyone needs to understand now two other concepts that merton presented they go hand in hand manifest functions and latent functions and basically he said policies uh institutions uh have what are called manifest functions it's an institution as a manifest fund and there's latent functions the manifest functions are the publicly stated reasons for an institution or a program or a policy or a strategy the publicly stated reason and the latent functions are the unstated but often equally important function of that institution or policy or program or strategy let me give you an example the example i often give in my classes uh the institution of higher education or educational institutions generally if you ask somebody what is the purpose of education in american society and they'll say this is like the manifest function this is the officially stated reason education is designed to provide citizens with the skills needed to be successful in a modern economy now that tells you almost nothing okay as a sociologist you would say that's just to some extent okay it doesn't tell you very much it's the standard party line right but you hear this kind of thing all the time now what are some of the latent functions of education the unspoken well sociologists focus on these because we focus on the underside the dark underbelly of the social world okay well actually uh educational institutions are designed uh to indoctrinate people uh to accept the conditions that exist in our society indoctrination this is not unusual in the united states it's the case in any educational institution within any society where the elite are able to manipulate the curriculum what's another latent function another latent function of education is it prepares people to operate in bureaucratic organizations and institutions which they will have to navigate through their entire life education is organized bureaucratically if you're successful in educational institutions you have sort of internalized this bureaucratic ability to navigate through the bureaucratic maze that is prominent in almost every organization so it prepares you for your life in a bureaucratic society nobody ever says that nobody says well the reason we have education is to indoctrinate students the reason we have education is to prepare people for no those are unspoken but they're equally important another latent function is education is a way to legitimately allocate resources unequally right legitimately allocate resources unequally people say well how come that person makes more money than me i mean that's not well they have more education oh well then that's okay right the inequality is legitimized it's legitimate because somebody has more education we could go on and on okay those are a couple uh functions so let me give you a great example now to apply this to um something that happens politically um over the last 10 years we have all these efforts largely by conservatives and republicans in different states to institute all kinds of additional uh voter registration requirements for voters like you have to have a special id you have to go someplace and get a form you have to register here you have to make sure you register every two years all kinds of new requirements now somebody asked them why do you have these new requirements they say well there's voter fraud there's a lot of voter fraud does anyone want voter for no nobody wants voter so in order to ensure that we don't have voter fraud we need to put in place some additional restrictions on voting that ensure that the elections are not plagued by illegal fraudulent activities that's the manifest function right now as it turns out people have studied this question and the level of of voter fraud is so so so so minuscule that it doesn't justify any of this right so then you ask yourself well what's what's the real reason that they they said the manifest function is fraud what's the latent function well the latent function is that they know instituting these requirements will place a greater burden on particular populations poor populations marginalized groups black and brown citizens in particular communities and as it turns out coincidentally those people who have the most difficult time meeting those requirements tend to vote for the democratic party not the republican party that is promoting this right that's a great example right manifest function latent function so latent function is essentially to reduce the ability of people to vote who would be voting for the party that you oppose and by the way there's been lots of research to show that this is the case in fact in the effort in north carolina the term was that the policy was designed with surgical precision to ensure that black voters in north carolina would be less likely to obtain the necessary requirements to vote okay so it's thrown out they're still trying this they're not going to give up so let me so that's an example so what i'm doing is i'm trying to apply these two concepts to one particular issue this issue of voter registration stuff okay now law of unintended consequences this is a beautiful story so as all this is going on and voters out there in these communities largely minority communities begin to become aware that there are these efforts to put in place these voter id requirements and restrictions that make it harder to vote we had an election and i can't remember which election this was when it was most um prominent maybe it was 2012 um the election anyway we had an election and something phenomenal happened there was an amazing amazing sharp increase in voter turnout in communities where voter turnout tended historically to be very low where people just didn't participate they didn't there's a lot of non-voting lines around the block of people waiting to get into the polling place somebody did a little research on this you know why are all these people coming out to vote and the reason they were coming out to vote some of them said look i don't normally vote i'm not crazy about the two-party system i'm busy but i heard that somebody is trying to restrict my ability to vote with these voter registration requirements so you know what even though i normally don't vote i'm going to vote damn it i'm going to show them i'm going to show up and i am going to exercise my right that they are trying to take away tell me that is not a beautiful example of the law of unintended consequences whether republicans and conservatives were trying to put these policies in place they were trying to minimize the level of participation the unintended consequence was that individuals knowing about the policies decided they would essentially act in opposition to what they could see was the intended the ultimate the latent intended purpose all right so you have the law of unintended consequences you have manifested latent functions you take these wonderful concepts and you can tell a beautiful story about the voter registration issue all right you're welcome okay i want to move into this question of political party polarization lots of people talk about that the polarization in the united states the parties are so polarized and um i've thought about this a lot and there's something to the polarization point and i'll get to that uh but i think it's been a misconstrued um and there are a lot of misconceptions so i call it the myth of political party polarization and i'm focusing primarily here on the political economic dimension okay the political economic dimension versus maybe the social cultural dimension we'll talk about that we touched on that a little bit first week so what i have here in this slide is theoretically what would polarization look like so i simply take an example in 1970 you have the democrats and our little left of center you can see that the republicans are a little right of center and people say that the parties are becoming more polarized okay this is what polarization would actually look like that is you would have the parties moving in opposite directions toward opposite poles you understand the democratic party would be moving more to the left the republican party would be moving more to the right so this is theoretically what party polarization would look like okay what in fact has happened both parties on the political economic dimension have moved to the right okay they both have moved in a neo-liberal direction hopefully you understand that now from last week and what that means is that rather than them moving in opposite directions they were moving in the same directions but notice here it's possible for the two parties to move in the same direction but become further apart you see that you can see that in my stylized little diagram here the republican party moved sharply to the right on neo-liberal neo-uh neo-liberal economic policies political economic policies the democrats moved to the right not as dramatically uh but still in that direction as you should know now both the democrats and republicans have both essentially practiced neo-liberal political economic policies promoted them and i have those dollar signs there because much of this is driven by money by corporate donations that's becoming increasingly important to both parties and of course those corporate interests prefer of course it's only rational neoliberal economic policies now the larger story narrative which i'm going to continually try to reinforce and hopefully you understand this um as the parties have both moved to the right on the political economic dimension and this started uh for the for the republicans obviously with reagan uh for the democrats with uh clinton okay clinton was pretty much a hardcore a neo-liberal uh democrat uh so as the two parties begin to move to the right on neoliberal economic policies the voters perceive less and less difference between the two parties and they also experience no significant differences in the quality of their economic conditions republican party comes in democratic party comes in right um and so the perception is that there really isn't that much difference between the two parties on the political economic dimension of political economic policies that affect people's livelihood and material welfare inequality continues to increase economic insecurity continues to increase right so what's left if they don't perceive much difference between the two parties on the political economic dimension then that increases point i'm making here the salience of social and cultural issues now on social and cultural issues the parties are further apart in more significant ways right and here's the problem for the democratic party and maybe as as uh frank has said they basically have abandoned given up on the working class um because they're pursuing these uh political economic uh policies that are neoliberal but the point is that if social and cultural issues become more salient more significant um a more significant reason for choosing one party or the other the case is that working-class voters uh tend to be on average um socially culturally um conservative and so by moving in a neoliberal political economic direction by not promoting what i would call more social democratic political economic policies which the working class supports they look at the two parties and they say well i guess what's different about them is the social and cultural issues and the republicans promote social cultural issues in a way that draws working-class people do that okay and this is what we you know immigration identity religion these are these are the issues that often sway right the working class to the republican party i don't see that the democratic party is providing any kind of economic advantage this is the dynamic we see in american politics this is a dynamic that uh frank highlights in all of his work from what's the matter with kansas all the way to listen liberal all right so think about what real polarization is um i mentioned that money drives it so i'm just going to put up this a little diagram here and um you can locate the document that this comes from but basically it's showing how uh over time uh the role of big money has become more and more significant both both for democrats and republicans both democrats and republicans two parties heavily dominated heavily dependent on the wallets and the checkbooks of the rich and the wealthy and we'll talk more about the significance of that when we talk about a investment theory of politics um partisanship in the trump era and i wanted to highlight something and now i see that what was highlighted in my slide is totally covered up in this slide so i will skip this slide and come back to it where you can read what i highlighted um this is another study the last one was by bartles uh what bartles was basically saying and what i wanted to highlight uh in that abstract uh was that there's much more unity within the republican party on social and cultural issues within the democratic party there is far less unity on these social and cultural issues in terms of a segment of their constituency who tend to as i noted before uh are a little more conservative on these social cultural issues um and this is another um angle on this question how do parties decide which issues to emphasize during electoral competition and the key point that i want to highlight here goes back to what i just said a couple slides ago rightist parties that would be the republican party will opt to emphasize values based issues especially in those cases where social demand in the electorate for values based representation uh is high and so when we talk about what they call um values-based issues we're talking about social cultural issues and the republican party has used these very very effectively to draw people who given their economic situation given their social status you would expect them perhaps to vote for support a left-leaning party if there was a left-leaning party right but if the issues that are going to be emphasized are not the economic bread and butter issues that affect their material life they will be drawn to parties that appeal to these other social um cultural what they call values based issues so there's lots of evidence for what i'm trying to argue here now i said polarization is a myth um there is a certain level of polarization and i give it a couple different names a few different names as i've thought about this uh first of all there is some when we talk about polarization let's talk about partisanship right let's not talk about polarization in terms of the two parties becoming one is going left the other is going right that has not happened okay the democratic party did not go left the republican party went way right and the democratic party went slightly right and the distance between them has grown significantly zero-sum partisanship this is the dynamic that exists that energizes this idea of polarization what do i mean by zero-sum partisanship in a zero-sum game a zero-sum logic what it means is that a gain for one side is automatically a loss for another you understand a gain for the democrats will be automatically interpreted as a loss a cost to the republicans a gain for the republicans will automatically be regarded as a loss and a cost to the democrats so when you have this kind of dynamic what i call zero-sum partisanship you're going to have increasing levels of polarization and in fact uh lee drutman wrote the a book on the a two-party doom loom he talks about this i actually came up with this several years ago this idea of zero-sum partisanship and it's nice to see that some of these scholars are using that term to describe what's happening so clearly this is the case we have a dynamic where one party believes that they have to stop the other party from doing anything positive because if they do something positive it will be a cost to them right a non-zero sum game we talked about this earlier when we talked about the labor capital accord is a gain for the other side yes it is a cost but it also benefits us because we get something done we get some policy instituted that kind of stuff right so right now we have zero sum partisanship which intensifies the polarization we also have this um what i call binary partisanship right you're either a republican or a democrat it's a binary system when we think about a binary system zero and one right you can't be somewhere you're they're zero or one right so in the binary partisanship dynamic if you criticize a republican you're automatically assumed to be a democrat and not only that but if you criticize a republican you're automatically put in the position of having to defend the democratic party because you're either a zero or one you're either a democrat or republican right so for example when i used to um which i did a lot i criticized george w bush during his um two terms horrible horrible president okay he's been elevated because trump is so bad that he looks relatively good i would criticize george w bush and people would say yeah but clinton i'd say why are you talking about clinton well you criticized george w bush so i'm going to criticize i said i'm not talking about clinton in other words you automatic by default by criticizing george w bush i was automatically assumed to be in the binary system a supporter of bill clinton i would say i'm not talking about clinton i wasn't a fan of clinton i don't support clinton doesn't matter okay this is the binary you're either in one or the other very very unhealthy dynamic for any kind of principled non-partisanship which we talked about earlier in the semester very difficult to sustain that kind of perspective when you have zero order partisanship you have binary partisanship i would criticize obama when obama was president okay i criticized both parties and people would say or let me put it this way i would make critical comments about obama in the classroom and students would automatically assume that i was a republican because they that's the way our brains work now okay we have this binary system then i talk about co-dependent partisanship codependent the parties are codependent as much as they hate each other they depend on each other you understand it's a dysfunctional relationship but it's co-dependent because the republicans can't mobilize their voters unless they could point to the horrors of the democratic party which they're doing now right they're saying that you better vote for us because biden is a radical socialist by the way i'm a socialist biden is no socialist okay but anyway that doesn't matter the point is you vilify the other party as a way to gain voters to support your party and vice versa right so democratic prices oh man the republicans i mean look at what they say you know look at look at the comments that their their leaders make uh how ignorant they are and horrible and you know you have to support us because you can't obviously support them right i always you know say like what would the democrats do if the republicans didn't exist right and vice versa right because that is the way they try to recruit supporters is by pointing to the and vilifying the other party right i would get calls from the um you know the democratic party would call me um and they would say it would always start out this way they would say you're not going to believe what the republicans are proposing now we have to stop them we have to stop them from doing this it's horrible the policies they're proposing could you please contribute of course could you please contribute some what i always say is well what are you proposing no we have to stop the rep i said but what are you proposing i know you want to stop what they're doing but what are you proposing what's the proactive policy platform program that you want to put in place not just stopping them right and this always threw them off because it's like well the main point is we just have to stop the somebody came up with a term for this called fortress liberalism fortress that is you you create we can't let them go any further right we can't let them go if they've gone far enough the right you know the neoliberal model the extreme republic they've gone far enough we're going to stop them here and a lot of people like me say well i don't just want to stop them what is the alternative policy program political economic system we're going to put in place okay all right before i said you don't have this polarization or not going to separate polls you have what's called and i didn't have the term for it here's the concept all by the way all of these are concepts okay if you don't know what a concept is we've gone through a bunch of concepts uh norm ornstein um andrew hacker and his colleague pearson i forget his first name um that's the concept that gets at what i was talking about they're both moving to the right but one is moving sharply to the right asymmetric polarization symmetric polarization would be that theoretical model symmetric okay asymmetric polarization is one is moving sharply the other doesn't have to move at all but the point is if one is moving radically in one direction even if one stayed where they were you would have polarization okay asymmetric polarization that's the term great concept thank you norm and andrew and his colleague there was a book written some time ago uh probably eight years ago now time is flying in my head it seems just like yesterday when i read this book um by norm ornstein and thomas mann it's even worse than it looks that's the title of the book these are two political scientists and this is what they wrote about the republican party um and this is what we mean by asymmetric um polarization republican party has become an insurgent outlier ideologically extreme contemptuous of inherited social and economic of the inherited social and economic regime scored full of compromise i'm persuaded by conventional understandings of facts evidence in science and dismissive of the legitimacy of its political uh opposition these are not radical political scientists they're actually very mainstream and norm ornstein is uh associated with the american enterprise institute which is a conservative conservative think tank uh but basically they're just describing what's happened to the american political party system and they point to the republican party because people say well both parties have gone to it no it's not true i wish the democratic party had moved further to the left i have to tell you that but they haven't the republican party is the outlier i wrote a piece you can go to my blog the title was of the republican party and the lunatic right and i basically said that the republican party today is a lunatic right-wing party um i don't take that back i don't care if people are offended by that that's a description uh there have been lunatic left-wing parties um but uh in the american context we've got a lunatic right-wing party and it is the republican party and that's my bias now the zero-order partisanship the binary partisanship the co-dependence partisanship all of this is a direct result and only possible okay that that form of partisan polarization i talked about that i think does exist is only possible because we have a two-party system and you really really need to understand this the united states is an outlier in terms of the number of parties that are viable and that people can choose to govern them it is ironic because in the united states choice is glorified the more choices you have that's what freedom is is to have lots of choices so when you go to publix you have 35 different types of orange juice isn't that wonderful isn't it amazing capitalism provides us with so much choice choice is glorified then you get over to the political system where you know to me it would be a little more meaningful only two why only two okay to understand why we have the two-party system that we do is not because the american people have demanded that we have two parties it's because we have an electoral system known as let's see if i have this on the next slide okay known as single member district and when you read druckman one of the articles and the book is look at this i'm using it right here there it is excellent book i just finished it over the summer and you are the fortunate lucky students to get to read a chapter so to understand his argument you need to read that chapter i'm not going to present it but the point i want to make is that we have a particular kind of electoral system it's known as single member district first past the post winner take all you get 50.3 percent you represent that district what about the people that uh you know 49.7 tough okay when you have a winner take all what's called single member district system you will inevitably have a tendency toward two parties and those two parties as they have in the united states we call it a duopoly think of monopoly duopoly those two parties will do everything they can to ensure that third parties fourth parties fifth parties are not viable okay but as long as we have a single member district first past the post winner take all we will have a two-party system and the point drutman tries to make is that the two-party system in the united states is the most significant source of the political dysfunction that exists as well as the undemocratic aspect of american society i know you think we live in a democracy okay he he offers a variety of alternatives and those alternatives let's suppose we had a multi-party system let me just give an example of proportional representation in a proportional representation electoral system you have lots of parties running let's say one party gets uh 15 another party gets 10 another party gets 25 another party gets 40 another party gets 60. whatever the percentage of vote is translated into the number of seats in the legislature so you have a range of parties they represent different political economic philosophies and programs and policies people can vote not for the lesser of two evils which is the standard way people vote in the united states they can actually choose for a party they believe in that represents them and it won't be a wasted vote because the percentage that party gets will be translated into seats in the legislature we don't have that okay we don't have that kind of system instead we have the two-party system think about all of the zero-sum think about the zero-sum think about the binary all of that kind of partisan polarization would be eliminated right would be eliminated because there would be multiple parties so it wouldn't be well a gain for the republicans is not for the democrats it's well republicans but there's all these other parties that might gain as well so that whole dynamic that polarization doesn't exist at all also when you have more parties uh turnout is higher uh people are more likely to vote there's plenty of evidence on this there's a positive correlation between the number of parties and turnout and this is a diagram i don't think this was in the chapter i gave you so when you read drudman take a look at this this is what he means by the two-party doom loop and you can see that it's kind of a vicious cycle if you don't believe me that we're an outlier and he has this this is a table again i don't think it was in the chapter you read uh what this shows you is uh american exceptionalism i'd like to use that term uh because we are so exceptional in ways that we don't want to be you can see that the united states has the fewest number of political viable effective political parties in the industrialized world okay this is the oecd organization of economic cooperation development the kind of countries we would be comparing ourselves to uh that have elections we have the fewest number of parties this has to change that is the point that druckman is trying to make in his book but he goes through the history of the parties the way they've evolved he talks about a four-party system when you had conservative democrats and liberal democrats liberal republicans conservative republicans he called that a four-party system there was a greater ability to not see things in zero sun you know so things have trans um uh trans the political party system has to a large extent um developed and evolved in a particular way over the last 40 or 50 years that has brought us to this highly polarized state i'm gonna have to stop at 60 minutes i'm getting close let me see what's next okay i'm going to stop here because i want to say a little more about this and i will be back the second half will be shorter because we're getting toward the end i think hopefully uh you have developed some new ideas ways of thinking about the american political system you understand polarization in a much more nuanced way and the issue of the relationship between polarization and the political party system okay all right i'll be back for part two i will see you then or i'll pretend that i'm seeing you then you

Political_Sociology_Lectures

Week_4_Lecture_labor_and_politics_part_2.txt

okay we are back for part two labor and politics and i only have a few more slides to show you but i didn't want to get abruptly interrupted when we hit the 60 minute point which is what voicethread does it automatically stops um so it is labor day again i just want to make that point and it's a good time for us to think about the situation for labor labor unions working class organization in the united states so i hope that you will find some of the ideas and information here to be useful as you think about this important topic nothing significantly will change the united states without some form of labor organization and obviously that would be even more ideal would be even more ideal if we had a labor a party more about that okay so let's continue um and i am showing you the cover of a report that was done and it was a way to i think challenge to some extent alec the american legislative exchange council and the effort by alec to expand the number of states that were considering a right to right right to work legislation now this was done some time ago there's actually more right-to-work states that have been added since then um but i do want to show you a few of the tables they produced um before i do that i want to mention again and this is something that you'll be reading about i made reference to it already in part one the significance of having a right to work law or not in relationship to how it impacts the politics of that state and these researchers from the bargaining table to the ballot box political effects of right-to-work laws uh they were looking uh at the difference between adjacent counties on the border trying to control for some of the characteristics of the population and one county was in a state that is right to work and the other county is in a state that is not right to work and looking at the differences between those counties in terms of the direction of the vote the presidential level and at the state level in state level races and also the difference in turnout that is the percentage of the population that actually participates uh and so this is a really really powerful demonstration of the mobilizing effect of having union representation of workers in your state and also clearly the effect that the right to work law has on diminishing uh that positive progressive political mobilization impact that union organizations generate so i wanted to share that with you there's been some great studies these are just a few that have been done recently on this question uh this is another one let me move my screen so i can see everything let's see i'll move it over yeah move it over there okay um and this just focuses on something that's a little different not necessarily the political consequences but the consequences of having or not a right to work law on the safety and health of workers in those states and again the point is that when you mobilize workers around issues that are related to workplace conditions you mobilize workers around progressive legislation that is the result of the education they receive in their labor unions it has a direct impact on the physical safety of workers in those states so this is just another connection we can make between having or not a right to work law the impact that has on unions and ultimately the impact it has on workers safety health income etc these are states that have passed laws 2011 2012 there was a surge of efforts by states this is the period when uh republican legislatures in the states were in high gear and it shows you where some of the efforts and action was taking place now this comes from that um a report by the center for american progress that also wanted to highlight some of the uh issues and consequences associated with the right to work movement and you can see that the top ten states by level of unionization versus the bottom ten states having a unionized labor force makes an enormous difference in terms of the level of social welfare benefits government assistance and forms of protection for workers here's health insurance coverage voter turnout which we've focused on which you'll be reading more about as well and what has been happening over the last four years under the trump administration uh let's remember that trump did run focusing on issues related to workers declining economic prospects de-industrialization the impact of trade pacts like the north american free trade nafta free trade act and the impact that has on workers losing their jobs and manufacturing etc so he promoted himself in some ways as a kind of populist um mobilizing workers around those issues it was a very very smart move on his part and the democrats uh have been out maneuvered constantly on this because they seem reluctant to talk about the working class but in any any case what has actually happened under trump not what the what was the rhetoric that he used to get elected but what actually has been the policy impact of his administration and again i mentioned the national labor relations board is the board that essentially enforces and ensures that there are in place protections for workers um this report and if you're interested in following uh all kinds of developments in the area of the state of the american working class definitely go to the american policy institute you will find enormous amounts of data and they have reports that they generate periodically one that came out recently titled unprecedented the trump's national labor relation boards attack on workers rights and they go into an enormous amount of detail highlighting all of the changes that have been made and by the way nobody ever pays attention to this they pay attention to all the crazy [ __ ] idiotic statements made by the president that gets all the press attention in the meantime while we're all distracted by that we have the roll back of all kinds of policies and regulations meant to protect workers they're being scrapped they're being rolled back and very few people are paying attention to this so the economic policy institute put out this report that highlights what's happening these are some headlines trump's anti-worker agenda president trump's regulatory rollback an attack on america's wallets because obviously translates into how much money people can make and a protection rules that exist under the national rate labor relations board and they have been also weakened wage theft the ability to form unions uh his trade policies have actually had a negative effect on workers uh despite the rhetoric and worker safety and health has also been compromised take a look at the report by the economic policy institute if you're interested uh very valuable stuff and this is just more connections on that same theme uh one thing to think about when we argue for government mandated minimum wage uh which i am supportive of and which has not been increased at least the federal minimum wage has not been increased in many many many years so it should be and i'm fine for a 15 hour uh minimum wage i don't really have an issue with that except that imposing that requirement is more difficult for some businesses than for others so you sometimes ask yourself you know should the wage levels of workers uh be mandated by the government and there's good solid arguments for that and i support the minimum wage but another way to think about this is would it be better if rather than we had the government mandating one wage for all workers if workers in every workplace were given the right to organize into a labor union to collectively bargain for themselves they could sit down with managers they could sit down with owners and they could determine what would be a fair wage in the context of that organization that business the amount of profit that that particular organization makes some corporations can easily easily pay workers fifteen dollars an hour without any loss to a significant amount of their profit there are other businesses where this might be more problematic you also would then have workers organized into labor unions mobilized educated raising levels of class consciousness participating more in politics than just the mandated minimum wage so i'm not saying that i'm opposed to the minimum wage i'm just saying we should think about other ways that workers in the workplace organizationally through political action can impact the amount of money they're paid the conditions of work terms and conditions of employment as we like to say okay i always like to show the um the vicious death cycle that we find ourselves in i pointed to this politically in terms of neoliberalism you can see the same thing here the death spiral you have declining levels of unionization that weakens worker political mobilization that means you're not going to have pro-labor policies being pursued by the parties because workers are not mobilized they're not energized they're not uh possessed with a kind of level of a consciousness these norms of equity uh that means that there's going to be an absence of progressive countervailing power against the concentration of corporate power anti-union policies will continue to be rolled forward and more declining unionization weaker political mobilization weaker pro-labor orientation apart you see what's happening here okay this is the cycle we have been in uh for the past uh 20 to 30 years and the only way this can be broken is to have significant serious labor reform that ensures and encourages and facilitates the ability of workers if they so desire to organize labor representation in their point in their place of work [Music] i just finished a book i came across last month fabulous unbelievable excellent unbelievably excellent analysis of labor and politics massive references to literature on almost every aspect of the labor movement in the united states the relationship between the labor movement and political organizations and what this writer does this researcher this scholar is he wants to explain why the level of unionization labor power is so much weaker in the united states than in canada so he's doing a comparative analysis which is a really fascinating way to approach his problem and he says you know canada and the united states they have a lot of similar characteristics so if you're you know comparing the united states with european countries that have 80 unionization it's a tougher comparison and so they have a similar kind of history in terms of the introduction of labor organization but why is it that today the percentage of workers in canada that are a member of a union is so much higher than it is in the united states and they have significantly more power labor unions in canada that's the question and there's been a lot of research done on this question a lot of hypotheses posed a lot of explanations offered and he goes through all of them and he shows with data those that seem to have support and those that don't so it's a rigorous rigorous analysis of this question and i'm only going to give you uh the punchline but if you're interested check out his writings pick up the book or i'm sure you can find some articles he's published that essentially have much of what he has put together in this this book that i just finished now one interesting thing that he observes historically the rise of labor unions around the same time in both of these countries how did the ruling class the political system what he calls the ruling part how did they respond to the demand for wider levels of labor organization and labor activity at the point of production in the workplace as well as in the political system now the united states has the lowest level of unionization and a much lower level than canada but here's the paradox he says that u.s labor was met at the time with a more welcoming pro-labor ruling party that adopted what he calls a co-optive response it brought labor into the party this was the democratic party under fdr roosevelt in canada on the other hand there was a hostile response from the ruling party that opposed incorporating labor organization into the political system it's interesting right now it's paradoxical because you might assume well if in the united states they were welcomed that should have strengthened them and in canada because they were rejected and resisted that should have weakened them okay so the first comparison is a co-optive response by a ruling party in this case the democratic party united states versus the coercive response from the ruling parties canada now the result of this was that in the united states labor is essentially defined as nothing more than another interest group among the many interest groups that have interests demands policy preferences but there's a whole bunch of other interest groups so just another interest group within the democratic party right in canada because it wasn't incorporated into the ruling parties labor unions developed a much more class conscious perspective on politics so labor is an interest group in the u.s in canada labor is a representative of the social class this is critically important and i am just just scratching the surface of the importance of this in the context of his book so he says the net result was in the u.s labor politics in the united states is framed in a pluralistic as a pluralistic idea we're going to talk about pluralism the pluralistic idea there's lots of different interests there's lots of different groups this group wants that this group wants that that organization prefers this policy just a part of the pluralist system okay in canada it was framed in the context of the class idea and that's part of the title of the book the class idea that recognized that labor represents particular class interests there are class divisions and these are integral to politics and so things are defined um in class terms in canada as opposed to the united states where we tend to avoid any conversation of class except the middle class and because it was rejected by in canada because labor was rejected by the ruling parties they developed their own labor progressive political party which we do not have in the united states that's the ndp the new democratic party in canada so it was the historical incorporation of labor differently in the united states than canada that has produced entirely different landscapes in the two countries with regard to the power of labor and the extent to which workers are actually represented by labor unions all right one final distinction and that is between social unionism and economistic unionism and this really reflects i think negatively on american unions generally almost across the board social unionism is more of a broad kind of moral philosophy about the role of unions in promoting working class interests across the board organizing workers everywhere whether they're in your union or they're not promoting progressive policies universalistic policies that will support and advance the interest of the entire working class that's called social unionism there's been lots of debates in the united states among union leaders where you have rank and file workers who say our union needs to be connected to the larger struggles of workers in other industries in other sectors of the economy economistic unionism is the typical form we have in the united states it doesn't promote broad-based working-class movements it isn't really interested with developing and forming any kind of labor party or working-class party universalistic social program that will benefit all workers the primary focus is advancing the economic interest of those members that they represent in that particular sector or that particular workplace economistic what that means is that you're fighting purely for economic benefits for your workers social unionism a much broader effort to promote progressive political economic policy across the board social unionism also in canada is represented by the new democratic party so they actually have a political instrument organization that promotes that okay let's respect today with some thoughts about labor and workers and the struggles that they have been under and to continue and we need to think about the importance of labor organization and political participation and how if we think about any kind of political economic changes in future what role those organizations would play and right now they are very weak in the united states and it's time we recognize that and do something about it and i'm not sure either party is interested in that we can say more about that in the context of what's happening in this current campaign so i will end there thank you i will see you next week you

Political_Sociology_Lectures

Institutions_and_Varieties_of_Capitalism.txt

hello students and welcome back to social change and international development it says globalization and development i guess that could be the title of this course that is the title of a course i teach very similar to graduate level and sometimes i exchange the powerpoint slide so that may have something to do with it all right so today this is the thursday lecture i'm giving it on wednesday and we're going to talk about institutional approaches to development which is a significant area of inquiry analysis and theorizing and then we're going to talk about comparative capitalisms which also highlights the variation in institutions that exist across what we call capitalist nations try to do this one recording we'll see how it goes i don't think there are as many slides maybe i can be a little less loquacious and excessive in my verbalizing as i go through the items that i want to talk about all right let's get started hopefully my allergies will not kick in and i will be able to speak clearly all right so two ways institutions have been examined in the development literature the literature that you are now much more familiar with talked about levels of analysis as ways to understand development we've talked about modernization theory this is a little recap dependency theory world system of world economy theory and most recently we talked about global value chains so two other questions related to this let's call it an institutional approach which has become more common in economics they've ignored institutions for a long time but they've learned something from the sociologists obviously we have to teach these economists what it means to do sound systematic empirically founded in reality not in these imaginary models they put together analysis it was a long-winded way of saying that we sociology my discipline is the broadest most comprehensive discipline for understanding the world the way it works in terms of social behavior and social organization and social change all right one obviously uh obvious question and a way to think about development and if you think about the levels of analysis think about the individual level the organizational level the national societal level and the global level this i would say you know kind of straddles between the organizational and the national level what kind of institutions exist political and economic institutions in particular because i think those are the most important as a political economist what role do those play in ultimately generating and ensuring development equitable development shared prosperity socioeconomic progress however you want to define it and some of you when i asked you that question the first week had identified factors which fall into this larger institutional arena so that's the first way we can think about institutions the other is what are some of the institutional variations that exist across what we call capitalist nations and i want to focus particularly on the major industrial sometimes called advanced that could be problematic capitalist societies because we call them capitalists but are they all the same notice on that first slide i said capitalism plural that means multiple forms of capitalism okay i'm getting excited i like this stuff all right so there has been an institutional turn this has been taking place over the last you know maybe 15 or 20 years where economists or sociologists political scientists responding to the assumptions of neoclassical economics have rejected many of those assumptions and said you must include institutions so we sometimes call this the institutional turn we had something in sociology called the cultural turn simply means that people are beginning to recognize the fundamental importance of institutions in this case so i like this quote that i take uh joseph stiglitz and his authors no example can be found in history of a process of development nested in an environment even vaguely even vaguely resembling this institution free i'm going to call it fairy tale of economic interactions that one finds in a good deal of contemporary economic theory so if you've taken economics courses microeconomics courses macroeconomics courses maybe the economics of development i don't know what that particular course you took included but the tendency has been to pretend that there are just markets and individuals or firms interacting in markets and that is how we understand development and obviously that is utterly utterly what what's the term i'm trying not to use too strong a language utterly inadequate let's just put it that way inadequate for understanding uh development so economy economics merges with sociology economic sociology that's a whole area of sociology one of the fastest growing areas and the area that i um associate myself with when people say what kind of sociologist is an economic sociologist all right so defining institutions in this literature and it's done in a variety of different ways depending on where people are coming from are they coming from a neoclassical economic position and then moving over to the institutional turn or are they political scientists considering institute or are they sociologists who are trying to promote the importance of institutions it really depends on where you come from how you conceptualize the narrative that you bring to understanding the role of institutions one way people have talked about this institutions are rules of the game in a society rules in the games that try to direct constrain sometimes we talk about guard rails people use this term all the time guard rails okay enabling incentivizing coercing requiring mandating those are rules they shaped the actions of economic actors right these these institutions are created by humans they're not naturally evolving okay they said can train constraints and they shape incentives as i said okay so when we think about economic institutions economic rules of the game one of the most fundamental is property rights a lot of conservatives say we need government to get out of the way and let the market decide free the market the magic of the market as if government has no role to play we've already talked about this obviously there are all kinds of institutional structures in place to ensure you know to put it bluntly that capitalists rule that they're able to dominate the most important is property rights the right to private property and the right to do with that property what you like and of course contracts have to be enforced uh etc okay so there's a lot we can say about that that's the idea that we're trying to promote here uh and economists try to promote when they begin to acknowledge the importance of institutions political institutions of course obviously what are the political rules of the game democracy we presumably live in a democracy we have democratic institutions yes we do we have democratic institutions now whether we actually have a democracy is another matter but we have democratic institutions we'll talk about a little bit about that distinction it's a big part of political sociology uh is it a dictatorship is it authoritarianism is it autocracy is it an oligarchy are there election laws are there ways for people to participate it's the big issue now with the republicans in various states rolling back various voting rights all of this stuff is contested i want that to be emphasized economic institutions how they work is contested we can go back to class conflict between capitalists and workers bourgeoisie and the proletariat and also in the political arena you have the same kind of contestation so these rules aren't fixed and they're not natural all right now another way to think about institutions is how should institutions operate is there a way that institutions should operate that would make them more effective efficient productive um and let's say in the varian sense max weber vaberian sense he talked about the ideal type bureaucracy people who are put in positions of authority decision-making positions should be placed there based on what they know what they've achieved what we call these we call this meritocracy where they got their degree do a lot of criticism of this stuff how they've been promoted right so one way people have thought about this is let's think about how an effective institution operating on what we would regard as the best acts aspects of bureaucracy would work immunity from bribe taking and capturing by special interests now this is the ideal situation right and i want you to think a little bit about how people have defined what an institution should look like right and that these are the institutions that should be in place in order for societies to develop like the united states or sometimes people hold up the united states as the role model you might ask yourself how important all this really is okay because this is theoretical practically how important this is or to what extent does what is regarded as the most highly developed society in the world we often think it's our country the united states deviates from many of these presumed necessary bribe taking campaign contributions capture regulatory capture we talked about that so i can't help but it's hard for me to talk about this stuff abstractly without saying that these things are violated but when people are trying to evaluate institutions and whether institutions have the characteristics that one would suggest or believe promotes development these are the kinds of things they're talking about no entrenched concentrated power sources that can either make new rules or subvert the rules embeddedness and embeddedness means that the institution has some connection with stakeholders the larger society technological flexibility openness to external innovation you don't have rigid bureaucracy that never changes you realize that conditions change and institutions must also adapt and this idea of countervailing power i think i've made reference to this before labor is a form of countervailing power to capital and is there countervailing power is there competitiveness within institutions that ensure that one group does not entirely dominate so when people talk about various characteristics of institute they touch on these kinds of things one book that made a big splash probably about 10 years ago uh i read it uh it's a massive tome it's one of those uh door stoppers uh i found it tedious because there was an enormous amount of really really detailed historical examples uh which i'm much more interested in the conceptual broad theoretical ideas some people like all of that historical detail i didn't i found it a very very tedious to read dense but the bottom line of this massive book that had enormous influence and was representing to a large extent the economists who are making the institutional turn by osama bleu and robinson and basically why nations fail right there's all kinds of books written with these kinds of titles why some nations are rich some nations are poor why some nations prosper and some uh you know stagnate etc why nations fail and their view was why nations fail well what makes a nation not fail what do they mean by not fail sustained prosperity and sustained prosperity requires this is the bottom line argument economic institutions are inclusive and political institutions are inclusive and you can see the definitions i'm not going to read them i'm going to let you take a look at this but i want to focus on something that is a major major emphasis here okay inclusive economic institutions enforce property rights okay you can see they're coming from a certain kind of economic theoretical uh background if they're emphasizing property rights which they do okay and that's you know betrays their background a little bit uh in terms of economics pro-market pro-capitalist and some of these other things make perfect sense create a level playing field for people to compete in a marketplace if you have a capitalist economy encouraging investments and new technologies and by the way you have inclusive economic institutions good and you have extractive economic institutions bad okay and i have a little definition of what an extractive economic institution is one of the problems with this as i was reading this book i began to get the same feeling i got when i used to read modernization literature well the united states is the you know most highly developed country and theoretically the most highly developed country is highly developed because it has inclusive economic institutions but then when you start looking closely at how they define inclusive economic institutions and then you look at the actual practices economically in the united states begin to see some some problems here right so sort of the glorification of the developed societies because they have all these wonderful inclusive institutions and the kind of degradation of poor societies because they must have extractive economic institutions you get that same kind of dynamic you find in modernization theory which i find a little troubling inclusive political institutions distribute political power widely in a pluralistic manner if you take my political sociology class we talk about pluralism whether it exists in the united states or not so maybe we should think about this as weber talked about an ideal type bureaucracy theoretically conceptually what it would look like and what inclusive economic institutions would look like what inclusive political institutions would look like all this emphasis on property rights i'll mention in a moment all right so i have some criticisms of the book i have lots of criticisms of this book am i going to spend a lot of time on it what i wanted to do was give you an example of a major piece of work that promotes the institutional perspective so you understand what it is and institutions are absolutely absolutely critical important to understand how societies develop but there are different ways of looking at these institutions and to some extent we've talked about aspects of this before now inclusive economic institutions property rights inclusive political institutions really should be based on citizenship rights capitalism property rights capitalism an economic institution democracy is based on citizenship rights democracy is a political institution i know there's this belief that somehow democracy and capitalism go together and they reinforce each other they don't they're constantly in tension property rights give rights to people on the basis of how much property and wealth and income they have that's the capitalist side democracy theoretically gives people rights on the basis of being a citizen and having the right to participate and vote and have an impact as a citizen on decisions that are made that affect them that's a lot different than property rights property rights often stifle citizenship rights and citizenship citizenship rights popular mobilization are often directed toward limiting property rights the ability of property owners and capitalists and corporations to do whatever the hell they want this is a very very important contradiction capitalism tension totally unrecognized in this book also china is a highly highly successful economic system the chinese system the chinese communist party rights right what about authoritarian capitalism there's some markets there but i wouldn't call the political institutions inclusive but the economy has grown over the last 30 and 40 years probably in the most prolific significant way of any single nation historically there's a lot more we could say about china so i'm just pointing out that it's possible to actually develop and create some level of prosperity if you want to call it that um without inclusive political institutions remember we also talked about bureaucratic authoritarianism right in fact often capitalists who want to invest freely propose that we limit democracy again this contradiction i'm trying to point out the contradiction uh there is a distinction here between what we call du jour and de facto the example i used before we have democratic institutions we have formal democratic institutions which would suggest that we have a democratic society de facto is well in fact what really happens what's the reality day to day how does the system really fundamentally work maybe deviates from the du jour the formal democratic institutions right there's a sociologist uh star he talks about civil oligarchy what he means by that is we have these civil democratic institutions but despite those institutions institutions the institutions have not prevented enormous enormous concentration of power in the hands of a very small number of people in the united states du jour democratic institutions democratic rules de facto i would agree with star we have an oligarchy in terms of the concentration of power creative destruction this is a big part of the analysis they say there should always be operating creative destruction what this means is that there may be some elites there may be some corporations that have dominated the economy for a long time have developed certain technology certain methods certain products but if you have an inclusive economic system there should be competition there should be ability of new corporations to emerge and for those corporations to be destroyed as new corporations and methods and techniques related to economic production emerge is creative destruction allowed to prevent it can you think of corporations in the united states that have a stranglehold on moving us forward because they are not able because they prevent the competitiveness that one would associate with inclusive economic institutions the iron law of oligarchy essentially says that over time there's going to be a tendency toward a small number of organizations or individuals power sources concentrating power over time and then preventing democratic participation and competition and finally one thing they totally ignore colonialism imperialism american interventionism where do the institutions that exist in less developed countries come from do they stem organically to what extent are those institutions a product of american foreign military policy i will point to one area that i've discussed before the northern triangle where many people flee and they want to come across the u.s border from el salvador in guatemala in honduras and somebody might say their institutions are characterized by the concentrated power of an oligarchy and they're corrupt well let's look at the long history of american intervention in shaping those institutions all right so if you wanted to create a typology i sometimes encourage students to just think typologically this is their thesis sustained prosperity requires inclusive economic institutions and inclusive political institutions so on the economic side you have inclusive versus extractive that's the dichotomy political inclusive versus distracted inclusive inclusive that's what you want sustained prosperity i don't know what they would call extractive and extractive i just said sustain poverty the more interesting cases would be the mixed cases where you might have an inclusive political system but a more extractive economic system or a extractive economic system but a more uh inclusive uh political system okay these mixed cases extractive economic inclusive political or inclusive economic extractive political once we broaden this we begin to think about uh cases that are probably more realistic as existing in the world today uh they try to do some empirical analysis these are economists they wrote a bunch of articles the book they wrote uh is written uh to be accessible to as many people as possible so they don't put a lot of quantitative regression analysis this is just a basic uh you know bivariate scatter plot and they're looking at the gdp so they're viewing that as a measure of prosperity we know all the problems with that measure okay but let's just put that aside for a moment and the protection against risk of expropriation now why the hell are they looking at the relationship between these two things i often ask students this when i'm in a class we have a nice discussion about that we can't do that what do they mean by expropriation well expropriation means the ability of the government to actually confiscate nationalize socialize property that's in the hands of perhaps the oligarchy land expropriation or corporate expropriation you nationalize so what they're talking about here is they do an analysis to see um the extent to which there are laws that make it illegal that prevent the government from in any way encroaching on property rights that's that's the broadest way to think about this and so what they show here is the greater the protection of property rights measured by protection against the risk of expropriation the greater the development so you can see they have a certain kind of bias here that they should review i reveal my biases okay i think you know what they are you can call them biases you can call them political tendencies preferences visions of a better world uh here's another interesting graph uh where they show the trajectory the developmental developmental trajectory of two societies and one of the reasons they do this is they call it like a natural experiment you have basically a country that was divided in half so you're controlling for all of the presumed cultural etc differences between these two countries that were at one time one right so they feel like they don't have to control for that and one has promoted a more let's say capitalist road the other as promoted in more communist road and they show that not surprisingly and certainly confirming their hypothesis south korea has done much better economically so uh it's interesting to see how they empirically analyze their thesis now you know we like i said we always hold up the united states as being the most advanced prosperous country in the world at least for some going to throw that in okay right i mean you know people certainly it's the most advanced developed society in the world and i i always ask the question well then it shouldn't be extractive it should be inclusive according to their theory and here's how they define the extractive states extractive the bad ones the ones that shouldn't develop extractive states are controlled by a ruling elite whose objective is to extract as much wealth as they can from the rest of society that sounds a lot like the united states today to me certainly under neoliberalism and i just give a small empirical tidbit during the economic recovery after the great financial crisis 2009-2012 if you want to call this a recovery okay recovery for some that's for sure in the us i think i mentioned this earlier 95 of the growth in income was controlled by the top 1 i think that pattern of distribution is consistent with their definition of an extractive state so let's not celebrate all right varieties of capitalism there is no alternative what we have in the united states is the highest most advanced level of capitalism that has evolved naturally so quit complaining there is no alternative you must accept what exists okay well clearly we know better than that don't we we all know better than that there are lots of alternatives the elite don't want you to ever imagine alternatives to what exists in your society but that's the basis on which people join social movements and make political progress so my point here is we have talked about capitalism as a system neoliberal capitalism i've tried to emphasize that neoliberal capitalism is most extreme predatory in the united states compared to other societies that are called capitalists so there's a huge literature some of you may be interested in this if you want to pursue it further called varieties of capitalism most fundamentally what explains the variation is the extent to which the society depends on the market to allocate resources or the state and it's not all state all market we talk about mixed economies there used to be a lot of talk years ago that was a term that was used it's a mixed economy what does that mean there are markets and there is state intervention state direction state confiscation of wealth and income redistribution market versus status that regulates the economy work living standards etc we talked about fiscal policy taxing and spending labor market institutions i discussed this in great detail in the sociology of work where we talk about the different kinds of experiences of workers in capitalist and when i talk about capitalism here we're talking primarily as i said about the major industrial society and social welfare policy these are three major areas where you see market versus state variations across the major industrial societies and theorists political economists because this is political economy have tried to categorize nations so you've got the anglo-saxon capitalist model some people say that's u.s and britain you have a communitarian that's sometimes a term used okay and you know we can quibble over this terminology and you know who fits into what i'm just trying to give you an example of how people have thought about these varieties germany and japan communitarian a little more refined market-based liberal as in classical liberal us asian communitarian often japan different different form of capital both capitalist societies the point here is let me let me just make a broader point here they are all capitalist societies to the extent that the means of production the means of production the corporations the factories the businesses are largely privately owned by capitalists okay it's the case in all of these countries i know people talk about you know sometimes sweden is a socialist no it's a capitalist country but it's a capitalist country that is very social democratic so you have continental capitalism social democratic capitalism mediterranean capitalism you understand and if you ask what's the difference between the countries in these categories one will automatically go to institutions and the extent to which market institutions rule state institutions the balance and the relationship between those two in the u.s we're the most extreme market neoliberal capitalist society among the major industrial capitalist nations liberal conservative social democratic liberal is uh should be actually on the right okay that's the most conservative social democratic on the left uncoordinated market the u.s would fall into that category very little intervention very little coordination lots of uh deregulation and coordinated market much more government direction much more government intervention and much more regulation augusta esping anderson a major major political economist major figure wrote some of the most influential works that have impacted the way i have thought about all this uh spent a lot of time talking about the worlds of welfare capital he focused on a social welfare state and he did make this distinction between the liberal welfare capitalism conservative welfare capitalists of social democratic capitalism examples us germany sweden respectively decommodification what does that mean that means that there are things available to the citizens that do not have to be purchased in a market but are given to them as a right think about health care just think about health care is the example that's the one most people think about today in conversations in the united states right has it been decommodified in a social democratic society it's provided to everybody for free you do not have to access it in a marketplace where you pay for it and where the amount you get and the quality depends on how much money you have decommodification is removing the fundamental requirements of life from exclusive market control for profit control okay social rights welfare provision the kind of benefits that people get there's a lot we could say about all these again trying to give you a little flavor sometimes say flavors of capitalism little flavor of the variations that exist one of the reasons i think it's important for students to understand this is they often assume that the capitalism that exists united states is just you know this is the best form of capitals if you show them that there are other capitalist societies that still have markets that still respect property rights but that are able to provide the citizens the higher quality of life that's important because then we can start thinking about what we can learn from these societies all right so these are all terms that come into play some people have argued that all of these capitalist countries are beginning to converge toward neoliberalism to what extent is there still a difference okay and people have been charting this over the last 30 or 40 years to see whether what we've seen developing united states is a general trend or whether there still remains persists fundamental differences even though all of these nations in one form or another have adopted some kind of neo-liberal political economic philosophy path dependence what that means is that the possibility of transforming political economic institutions today depends on what has come before what path have we taken historically which constrains us makes us dependent upon certain political economic structures corporatism is a term sometimes people use corporatism to describe the fact that corporations rule in the united states but corporatism is a term in political economy political science that historically has meant that in this society political economic policies are determined by a kind of accord where labor business and government meet negotiate and put in place policies trying to represent the interest of all groups we don't have a corporatist system in the united states it's much more common in europe i've talked about commodification decommodification a social wage means that you receive income for not working that's what a social wage is that is you receive benefits from the government independent of selling your labor power and we know this is a big debate now because they can't find workers and they think you know we got to push people back in the labor force therefore cut off their social wage so they have nothing to sell with their labor power to survive flex security is a term that's been used in some of the nordic by the nordic i mean norway sweden finland okay flex security it's kind of a compromise on the one hand you give businesses you don't restrict the ability of businesses to restructure lay off workers downsize that's the flex you give them flexibility the security side comes from security is you have in place protections for those workers who are subjected to those fundamental changes sometimes i talk about this idea of how does a bumble bee fly and that is i don't study insects okay i don't know anything about them but actually structurally you would if you did study insects the movement of insects what what you know um structurally would be required to be aeronautically able if that makes any sense i'm trying to talk in a sophisticated way people said the bumblebee is structurally shouldn't be able to fly well people have used this kind of analogy to talk about look at the nordic countries look at how much taxation there is look at how much redistribution there is look at how much decommodification there is how do those countries grow and expand and how come they're so prosperous how do those bumblebees because they deviate from the cl neoclassical assumptions about what's required for economic growth how do they fly how do they grow and again this introduces the idea of varieties of capitalism there was a period when you know people would say the united states it's like the workers united states have had a very very difficult time over the last 20 or 30 years with outsourcing and offshoring and globalization they've been beaten up and you know the democratic party has you know which totally abandoned concerns for most of the working class look you know these are conditions beyond our control globalization is inevitable and it's going to have negative effects on workers in all countries and some researchers some really really great researchers by the way who wrote one of the best books on global value chains outsourcing economics is the name of the book they did this study they said well let's look at all these countries that are equally exposed to globalization like the united states has the effect on workers been the same in all those countries you probably know what the answer is no it hasn't and i will let you read this quote because it's very important the effect of globalization on the workers in these countries depends on the labor market institutions the social welfare institutions and the fiscal policies of those capitalist countries some countries have done a much better job of protecting their workers from the negative effects of globalization than others very important research that debunks this argument oh there's nothing we can do it's just globalization get more education that's what democrats always say just get more education that's that's the answer to all problems that's why all of you are taking classes now you assume it's going to okay i'm not going to go down that road okay american exceptionalism all right [Music] this graph table shows the extent to which the benefits provided by the government addresses poverty that is reduces the poverty rate okay so here's the point capitalism produces poverty even in the advanced countries there always are is going to be a segment of the population we can talk about which segment that is i don't want to go into that so the question is after the government intervenes with government benefits social programs social welfare to what extent does that impact the poverty rate and this shows which countries the intervention does the most and the intervention actually does the least who's on the bottom the united states so the next time somebody tells you that the welfare system is too generous what you need to tell them because that's utterly i mean they might think it's absolutely but relatively the united states is the least generous social welfare state among countries we should be comparing ourselves to as i mentioned before organization of economic cooperation development here's a relationship between social expenditures and the relative poverty rate all right notice the u.s has one of the highest poverty rates and of course it has one of the lowest levels of social expenditures so there's obviously a correlation between these things i'm going to go through these quickly you can study them in more detail child poverty something people might be concerned about well adults you know it's one thing but you know children the most vulnerable and we have the highest level well just about we the highest level uh other than turkey mexico and israel and i think israel must be there maybe because they're including the palestinian population high levels of child poverty now biden's uh proposals and child tax credit some of his other proposals would make a significant difference here but they have to be permanently in place not just a one-year policy that expires after the crisis has ended of course this crisis is persistent so this is not the poverty rate this is where always year after year after years and nothing to do with coven the u.s has the highest child poverty rate government mandated leave and holidays paid or unpaid paid holidays paid annual leave zero we are exceptional i will say that per capita expenditures on health many of you may be familiar with this because there's been a big debate about medicare for all and you know we said universal health care united states spends more on health care than any other country by far and we have subpar public health outcomes now if we spent the most and we saw that the population was really really healthy compared to other countries we'd say well it's worth it but remember the reason you see this is because the u.s has a commodified not decommodified commodified health care system we've already talked about inequality the united states has basically the highest level of in inequality except for a color co a few other countries that we typically would not want to be comparing ourselves to all right i'm just showing you a bunch of these little graphs this is one that tries to categorize countries on the basis of the relationship between the size of employers and the genie coefficient the higher the coefficient the greater the inequality you see the united states always off the map as well and then they try to categorize the countries that fall into this bivariate relationship the link we always talk about social mobility right some people have assumed that you know in the united states anybody can be a millionaire there's lots of movement there's lots of social mobility well if you've taken my intro to sociology class or probably sex race and social class uh you should know that in fact of the united states does not have the highest level of social mobility and what this shows is what's the correlation between your parents socioeconomic status and yours as an adult the tighter that relationship obviously the lower the level social mobility and this graph shows the tightness the higher the bar the greater the tightness where parents income is what predicts child's income collective bargaining this gets us into labor market institutions the united states has very very weak collective bargaining coverage that means people who are members of a labor union and that translates into poverty so again you can look at all these graphs you can see that there's wide variation across these nations i often ask students as i have before right what's the hypothesis supported by this graph the greater the inequality and by the way the united states has the highest level of income and wealth inequality among major industrial societies we want to compare ourselves to the greater the i'm sorry yes the greater the immobility okay so that vertical axis axis the vertical axis okay the higher you are on that the less mobility exists that's why it's immobility okay the higher the score the less social mobility so the policy implication is if you want to address this problem where the united states seems to have less social mobility than we would like you're going to have to do something about the distribution of income and wealth that's the policy implication health expenditures as percentage share of gdp again we spend more than everyone else and we have one of the highest infant mortality rates look at all the countries that have lower infant mortality rates in the united states life expectancy we lag behind we're not getting our money's worth this is an interesting one this is the relationship between church attendance and welfare spending and there is a hypothesis here let me move this so you can read it i just think this is interesting something to think about uh introduces a whole set of new issues uh there is something called the ingleheart thesis it's ronald ingleheart his thesis is that when countries provide a lot of protection to their citizens remember the us provides very little protection uh there are many european societies that provide much more there will be less of an attachment to religion so the idea is that religion somehow is a substitute spiritually emotionally for a lack of security so the greater the state provides the basics the fundamentals the requirements of life the less likely one is religious okay now there's a lot of mechanisms that make that connection we're not going to go into it you can read this piece um this excerpt i took from the article something very interesting to think about all right let's see where we are i think we're almost done we're going to get this all right so take a look if you like at the um these videos i think i did ask you require you to look at george lackeys because i think or leki because i think he does a good job of addressing some of the questions people have when one promotes the i will say the political economic performance superiority of nordic countries uh there's a little excerpt from michael moore's film where to invade next he talks about some comparisons between the united states and france it's quite humorous quite entertaining so these links are for your intellectual edification to get a better understanding of some of what i'm trying to communicate here um well we are still you know we are the most democratic country in the world right we are the model democracy for no we're not okay i guess i'm on a roll here i'm trying to just burst the bubble uh some of you may know this because i make reference to it in some of my other classes there's something called the electoral integrity project and these researchers look at all of the major western democracies on a bunch of factors and they put them together into a scale to determine which countries have the highest level of electoral integrity electoral integrity when you ask people in the united states why do you say you live in a democracy they'll say well because we have elections because we can vote sometimes some of us can more easily than others we can vote we can participate in the electoral process well turns out on that dimension of democracy we are at the bottom it's shocking to many people right how is it that the united states has the lowest score on electoral integrity well we have gerrymandering okay we know about that these uh you know congressional districts carved out by republican or democratic legislatures the system is entirely decentralized voting rules here voting rules there voting rules in one state voting rules in another that's absurd these other countries have a national system you cannot allow different states to be determining who is eligible how to vote on what days and what times we restrict third parties we only have two parties that's a huge undemocratic aspect to our society you have partisan control of election administration the supervisor of elections in duval county he's a longtime republican and he's in charge of determining how elections will be conducted in duval county we have all these voter registration problems rather than you're a citizen you have a social security card you're a citizen you can vote period election day is on tuesday that makes a lot of sense again two parties versus multi so you put all this stuff together sorry folks we're at the bottom we are number one on military spending okay so if you think that you know we're on the bottom of something we're number one okay all right let me finish with a quote i take i put in my book on social economic development i think it's important because sometimes we get into this mindset this is the best system in the world that's the best system in the world so this is a great quote from alec novi here and elsewhere we must never forget that perfect systems exist only in books the real world east west irrationalities misallocation misemployment of resources waste in the real world socialist social democratic capitalists there will always be intractable intractable problems contradictions and i like what he says here it's a good thing these contradictions exist because if they didn't first of all the world would be boring and intolerable and intolerably dull and people like me would be threatened with unemployment because you know who would turn to sociologists if everything was so predictable and work beautifully but we must learn from the things that go wrong in the hope that by doing so we will diminish the ill effects of predictable troubles and that is the most important point we know what kind of system we live in we know what the consequences of that system are we know there are alternative ways to organize our political economy to diminish the ill effects of predictable troubles but there are vested interests okay extractive political institutions if you like that thwart the ability to fundamentally transform change modify reform whatever term you want to use american capitalism all right on that note thank you and i will see you next week

Political_Sociology_Lectures

Week_8_Lecture_Capitalism_and_Democracy_part_2.txt

okay i wanted to finish up the discussion of capitalism and democracy and say a few things about one of the readings in this last segment so let me move on to this slide so i think merkel does a nice job of uh also uh highlighting the points i made for bowles and ginthis i used to have the students read a article by bulls and give this that did touch on property rights and citizenship rights but it was a long and excessively complex dense piece uh so i thought i would simply present it myself and then i found merkel's piece which i think is uh quite good uh in touching and highlighting touching on and highlighting some of the main uh issues related to this political economic issue at one point he says capitalism is not democratic to understand that based on property rights and democracy is not capitalist um and again we can think of that in the context of property rights and citizenship rights it's one way to understand that statement he also makes a reference to varieties of capitalism and democracy um i think i might have made some reference to this earlier in the semester i can't remember but in my social change and international development class we spend a week discussing the varieties of capitalism framework the main point here is that capitalism as a system is not monolithic in the sense that there are wide varieties across capitalist regimes in terms of this kind of balance if you like between uh the privacy of property rights or the i won't say the primacy but certainly the role of citizenship rights and the balance between uh democracy and capitalism so um the united states if we want to think about uh its form or type or variety of capitalism i would describe as the most extreme form of neoliberal capitalism among major industrialized societies i don't think there's any question about that one of the measures of this would be the extent to which labor is decommodified what that means going back to what we discussed in part one is the extent to which labor actually has some protections from the market and has the ability to receive a social wage or citizen's wage which allows workers labor to decide maybe not to enter the labor market if they can receive support over an extended period of time to meet their economic material needs so if you just talk about the decommodification of labor we have the least decommodification or we might say in the united states workers are more commodified than they are in other industrialized capitalist societies when i talk about these other societies we're talking about france uh italy sweden finland norway other there's nordic forms of capitalism um there's anglo forms of capitalism there's communitarian forms um there's continental capitalism asian so there's people have tried to break the capitalist nations of the world into these categories and much of the basis on which they're placed into these categories involves the extent to which there is a significant role for government in intervening in the market uh impeding you might say property rights or not so we have the most predatory extreme form of capitalism neoliberal capitalism in the world among the major industrial societies and this explains a lot of why the united states is an outlier on almost every measure of the socio-economic quality of life market justice has prevailed over social justice to a much greater degree in the united states than in other countries he makes reference to social welfare capitalism the golden age of coex coexistence and i just like to link some of the references and merkel back to things we've discussed throughout this semester this labor capital accord this period from basically 1945 to the mid-1970s that was the keynesian new deal a period that we discussed now there is an aspect of neoliberalism that merkel highlights uh it's very relevant for the united states and it involves the financialization of the economy which we've touched on briefly and this idea of shareholder value and it's really important for you to understand this principle of shareholder value and the value placed on this principle in american society among the corporate elite among business schools and so i want to say a little bit about the relationship between financialization shareholder value and the corporate restructuring of the american economy and in some ways as a result the impact on the globalization of capitalism so let's touch on a few of these points briefly so this relationship between financialization and shareholder value does have its origin in two theoretical models observations that were made in the 1960s into the 1970s and it's important to link an existing political economic dynamic uh shareholder value capitalism if you like uh with these theoretical developments now one uh for people who study american business the american corporation one thing they observed back in the as early as the 1950s and 1960s uh was what they called the separation of ownership and control what that means is that you have professional managers who essentially control the corporation they make decisions about investment and disinvestment and in decisions about production and those sorts of things so you have this kind of professional managerial class a managerial class that is controlling the organization by making decisions but they don't own the org they don't own the corporation the corporation is owned by shareholders and there might be three or four that have the dominant shareholding power but the point is ownership is by shareholders control is by managers now this raises what's called the agency problem in the large corporations that is the agency problem we talk about principles and agents okay so the principals let's say are the owners the the shareholders and the agents okay are the managers now the question is how do the principals and in this case the people who own the corporation own the shares in the corporation know that the principals that is the managers are acting in their best interests because you've got the separation of ownership and control shareholders on the corporation by owning shares but decisions about what the corporation should be doing how it should be operating what its responsibilities are are made by managers managers presumably trained in business schools is there a divergence in interest and in agency theory there's an assumption that there could be and in this case that there is that is that managers are making decisions that are not necessarily in the interest of shareholders therefore need to develop a system that ensures and this is like the agency problem right how do you ensure that the agents are representing your interests what do you put in place to ensure that the managers are making decisions that are in the interest of shareholders now remember the shareholders the owners have an interest in the stock price increasing managers may have a whole range of interests a whole range of priorities so in the 1980s you had this movement toward placing shareholder value at the top of the list of priorities or as the single obligation and objective of the corporation enhance shareholder value i used to teach in a business school at an institution before i got to the university of north florida of course they wouldn't let me anywhere near their students today over in the business school because i might contaminate them which i would make every effort to do but at the earlier institution i used to teach a course organization theory i thought of the business school and i would ask the students you know i say you know what are your you know interests what are your par my number one priority is to enhance shareholder value they just kept saying this over and over again it was a mantra it was like a kind of um form of indoctrination shareholder value that's the only thing i should be interested in as a manager now this has very very significant and devastating effects on the american economy if you think about it if you think about the fact that managers primary or singular objective mission goal is to enhance shareholder value that is the value of the stock of that corporation the price the stock price what kind of decisions are they going to make are they going to make decisions that are in the interest of the communities other stakeholders workers probably not so you can link a lot of what has happened to the american economy and do income distribution to this emphasis on shareholder value that's what i want to discuss briefly if i can do anything briefly okay so one of the best writers on this question if you're interested is william lizonic he's an economist industrial economist and he's written many many papers if you go to google scholar and just type in lasonic you'll see lots of papers he's written on shareholder value and he points out that there was a time when a profits of a corporation were retained and reinvested and in fact when we think about why do we allow you know the capitalists to retain the majority of the profit rather than distributing it to workers many people will say well we do this because they are in a position to reinvest the money and expand the economy and expand employment and expand production well under shareholder value as lozonic observes you see a shift in managerial objectives from retain and reinvest to downsize and distribute what does he mean downsize and distribute downsize means outsourcing all kinds of activities that at one time were performed by the corporation outsourcing and let's just say outsourcing and off shoring okay and distributing dividends to shareholders so as i said earlier when i talked about neoliberalism under neoliberalism the rise of finance becomes increasingly increasingly important and there is pressure on corporations to restructure themselves and that is to essentially outsource shed what might be regarded as unnecessary activities let somebody else do it that's the outsourcing place those activities where possible or subcontract them to locations where wages are low regulations are minimal unions are non-existent and this is all part of the shareholder ideology so when corporations started restructuring during the 1970s during the neoliberal shift they were rewarded by wall street because they were reducing the hard assets they owned that they were responsible for they were outsourcing that and having lots of different suppliers compete for their business so you had the what's called the nike vacation the transformation of the corporate structure mean and lean that's what they like to talk about back in the business school we want the corporation to be mean and lean only focus on your core competencies get rid of all of the extra activities that you're engaged in that aren't absolutely necessary let somebody else do it outsource it and let people who are interested in handling those compete with each other which reduces price for your business so we have a long period of this process and again i want you to understand corporation restructures the stock price goes up it's positively reinforced managerial behavior positively reinforced by the financial sector and if shareholder value in enhancing shareholder value is your main objective then managers will continue to do these sorts of things now here's one other aspect i want to point out how do you get managers how do you ensure this is the agency pro how do you ensure that the managers are actually going to be doing what you want them to do yes you can create this ideology of shareholder value but is that going to be enough i mean maybe in business school they actually learned that there are other stakeholders that they have responsibilities to other than the shareholders so here's a key aspect you provide managers that you hire at your corporation with stock options think about that stock options yes you've been hired and here's your salary and by the way we're giving you some stock options what does that mean um you have the opportunity to be compensated with shares of stock let's say you hire a manager and you give him 100 000 shares of stock as a form of compensation the whole point of that is to align the interest of the managers with the interest of the shareholders now when managers make decisions what are they thinking about they're thinking about their own stock portfolio which contains hundreds thousands of shares of the stock of that corporation so they will make decisions which are positively reinforced by the financial by the market by wall street and those will be decisions which largely are reduce costs across the board reduce costs in any way you can if it means closing a factory close a factory it means laying off workers lay off workers if it means outsourcing something in china outsource it to china beginning to see the picture here so i say down out off none of those things sound very positive down size out source offshore all of these activities were rewarded by wall street enhanced the value of the stock reinforced that kind of managerial behavior and you have this spiral of the american economy de-industrialization [Music] unemployment people leaving manufacturing and finding work in other sectors that are much less rewarding materially and economically this gives rise to higher levels of income inequality less social mobility wage stagnation all of these things so you know when people try to understand how we got where we are today in the political economic sense uh shareholder value plays a very important role and that's why i wanted to emphasize this okay um i think this is the last slide so this is basically just making the connection between some of the things i've already talked about neoliberal corporate restructuring right remember the crisis of the 70s corporations felt that they weren't making sufficient profit cut taxes cut regulations beat up on unions slash welfare and let's restructure the organization in ways that will also maximize profit because regulations on the shifting and movement of capital were also eliminated made it easier for corporations to outsource and offshore so you have what though some theorists have called the nike vacation uh nike is a classic example one of the classic examples there's many now of a corporation that sells uh sneakers sportswear but owns no factories none zero every single thing that nike sells is produced by subcontractors in another country nothing as far as i know is produced here in the united states so you've got a company that is factory-less wall street likes this they don't own factories they don't have all these obligations all this property that they own that they have to manage that they have to maintain you've outsourced that to somebody else somebody else's problem and those outsource you know independent contractors are competing with each other for your business that you know any factory wants to be putting together nike shoes because the market is massive right okay so if you look down the left side neoliberal corporate restructuring you have the nike vacation of the corporation the factory-less corporation a corporation that owns no factories now what do they do with all their profits well in the old days somebody would say well they retain the profits and they reinvest them they reinvest them in one they reinvest them in factories or they reinvest them in upgrading the technology they don't own any factories they no longer own any factories there is no reinvestment as lazonic argued was the pattern in the past retain profits reinvestment production expansion maintaining your facility they don't own any facilities they design and they market everything that happens in between somebody else takes care of wall street likes that mean lean profitable so what are they going to do with their profits if they're not invested in production they invested in financial instruments one thing corporations have been doing and alarming rates is taking their profits often the profits they've been getting from tax cuts because presumably tax cuts provide capitalists with money that then they reinvest and they hire workers and we all benefit we don't because what do they do with the profits and you can do a little research on this corporate buy back shares just type that in they take their profits and they actually buy shares of their own stock what does that do well if you're increasing demand for your shares the price goes up totally totally unproductive this is the degenerate form of capitalism we currently have in the united states right and the arrow from the bottom upward to the right is simply this is great for the financial sector it enhances shareholder value it is rewarded and we have you know somebody talked about the two-party doom loop we have the financialization shareholder value doom loop this is where we are when elizabeth warren was running in the democratic primary because she is an expert on the financial sector she spent a lot of time talking about this issue this problem and what had to be done to shift corporations from shareholder value to stakeholder value and i think she was a little naive to believe that you could just convince them that it was for the general interest the public good you had to put in place some really tough regulations and this was why earlier in the semester i showed you some of the headlines wall street sigh of relief that kamala harris was selected rather than elizabeth warren because they know elizabeth warren has an agenda and its agenda to address this very problem in corporate america does not want this problem addressed and i doubt that biden has any interest in addressing it either because he's getting massive amounts of campaign contributions from the financial sector as obama did which explains why obama did not prosecute the criminal banks that produced the financial crisis of 2007 2008. okay it all circles back to the current political situation that's the point of this course is not to memorize a bunch of stuff and spit it out an exam it's to develop the conceptual framework that allows you to understand what the hell is going on at the same time enriching your intellectual horizons with bulls and guintas and merkel and lazonic and erico and wright thank me later that's it for now you

Political_Sociology_Lectures

Week_13_Interdependent_Power.txt

good morning students and welcome back to political sociology and this week we are going to investigate some of the work that's been done by various theorists and analysts and the message is little change of pace you are not powerless you are not ultimately the victim of domination in all instances in all cases the oligarchy the capital's class and the forces of darkness cannot always win and cannot always control you and this brings us to the work of francis fox piven who wrote a very important piece when she was president of the american sociological association on interdependent power something that has been a part of her research analysis theorizing and general sociological work for many many years so you are fortunate to be able to be exposed to this you can thank thank me thank me later it's okay no one ever gets in touch with me anyway no one shows up for office hours nobody comes to talk about the election not sure what it is with these classes but uh they become quite impersonal nonetheless this is important stuff i hope you pay attention and i hope you're still tuning in uh if you're not uh then you're obviously not learning anything uh and that by the way uh that is the purpose of this enterprise i know sometimes we forget that and you think well the purpose of this enterprise is simply to collect credits and get a degree or manage your education as efficiently as possible which sometimes means i hate to say this but the least amount of work producing the best results so um okay i'm in a little cynical mood this morning in any case let's talk a little bit about francis fox piven uh to begin with because she is a major major figure uh in sociology and she's been at it for a long time and um i thought you'd be interested in knowing probably the work that she is best known for is poor people's movements so she wrote with uh i think it might have been her husband richard clower uh he's a political scientist they wrote numerous articles uh several books together this might have been the most uh noteworthy early in her career and it made a significant significant uh impact on how we study political movements social movements and what kinds of tactics ultimately are successful why they succeed and how they fail as the subtitle of the book indicates and one of the more provocative discoveries that pivot and cloud made during their analyses and observations and study of social movements in the united states was that disruptive social movements that sort of undermine the social order in either locations or more broadly uh tend to get the most positive results ultimately so some people view this as kind of an insurrectionist approach to thinking about social movements that there has to be a kind of insurrection that disrupts creates civil disorder direct action that ultimately brings so much chaos uh to the normal operation of society that the officials the uh elite have to respond in some way to bring the society back to some form of predictable order um and in fact uh that is the case that's why many of us who when we talk about real social change we say real social change only takes place when you have a very active social movement that is able to uh disrupt engage in civil disobedience and ultimately threaten the operation of uh whatever the system depends on capitalism the movement of goods whatever it happens to be uh so we have learned a lot from their work uh pivot and cloud uh pivot uh got a lot of media attention probably about ten years ago when uh glenn beck i don't know how many of you know glenn beck glenn beck was a um commentator on fox news he decided uh to have a continuous series of uh programs uh highlighting the threat uh to american society posed by piven and clower you could probably look up some of this stuff on youtube and see some of his broadcasts and it um of course the people who watch fox news and who were glenn beck fans were listening to this and he was saying you know she basically is arguing for a revolution in the united states uh to overthrow the way of life that we all love as americans and um she started getting death threats uh and so there was an article in the guardian that's what the photo on the left is taken from of francis fox piven she's quite toward the end of her career at this point she found the whole thing ridiculous but then she realized that some people were actually believing uh some of the outrageous claims that glenn beck was making on the other hand uh she has studied and social movements that have been informed by her work definitely would take a more radical approach to protest so that's a little background on uh francis fox pivot i encourage you to look at her work uh wonderful scholar sociologist i hold her in the highest regard i believe uh richard cloward passed away uh some time ago all right so what's up pivin's of basic fundamental argument um and the point is one that sociologists are quite aware of uh when we think about any kind of institution sociologically we think about interdependence it's a fundamental sociological concept that is closely connected to the work of the classical social theorist uh emil durkheim but for uh piven we need to think about how this uh institutional fact of interdependence translates into uh potentials for uh power so if she says uh the institutions can be quite hierarchical uh they can be unequal they can be oppressive uh you can have managers you can have capitalists who obviously possess enormous amounts of power vis-a-vis the workers in any organization for example but every organization every institution relies on compliance and cooperation uh no organization or institution can accomplish its goals can be effective if human beings do not in some way submit uh comply and cooperate and so if that is what organizations or institutions depend on uh then the withdrawal of compliance and cooperation uh will bring those institutions to a standstill and is a source of what we call what she calls there are different terms for this the term she uses is interdependent power um notice that when we start thinking about this form of power we're thinking about it in a more um institutional uh organizational context than the entire broad political system though it certainly can apply to that as well uh as i'll mention in a moment so withdrawing or threatening to withdraw cooperation your labor power okay this is a very marxian a concept so here we have a nice integration of durkheimian and marxian theoretical threads and insights labor withdrawing their labor power not participating this will render institutions unable to achieve their objectives their outcomes their mission their purpose very important way to think about power uh you're all students uh the institution of the university depends on all of you to actually show up attend classes and essentially cooperate and comply if for some reason you thought the institution was operating in a way that you found undesirable and it's obviously a highly undemocratic institution uh at almost every level uh you could easily uh shut it down uh you can't do this individually one student isn't gonna you know says i'm not gonna show up for classes you will be penalized this is the problem with individual versus collective action but if every student or at least a large majority of students at the university of north florida decided that they would not attend the university at all the institution would um obviously come to a grinding halt as would be the case if all the faculty did that or all the administrators right okay so i'm not suggesting you necessarily do that although i have to say i'm always surprised at how weak the student level of organization is considering this amount of let's call it a latent power we'll come back to that that you possess um okay so these are just some points that are made uh along the way or things that i think about when i read about interdependent power we think of it as a a basis for disruptive action that's another way to think about the exercise of interdependent power you're disrupting an organization's operations um power not based exclusively on material resources again this is an important point because throughout this semester we have talked about the fact that the power of the elite the capitals class uh the plutocracy uh the oligarchy whatever you want to call them uh is based largely on the ownership of property the ownership of material resources well that is not the case in the example of the process of exercising interdependent power it's based on your position in institutions and organizations and the extent to which those who have material power rely on your positional power these are all terms that tend to be used to describe the kind of thing that pivot is talking about all institutions depend on cooperation compliance as we said uh capitalists depend on workers to produce surplus value uh one of the points i made uh early on during the uh covet 19 uh pandemic was that you know why are people trying to push all workers back into the workplace when uh it could be dangerous it could be unsafe they may not be protected they may not have a sufficient ppes uh well capitalists can't make a profit if they're not exploiting workers so you had a very interesting phenomena taking place it was like a natural experiment in terms of the nature of the relationship between the owners and workers we'll talk a little more about that in a moment uh and it can take unruly forms that's a quote there should be an end quote there uh as well and i have a piece that is posted in canvas that talks about uh pivot advocating unruly forms uh direct action uh often when people propose a direct action what they're saying is that yeah we're going to protest we're just not going to march down the street and enchant you know solidarity slogans we're actually going to engage in some kind of action that is disruptive that's what direct action means you're shutting something down you're disrupting something um you're intervening in the normal operations in my review of the election that hopefully you looked at it's my post-election analysis commentary at the very end of that i mentioned that there are organizations currently that are preparing for the possibility that uh donald trump would um not leave office or use some legal or extra legal mechanism to steal the election so to speak what are people gonna do about that and one of the proposals was uh direct action uh shut down the country engage in a general strike so the strike is one of the most obvious forms of the exercise of interdependent power that labor has that workers have that those who have relatively less power can exercise the general strike rarely exercised but uh sort of the utopian dream of the left is that all workers will walk off the job across the entire country or you could have a general strike in a city i believe there was a general strike at one time in oakland a very powerful general strike where you were able to it's not easy to do this to get all the workers to agree to walk off the job on a day or during a week uh and just literally shut down the system and this is often a way to gain concessions so people have mentioned the general strike as a way to address uh for example an action like a coup where the president would you know essentially illegally seize uh power all right the walkout we know that workers walk out that's another uh exercise this has happened in the fast food industry uh as people are fighting for 15 an hour because of minimum wage workers uh the slowdown this is a classic case where workers begin to work a little slower uh not really exercising all the effort and energy uh that they normally would or that managers assume workers are uh capable of and there's lots of subtle ways that workers in organizations and again notice that when i'm talking about this interdependent power i'm talking very much about the ability to exercise this within the workplace within an organizational or institutional setting and of course sabotage you can sabotage an institution organization in a variety of ways so people who study worker actions worker resistance i'm going to teach the sociology of work um on occasion uh they will use various techniques within the workplace uh to undermine the power of managers and to exercise some level of resistance in any way that they can uh latent and manifest power so one of the points that those who study interdependent power make is that all individuals who work in an institution organization exercise latent power by virtue of the fact that the organization or institution could not operate without their compliance and cooperation and in most cases people comply and they cooperate and there is no issue right so that's the power is latent in other words it's always there but not necessarily exercised manifest power is when you're actually exercising that power and rules and rule breaking uh which occurs routinely in any kind of organization or institution where you have bureaucratic rules this is another way to exercise power in small ways but ways which often give people the sense of having some level of agency and dignity in their lives in their work lives uh collective action obviously as i said is the best way to achieve this rather than individuals striking out it's much more effective if all workers agree to engage in whatever action is proposed as an exercise of manifest interdependent uh power and this works against uh the kind of ideology that is embedded in our brains and that is that you know the people who are really indispensable the people that really make things happen uh the ones who ultimately are responsible for the efficient and productive operation of an institutional organization are the managers right because they're the they do the mental work uh often in organizations where other people do the the um manual work uh and so this ideology of superordinate that simply means those above indispensability is a very powerful ideology that has to be overcome for people to realize their own indispensability okay i just want to touch briefly on a debate within um sociology sociological theory if you take a social theory course you may be introduced to the idea of um what's called the structure agency debate and if you've taken sociology classes and this class is probably no significant departure uh from this emphasis we tend to focus the way in which structure shapes controls and ultimately determines what humans do now this poses a little bit of a problem because some people say well don't workers have any agency don't they have any ability to exercise some autonomy and therefore impact the larger organization or the institution or are people just ultimately controlled and determined by the structures of power now clearly interdependent power brings agency back into the picture you understand and that's important there's a quote by marx that sort of gets at this structure agency question i'm going to paraphrase but basically he said let's see workers make their own history but not under conditions of their own choosing so worker actions shape the society shape history that's the agency side but they don't make and take those actions under conditions that they created they're responding to conditions that are imposed upon them so in other words the kind of action that workers would take is highly dependent on the particular structure in which they find themselves so this is some way and there's an enormous literature here i'm not going to go into because it gets into a kind of esoteric theoretical debate but it is important because sometimes uh i think sociologists tend to focus too much on structural determination uh rather than what else is called a gentle agency like human autonomy um power piven obviously is making a kind of contribution to this but she's sort of sidestepping in her article all of the heavy theoretical arguments that have been made much of this gets into phil philosophical questions uh as well uh about free will um but let me say something about this question very briefly um because you know how can you reconcile structure and agency this has been sort of the challenge for uh social theory to understand what's happening in the world not taking a fully and what is typically maybe the tendency in sociology to take a structurally deterministic view anthony giddens is probably one of the most prominent sociological theorists uh in the world and uh he has struggled with this question and he developed this term structuration now what's interesting about that term is it essentially implies that structures are in the process of being produced structuration how our structure is shaped and produced and in his elaboration of this question he came up with what he called the dialectic of control and i think this is very closely connected to piven's interdependent power i'm not sure she makes sufficient acknowledgement uh of gideon's uh contribution here the dialectic of control and i'll simply read this no matter how great the scope or intensity of control super ordinance possess those with power managers owners since their power presumes the active compliance of others capitalists can't make profit without the compliance of workers or the exploitation of workers those others let's say workers can bring to bear strategies of their own and apply specific sanctions and so it's out of this interchange that structures emerge and that structures change structuration a process of creating and producing structures all right so what is this figure um many years ago i used to teach organization theory i wrote a organization theory book that was used in courses and i was trying to lay out a way to understand the evolution of organizational theories and management strategies uh historically how they've evolved and to some extent um the argument i made is very consistent with what both piven and um uh guidance are talking about so just to highlight this let's see if i can use the pen since i'm sitting here all right let's see i'm going to use a pen all right let's see if i can do this all right so managers devise some kind of organizational technique mo usually these are designed to structure the workplace in such a fashion that they will get workers to do what they want them to do these are systems of control we can call them systems of social control these are imposed on what i call the human factor of production and remember all of this interdependent power depends on human subjective understandings consciousness reactions the things that are happening to them so you impose some organizational system on the human factor production humans react to this they respond to it and they respond in ways that are not necessarily anticipated by managers we sometimes call this the unintended consequences and therefore managers have to somehow adopt and change and revise what they're doing because of the way that humans respond and that has if you follow organization theory has actually resulted in changing the way managers and organizational theorists managerial consultants think about the next strategy that should be used because one strategy wasn't as successful and effective as another so that's just a one example in the context of organizations uh that i tried to outline uh in my book and what i was trying to argue in my book throughout the entire uh every chapter uh was that we can understand what happens in organizations very much by this interface between structure managerial structures imposed on workers and the human factor response to those and i outlined this in a broader way for students to think about a paradox model generally if you take classes you know that i like to emphasize the role of paradox and so i try to lay out here what i call a paradox model and you have the action um here and the action that um someone is engaging in it could be a um effort to accomplish some particular goal uh has an intended consequence i'm going to put this in place in order to achieve this particular goal that's the intended consequence of course in sociology we also emphasize the unintended consequence um so if you have the intended consequence if you follow it down at the bottom uh that then in turn has a positive effect on the ultimate organizational objective you are trying to achieve but if there's an unintended consequence often the unintended consequence can undermine your organizational objective let me give a classic example of this um at one time workers were sort of there was something called a putting out system where capitalists would give workers the materials the tools and then workers would produce some product some commodity in their home and then they would turn them over to the capitalist and the capitalist would sell them uh the capitals thought this isn't a very good system because you know i don't know how hard that person is working at home maybe they could be working harder maybe we could intensify the level of productivity uh so the action was let's bring the workers together into a factory now what i'm describing is a long drawn out system historically but the action was let's bring all the workers together in a factory where we can supervise and monitor and ensure that they're working as hard as they should uh so the intended consequence would be uh that workers would because they're working in a factory they're being monitored and supervised they will work harder and the organizational objective is more will be produced more productivity more profit what's an unintended consequence maybe we could say an unanticipated consequence well when they brought all these workers together the workers started communicating with each other before they couldn't communicate with each other because they were all isolated separated fragmented geographically spatially now they're all together under one roof and they turn to each other and they start talking to each other about the nature of work and how hard it is and how they feel like they're being exploited and abused and disrespected and then they organize the unintended consequence and then there's an organization of workers demanding this demanding that walking out collectively obviously that has a negative effect on organizational objectives so that's the paradox model i was trying to use for people to understand one of the significant one of the most significant dynamics i would say the most significant managerial challenge in any organization is the reaction of the human factor to what managers impose on them that's the purpose of this diagram essential workers essential workers think about this this is a new term isn't it it's very interesting i'm highlighting this because there's been so much written so much said so much media attention devoted to essential workers who are the essential workers the indispensable workers essential workers workers that were doing routine jobs that were low paying and insecure and precarious and now we call them essential this is testimony to what pivot is talking about and now if they're essential they're interdependent i'm sorry they're indispensable right they can exercise inter-dependent power suddenly these workers may realize they have they have always had latent power but now they can translate that latent power into manifest power very interesting so this is how a crisis in this case the covet 19 coronavirus pandemic and its impact on our economy and how did we keep the economy going how did people stay alive who did they depend on the most for survival for some minimal level of functioning to keep things going sufficiently so that people could get by we depended depended on essential workers and we have some nice headlines here i always like to show the significance of these things with the headlines the one on top workers are more valuable than ceos i always knew that but the average american doesn't believe that because they've been brainwashed to think ceos are indispensable and workers are easily replaced and as far as ceos go they want workers to be easily replaced the subheading here sub uh heading of this article the coronavirus pandemic has revealed a simple fact it's low-wage workers cleaners cashiers care workers that make the society run if they withhold their labor power if they walk off the society can't run it's not the bankers it's not the hedge fund managers not the landlords it's not the ceos what is the lesson we will learn from this politically my fear is once the coronavirus passes we will be told that everything can go back to normal the collective power the pandemics essential work collective power notice that word is used both of those articles it's the key good stuff sociologically we need to use that sociological lens to understand the way in which these events that take place have enormous socioeconomic and political consequences and opportunities is this the same one oh i know what i wanted to show you um the top item is a article by piven where she's applying her interdependent power theory to strategy for the occupy movement so again we sometimes talk about applied sociology we don't usually think of applied sociology in this way but the point is how do we take the lessons we have from sociological analysis and apply them as piven and cloud have always done apply them to the real world of politics in order to organize for progressive social change i love this stuff it's a wonderful book written by the political economist beverly silver check out her work he's written books on globalization and the history of workers movements and one of the most significant is this book forces of labor she makes a distinction which um she can attribute to the work of eric owen wright who i've spoken about earlier in the semester and you read one of his uh articles and it's a difference between associational power and structural power and this is importantly related to this idea of interdependent power now associational power is the power that workers gain from joining together in a labor union that would be the best example of associational power structural power structural power is the power workers have by virtue of their location their structural location in the production system of capitalism that gives them some enormous or significant leverage so one of the things that silver does in her book is she talks about how associate associational and structural power uh have been used and can be used and i want to give you an uh example of this and this is something she discusses in her book so globalization and i haven't talked a lot about this i do in social change and international development but one way to think about so uh globalization uh is that it involves corporate restructuring which occurred during the crisis of 1970s as we moved into the neoliberal era and we've talked about some of the characteristics of neo-liberalism one of them was corporate restructuring and the roll back of any kind of regulations that would limit the movement of capital uh the restructuring of productive operations and so you had a continuous process of you know through the 80s and 90s into the 2000s outsourcing certain activities of the corporation to contractors and offshoring that activity to locations where wages are low regulations are few and taxes and other kinds of costs are minimal so you can think of this as a kind of extended division of labor at one time the corporation was encouraged to be vertically integrated you need to control every aspect of the operation you can't leave things to chance you make yourself vulnerable but under neoliberalism there was a movement toward what we call vertical disintegration where you essentially take the corporation and outsource those things which aren't core competencies the things that you do that are most central and important and valuable and you can outsource the other stuff to subcontractors who can compete for your business so you have this movement or vertical disintegration also we have what's called just-in-time systems i'll talk a little more about this um in a future uh lecture but the idea is that you don't want to be producing large quantities of something and storing it in a warehouse and hoping that people will purchase it or then you have to market it to get them to purchase it what you want to do is keep track of exactly what people want what they're buying at the moment and be producing and having delivered to your distribution center stuff that can be sold immediately you can't make a profit if you have lots of goods that are sitting around waiting to be sold you want to sell the stuff immediately to realize the profit that's embedded in those commodities so we moved from a just in case well we better stockpile those because we never know if there might be a cr to a just in time deliver it just in time so i can sell it immediately and make immediate profit and i'm going to talk about choke points so think about the global system now okay the way we've restructured production globally uh here's a little stylistic way to think about let me move this up to think about vertical integration versus vertically integrated vertically disintegrated so vertically integrated firm is one research and development supplies um components parts inputs are produced by a factory that is part of your corporation branding and marketing is run by your corporation manufacturing the actual production you control all of this right so the idea was the vertically integrated firm has total control of every aspect of the operation then we then had this movement vertical disintegration outsource right so some of these activities might be better handled outsourced subcontracted by other firms you can tell them what you want and how you want it done but they can do it and maybe they can do it cheaper particularly if they're located in another country so you had a movement from the vertically integrated to the vertically disintegrated all right uh this is the kind of uh image i sometimes create with powerpoint that just confuses people but here's what i want you to think about um think about a company like um apple or nike apple produces nothing nike produces nothing apple and nike have research and development they design their product they market it they obviously brand it it's the brand it's the icon it's the image they put on it that actually brings the value to it in the way that the current global system of value capture is organized so what do they do they're going to outsource like the parts and the inputs whatever goes into the final product being a sneaker or some electronic good you know we're going to let some other company produce those okay so i have the parts and inputs maybe that's a company you see that exists in the united states that's okay uh but the manufacturing the assembly the work is done in china because you can pay workers in china chinese factories uh with subcontractors that run those factories uh for virtually nothing so all of those materials are sent over to china the assembly the manufacturing of the electronic good or the sneakers is done there and then it's exported from china and imported back to united states now you see that this may be a very profitable arrangement for these companies they don't have to any longer invest and have money and capital tied up in factories and ideally not a lot in warehouses because they want the stuff delivered and produced just in time to get on the shelf and be sold to realize profit this is the global so we call these we call these global value chains this is a very typical arrangement the nikefication of the corporation now notice there's a greater division of labor there's greater interdependence the company the corporation doesn't have full control they're heavily dependent on certain things happening in order for this to work you understand so the point his point is they are more vulnerable that's the unintended consequence on the one hand they did this to maximize profit on the other hand they have created a certain level of vulnerability and what beverly silver talks about is the way in which workers can exploit the vulnerability of the production system globally to their advantage now when i talk about this in my classes i emphasize the importance of transportation logistics the reason transportation logistics has become so important in business schools um and has become a huge industry of its own by the way an industry uh and an area of study and an academic department that didn't exist 25 30 years ago why is it so important now because goods are being produced in china for example and they have to be brought back to the united states to be consumed the separation of production china consumption in the united states elevates the importance of moving the crap the goods from the point of production to the point of consumption this has to be done moving imported goods as quickly and as cheaply as possible from the point of production to the point of consumption is from my perspective the primary objective of logistics or what they like to call over in the business school supply side management this creates vulnerabilities and here's the point this creates enormous amount of interdependent power for people in transportation and logistics let me just give you one little example the image at the bottom here is a poster that is up in my office because i spent um probably seven or eight years devoting my energy to studying uh the global movement of commodities through ports and we have a port here in jacksonville you can see that in the image now the head the heading of this um poster that i have in my office choke points now if you go over to the business school you know they would give this a different title they would say these are the portals by which commerce meets the needs of american consumers consumers right okay yes great wonderful but you know we sociologists we think about it differently these are choke points these are points that can be choked off if you want to exercise power and notice the headline from a business public publication how fourteen thousand workers manage to slow down the entire economy who are those workers long shore workers they're the ones that take the containers off of the container vessels and move them off of the terminal the container terminal to rail or to a truck to then be moved to a distribution center never use the word warehouse distribution center which then will disperse those goods to the retail outlets that all of you love to spend time walking through touching all of the objects at walmart don't shop at walmart by the way that's just just don't shop at walmart enormous amounts of power you know we talk about the essential workers getting the goods getting the goods how do the goods get to their destination and how much power does transportation and logistic workers have they can shut down the entire economy because now almost every good that we consume tangible goods durable goods is imported is brought through these ports those circles at the bottom identify the size of those circles identify the significance of the ports you will notice that la long beach is the largest port in the united states one of the largest ports in the world lots of stuff coming from asia crosses the pacific ocean and enters there uh but there are also ports as you can see on the east coast and you have the international long shore workers longshoremen association on the east coast and the international longshore and warehouse union on the west coast these are probably the most powerful unions in the united states and the workers in those unions receive very very high levels of compensation now capital capital's class has done everything they can to circumvent the power of those workers by automating by creating cranes by using containers which has minimized the workforce significantly all kinds of automation has been applied to container terminals so the number of workers who are employed by the longshore workers unions has dwindled dwindle dwindled but the point is they can shut the joint down and i can tell you that the union that represents longshore workers on the west coast is much more militant much more progressive and much more willing to take action than on the east coast and they have done that in solidarity with various movements that have taken place in the united states all right interdependent power very very important stuff that concludes this lecture i will see you later or not

Political_Sociology_Lectures

Week_12_Lecture_Identity_Politics.txt

good morning students and welcome to another episode of political sociology and the topic this week is a vexing i would say complex topic related to what is sometimes described as identity politics and i should say that as i'm putting the slides together thinking about what i'd like to say about this topic i find it difficult because in some ways we've skimmed the edges of this identity politics issue and now we're going to address it more directly but it's not an easy topic to capture because there are different ways that people have defined identity politics and uh in particular critiqued uh identity politics from both the left uh and the right and identity politics is often used as a term to denigrate a particular style of politics again from both the left and the right so it's a little tricky i'm going to touch on various themes i'm hoping that you will read the three articles i think that will clarify some of the issues and debates uh the ben michaels piece is very important particularly important i think for sociology students who take courses in race sex social class uh courses that uh spend enormous amounts of time talking about a levels of inequality how we think about inequality and ben michaels points out a very very critical distinction between different forms of inequality and how we think about addressing those and in some ways you might find his position as many people often do when they first read his work uh and i would put uh with him uh the work of adolf reid uh and i would encourage you to look at his work if you're interested in a certain critique of left liberal political strategy and political action and often when people read pieces uh by ben michaels for example uh there's another article i think it might be in the readings canvas um some kind of title like what's so great about diversity what's so great about diversity and the typical sociology student of course is to some extent socialized in valuing uh diversity uh for diversity's sake and they might often think well somebody is you know questioning the value of diversity maybe they're coming from some kind of right-wing reactionary position when in fact uh ben michaels as well as adolf reed are both actually coming from the left so that's what complicates the conversation about identity politics i just wanted to start with that it's early morning and my throat is going through its usual congested state so excuse me if there are moments when i am coughing clearing my throat trying to drink some water to help it is an early morning lecture i'm giving today all right let's get started with this topic and of course my remote only works sometimes there we go uh you could spend a whole semester talking about the concept of identity and social identity both in uh psychology and sociology there's social identity theory so everyone has to some extent some kind of identity we often talk about the statuses that people occupy and the extent to which those statuses define who they are and that in a sense is identity and that's a important very important sociological a concept and your identity often shapes how you think how you act you think about your relationship between a reference group other people that share your identity all of these things are critical sociological uh phenomena so you know when i think about identity generally it's not surprising that you would expect uh identity to have some impact on politics clearly some people think that there has to be some level of identity among a collective in order for them to act politically that they have something in common some identity in common that they share and that in turn shapes maybe the political ambitions and goals and visions that they have uh and clearly identity is used as a way to mobilize so we have to think of this both in terms of how we identify personally and how organizations and for political sociology the most significant is political parties how political parties shape our identity um so a lot of identity in the political realm has been uh aimed at uh both asserting one's identity as a legitimate um form of membership that should be recognized and that should be a source of pride so we have a long history of social movements and social movements again one of the structural factors that shapes identity and obviously social movements are politically directed and they have political aims performativity is a term we use often and that is you know how does one express themselves in ways that communicate their identity or that it communicate their respect for particular group identities and this has become a major uh issue in terms of i think something i might have referred to before uh woke culture wokeness the black lives matter movement uh people uh expressing themselves in ways that endorse and recognize uh those movements as well as the actual participants within the black lives matter movement the actual participants that are behind the movement so you define and construct your identity by doing it's an important way to think about a one's identity you become uh by doing if you if you define yourself as a a political uh activist um you validate and reinforce your identity by actually doing things which you associate with political uh activism uh that's another way to think about this this concept um of performativity which is a term that's widely used now in the social sciences and particularly in sociology another way to think about identity is very sociological a lot of sociological work has been done on this is reactive reactive uh racial ethnic uh gender uh identity so if somebody identifies themselves plus let's say i discuss this sometimes in my intro class when i talk about status right like what is somebody's master status what does that mean that means if you ask them you know who are you how do you define yourself does someone when you ask them that question respond for example i'm a latino woman that's their identity now often what people will say and you hear this often from sort of mainstream dominant um racial uh ethnic groups white americans is you know why do people always emphasize their you know racial and ethnic identity uh you know that shouldn't be important that shouldn't matter right and the point is that people identify themselves in particular ways based on the extent to which that racial ethnic membership for example matters whether it's significant whether it has any impact in their life whether it shapes the way people react to them or interact with them or whether it affects their life chances right and often this depends on where you are what the situation is uh who you're surrounded by so when we talk about um the extent to which one has as an identity a particular gender uh a particular sexuality a particular ethnicity a particular racial uh membership or category it's significant and its significance will shape whether people view it and view themselves within that social identity group you don't typically if you ask you know let's say white males in the united states it's unusual that somebody would say i'm an irish male i'm an irish man you might say well you are irish right you have irish ancestry why don't you mention that because it's insignificant it doesn't matter it has no significant impact on how people react to them their life chances any aspect of their day-to-day life and therefore it isn't an identity right so identities are activated so to speak by the way in which the largest society treats people who are in particular kinds of and in our society most significantly racial and ethnic categories so this is just ways to think about identity generally but you can also think about that reactive identity in terms of how it galvanizes political orientation and action okay what do we mean by identity politics there's lots of definitions and again because it it tends to be a politically charged term uh coming from the left sometimes from the right um and it's often used as a negative way to describe politics uh there are lots of definitions okay um this is one political action based on advancing rights respect representation for various status groups racial groups gender groups ethnicity groups sexual orientation what have you right and we can think about movements that are related to these particular social status categories and when people study identity politics they're studying the experience of people are the cultural features of that identity and how all this translates into political action political mobilization and power or powerlessness right so political action that's based on membership in a particular status group okay a little background here these are just a bunch of terms and issues related to this larger question i didn't want to go there okay uh something that came up in a readings on race um and politics this is an ongoing debate on the left when we think about what to emphasize politically should we focus on race or should we focus on class is one primary is one secondary does promotion of anti-racism address the class inequality issue does the emphasis on class which is the more pure left orientation incorporate race so this is a ongoing uh debate again uh you had a reading by uh i think cedric johnson adolph reed is heavily involved uh in this conversation as well uh to the extent that in some ways he criticizes a black lives matter from a left perspective because he thinks it's emphasizing anti-racism which doesn't get at the source of the problem in the capitalist society all right so when you see critiques of movements that you think are progressive it's very important to look at exactly what the basis of that critique is because you might often assume it's from some right-wing reactionary position and that doesn't mean it's necessarily entirely unfounded or wrong but you need to know what the politics are that motivates these criticisms uh failure of class-based political action so the general kind of critique and we're going to touch on various aspects of this uh is that if you're focusing on promoting particular groups advancing their interests you are producing a certain level of fragmentation rather than collective solidarity what should collective solidarity be based on if you are coming from the left where much of these debates are taking place it should be based on social class first and foremost so this is a legitimate um debate reasonable people on the left uh disagree uh on the primacy of one versus the other and how to think about it politically and of course how to think about it strategically uh the personal is political is a term that uh emerged back in my day uh with uh the women's movement and other social movements and the idea is that your personal experiences your personal day-to-day experiences your private experiences are in and of themselves political because they are shaped by larger structures of hierarchy and power patriarchy etc right so often what happens is um this produces a a way of thinking about your personal experiences and the personal interactions you have day to day and focusing on those and again in terms of the criticism some people would say uh that often distracts from the larger effort to mobilize people against a transformation of the political economic system privatized activism is a term i sort of invented back in the 60s when you had at first people who were on the left and wanted to hopefully mobilize working-class people around some kind of left-wing revolutionary movement over time they became a very disappointed and very alienated the working class wasn't meeting their historic mission if you're a marxist and in fact many of the working-class people seem to be rather conservative in supporting you know the vietnam war and they weren't getting behind major social movements and so many of these activists became estranged alienated and they decided what they would do was rather than try to mobilize some mass nationwide political class-based movement they would live a certain kind of lifestyle so the the sort of caricature of this is uh you had all these leftists uh maybe bernie was a part of this but bernie didn't just go privatized right they all moved to vermont and they live out in the country and they grow their own food and they compost their garbage and they live a particular lifestyle but it's privatized activism you understand it's their private life is lived a certain way they have a certain identity they practice certain things but they are not promoting any kind of widespread public political agenda and you saw a lot of this uh on the left and then a lot of this ends up getting into the you know personalist political and a focus on uh certain status groups and advancing the interest of particular status groups uh cultural politics versus class politics to a large extent what we have in the united states is a cultural politics division between you know if we talk about the division between the democrats and the republicans it's not class-based uh it's more culture-based and so many people believe that the uh identity politics has kind of played into this cultural politics and for those people they want a class-based politics they don't want a culture base but they don't want a culture war they want a class war they want class struggle they want class conflict class warfare uh not cultural warfare uh and i'm working on an essay on this topic uh the title is that a class war is better than a culture war uh unless you're a capitalist and the idea is that um the capitals class the corporate are perfectly happy of seeing the american population divided along these cultural lines uh one way to think about the cultural divide is cosmopolitan the educated um versus the traditionalists right the traditional people have traditional values maybe they have less education maybe they're less sophisticated less cosmopolitan you have this kind of split between cosmopolitan and traditionalists that's not a class split uh and in fact uh it doesn't necessarily align in any significant way with political economic policies or ideology so some people worry that identity politics plays into that right that's a form of identity in sociology we talk about universalism versus particularism if you're focusing on a particular group or you're a member of a particular group and your primary emphasis is to advance the interests of your status group the group you identify with that's a form of particularism universalism is trying to find a basis that connects all of the different marginalized groups into a cohesive social movement this is the big challenge that people find on the left a biracial multi-ethnic multi sexuality movement of working people working people that would be the common element so people who critique identity politics often say it's a form of particularism rather than universalism rather than having policies that focus on advancing the interest of a particular group we need to have universal policies so this has enormous implications for public policies means tested policies that focus on advancing the interests the inclusion the representation of a particular status group or do we try to have universal policies that include all citizens as eligible for certain kinds of benefits going back to adolf reed one example here would be reparations that's a huge issue obviously reparations uh the reparations debate the reparations proposal is directed at black americans as a compensatory policy and um reed actually opposes that from the left and i've written a few things about that as well for read the issue is what are the political implications of that policy how does that advance the larger vision we have of a social movement that is biracial multi-racial multi-ethnic etc okay fragmentation versus collective unity you understand what i'm saying so when you begin to think about identity politics you begin to think about all of these particular status groups that in some sense us are operating as interest groups and the critique with on identity politics is largely directed at the democratic party which has cultivated identity politics before i said that people have an identity themselves but that identity can also be galvanized and energized and cultivated by political organizations most important political parties and that has seemed to be and i'll i'll say more about this of the position of the democratic party uh uh richard rorty a very prominent philosopher who did a lot of writing on american politics political dynamics he's no longer alive but he talked about the cultural left and the cultural left uh was a description of a tendency that rorty was criticizing and it falls into many of the identity politics issues we're discussing here so if the uh personal is political many people have focused their attention on microaggression you probably heard this term uh you know we need to be fighting microaggression the way people interact with us at this micro level they interact with us in a way that is oppressive uh and reflects maybe some kind of hierarchy and some people say the emphasis on microaggression often distracts us from the larger issue of class exploitation in other words if individuals are treated well interpersonally and they do not detect or perceive aggression that somehow this is the most important accomplishment from the left people would say well that doesn't really significantly solve the larger problem of the way the political economic system of capitalism is organized that doesn't get us very far right white fragility white privilege getting people to acknowledge their privilege and the fact that they are unable to acknowledge racism racial oppression right so once people are aware of that once they have gone through the training sessions where does that leave us i'm not saying these are insignificant accomplishments politically socially but what i'm pointing to is a larger debate that's taking place where does this get us in terms of addressing the fundamental basis of power and exploitation within the political economic system these are the ways in which people are discussing arguing debating and often disagreeing about where our priorities should be placed politically so you can define identity politics you can describe it here we are also talking about the political implications uh as a strategy i did want to make reference to william julius wilson will will william julius wilson uh i believe he is still alive uh was a uh is a sociologist very prominent sociologist uh had some major writings back in the 80s uh in the 90s early 2000s uh focusing on uh the political economy uh of race um de-industrialization uh inequality etc and william julius wilson wrote a book called the declining significance of race and this title alone created an enormous kind of backlash against wilson and what wilson was trying to argue was historically much of the disadvantage that black americans experienced were a result of kind of explicit forms of racial discrimination however he believed that when he was writing the current state of what he called the underclass and of course that became a very controversial term but he was talking about the percentage of the black population uh that was essentially in deep levels of economic deprivation and poverty and unemployment joblessness etc he said much of this has less to do with being black according to julius wilson then being working class and because blacks are disproportionately represented in the working class the neoliberal policies that were being enacted during the 80s and the 90s when he was writing have a disproportionate negative impact on black americans and so his point was you know attacking racism and discrimination is not going to get at the source of the problem because the problem is the way in which under neoliberalism the political economy has been reorganized in ways that disproportionately affect black americans now historically the racism and discrimination placed black americans within a particular occupational location disproportionately then when you have neoliberalism which is a capitalist class versus working class obviously the capitalist class imposes their power the working class suffers generally and of course blacks disproportionately so his arguments were if we want to solve the problem we need to promote what he called universal policies that is policies that are directed not particularly at blacks that particularism universally for all working people and if you have working-class policies that is policies that promote the power the bargaining power of the working class blacks will disproportionately benefit because they are disproportionately represented so he took a lot of heat because he was essentially saying race is becoming less important as a factor and class is becoming more important as a factor he was coming from the left in his analysis and people responded to this uh so again uh this is just one example of how this uh debate over the forms of stratification right we have this class in our sociology department sex race and social class right the three dimensions of stratification uh is one more important than the other uh when you think about political strategy and wilson uh focused on uh class okay um so what is uh social justice again debates about this also revolve around identity politics um nancy frazier you uh read some of her work earlier in the semester i've made reference to it she talks about progressive neoliberalism we'll talk about that in a moment uh is it the politics of redistribution that's material redistribution redistribution of resources redistribution of wealth redistribution of income or is social justice the politics of recognition that we simply need to acknowledge and recognize the rights of all groups that is the sum and substance of social justice or does social justice involve something more substantive from her perspective okay now she has a position and her position is that the politics of recognition doesn't get us very far the politics of redistribution does why is one more acceptable than the other i want you to think about that to the larger population including elites so identity politics the politics of recognition inclusion acceptance representation that's the goal presumably of identity politics we have a term identitarianism identitarianism okay i can't say that but that is a term that describes the tendency for politics to focus more on the identity of particular groups mobilizing those groups around the identity then around a fundamental movement to restructure the political economic system and so many people are uh critical of identitarianism all right so just to go back to something we discussed earlier in a semester so you can get a feel for this because to some extent she's touching on the identity politics of let's just say the democratic party okay uh what she calls progressive neoliberalism the key point here is that we're retaining the neoliberal political economic system but we're approaching it in a more progressive way and she says this is based on an alliance of mainstream currents of social movements this is the identity feminism anti-racism multiculturalism lgbtq plus rights democratic party is associated with all of these movements and promoting the inclusion the representation of these groups recognition the politics of recognition at the same time progressive neoliberalism of perhaps the democratic party or a significant segment of democratic party is closely aligned with particular business sectors she calls them high-end symbolic and service-based business sectors wall street silicon valley and hollywood in this alliance progressive forces are effectively joined with the forces of cognitive capitalism this is a term to describe um sort of digitally energized sectors of the economy which also include the financial sector obviously it includes a silicon valley she says especially financialization because financialization is a significant aspect of neoliberalism and the democratic party has very strong connections to and receives enormous campaign contributions from the financial sector so you have right neoliberalism reactionary neoliberalism versus progressive neoliberalism the point is they are both neoliberalism and so the left we like to think of that as the oppositional is to some extent a form of what we might call artificial negativity now [Music] what is identity politics promoting as some people say equal opportunity exploitation we're not eliminating exploitation of the of the capitalist system that's inherent to the capital system or simply saying that everyone should have the opportunity uh to be exploited across the entire stratification system people should be represented social justice is equal representation so equal opportunity exploitation is just a term to get at the problem with identity politics as it's been practiced social justice as equal representation period all right um i want you to read the ben michaels article obviously it's important and there is a lot there to digest i just want to touch on one aspect of his argument and do it in graphic form sometimes this helps people think about the issue um so you can think of these shapes triangles squares circles diamonds as racial ethnic groups and the image on the top that's individual inequality with horizontal inequality and he makes this distinction so you need to understand the distinction between individual inequality versus horizontal inequality horizontal inequality is often associated with people who point to the disparities between groups the disparity between you know median white income versus median blacky what's the disparity right so just like we have the identitarianism we have disparitarianism people who focus on disparities and what we need to do is eliminate the disparities we need to eliminate the discrimination on which those disparities are based so in the top image what you have is the diamonds are obviously the dominant the i'm sorry the triangle is the dominant group the diamond is this the ultimate subordinate group this is basically looks like a class system in a sense that if you're a diamond you're basically at the bottom if you're a triangle you're at the top it's a cast system so you have inequality here but you also have horizontal inequality what would individual inequality look like without horizontal inequality because what ben michaels is getting at is people want to eliminate the disparities they want representation some groups are more likely to be in a particular social category than another that's a disparity in the stratification system how do we solve that well we solve that through equal opportunities to enter into whatever occupation educational level social class position you want so what you have now is at the bottom uh you have indiv you still have individual inequality that hasn't changed at all you still have a highly stratified system but each group is represented you see that right so to be a diamond does not determine what social class you're in to be a circle does not determine what class you're in you could say these are cross-cutting cleavages that diamond in the top left does that person associate with the upper class the top strata or that would be horizontally or do they identify with the other diamonds the vertical dimension so the point is distributing the population equally and this would eliminate disparate disparities distribu distributing the population equally so if we have uh if 12 percent of the population is uh as black in the united states if uh 12 of the capitalist class are black uh you solve the problem from the horizontal inequality problem but you haven't done anything about the individual inequality problem you haven't done anything about the exploitative nature of the capital system the inequality could be exactly the same could be as extreme as it is today income inequality wealth inequality the only difference is is this what we want to accomplish that every group is represented proportionally within the stratification system this is what ben michaels is trying to get at and for him we need to address the individual inequality as importantly as the horizontal all right here's another way to put it in the context of the disparitarian the problem becomes not just not the unjust exploitative system itself but rather the lack of minority representation within it when you read the articles on um gender and politics on feminism often there is a celebration of the fact that some woman is now occupying a position in the corporate elite right and there was a criticism of that at one time that was it this aspirational idea or what i think um featherstone or maybe a sarah jaffe called trickle-down feminism right that is that individuals who make it into those positions represent it's a victory uh for a feminism but it does nothing to address the larger experience of the vast majority of women right and it glorifies a particular position at the top of the hierarchy and that should trickle down right in terms of uplift and aspiration so ben michaels is uh promoting of something uh more radical and it's important to understand why it is that almost all corporations corporations that are highly exploitative in terms of how they treat their employees in terms of the disparity in income between the corporate elite in that particular organization the capitalist class the owners the ceos whatever you want to call them and the rest of the working population but at the same time they're very open to uh diversity plans diversity policies they have you know ceos that are responsible for ensuring that there's opportunities for women blacks hispanics etc to occupy positions within the organization they support diversity they support inclusion but then they turn around and they impose minimum wage legislation living wage legislation progressive taxation because diversity and inclusion as praiseworthy a political goal as it is does nothing to challenge in any way the sources of their power and domination it goes back to the term by frasier politics of redistribution of the politics of recognition uh we'll recognize we'll include we will allow different groups to be represented that's perfectly fine we need more diversity we're all for that but redistribution redistribution of income and wealth and property no so politically how far does this get us um and this has been the critique of identity politics uh that it is essentially hurt um the democratic party assuming the democratic party is interested in actually promoting any kind of broader class-based politics which i don't think they are but the point is that it's hurt them electorally because it looks as if the democratic party is simply a party that's interested in promoting the rights the inclusion or representation of particular groups and that's going to solve the issue and this becomes a substitute so over the last 20 to 30 years as inequality has increased inequality and income inequality and wealth of the democratic party has not focused on the question of inequality it's focused on a question of representation recognition equal rights access equal opportunity and particularly equal opportunity meritocracy what that means is every person should have equal opportunity to obtain an education to make it make it some level of mobility none of this questions challenges the level of inequality in wealth in income the stratified nature of capitalism the social relations of capitalism none of that is discussed none of that is challenged just want to make sure that each group has the same opportunity the same rights get an education you know use their skills and energy and talents to go as far as they possibly can as far as they desire you've heard this kind of language over and over again mobility they want everyone to have an equal opportunity to be unequal another way to put it there are currently a number of books that are coming out there's a backlash against the meritocracy basically meritocracy is a concept meritocracy is the way it's been used often this is directed uh at the democratic party that tends to elevate educational credentials um to the highest level in terms of evaluating the contribution of individuals the value of people and what you ultimately need if you are making insufficient amounts of income suffering from economic insecurity what you need to do is you need to get more education this is the answer this is the panacea right so people have been writing about the fact that meritocracy has become a way that people essentially justify the positions they're in well meritocracy presumably meritocracy means that you allocate positions and income and wealth and influence and power on the basis of what people have accomplished based on their merit so if we have a meritocracy and if you promote meritocracy as a principle uh how do people feel that are economically deprived that don't have much political power much economic power are living with economic insecurity well we have a meritocracy and we have equal opportunity to educational resources so if somebody is suffering economically they have no one to blame it themselves so we have several books i'll show you one i'm reading now it's outstanding this is let's see if i can put this there we go okay the tyranny of merit meritocracy used to be a principle that everyone agree of course we should be you know hiring people on a basis of merit but we know that the ability to get married is obviously unequally distributed and the question is whether that should be the single basis on which one is able to move up or down or experience economic security and what does it say to the losers what does it say to those who don't have a college education so sandal michael sandell he's a very well-known prominent political philosopher just wrote this book the tyranny of merit and he talks about the rhetoric of rising much of the book is a critique of the democratic party or the liberal class as um thomas frank would describe them and how they glorify credentials education and because of that they totally demoralize a significant portion of the population this creates resentment in hostility toward the highly educated who are presumably the experts the ones who are in the know the ones who should be making the decisions and of course this plays into the hands of right wing populism for sandel much of what we're seeing today the culture war between particular segments of the american population is attributable to the way in which the democratic party liberals have approached the question of inequality and meritocracy and um let's see here's another one i can't read too many of these books you know but the meritocracy trap i said there's a backlash against meritocracy another book that was just written uh called the cult of smart the cult of smart and that individual makes a somewhat more controversial argument and the argument is that not everyone has the ability to excel in an academic institution doesn't mean they're stupid doesn't mean they lack intelligence it simply means that not everyone has that particular talent that particular skill so why do we as a society hold up as the single most legitimate basis on which to allocate money resources prestige status wealth income how well you do in school or how many credentials you have now the controversial part is the claim that people have different inherent levels of skill and talent i don't know if this author uses intelligence i haven't read the book but the point is that's just another recent publication coming from somebody who historically has been to the left and when you start talking about inherent differences between people in terms of their abilities and he's not saying it's correlated with race or ethnicity or anything like that or religion or gender he simply says that it's a fact that there are these variations and so we can't just assume that everybody is going to excel in a particular institutional environment and that institutional environment and what that institutional environment gives people and those are credentials should be the single basis on which we allocate and justify how much income and wealth people have so sandel talks about the rhetoric of rising right everyone can rise everyone can make it there's social mobility there's equal opportunity we just have to open the gates and everyone will have a chance equal opportunity to be unequal does nothing at all to address the unequal distribution of income and the unequal distribution of wealth that is the key point here okay i'm hammering this home uh what about white voters are they an identity is that identity politics of course there's been a lot written enormous amounts written about the white working class it's funny when people use the term working class it's often assumed that they're talking about white people even though a blacks are disproportionately represented in any measure of what we would call the working class whatever way you want to determine who belongs or who doesn't um and clearly it is an identity politics that is now being used certainly by donald trump uh and by the right uh the right populism uh is definitely based on a kind of cultural chauvinism if you like uh ethno racial nationalism and uh this notion of whiteness has been a lot written about that over the last 10 years and particularly over the last four years and if we think about reactive ethnicity you might say well you know people may not identify as white if it is insignificant right if if they have white privilege if it has in no way if it in no way affects how people interact with them or the opportunities for them like we said somebody you know identifies as black or somebody identifies as latina hispanic that is something we would probably expect given the nature of the society in which we live and the kind of experiences that people who are a member of those groups daily face but we now see this notion of whiteness and it's reactive because first of all you can gin people up by telling them that you will be replaced they are taking over the country is changing this was a white christian nation you hear all this kind of um language being used and it's being transformed and one of the slogans of the white supremacist marching in charlottesville was we will not be replaced so clearly this has become another aspect of the culture war dividing working-class people as i said what i'm writing about and really advocating is a class war i don't know what form that would take right now we have an asymmetric class war that is a capitalist class is beating the hell out of the working class i'm talking about class conflict class struggle between working class people and capitalists is better than a culture war which is the source of the polarization that people bemoan today in the political system okay so um this is a diagram i put together some time ago talking about the nature of the democratic party using the concept of nancy frazier progressive neoliberalism and you can see that the neoliberal aspects as well as the what are called progressive uh aspects uh dim at one point i was using the acronym dim diversity innovation and meritocracy and you know what is progressive uh it's progressive to promote uh innovation and cognitive capitalism particular industries that are on a cutting edge this was viewed as a kind of progressive thing but what i'm pointing to as you move from the left to the right as you move to the right you see that promoting these kinds of presumably progressive factors movements developments has not produced anything that is ultimately progressive or transformative yes we have enormous innovation in the technology sector now what has it given us we thought it would be the great leveler everyone would have access to the internet it would be sharing information widespread participation uh district this distribution of power what do we have instead we have surveillance capitalism we have technology being used to monitor levels of productivity of workers in factories and offices of course the question has always been who controls the technology it's not neutral who controls it and how is it used and for whose purposes and within the context of a capitalist economy obviously it's the capital's class that has the greatest ability to use the technology for their own interests so we have surveillance capitalism if you haven't read that book shashana zubov wrote a book called surveillance capitalism it's a magnificent i would call it a magnum opus explaining exactly within a kind of um jesus marxian concepts although i don't think she's a marxist to describe what's happening in the social media high tech economy google um facebook we have toxic financial instruments the innovation in the financial sector actually created instruments investment instruments and vehicles which contributed to the great financial crisis we're not benefiting from that and of course they've used their technology their technological innovation to extract money from people's accounts in very underhanded ways and we have oligopoly within the tech sector of the tech sector is essentially dominated by uh three or four firms and so were at their mercy so that's the innovation cognitive capitalism which presumably was a kind of progressive capitalist development didn't turn out that way education and meritocracy we've already talked about that what is it produced what we have today is hereditary meritocracy what that means is the amount of merit you have the amount of education you have the quantity and quality of education you have is largely a result of how much wealth and income your parents have the correlation is extremely extremely strong it's created this idea of professional managerial privilege this is discussed when you read the article by thomas frank from listen liberal and the democratic party has essentially become a party of professionals highly educated professionals and we have massive student debt for those who have been told that that's all you have to do is get an education we live in a meritocracy the key to success the key to rising the key to upward mobility is getting a college degree etc this has produced student debt so that's where that's taken us so the three elements of progressive neoliberalism certainly have not challenged neoliberal political economy and they have turned out to be somewhat regressive in their impact and diversity we've already talked about uh how identity politics disparitarianism creates in many ways bigger problems than it solves so out of all this we've had growing individual inequality all of these features the innovation right because people say the key you know the key to uh ensuring that we have uh an economic system that benefits people innovation is important so they glorify innovation right and if you think about it over the last 20 years the level of innovation has probably been as high as it has ever been in terms of technological information technology innovation probably been nothing like it over the last 20 years during that period of massive innovation what have we seen record levels of inequality of income record levels of inequality of wealth and let's move down to education meritocracy the millennial generation the most highly educated generation in history the most economically insecure diversity let's encourage people to strive let's eliminate the obstacles let's eliminate any uh forms of discrimination inclusion representation this is not in any way addressed the inequality issue as well all right this is the last item i want to show you how the um identity politics issue plays out i don't know if you can see all this see if i can move this back it up all right there was a interaction that took place i think this is probably back when bernie sanders was running in 2016. and um a woman got up uh and she said you know how could she become the you know the second latina senator the u.s senate she was asking sanders this and people were cheering and everything he said you know let me respond he said in a way you may not be happy with and this sort of gets at this tension right because his point is sure we need lots of representation of all groups in society but he says it's not good enough to say hey i'm a latina vote for me it's not good enough i have to know whether that lefting is going to stand up with the working class of this country is going to take on big money interests one of the struggles that we're going to have right now we lay on the table the democratic party it's not good enough to me he's saying okay well we'll have x number of african-americans here we'll have one number of latinos there z number of women here you see what he's getting at his response is reflecting this issue of representation versus fundamental policy change okay so he says um that's not good enough to have a certain number of african-americans certain number of latinos certain number of women he said that's not good enough he said we need that diversity of course that goes without saying that's accepted right now we've made progress in getting women into politics he goes on to say but here's my point this is where this is going to be division within the democratic party this is the debate exactly this conversation this response to this woman reflects the very issue and the fact that sanders responds this way in a way that probably was not what this person was expecting nor maybe the audience that was listening to this points to the fact that he's actually coming from a left position where he believes the point is that we need to promote policies that benefit the entire working class and it doesn't matter what the representation is ultimately of the people who are in the movement obviously you want as many people as possible you want as much diversity as possible but that alone does not in any way get you where you want to go politically so he says you know if that guy is going to be shipping jobs out of the country and exploiting his workers it doesn't mean a whole hell of a lot if he's black or white or latino so the politics of representation versus the politics of redistribution to get at frasier's point okay as i said this is a difficult topic to easily capture in a single lecture but i've tried to point to the political implications of what is regarded as identity politics i hope you have an understanding of that of course any questions you have please let me know i'm no longer using voicethread i'm just using the other system so you can direct your questions to me okay thank you you

MIT_6S087_Foundation_Models_Generative_AI_2024

MIT_6S087_Foundation_Models_Generative_AI_CHATGPT_LLMs.txt

all right welcome to the third lecture on Foundation mulative AI So today we're going to cover chat GPT um and um right I mean I think for a lot of people chat GP was the the tool or the the AI that really made people understand this is different now we're able to do things we weren't able to do before and and definitely uh created some kind of hype uh so hopefully after this lecture you'll you understand kind of the basic idea and also somehow understand the BET right the bet that open Ai and Ilia the head researcher did in terms of what actually would lead to CHP and how in hindsight it might be quite I mean easy but it was a really daring bad not obvious at all at the time that this would actually work out um so should be be a lot of fun and just to quickly go through our course schedule as well a little bit right so today is January 16 uh and next time we'll talk about stable diffusion image generation and then we'll talk about emerging Foundation models basically Foundation models generative AI in the commercial space H we'll have two guest speakers and then we'll end with the lecture on AI ethics and regulation as well as a panel okay so what have we talked about before we started off H with an introduction a short high level intuitive answer to what is foundation M generative AI we went a little bit on a philosophical digression and asked about how's the world structured because that allows us to think about how we should learn in the world then we on the second lecture went through all the different algorithms um and yeah today we'll we'll dive in more specifically into chpt and kind of uh pull everything together um and to reiterate right so what do we do in uh Foundation models geni well we apply this self-supervised learning where we learn without uh label data so we can we can get you know as much data as we want because there's no human being in the loop so there's no limit how much we can scale this up and and what we get from this you know by learning from observation and learning from the data directly is a very contextual and relational understanding of meaning and we gave this example before about you know from a supervised learning perspective you learn what a dog is from seeing you know labeled uh examples of dogs and in reinforcement learning you focus on optimizing certain goals and you understand a dog in relation to how it makes you happy or fulfilled in some sense or optimizing your goals but in self supervised learning right it's the foundational technology behind uh Foundation models you learn from observing dogs in different context and you get a very relational definition of a dog so it's something that's walk by an owner with a leash it has an anistic R with cats it chases fris with oone right this is your definition of what a dog is and today we'll you talk about something that's extremely engineering heavy in you know chat GPT uh relies on a lot of tricks and Engineering insights and breakthroughs that we're not going to cover and I think still though you know like it's like talking about a car you can understand the high level perspective of a car and get some insights how to work how it works and how it's going to be useful for you without getting into all the engineering details but of course in real life those engineering details really really matters and are very very hard to get right and that's something that we won't really dive into in this lecture because that's just when you bring something up certain scale and you have to paralyze a lot of machines Etc and think about high parameters it's a whole science so it's not trival at all but it's kind of hard uh to teach in a course like this and and you have to learn by just actually building this stuff um okay so um also a little bit of philosophizing in this uh class as well um I think that again like we talked about a little bit of a theme here right is that the why this new AI is so powerful is because it doesn't Force things to comply to Simple Rules right it kind of abandons our ability to understand and compress what we're seeing and deals with that chaos directly that's why AI is so powerful and so humanlike um so also like when I talk about this in CHP we try to make very high level um statement but of course the nuances matters and I think it's quite interesting uh I took this quote from a general from the 18 and 1700s and he says this uh quote that P Theory which sets itself in opposition to the mind and what he meant was that he's a general so he fights in battles and War and at the time people loved to come up and theorize around War like we should have certain rules and how soldiers should behave in fighting and stuff like that but he's like well I've been in War uh and Wars don't comply to rules first off so you know everybody has a plan before they get hit in face basically so you know as people start shooting at you and you have this fog of War of you don't know what's going on there's no simple rules to help you there and also what he says this in terms of the mind he says like well actually he's realized by working with soldiers that soldiers and human beings our mind we're not good at acting according to rules that we try to memorize we're very intuitive and very kind of quick to react to things by our intuition that's what really really matters and that's what we're strong at so if you force a soldier's well Al try to memorize a lot of rules and that's how it should act in a battle you're kind of screwed and very limited in what you can do uh which also is something that I think AI uh in a new type of AI leverages okay so chat GPT um right this is a really amazing breakthrough that uh has some very humanlike Mastery of language that we can communicate that can basically solve a really wide array of tasks for us anything that can be phrased in terms of text language it can it can basically solve and now as well when with gp4 ET becomes uh it's able to handle multi modalities but it's it's extremely powerful so let's try to break this apart well first off what does this name actually stand for well the chat part is obvious it stands for chat and then GPT stands for generative pre-trained Transformer and this is a I mean a good description of what this uh actually is um and I think also if you look at the the two different three different concepts here they're also almost corresponding length in terms of how important and influential they are in making chat GPT work so chat part we we'll cover last it's the kind of the least important one in some sense H the Genty pre-trained is the self supervised step of how you train this and arrive at this uh model and then the Transformer is the basically the engine behind it in some sense and so let's start with this generative pre-train what does it mean how do we pre-train this model and that's basically where openi spent 99% of the compute was to do this pre-training step so it's it's very very important okay so what we're going to do is that we're going to uh just take some random text from the internet so we have a sequence of words and and then we're just going to try to predict uh the next word based on previous words so let's say we have uh we start with i here as input and then we want to someh predict the Target right so we know we know or the computer knows somehow by just downloading the text that what this whole sequence is but when it trains this AI model it hides part of it right so it just inputs I to the AI model and then the AI model is supposed to do something with it right so it's supposed now to make a guess or what's the next word so you you basically would allow the model to uh guess and then maybe it's off right and then you can give some negative feedback uh and then when it gets it right you can give some positive feedback right so this is the high level uh what we want to accomplish so the first thing actually that you start thinking about is well there's multiple you know giving a sequence there's just one ground Tru correct prediction there's still in this example just one single word that actually will follow but there's tons of words that are not the correct guesses so maybe you want you know you want to allow the model to make the best use of this example as possible so you can basically make a lot of guesses and you can you can give them information a lot of aot about a lot of different uh guesses that are actually wrong right so you're able to give more information to the model here like hey actually go and gone here are wrong umbrella is also wrong right and then it kind of gets it right and then you give some positive feedback back and we're going to kind of do this um we're going to maximize this so uh we're going to create scores or predictions for all words in the L like in the human you know vocabulary in the English vocabulary that sounds extremely expensive and it is quite expensive and so I have different tricks to make this work but you're going to make a guess and a score like a prediction for every word in English in the English language and only one would be correct but it gives a lot of feedback as well because there's a lot of information knowing which ones are not correct and and how we're going to do this is that we're going to create these U probability scores meaning that these are just non- negative numbers and they all sum to one so they actually corresponds to the the models guess at the likelihood of like the likelihood of a word coming next so you make a distribution of all possible English words and then the score corresponsive likel of this word coming after the what is seen so far uh okay so here you know at this point the model is in i12 it creates a distribution you know here's this four words but all words in English language then um you know you Rel you you reveal which one is the correct one so the is the correct thing to come after and then H you give this feedback to to the model it's called back propagation so you give some feedback through model it should push the uh the score or the probability distribution for the correct one to be bigger or larger and then reduce all other ones so you know the next time it sees uh the same example or a similar example it actually does better and you know this is just one single example but you accumulate all of these directions and information across a batch of examples that you see at the same time so it takes small small steps to getting a a better and better distribution and a more itic distribution of what word will come next given previous words and you do this in a batch on tons of examples and of course you know we have unlimited amount of data uh because we can just get text Data from online of course we do this on the whole um uh sequence so we can make the most use of this sequence so we predict the uh the next word for every possible uh combination here all right so now that we've trained it we have a model that's able to uh predict the next word given previous words so again we uh have some starting point I for example it's not a very interesting prompt to model but it's a starting point it gives us a distribution now over all possible words in English language and then we just take the argmax we can sample where you take the argmax which which the most likely word to come after this we we take argmax and we put it into a sequence and we get a longer sequence and then we can run the model on this sequence and then we do the same thing and we just keep going H we can make our sequence longer and longer and we can continue this uh you know for example till we reach a period or something like a a specific token that says we're happy until we complete a complete sentence for example um and this is kind of expensive uh to do because you have to generate one thing at a time but of course training is is much faster because then you can just you don't need to generate and run on your own input you just run look at the input that you get yet but here you actually have to look at the prompt generate the next word add it and run it again so it's kind of expensive and it's sequential in that sense but you don't have to do that during training only in evaluation and training is what's most expensive so that's fine somehow and okay so you know if I went home it's not a very interesting prompt uh what type of prompts would be more interesting well we talked about this a little bit before and now you know if it's really good at predicting the next word based on previous words we can give it interesting prompts and it can start solving interesting tasks for us by just being able to predict the next word based on previous words um so here we see that we we uh basically have this different language task we just give this to uh the model and if it's really good it should be able to generate the the sensible things that we're looking for and if you try this for CHP right it does so it basically has killed a lot of different research Labs that focus on a specific task because now it does all of this uh really really well and I mean this is basically from a modeling perspective this is chdp in a nutshell okay so what's the I mean it sounds maybe sensible and reasonable but of course the what set CTP apart was a tremendous scale this was trained at a scale with an amount of data and parameters that we never seen before so this is a a year old now but this is I think this was 3.5 or something the first version it was using 175 billion parameters and just training the the final model like I not including all the iteration that you have to do to try things out but just training the final model cost around $5 million just in in Compu electricity bills right that's how huge and much compute they spent on this and uh again so it's a very very simple approach but it's it's a certain scale that's that's never been seen before and really that's what a big part of open eyes bet and and the research there is like well you know we've been doing this language modeling for quite some time trying to understand Language by predicting the next word based on previous words and we're doing you know we're using it for certain things but I mean know very few thought and were convinced that if you just scale this up big enough it will become a multitask Sol and show humanlike Intelligence and that this actually really will work and people talk a little bit of this emerging abilities because also it's not linear right like you start adding and putting putting more and more uh compute and parameters and like oh it's still not very useful but then at some point just start being like extremely useful so it was it was a huge kind of leap of faith as well for open eye to say like well we're just going to go all in and just make this bigger and bigger and bigger and and and then like in hindsight like maybe it makes sense but it could have been a case like it wouldn't work and then people like oh that's a stupid bet like why would you think such a simple idea and approach would lead to to such sophisticate intelligence but it did okay so we covered thetive pre-train part right so uh you know we've now said how we basically are going to uh train our model but how does this model look like like how does this kind of engine look like so if if this whole thing is a car then basically g pre-trained is how you teach a driver to drive and then the Transformer is the engine that that it leverages and some people definitely claim that this Transformer part is extremely extremely important so there's a debate a little bit what was the most influential part of making uh CHP and large language model possible uh Transformer is definitely a significant part of it and and I'll let you judge for yourself but uh I think it's less important than the actual modeling perspective that we've we've come up with okay so in order to understand the Transformer we're going to start to thinking about how we can process sequences so text is just sequence of words so we just going to think about how we can process sequences okay so let's say we have this uh uh sentence that we downloaded from online and we want to process uh one word at a time we want to predict the next word so we'll have our model and it basically uh looks at the first words create some intermediate embedding or feature here one and and then it uses the that in a second step to uh predict the next word okay we go on and then it uh looks at the second word so I I lookas went but of course to be able to do a good job you also wants to able to incorporate the previous word and the features from there so you kind of also processes and includes into the second Vector both the previous word and the current word to create a new representation of the whole sentence so far and then it uses that to to kind of predict the the next Target okay we go on and we do the same thing um and uh of course we do this for whole sent and I think what's this sounds maybe trival but the thing that's important to notice here is that for every step here that's label with the same uh digit you know they can all be done in parallel so everything at step two here can be done in parallel they don't need to wait for anything right step two needs to wait on step one to finish right but all the two can be run in parallel they don't they don't rely on each other and step three can be run after step two is been run because step threes don't rely on each other and so on and this is key because in in deep learning we use this uh um computer is called gpus and basically the cost is you know if something can be done in parallel it's a single cost we don't care how much they do in par if it's it's done in parallel it's a single cost so if we can make multiple steps into single step in parallel this is a single cost we want to run things in parallel as much as possible so here basically you know this be a cost of four because all these different numbers can be run in parallel uh so this would just be a cost of four and then of course uh processing this whole sequence will be a cost of nine and this might be you know seem like a pretty good job and reasonable because this is a sequence we have to process process it somehow and we're paralyzing most of the steps so maybe that's the best we can do and this is called a recurrent new network when we process things this in a sequential way H we try to Pary as much as possible but your current process depends on the previous the previous step and these are extremely extremely popular and a version of them called uh lstm long short-term memory networks um it performs really really well and some people say it performs you know almost better than Transformers a lot of times but they just take them longer to train because we're going to realize why it one a point but they work really really well and and also notice here somehow that uh this was very very intuitive for researchers to say like well text we read text from left to right we process words one at a time and therefore our models should to to to tble to learn effectively from them okay so we're going to simplifies a little bit and just think about how information Flows In in these uh uh networks or models so again for rec Network we basically have this very simple information flow where uh things flow forward this in kind of this sequential way right so to get from uh you know for the information from I to go to the uh information prob being processed step number nine basically right when you want to predict the period has to travel eight or nine steps here to to to uh be used so let's think about this a little bit start we start off now in a Transformer which basically starts off the same way so we we let the first we know we just process the first word uh and we prict the next Target based on that uh but the the difference happens when we look at the second word so instead of doing sequentially and saying that we're going to use the process that we've been used that we used before we're just going to directly incorporate information from I and when to the Target so we kind of we're we're not going to enforce this quential structure we're just going to directly let the information flow from the previous word to the current word Etc right what we've seen so far and then if in the third step as well we also just going to let information flow directly and have a direct connection to the previous words and not force things to happen sequentially um and the important thing to notice here right before when we were processing things sequentially you had to wait for the previous step to finish to do the next step right but here you don't because here everything is processed independently so you don't have to wait because every Target somehow has a a node or sorry an edge to the previous word so they can all be run in parallel they don't need to wait for anything they basically kind of re redoing all the work somehow for every step because they all have this added to the previous words so all of these steps now like none of the steps have to wait for each other right there's no independency they can just go Direct to the the the source and use that information and of course you know we do this for the whole sequence and um again to reiterate right so for the last Target all of these computations can be done in parallel right they they're somehow aggregated at the Target and they can all be done in parallel because they don't rely on each other but also right this can also be done at the same time in parallel because eight doesn't depend on Nine and Nine doesn't depend on eight so so they can also be done in parallel it's the same step and this is true for all of these steps yes this sense when we compute a output distribution over all the words later like as a prediction we talked about the first thing know oh yeah yeah sorry sorry so exactly this this works only during training now right okay and I'll come to that actually later so this is only during training where we can optimize this way okay but that's also yeah that's a great question but that's also like something that's uh in uh in deep learning we basically almost I mean the training is the most expensive part uh because then you op you optimize and and you do back propagation to update update your parameters which is very very expensive when You' when you kind of uh when it's done you freeze it and you don't update things more and it's going to be much faster to run so it's much less uh uh well that's a modification but uh it's a little bit less sensitive in a sense uh we care about about I mean we care about both being fast uh and yeah I mean but somehow H this is going to be much much faster to train than a recurr network so you're going to be able to get much much better performance and then the difference in deployment is going to be less uh significant okay but during training we can do this because we not upend the words we just see them in the sequence we can do this um yeah actually this is my not this is of course only training um all right so now you know again we look at this next to each other uh like what's I mean maybe I mean what's the biggest difference somehow right one of them the top Rec Network looks very structured right this has kind of a strong bias of of processing things sequentially the bottom looks very chaotic and it looks like Ah that's just a lot of connections it probably is pretty hard to make sense of the sequence given that it's all FedEd you at the same time uh so it's may be surprising that one works better another one you know one that definitely kind of needs more data to start learning useful things so Transformer needs typically needs more data to start doing a good job but there's also another thing that we're uh you know really forgetting here right so in a recuit network things are processed one at a time so the the model can figure out that you know Financial comes after the because it sees the first and then Financial but in a Transformer you know in the below here like if the only thing you see is the word and they're all F to you know for for if you look at the prediction we going to do at at step number nine if you see all these words the same time right there's some kind of comp like you can permute all the words and you basically see the same thing so there's no sequ there's no sequential structure and force and Transformer whatsoever actually right everything is being fed at the same time so there's no sequence anymore right you're seeing everything at the same time so there's no sequence um so how do you solve this well you do the simplest thing you can H again just we remove the numbers here and make make it more obvious that there's just you know words there's no sequence anymore because everything is connected in Transformer but recur not there is still the sequence by how by virtue of how things are processed so how we solve this that is that for for the Transformer we're just going to add to each word a positional encoding so we just add the position again it's like the the Transformer has to figure out if the sequence matter it should use that information even but it now has that information at its disposal because we're going to encode a sequential structure not by how things are processed but by just appending a positional encoding and this is actually you know quite almost contra inuitive it's you know it's it's like seeing all the words in a book at the same time like it's fast but it's very confusing and then you have to like figure out by a small you know number how things are actually oriented so like if you go to the movies and you think in terms of of frames you can sit down and digest the whole movie in one second it's like super efficient but you're seeing all the frames you know flash at the same time and then after like in your own head you have to put them in a sequential order if it's useful as you know understand the plot of the movie which typically is right but there not like the transformer has to learn that implicitly because it's not happening directly okay so why is this good as well well it's good because another like it's fast but also it's good because for new networks memory is very hard so it's hard for networks to remember things so let's say you know that that if you read a book or you watch a movie if you want to understand the end part it might be good to kind of go back and look at the the start starting part of the book or something you know or or it's good if you remember that information but probably you know if you don't remember you have to go back and look it up so in a Rec Network Rec Network because we're processing things uh sequentially here so to for something to be used like to for for information about the first uh word in the sequence to be used the last step here so you know to have information about I at the step point you digest money you have to somehow remember that information through throughout this whole processing step right and that's very very hard for networks to do and a lot of work was done to make that work better but the nice thing with Transformer is that it has a direct connection so if you know like it has a lot of connections but if it's a very strong kind of recurring uh thing that the first word and last word correlate somehow it can pick up on that very quickly and make that edge very very strong and kill out other edges so you can incorporate information very quickly or very efficiently so you can basically incorporate long distance information in this sequence very efficiently because there is no real sequential structure but when we force the sequential structure and Curr Network it's much harder because then we need to remember that as we process things okay so to summarize Transformers um we do everything I mean everything we care about in when it comes to deep learning that we want to do things in parallel uh as much as possible because if everything is done in parallel it's a single cost and and a Transformer is doing that optimally because it just process everything in parallel and then it removes sequential structures we have to you know give that information to the model by appending positional encodings um and also it's good that it's you know it turns out that this long uh distance information in data is typically very useful to be able to incorporate efficiently and memory is hard so Transformer are good at incorporating this information and and is able to do better by it and they replace these Rec networks and again right this is we talk especially during training but this is uh uh the difference is less severe when it comes to uh inference or when we deploy them and uh again like since now we have St super plus learning we can train by just downloading text from the internet and there's no human being Loop the scale of data is so much you know so big that we can learn a lot by you know we can learn we can learn and afford to use a lot of data to learn basic things so Transformer has much less structure and has to relearn a lot of this structure but since we have so much data and we don't need to have labeled data we have we can afford that right we can afford a train on a scale that we haven't seen before so that's also why this works so well okay so uh now we have a language model right we know how to train it uh we know what kind of engine that it can use to uh be efficient and work well so now we're just going to look at the F last part which is the chat part um so you know you you you train this model now that you call DPT 3.5 or something and now you want to turn turn into chat GPT right so we have a uh a really uh good model basically we've done 99% of all the the work that's that's required and a lot of people still kind of debate how important this last step is but open is it does a difference uh but you know when we have this model we see we we see that kind of oh it it works fairly well but H we want to be able to improve it there has some stupid failure cases and we just want to make it a little bit more sophisticated so the first thing we we we're going to do is that the model now has been trained on a vast amount of data from you know any Source on internet you can imagine right so novels Wikipedia Facebook posts uh you know anything basically but how you how users are going to use this is through some chat bot right so it's dialogue like human dialogue is what they call it and of course it's been trained on a lot of text that's not human dialogue and now we're just going to say we want to be able to hone in and focus and adjust itself a little bit by training only on human dialogue so we going we going to go and collect the best data we have of human dialogue from whatever source that we have and we're going to train for a little bit longer on only that data so we're going to find T the parameters only on human dialogue data so it can hone in its parameters and focus on a specific use case okay so we do that and we're even one step closer and so now we're even working we work even better but then opening I wants take it one step further and say like well there are some observed problems in this model we're going to address um and uh one one thing is that right now when we've trained on on this text data right each each uh Target is worth the same somehow and we don't we don't really uh separate good or bad dialogue so the model doesn't really know what's good or bad dialogue it just knows what's plausible dialogue from the internet but somehow we just want to say like well yes you know what's plausible dialogue but it would really good if you understood what's not helpful dialogue and what's helpful dialogue so you can just give us helpful dialogue and okay another another problem is that we're somehow too greedy so when we train things to predict the next word based on previous words all we care about is to give the most likely next word but if you want to generate a sentence right you don't really care about optimizing the likelihood of the next word you you care about optimizing the accumulated likelihood of your whole sentence and a lot of times you can sacrifice you know short-term profit for optimal long-term profits and exactly the same thing when it come to generating these these sequences so what we really care about is the the score of the sentence at the very end when we're done with it we don't care about picking the best you know the best step in at every you know every step of the way for example here right if you go down a little bit at bank you can you can reach out reach a much higher optimal score at the end of the sentence and that's of course what we care about so somehow we're too greedy we should be a little bit more long-term optimizing because if you give a whole sentence we care about the the quality of that sentence that's what we want to optimize for okay and then the third Nuance or or kind of difficult with the current model is that somehow we would like this model to be a little bit bit more robust let's say so it turns out that this model now has been trained on uh text online and of course it it works really really well but people are going to use this and interact with it in ways that maybe it doesn't really correspond perfectly to its training data so it's going to uh see things I haven't seen before so there might be a kind of a distributional shift between how people use it and what it's been trained on and also right as we said when we Deo this model they're going to you know generate a word add it to its its own uh add it to like its current sequence and then rerun itself on its sequence right so it iteratively create a longer and longer sequence by running it it itself on its own output so it adds a word after word and no AI model is is perfect so maybe it accumulate some error as it start adding words and it's just going to go off a little bit and it just goes off a little bit like here for example when you say you know I went to the financial and then just you know some small error happens and it goes off the road to restaurant like somehow you know they started seeing that okay now it's basically go Haywire because it went off and it's in a different space than it's been trained on and it's just going to generate nonsense so somehow we want to be able to say like well if you find yourself you know a little bit off the the the path you should be able to find your way back to be as as robust as possible to any kind of use case and also when you generate things you don't want to you know go off the road you want to be able to find your way back as as much as possible to be as useful as possible okay so uh these are the the the three different things we want to address right what's good and bad dialogue uh we don't want to be that greedy and we want to be more robust and learn to solve correct and this is where we're going to do reinforcement learning from Human feedback uh and that's what open AI does on chtp and this was very very hyped for a long time but now people talk less about it uh okay so what do we do well we have a great model that's been fine tune on dialogue and it's able to generate really good answers still uh to different prompts so uh we're going to run this model now on a collection of prompts and we're going to generate four we're going to sample four prompts for any as four answers to any prompt that we have so let's say we and this is very cheap to do so let's say we have you know a million prompts that we found online now we run our model four times on each prompt with different random seeds we we sample uh four different answers so now we have 1 million prompts with four uh candidate answers okay and then we're going to say that we're pretty rich so we're going to pay people to actual human beings to label these they're going to rank this this uh prompt or the answers that these models produce to these prompts so we're going to pay pay actual human beings to score them and say are they good or not they're going to rank they're going to rank the the quality of these outputs okay but again human beings are very very expensive H so we don't want to use them too much so here again we're going to go to reinforcement learning and say to deep learning in Ai and say well we now have 1 million prompts with four ranked answers but why don't we train now a AI model a new AI model basically to simulate a human being assessing assessing the quality and ranking these prompts um so um we're going to now take this uh model this robot basically right to look at a prompt and generate an answer and then and then it tries to uh predict the score that a human being would give to this uh answer this prompt right so just learn to imitate human beings ranking these answers okay why is this good well it's good because now we can basically uh Rank and score answers as much as as much as we want because the computer is very very cheap so we can scale this to as many settings as we want um right so this is this is much much cheaper okay so what do we have well we have a robot or AI model here that's able to take a prompt and an answer and then give it a score like let's say between one and five and say how good is this so what we solve now is that we we know what's good or bad dialogue because a robot or a computer has learned to imitate human beings that clearly know what's good and byad dialogue so the robot also knows now what's good and byad dialogue so now we can run this robot on any prompt and answer and get a score of how good this answer is so now we suddenly have at least some insights around what's good or bad dialogue so we've solved that okay and the last two problems we are going to solve by using reinforcement learning so what is reinforcement learning well we talked about this a little bit before but uh something is very important and characteristics of reinforcement learning is this delayed feedback so uh in reinforcement learning we're going to have our our starting point of a really good model but we're going to allow it to start generating things right it generates a word puts it in its own uh input and it reruns itself so it becomes a longer longer sequence one over at a time so we start off with this I and we now have a probability distribution and we decide what to go for next and we go with went and then we have a few options again and we we take a next step two and again there's no at this point there's no feedback we don't know if we're if we're doing a good job or not before we had instant feedback because we had a Target and we could learn to do better here here there is no instant feedback so we're on our own and only when we you know we reach some uh predefined token like a pier for example then we stop and then we give uh our sequence that we produced to this robot and then it tell robot like hey is this good or bad so only at the very end when we're like Hey we're done we give it to the robot and he scores it then we get the feedback okay and why is this why is this difficult well it's difficult because let's say we do this again so I mean when we produced I went to a walk period I mean uh at least it's a pretty good sentence it's like medium score at least but let's say we now generate I went to lip I row row period I mean that's not not a very good sentence it doesn't make any sense basically it's is a very very bad score uh but you know a big part of reinforcement learning now is how do you make sense of this information you have two signals you start off doing the the same Step at the first step and then they diverge what would actually caus one sentence to be better than other one right how do you in incorporate this this delayed feedback to actually learn to generate good sentences um because we I mean we don't really know right what what did CA us to make better that's what reinforcement is is is about like how do you figure out what actually helps you reach your goal and optimizing your score function even if it's delayed um okay so another thing in doing this that's very very important it's exploration versus exploitation so let's say now basically that our model has seen these two different cases and have received two feedbacks right and in one of these you will got a pretty good score and in one uh uh you know in the lower here you got a pretty bad score so let's say we rerun this model again and it it went from I to went and then it let's say it kind of it remembers a little bit what we've seen so far and then it can say either here we can uh say we can just try to be kind of greedy right and exploit what we've seen so far so if we go down this path to I went to a walk then at least we know that going to do a decent job and better than this alternative that we've seen so then basically that means that we just exploit the information that we've seen so far and do the best based on the current knowledge that we've received right but the problem with this is that if we do this you know we're not going to see anything new we're just going to explore and explore the the things that already have received feedback on they we already know St pretty well but we're not we're never going to do really good to do better because we're just only going to explore the sequence that's we already have seen H and this of course is not good because if you start exploring a different rout for example you might again find a much better solution that's much more optimal and that that's what you want to accomplish you you know you want to explore your your space to be able to do better than you've done before and see and see parts of the data you haven't seen before to get more you know useful feedback back and and and scores from the robots um and also what I think is quite important to to emphasize here is that and this exploration uh cannot be completely random right let's assume that you would just uh generate a sequence you know of 50 random words I mean that would be completely nonsense and you would you wouldn't be able to get good feedback on this at all right you just be complete nonsense and random and you would not be able to get any useful feedback and it would be very very hard to improve so in order to generate you know this exploration you you know you want to be a do a very very targeted exploration around language is still kind of make sense so the robot gives you good feedback and you actually can start you know making progress so that's also why you uh open is able to use reinforcement learnings because they have a really good model and St language already and they're only really exploring the fringes of the knowledge this model model already has so like they basically only explore good prompt or good answers but they still do some exploration there but they're not doing a random exploration they're using the current knowledge to do an effective exploration of the space okay um so this now the reinforcement is is forced to balance exploration versus exploitation to optimize delay gratification actually leads to very non- GRE and and independent robustness so these are the consequence of applying reinforcement learning where you only get feedback at the very end so there's less you know supervision right you're more on your own and you have to H deal with an uncertainty of not having constant feedback you have to figure out things by yourself which leads to you being more robust and also again in reinforcement learn here the only thing we care about is the signal at the end so we don't care about making the best next step we care about optimizing the whole output so we're now addressing these things and somehow this corresponds you know if you raise a child for example h a lot of children will do better if you give them some space on their own to figure out things as well and not just constantly doting on them H it leads to kind of more robust uh people Okay so we've solved our problems uh we used human I mean we actually paid human beings for label data which is not maybe that like goes against our principles here but but uh we still did it because open AI is Rich so they paid people to label things but uh then they want you know they didn't want to spend too much money so they created an AI to replicate the the job of the human beings H but then at least they got a you know robot or computer model and now is able to say what's good or bad dialogue okay we still want you know we still want to be uh optimize you we don't want to be too greedy we want to optimize the the complete output and we want to be more robust and and and be able to self correct so we use reinforcement learning to optimize and make this robot happy so we've we've now addressed all these things and and we got an even better model okay but now we have an even better model but we already had a good model to start with so why stop here right why don't we just use this model now to uh generate uh you know for each prompt generate uh new answers that are even uh better answers and then you give these answers now to human beings to score them right and train a new robot to imitate this scoring and then you use reinforcement learning to train on this to get an even better model right it's not nothing stopping you for just rerunning this whole cycle again and someh this makes sense as well because if you now have a better model you kind of want to you want to go to the human beings and get more feedback that's more relevant to this model because this model now is is doing better than the previous one so you want to have feedback that's maximally useful for your current ability is right if you're a kid learning to write or whatever you know you want to get more sophisticated feedback as you become better at it and exactly the same thing we can do this models we can we can just run this step uh and do it all again and you know you can done you can do this as much as you want of course uh maybe with some uh you know decreasing returns I don't know exactly I think open a runs this two or three times um okay cool so to summarize what's the big fuss well just predict the next word based on previous words that's basically it uh who knew that this is going to work at the scale that uh you know and reach reach this kind of intelligence that we're seeing uh was quite hard uh but it did uh Transformers allows us to leverage more data and train quickly because we can paralyze paralyze all these steps in during training and uh then uh when we've done this we have a really really sophisticated model and we spent 99% of our time in computer on on this preaching this Transformer to predict the next word based on previous words then we can adjust things to make it a little bit more nice to interact with if you're a human being so you fine- tune on some data set around human dialogue and then you incorporate human feedback and reinforcement learning to get even uh a little bit more of performance out of this okay um again uh obviously self super learning in Foundation models are at the core of of CHP um uh also maybe I mean uh generative a versus self-supervised learning U maybe it's useful to kind of um I mean the difference between entive Ai and self super learning is not clear uh and people use it uh typically I mean there is more the difference is more clear when it comes to uh the research space but people say that CH P generative AI or ex Strang uses s super learning I mean generative AI somehow puts more emphasis on the ability to be able to generate some output right to create something uh but obviously catp knows both how to read and to write they are very related skills and self Suess learning care more about both aspects somehow uh I mean this is terminology uh okay awesome so next time we will talk about uh we'll do a similar Deep dive into stable diffusion there will be self supervised learning and Foundation mod an AI but uh I think it's going to be slightly more conceptually interesting um so should be a lot of fun and yes please go to the website for more information Etc and if you have any questions feel free yeah can I something can I assume that probability on that after there changes based on the subject of the totally yes yes yes great question okay so the question is uh does the distribution or the probability of over the next words given the previous words change given the prompt yes that's the that's the you know the whole point right because you train to generate the distribution uh given the previous words and they prompted just the previous words so Al like yes the more the longer sequence and the more context or the longer prompt you have the more specific prompt the more Peak your distribution will be because the more information the model have around your specific context in use case the more it knows how to collapse into a space that you want to know about right so if you just say you know if you start sampling the model with no prompt it's going to generate so the most uh common starting points on the Internet or something just like random text but if you start saying like hey I'm interested in history this this this then it's going to say like well okay I know history I'm going to generate things from Wikipedia history blah blah and it's going to be able to collapse and create distribution that's much more targeted uh so is if it doesn't have data run a context or about you and your interest as a person it it won't be able to tailor to you right they cannot create Magic out of thin air it can only do the best of The Prompt and knowledge has so far it's also why data is so important to have around you right and good prompt Etc and it's also why this prompt engineering to be able to like how do you create the best prompts to get what you want um yes so um just to like double check so for the reinforcement learn with even feed back um the the part that the robot that learns to rate the responses that's supervised learning because yes it is supervised learning and then the actual model generating is reinforc learning because itates the response yeah that's a great Point actually okay so yes in this what we talk about right now basically we had a few models involved and and we talk about reinforcement learning supervised learning and self-supervised learning right so exactly the robot that just H try to replicate the the human beings putting scores on the um uh answers that we generated prompts yes that's train using self uh supervised learning right because you have these labels now that you want to replicate so it's train using supervis learning but you know how it actually works that it that that model also leverages the pre-train model that we have so it's as a starting point so again like you can see now as well that like supervised learning where you have limited amount of labels ET only becomes really useful when you have a starting point from superv learning right so you already have a world model that you can leverage and then you you can use your your labels more efficiently right so the first step of preaching the next word based on previous words that's selfs supervised learning right that's 99% of the work and then we have supervised learning what we learn from the human beings but we already we have a starting point already so it's easier and then we do reinforcement learning right so we try to generate so we have a model do reinforcement learning but again the starting point of the reinforcement learning is this self-supervised learning you know model that's trained already using self Su learning so Al that's what's so fascinating with self suus learning because now it's the building block that makes all other Technologies in AI actually fruitful yes wondering language models are they actually models based on how a child learn is it any is it related to any sort of like cognitive science type of like learning because I saw like Transformer that's kind of how a normal person learns right when you're learning a new content you to Rel it with everything else you know yeah so I'm just I mean this yeah okay uh so the question is basically is the Transformer inspired by research about how we our kids learn right how the brain works yeah I mean you're going to find a lot of work around you know making those connections and uh then there's a huge debate in in like the Deep Learning Community is that is that actually true is it kind of wishful thinking and in hindsight we make this connection right and I mean and a lot of people throw around I do too right you for around similarities to how our own brain works and some people are like oh that's not you know we have to be very very careful because we kind of compare you know we're anthropomorphizing AI in some sense I think that there's strong connections right but like what came first I actually think that uh people have some intuition they tinker and they try things and then suddenly something work and they work on intuition and actually theory is very much hindsight so some engineer played around with things actually theer was from Google right so we build a Transformer they play around things they want to accomplish things and then like oh this makes sense this is probably how I would do in just have some intuition and then it works and then when it works the theorists come in and say like well this reminds me of this and this uh so there was definitely no strong like this is how kids learn and then we replicate that it the more I would say the more likely description of how the Transformer came about was just Engineers tinkering out and trying things and then it it just like oh this works now and they are not prbly completely conscious themselves about what they were inspired by you know so does that mean that we not have collaborations neuroscientist yeah it means that the collaboration with neuroscientist in deep learning is is quite quite rare actually yeah that's a good question yeah it's it's I would say it's quite rare yeah next based on I want to yeah um yeah like the question is why do we only PRI the last word based on previous words why don't we just take a random word mask it and try to predict it based on the surrounding words um yeah the answer to that is we used to do that way we used to do mask R and predict that and it works better uh but due to engineering you can Bas basically kind Transformer you can maybe this can be like a homework for you but if you look at Transformer Works uh if you mask a word uh then you will like then you will only um okay I if you do this you know if only predict the last word based on previous words you can make this okay attention that the Transformer does it's called attention when it tends to previous words uh you can structure it in a certain form a kind of a triangular form so you can process a whole like you can basically in a single goal so you can okay this maybe any like you have this tensor product you can Define this m Matrix High dimensional matrix product in a way that's efficient so you can basically uh in a single goal predict the target like you can you know you have a sentence you can you can run the whole sentence and predict the next word based on previous word for all the words in a single goal if you do um Mass language modeling you cannot do that then you basically have a full attention you can attend forward and backwards but you can only run the targets that you you masked uh so this means that uh in terms of in like in terms of signal to the model if you have a a sequence of uh length 2,000 you have 2,000 you know signals for auto regressive because you can use each Target by itself but if you mask then you maybe mask like 15% of the words and then it's it's only like that's only 50% then in terms of getting that feedback uh so it just in end of the day it means that uh when you try this empirically doing it aut agressively and this trick to be able to use the each you know each word is a target itself just leads to better performance at this task of generating export based on previous words uh so yeah so it's I guess it's engineering empirical okay it's not as in an Ideal World if you have unless compute you would have you know you would rather do it in a clean modeling way uh but now because of these restrictions and Computing stuff you just try something like hey this a speed up here and they see oh actually works better given the same amount of compute yeah does that give you some sense of answer we can talk more about it offline as well what are like the main challenges of these models nowadays and are there other like language models that are like better but they do like more expensive than or like this is like the best um yeah I think so what are some what are some challenges of these large Lang models well uh one challenge is to uh make them behave like we want them to meaning like okay we want them to be you know a little bit politically correct at least right and and and kind to us we don't want to make up things right we want them to be I mean we want to able to rely on it as much as possible and if they if you like ask a factual question and they give us something that's wrong and they're confident about it maybe that's bad so like I think what we're starting to see is that they um are very humanik even it's in its mistake they're like well they're somehow they're biased are we going to talk about this like they're biased and they have stereotypes around things uh they also like we do I mean they suffer from wishful thinking and some type of imagination where they rather be you know make you happy than being completely truthful uh so like these are things that you then somehow have to balance uh yeah and then so that's like some problems immediately arrived and then what else people definitely working on and open is working on autonomous agents basically how can you get planning into this so if you if you now you know let the model be able to uh generate you know digest some input generate some output and then you know digest that input generate some more output and stuff like in this kind of planning step uh it becomes has much much better abilities so you can basically instead of having a single go at your prompt if you can you know have a few tries and improve on itself and only get give you the the response after it's you know done this internal uh iterative uh um processing it will be to do much better and then like again if you throw in some tools there it can search internet to get more information you can retrieve information and be able to do do much much more so this kind of planning is uh definitely something that people know that these models you know very very helpful of course it's more expensive again because it has to run for longer and stuff but it's very very useful so then what you have is like open a talk about a star and stuff so these are reinforcement learning techniques of how to do planning well so how to incorporate planning is something that people talk about a lot and then of course multimodalities so it's not hard to see that this idea of predicting next word based on previous words corresponds really well to videos just to kind of predict the next frame based on previous frames uh and why aren't people doing it well because it's videos are suddenly a next level in terms of compute what it needs because they have tons of videos a frame is very expensive because it's a high dimensional picture or image uh but clearly you can learn a lot about the World by looking at videos right you can even sort to understand how human beings work even better because you can see people being upset or sad or happy whatever right in in a video and start picking these cues up and you can connect the vision part to the text part and get a multimodality model that's able to do both in a really really sophisticated way uh also something that I think these these people are working on all right thank you

MIT_6S087_Foundation_Models_Generative_AI_2024

MIT_6S087_Foundation_Models_Generative_AI_ECOSYSTEM.txt

uh welcome to the fifth lecture on foundation multii and today should be especially fun because we have two guest lectures so uh uh Professor M Kellis from MIT will show up and give a talk about biology in Ai and the AI Frontiers Frontiers in computational biology then artam is here he's flown in from uh Silicon Valley to talk about autonomous agents so it should be a lot of fun and and I'll start off talking about a framework for foundation models right so will it be a single Foundation model like a single brain to rule them all will open AI have a monopoly on AI or not like what will be if if not right what kind of foundation most will exist what can you leverage and I think it's going to be useful if you're a researcher but also especially if you're in business understand what kind of foundation models technology technologies that will exist and how you can uh use them and leverage them and this is based on a talk that I've been giving to uh priority firms and kind of C Level of bigger companies around uh how to survive Daya explosion okay like a lot of things are changing now and and people feel it's a different type of technology and it's changing the landscape and how do you survive survive and prosper in this new age so that's what we're going to figure out today um okay so You' seen this before but a primary so what was the Breakthrough that happened that allowed us to do all of these different ad advancement that that we've been seeing well we've been using ourselves as a reference frame right to to ask the critical question how do we learn about the world how do we go from being a blank slate baby with no knowledge about the world fairly useless to becoming a useful knowledgeable adult what what enables us to learn from the world that we uh interact with right this is a key question in AI uh what we've been saying is basically that what's responsible for giving you most of the knowledge that you have about the world it's not your parents it's not your teacher it's not Academia so it's not supervised learning you not you don't learn from experts okay so that's not that's that's a technology we that we tried for a long time but it's not the answer it's part I mean it's helpful but it's not the answer also it's not a DNA right it's not your genetics it's not your immediate environment and you're trying to optimize your goals in that environment so it's not reinforcement learning reinforcement by itself is not the key answer to how we learn about the world right it's helpful but is not the the main uh uh responsible part so what is it right well it turns out the most things that we know we learn by ourselves so this is the key you know Insight that allows us to do all the things we're doing right now and this is possible by defining company like meaning by the company it keeps so if you take a dog right you don't know what a dog is from your parent telling you or your emotions getting you you learn what a dog is by observing dogs in different context correlating conting dog with other cont Concepts so what you get what how you understand a dog is a dog is something that's walked by an owner with a leash it's something that has an antagonistic relationship with cats it's something that chases fris with those theone this is what allows you to understand what a dog is and this is how you define what a dog is and what you get by this is a very relational understanding of meaning right and as you learn about dogs by correlating contesting dogs with other Concepts like cats you intern learn what cats are so it's kind of self-referential and and very very powerful you can pick up know across modalities of course because the word dog will be Ed or named more in context where dogs appear so it's extremely powerful so what does this lead to well it leads to that the more relations that you're able to understand the better understanding you get of meaning you understanding what a dog what a Love Is Right you understand what love is helps you understand what a dog is because an owner loves his dog so it's like a lot of synergy networks affected play okay this is the key Insight here so why not take a you know huge model with as many parameters as possible and train train it on as much data as possible to learn all of these different relations to get the most precise and Powerful understanding of meaning and then use this model basically everywhere right and this is what a foundation model is right is it a key breakthrough you know throughout generative Ai and everything that we're seeing right now and also of course your brain is a prime example of this okay so here kind of going divert a little bit I've thought about this before so what does this actually mean right what does it mean now that we have these Foundation models that learn relational meaning and the more data you're training on them on the better they get somehow and just get bigger and bigger so one thing that you know that's happened right is that there been a complete change and transformation in the research world so six years ago when I was at Stanford there used to be you know one research lab data set and and AI model for different language task right so this start this start off initialing language you would have one data set one model one research team working on translation another one working on question answering a third one of sentiment analysis and then a fourth one of predictions right in isolation isolate effort but they ask them hey you know this seems to be a shared perspective here and a shared Intelligence being language modeling or language intelligence can we optimize that instead and know pull our data and efforts together and optimize single intelligence that synergetic because we don't have a you know separate brain for each language task so real intelligence should be General and and and synergetic right so that's what I did we're able now to build this language models that are trained and really understand language in a deep intelligent way so all the different tasks that we care about in language are just downam tasks of this real intelligence this real Foundation model okay and of course even though you know a lot of companies are behind in this the same applies to companies and businesses if you come to a company today right how the think about AI or data or Technologies is typically very isolated efforts you know solving something right so if you if you go to a retailer for example they may might have a SE separate service and data data set and team working on product search another one for recommendations a third one for assortment planning right campaigns marketing but these of course are not separate intelligences these are very very synergetic right if you're able to recommend the right products at the right time to user recommendations that should influence your marketing like what product should you Market who so someh you want to build right this intelligence around a company as well and this what you can do right so you can build a single brain around an a company to really get at all the synergies and get the most performance okay so what this leads to is basically right so at Stanford now the only research team that survived and prospers are the one focusing on building a foundation all around language right like the key core intelligence and also the businesses that build a key core intelligence around their own business space also the one surviving so you're seeing really kind of a major Extinction happening now the AI hits us like how do we survive and prosper with this changing the the space completely and that's what we're going to try to understand today right so the first question that comes to mind is okay if this is the case where does it stop will it be a single AI model or brain to rule them all right will open AI have a monopoly on on intelligence okay well first of like who knows in the in a few hundred years but what we're seeing now is that in the foreseeable future that's not not going to be the case right it's going to be a collection of foundation models and one way to kind of see how this happens is uh let's say you have like a large language model for example and you know when it comes to different applications of of large language models in language settings you might have you know uh law you might have marketing you might have uh writing on parles right this is still going to be one core intelligence right this is not sufficient to split and build completely separate Brains it's going to be the same similar Foundation mod technology around this so they're going to subscribe to chpt maybe fing a little bit but they're going to share the same model right but let's say now use this language model you come to a new setting like Logistics maybe you work with DHL or FedEx or something and they have like 100 billion data points around these different movements okay and you try to apply this large Lang models but it intuition fails it like first okay data is different it's a complet different setting and it's understanding from what is learned from the internet language generally it might not apply and it might come up with a bias is actually hurtful right you want to relearn in these things from the data itself right so then you start seeing that okay here there's a different applications and adding this data and this potential to the big model kind of deludes his abilities almost right it's not helpful so you can see you know a New Foundation model model technology model arriving and being able being enbl right around Logistics and that's kind of defensible has a mode and a large language model is not able to compete right so another way to see this is that yes we want to share edes to New Concept to get a more precise meaning understanding of meaning but also want to make sure that the cost of adding that edge and the delusion that you might get is is you know out like the value is outweighted somehow right you don't want to dilute the model and you don't want to you know add so much cost that the Val gets is is not justified right um okay so what we're seeing is that there's not going to be one single AI mod rule them all so there'll be a collection of different types of brains so what how will they look like well uh I think it's going to be quite hard to say we're going to try to think about it build some kind of intuition and framework around it and I think uh we can take inspiration from our own brain in a human setting and it's going to be a combination a little bit because right we have humanlike intelligence that we want to replicate that we can be inspired by our own abilities but also now in a modern economy in Civilization we have so many unnatural and unhuman settings like logistics for example that might that are our own intuition might fail us also this also enables new types of foundation models around settings that we haven't seen before because that you know we live in a new kind of world okay yes so in asking what kind of AI models that or Foundation model technologies have a unique defens ability a mode and a unique selling point uh it's important because that will allow you as well to understand what a models you can leverage and invest in and it's also going to be key because if you're right if you're a researcher you might want to invest in certain Foundation model technology and build it you might interact with other Foundation models but you want to invest in a specific one and also as a company as a business it's going to be key to be able to identify what type of foundation model is most important to your business cases so you can incorporate your own knowhow data and secret source and structure your own competitive advantage and so what have we seen right well when the large language models in ch P was released there was a lot of you know attempts to just build separate brains let's say around specialization uh so there was a lot of companies for example that would take a large language model and find you into different settings just in language but you know what we're see as well you you won't have again a separate brain for history language for legal language for customer service language those won't be separate right this this doesn't justify kind of I mean a lot of companies that went hyped and focused on specific settings or just language didn't survive in Prospect because if they don't have the data right if they don't own the channels that's not sufficient mode to kind of justify their own existence a big company that's a custom customer service company will just subscribe to CHP find you a little bit own setting right it's quite simple and then use that right so it won't be like a separate B brain built from scratch in just the specialized settings um yeah right then again so like techniques here like prompt engineering writing a the best prom prompt possible for your setting doing some kind of prompt tuning and and fine tuning even like these are not you know these are just taking a foundation model core technology adapting to new setting which is very very powerful and a lot of companies of course going to use this but it's not you know there's not going to be multiple companies just focusing on this right um okay so these are not separate Brains it's just fine two inversions of the same brain and there might be multiple providers like open AI coher entropic you know but they're very similar these are same language large language model right okay another way that we initially thought that we would see different types of brains were in terms of different data types almost like senses for ourselves there would be one really big foundation mod around uh language another really big one separate one around Vision a third one around audio C this is also not sufficient to justify its own Foundation model or technology because there's a lot of synergies again right you can you can hear a dog you can smell a dog H you can read about a dog and somehow these are synthetic things and you don't want to separate them too much so it's also not like that's not the proper way to think about how this works so I think a really good inspiration about thinking what type type of foundation models that we will see exist and that we can Leverage is taking inspiration from our own brain because actually our own brain is a collection of multiple brains interacting collaborating um and this is a combination typically they evolve throughout Evolution a combination of going for both specialized data data and specialized application right so it's key to have both and a lot of times we look at this large language models that were able to interpret and they do some kind of reasoning and planning you know when it comes to kind of Consciousness and and rationality that's in our own brain it's like we have a brain for that right but it's actually a very very small part of our brain most of our brain is you know quick intuitive automatic system that show real deep intelligence but they're not as kind of conscious as maybe you know large language model might seem to be right but Consciousness and planning rationality is actually very very small part of how we live and make our decisions is almost in hindsight mostly we maybe make rational arguments and and stories around what we do uh so it's going to be a lot of these Foundation models as well that are not going for this kind of Consciousness they're going for these automatic systems in intuition and that's really really key as well okay right so if you want to be able to build a defensible brain in terms of foundation model technology right a large language model model work versus like a large behavioral model it has to be unique both in terms of specialized setting and a specialized type of data and right what this means basically that you need proper data and proper proper algorithms to make this work and for a lot of companies they won't be able to provide the proper algorithms and develop that because it might be too expensive so they're going to they going to subscribe to some big model that's already pre-trained with really good algorithms etra for their setting and then incorporate their own data and build on top of that right and again I mean we we these all these different models are going to communicate with each other right so they're going to interact Etc but they're going to be specialized in own ways and then maybe there going to be different models like a large L mod can connect all of them in certain ways but they're going to uh interact be specialized but they're definely going to be able to work together okay so to recap right so we know by now that meaning is contextual relational and Foundation mods are basically leveraging leveraging that as as much as you can okay so it means you know the more relations able to understand you get a much more precise and Powerful understanding of meaning but there's a limit somehow and there's more Nuance right so we try to scale this as much as possible see like well there actually there's justification for having separate Foundation models around certain areas H and those Foundation models that are a will exist will have both propietary algorithms and data right that's key to be able to be defensible as a foundation model technology okay so now we're going to jump and talk a little bit more what this means commercially right so now we talked about Foundation mods and this I mean applies a little bit in terms of research and what you want to focus on it as well but especially when it comes to businesses now and so for uh the commercial setting right so human like companies are not really babies but they seem to be almost like human babies in some sense um and what I'm convinced about and I think what we're starting to see is that every company that exists right now I mean every company that will be able to exist in the future in the near future and to be able to have a mold and defensibility we we'll have to have one single core intelligence that it leverages right one single core model and Foundation model around its business that leverage leverages all of its uh secret source and synergies in its setting right if it splits those things apart it alludes itself so it has to have one single core model that builds on to structure this intelligence and knowhow to be able to be competitive and this is going to be you know summarizing it's it's kind of intelligence around its space okay um and it's going to have this core brain so this is going to be a foundation model that it's it's going to say what kind of foundation models exist there and it's going to subscribe to one of them which is going to be it key Foundation model technology incorporate all of his own data and secret SS and knoow and build on top of it and then of course this model itself will communicate and and and interface with other models that exist Foundation models but won't be as Accord to it right it won't maybe won't be able to contain it secret Source right so you might have this core Brin that interface with a language wall and and other Foundation models maybe weather or something like that or vision and incorporates that and use that right it's like using a computer it's very very useful and useful assistant but it doesn't contain its secret source and competitive Advantage so it has to build that structured intelligence to Foundation model and of course and other uh companies will be also building Their Own Foundation models that will summarize its own competitive advantage and setting and they can collaborate and communicate with other companies Etc um all right so if we starting to see a little bit that uh these uh brains right are built um across companies so like if you're not Google Amazon you won't be able to build a model from scratch so you'll subscribe to some Foundation model technology and incorporate your own data and and build on top of it right so now we can think a little bit about how this is going to define indues a little bit right so what indues is going to share a similar Foundation model technology and right and again so each company will leverage and build a single core intelligence around this business but the foundation model Technologies may be shared across companies right so uh you we might see that when it comes to large language model models like if you're in publishing or in the legal sector that might be the key Foundation model technology that you want to use to incorporate your own data and know how and leverage you know that's going to be your core intelligence but if you're in retail right then you're transactional data is much much more important and you know it's the like 99% of the data that you have and then you want to have a model is able to understand that behavior behavior of consumption and incorporate that effectively right and build around that so that was going to be your core foundational technology and then again you have like Logistics and then you know Financial Time s is something as well that doesn't isn't really captured well online on the internet or or really complies with human intuition so there be a separate Foundation mod around those data sets which are huge but hidden in in in company's databases and not online right so that that also is a foundation model technology we're seeing emerging um and again I want to reiterate that of course Ai and this new AI is going to be everywhere like a computer I mean it's going to be super useful technology for everybody to use and that's not really what we're talking about now there's going to be a difference between having a super useful assistant and also you know building your competitive Advantage structuring your intelligence you know how and secret Source around your business and leveraging that right that's going to be your existential kind of competitive Advantage right because if you're just using the other models everybody else is using you're not really incorporating your own unique setting right this is useful assistance but it's not actually your own unique intelligence right that that you need to build um and I think as start thinking about uh understanding what type how to build your own defensible brain for your uh business it's very very important to look at the data okay so if you're running your business right now start thinking about what are your competitive advantages and how can you get the data to learn about them right so you can structure and build this intelligence at scale right do you have the data channels to to track the data that that summarizes you know what you do well and if you know if there's data out there that you are not able to track but some other company might be able to track that you would like to use because it's very stic to your setting and it's key right to understand how you can get your hands on data because that company that has the data that you would really benefit from is either your best collaborator or your worst competitor okay so we're going to jump now into some more actual specific examples of how we can do this but also just to kind of quickly mention in a business setting that there's a lot of hype around AI for quite some time so there's a lot of companies say that they do AI but they do the old type of AI and maybe not real AI in a in an intelligent way so again if you know be Varan skeptical of companies that set to do AI but they peaked pre-2020 right these are new technologies h and a lot of the companies that that became big and successful before 2020 are not leverages these new technologies they're Legacy see okay if they come in and say hey we're going to help you out oh yeah but we don't need your data we we'll fix it anyways well that's that's BS right the data is the key that's how you incorporate and and help you out uh so then also they're not doing the real type of intelligence that needs your data if they require a lot of manual tuning right you're not supposed to do the job your data is supposed to do the job so if you're a retailer for example your customers is supposed to tell you what they want by acting your channels you shouldn't do that by doing a lot of manual tuning and guesswork okay statisticians and pretty user interfaces again like people in data analytics CER love to come in and look at certain statistics like the you know summaries like mean standard deviations is just too uh uh compressed right it doesn't it's not nuanced enough you need to build actually need to train this billion parameter models on your data you can't just look at the averages and do some kind of Wishful storytelling around it uh so also be skeptical of that right these are not very explainable Technologies you'll train them and then try to figure out what they've learned and also single function AI companies right again if somebody says hey you know you're a retailer oh we going to do your search only we don't do recommendations we don't do search it's like wait does doesn't make any sense you splitting your intelligence apart don't have separate services and don't trust these AI companies just focusing a single function because they don't they're not building real deep general intelligence they just specialize in building complicated systems okay so uh I'm going to go some through some examples that I've been doing with my startup unbook AI so we build um Behavior to behavior Foundation models that generates human actions and wants so what does it mean well again like if you're I mean so retail is is a one of our core areas because consumption is behavior so if you're in retail right catp is a super useful assistant okay but it's not revolutionizing retail at its core because retail core is not a text based language from the internet and it's very hard to incorporate data right retail adds core is consumption and consumption is behavior right so unboxi and behavior GPT is that kind of chhp Revolution but for retail and behavior driven businesses so they can incorporate their own data and build their own competitive Advantage incorporate that right it's not only about using that oh hey we have a helpful CHP assistance that helps you buy things easier like well that's everybody can incorp build that quite easily but you won't have a big huge model that incorporates billions of your transactional data points that learns from your customers and and structures your your secret source and and intelligence okay so let's look at one example from retail and again what we're doing is that we're building like not you know one single model for everything but for everything for this business which is in retail right so behav driven we try to build one single core deep intolerance around this business and we don't you know we don't think first about the downstream tasks right we actually don't care about the downstream tasks initially right we just care about building a deep intelligence around the business the performance on the downstream task are just proof that we're actually making progress in solving real things so it's about building this deep intelligence rather just solving specific problems and so the first thing you can do when you have this right is to Al start applying where it makes the biggest impact so we started applying this model for making navigation on the e-commerce s to this is a company that sells uh wallpapers and posters and canvases online so made the navigation extremely personalized and tailored because we can Now understand consumers and and their behavior so we can make their experience much more automatic and pleasant okay uh and this makes a huge huge impact and of course like there's a lot of air companies that been doing personalized stuff navigation before but there's a huge difference like it's comparing a a horse to a Ferrari and you need to put your Ferrari where it really matters and the cool thing as well right we had we had chat Bots before chat P but when you start playing with CHP you understand this is intelligence a completely new level exactly the same thing with these new technologies just because it's a similar setting it has a much better performance and outputs uh again right another thing just illustrate that is important you know if you're you come to a site and you search for something H they have like a thousand different motives and and different versions but but H they only have real estate to you know this are top real estate of six uh products right so here we can see that we give completely different experience there's like there's nothing in common between these two results here though they both make sense in term of ter searching for blue which an abstact search up somehow in one of the cases right this it's a family probably with kids looking for something for the kids room and we can give results for that in other case it's probably some teenager looking for something the room but it's extremely important to make Unique Tailor experience for for people and getting to know them at a very very you know deep level to make it a site is uniquely theirs which is possible to do in e-commerce setting and it really you know makes a difference in Drive sales again and this of course you know leads to increase in terms of Revenue and sales but also applying deep intelligence you know might not always just lead to short-term increase in sales and that's not might might not be the most important it gives other metrics that also extremely important for long-term Prosperity Revenue like people being happy coming back H spending more time you know Etc uh so it's these are also extremely important uh kpis anyways with this just by doing a more personalized navigation we're able to increase Revenue with 14% right total revenue with 14% which of course is is huge again this was done with 12x less work so now there's no need for experts going through looking at misspell search queries and tagging things up because it's done automatically and we let the customers do the job right they tell us what they want we just need to find it and learn about it okay so some um interesting uh quiz here so here for example again one of these models that look at similarity of products right one is a decent model is trained on Behavior online generally right so like some kind of General model for online behavior and another one is is highly fine to specialize for this uh you know European e-commerce shopping of posters and wallpapers and canvases right so in one of the cases when you look at similarity if the skylines one care more about geographical location so Durham here is closed because it's the same place another one when you look at similarity care more about the color palette so it's like well these both of these are very reasonable but which one is actually the true one for a typical customer at the site right it makes a big difference what do you think well so it turns out that actually uh for customers that buy wallpaper wallpapers and canvases online in Europe they care much more about the colar pallate than they care about the product being from the same place makes you know but both of course makes sense of similarity but one drives sales and one doesn't similarly we have you know a model is train and is working reason well on in data and it looks at similarity the old one so you have Clint Eastwood here with a gun it's a black and white you know old movie picture in one setting we're recommending other old black and white pictures from movies right H they seem to be more romantic something right but in other case when you actually look at at customers interacting with this product similarities and it's different if you let customers Define similarity here you can see that people actually care about the done and like in a cool old movie and and it seems to be more cleanwood focused and maybe it's uh a young man or something looking for something for his room right and of course they have nothing in common both make sense from just an abure perspective but one is given by looking at user behavior and one is just looking at internet overall okay similarly we can look at the search end search engine optimization and keywords tagging for products right so a lot of these companies have used teams that go through all the products to add keywords and and categories to maximize interaction SEO so here on the uh top we have a picture of swans flying over a some with colors here right so if you look at an expert a company look saying like Okay how should they start adding labels to this picture uh that explains it right so they add nature Force Woodland trees Etc and then you actually look at the AI that's trained on how people interact with this partu what they search for and learn how actually the language that customers use you can see that it says the most obvious thing in this picture that stands out from a user perspective is not nature forest trees Woodland it's a sunset but that's missed by an expert it's very very hard of course like how do you know that s is the thing that this is the most important thing in how people think about this product it's very hard but you know and it's it's not very intuitive but if you look at all this data it picks it up and learns what what what makes these things stand out and similarly in in the thing below here right we have a lot of different you know human expert putting uh categories and labels here but they're missing the key words that people would want to have when they interact with this product right and how they how they think about segmenting products right Temple and Japanese and S and they're missing all these words and this like makes a huge difference okay so and again this is very very nuanced this company has uh you know 21 countries they're active in 13 different languages and you know you also want to be smart because in a lot of different places there's multiple languages being used like in Sweden for example we love to throw in English words in our everyday vocabulary so then you want to be able to learn you know language from scratch and how people use them to describe wallpapers and canvases in different types of languages so this Model start picking up looking at all this data start picking up and learn languages and the nuances difference between different markets okay and then of course this is just kind of consumer facing things this model was then used for a lot of different settings because there's a deep intelligence around your business so you can use this for for store planning in physical stores it can do again quality insurance and master data to find mistakes and make sure everything is legal and correct and also it can do business intelligence like okay you understand your different markets in Europe okay how does if you want to go into Belgium a new market how does it relate to other markets what's the best strategy to go in there and succeed what should you what product should you focus on and so it's a very deep intelligence now that you can use everywhere okay another example is from Workforce and here again we try to build you know a model for everything around this uh Consumer Service Company okay the workforce we call it they uh are do customer service they have 50,000 employees 30,000 employees leave every year so it's a huge cost it's called attrition so people leave fairly early in customer service when they do phone calls and customer support uh it's a huge cost they interact with half a million candidates that they want to hire every year so this is a huge cost for them so if they the outline was like Hey if they're able to understand better who's going to leave and when it's they G it's going to be able to reduce cost dramatically okay but when we came to them it's was like well we don't care about people leaving right there's attrition we don't care about the downstream task we care about building a core intelligence around your employees and your business and getting to know your employees right so then we build and look at all these behavioral data points around employees to get to know who they are and how you know based on their behavior and we start looking at that first and then when we solve that and we able to build an intelligence that understands and able to prct what employees going to do then we look at the specific case of attrition and to our surprise when we built this model and we then started fineing it on attrition of predicting who's going to leave and when the model was able to predict if a person is going to leave the next two weeks or not with a 91% accuracy which is unprecedented in this setting and we came there it's been a huge effort right like big Consultants firms coming in spending you know millions of dollars you know doing surveys and and data analytics with no no basically no results they came in as well as like hey we have tons of data we have 10 to 20,000 service and questionnaires that we done on employees we're like 20,000 data points are nothing in this setting right and then we Lo we looked at these servers there was like zero correlation between people answered on these servers and how they actually behaved so what we did in stud is that we looked at at the behavioral data because actions don't lie and then we look at Behavioral data we can look at data being tracked which is much cheaper and get you can get to like 100 million data points quite quickly okay also I've been talking a little bit with different uh insurance companies because they rely a lot on data and predictions this is I think it's an interesting industry where actually I wonder if it's going to be able to survive and prosper uh is it possible to build a single model around insurance and the problem for insurance for example is that it's span so many other Industries and they don't typically have control over the data channels so I think it's like it's going to be very hard for an insurance company to provide the both the best health care insurance and the best real estate Insurance like if you if you're not a real estate developer and you're able to gather all the data around how people you know how properties behave and how people act around them and how things actually turn out it's going to be very hard to build a real competitive advantage in intelligence because you don't own those data channels so for insurance companies for example I think it's going to be interesting space to see how they developed and if they're able to still provide the best predictions if they don't control the data and how they cannot you know spread themselves too thin by saying we're going to do insurance in a lot of different areas right you want to do that in a perhaps a single area and build up some intelligence there okay and lastly again this isn't only apply to companies this applies to any entity in a commercial space so if for example if you're a private Equity Fund as well need to think about how to leverage these Technologies and so what something that we've been working on is to build a single model intelligence around a private Equity company so uh typically if you're in private Equity you think about horizontal and for example vertical integration you want to pull Talent across your companies that TR in production but it's also so important now to think about you know sharing intelligence having synergies intence across the companies data and AI right like AI integration so that you know you are able to build this core model around all these different companies because you own a majority in them and they can then get all the synergies of actually putting the data and they know how together and you can leverage this and now when you add a new company you know exactly what they're going to add to this intelligence that they fit in and that they can you know benefit from the synergies network effects immediately right and it's also very important to structure your intelligence because you don't want to rely on a single brilliant person or like some fingertip feeling you want to build in structural intelligence you're more robust to these things okay so to summarize um right this new AI Revolution we know it's Foundation models uh so understanding what type of foundation models that will exist out there so you can Leverage is going to be key and that the process of doing this is understanding the core intelligence that drives your business area right and you need to capture that and be able to structure what makes your business unique and successful into real structured intelligence like Foundation model right and it starts off by identifying the data that's key for that and you need to do it early and do it really well and do it now right because people starting to do doing this and it's being basically a lot of extinctions because they're not actually uh capturing the intelligence that they need and they don't have the data channels to do it you need to make sure you do and I think everybody can agree for a lot of businesses right basically a key competitive advantage and your extens existential justification that you have some unique intelligence around what you do um and before I think it was more indirectly but now it's more and more directly that you need to capture this intelligence in a structured way more and more right through these Foundation models okay uh all right thank you so much that was it and if you have any questions I can take them out

MIT_6S087_Foundation_Models_Generative_AI_2024

MIT_6S087_Foundation_Models_Generative_AI_HOW_IT_WORKS.txt

all right uh okay welcome to the second lecture uh on fation mods generative AI this one should be a fun one we're going to dive into all the different ways we train and arrive at this Foundation models and generative AI um and if you ask me I think that that this is kind of the key breakthroughs and it's going to give you a wide understanding of what's going on I mean some people focus more perhaps on certain engineering trick that that's happened in the last few years but I think these are the conceptual breakthroughs uh so it's going to be exciting to to talk about uh all right that's so today we'll go through all different algorithms meaning how we Define objectives and goals for for computers to interact with the world and data uh to learn from it so quickly recap from uh last class right we provided a short suin answer to what is foundation models dtive Ai and how you learn from observation and that meaning is contextual and relational that went we went on a little bit of a philosophical Journey where we asked how's the world structured right somehow uh the world is very chaotic and we need to deal with that chaos because math won't save us so that's where new networks and and the new type of AI comes in and and helps out and that if you want to learn from the world like supervised learning when you learn from an expert doesn't scale well because you rely on human beings that have to label the whole world the whole world cannot be labeled so it doesn't generalize well and reinforcement learning also doesn't work because it's too risky and too slow if you have no starting point and we going to talk about this in this class like if you have if you have some starting point you can do it but if you have no world model on a standing off the world what server you cannot do reinforcement learning because you don't even know where to start you'll make no progress and you unfortunately die way before you make any progress whatsoever that's why the technique behind Foundation models generative AI generative AI called self-supervised learning that's key right some people call this unsupervised learning the the correct term is self-supervised learning but that's how we arrive at these these Technologies okay um right so we learn from No Label data we learn from this data in general which means it scales really well we just needed data and then we can learn the structure for from that uh all right and again we said how do you learn what a dog is well you learn what a dog is from observing dogs in different context you correlate and contrast dogs with other Concepts like cats and then in turn you also learn about cats you get this very relational understanding of meaning and that's what we're leveraging here so H today we're going to talk about uh these different approaches more in detail so we'll talk about natural language process processing in language uh basically the the what happened in the beginning of early days of natural language processing and then how we arrived at chat GP type of Technologies and this includes Cal language modeling CLM and mass language modeling MLM we'll talk about contrasted learning which is uh very popular when it comes to vision and images we'll talk about puzzles and games uh the noising diffusion uh also very popular in text image generation and like stable diffusion order encoders Gans so generative adversarial networks new networks and then we'll talk about a little bit about generative approaches and repres versus representation learning and then we'll talk also about autonomous agents a little bit uh all right so let's get started so we're going to start with language which is also basically where a lot of these uh kind of conceptual breakthroughs actually started and so langu language is a little bit special and that's what I'm going to argue as well actually the language is kind of special it's uh it's man-made right we created it for some kind of purpose uh we don't only communicate in terms of language we also think in terms of language and that might even be the more interesting and important component of language is that we think in terms of it uh rather than it it allows us to talk to other people and I think if we if we came across a intelligent other life form even if they weren't able to communicate with each other they will still have a language to able to think and plan Etc and we're going to talk about this later and really it's an efficient Universal Medium for transporting and verifying ideas and we'll try to make this more um tangible later but this also kind of hints at how we can use these large language models to understand language to create kind of autonomous agents and even more humanlike intelligence because a lot of this is hidden in language itself Okay so so now it's several several years ago I'm getting old but when I started off my career at Stanford uh 12 years ago I think it was don't quote me on that but uh then there was a specific research team data set and like model for each specific language task so you would have one research team H one data set and one model that they were optimizing right an algorithm for translation and then you would have a separate research team model and data set for question answer Ing and then another you know isolated project and and data and model and and researches around classification and prediction and R Etc right so these are kind of isolated efforts that people were optimizing specializing forign Building Solutions and collecting data but you know we started asking ourselves like hey is this actually good are we spreading ourselves thin here we're all working on Solutions around language and understanding language you know this seems to be very related task doesn't seem like human beings have separate brains for each each different language tasks so maybe there is some objective or some something in language that we can optimize for and learn they kind of get the underlying problem of understanding language and then we just see all this all of these different tasks as just kind of Downstream tasks that we use this big good language understanding brain to to to solve right but the maybe we can optimize that instead so that's what we start asking ourselves right um and right let's say we want to accomplish this right let's say we want to optimize and learn some type of language how could this look like well um somehow we want to be able to uh digest like our model AI model or computer model right to able to digest language and then put it on some kind of representation space or feature space right into some useful format and that we can use that format and and kind of send it to other task right so let say you know let's say we have this type of AI model is able to digest language kind of encode its meaning right into some representation like numbers for example and then we can then feed this to all the different tasks right that's a really good starting point because if we're able to kind of featu and represent language and and we also get it the real meaning of the text it's very very useful as as useful tool for these Downstream tasks and so this would be nice to have that's what do people start working towards basically featuing and representation learning on language where sentences and text that has similar meanings are mapped very close in this in this High dimensional meaning space um and that it's like a very very nuanced granular way of representing meaning um okay so let's like focus a little bit on the meaning part here which is very obuse and and fussy so how can we actually represent meaning in some you know how can we let how can we if the mod is able to decipher meaning how should I represent meaning of words so let's say we have four words cat kitten dog and puppy and let's just for uh like teaching uh purposes also present images just to kind of show that these are nuanced concept right we just focus on the words but let's just add some images here just to see that these are new nuanced Concepts uh and clearly you know there is a strong relationship between a cat and kitten because they belong to the same species right and also it's clearly a strong relationship between dog and puppy because it belongs to the same species right so that's great that's a good starting point if we understand the relationship these words but of course it's actually much more nuanced than that because there also a strong or like at least some relationship between kitten and Poppin that they're close in a sense to describe baby animal animal or cute baby animals while dog and cat perhaps you know pertain more to adults or grownup animals so this also this Nuance to meaning here that we need to represent and if we were to try to represent the meaning of these words by just mapping them to a single digits for example you know computers love numbers and digits but if we if we map these words to single digits we would have to pick you know should you know should represent that these things are closed because they're in the same species or or should we focus on you know what kind of age they correspond to so how we'll solve that it's quite simple and we just say well we're going to map these words to actually a combination or an array of numbers right so it's high dimensional Vector it's called so we're able to represent you know a lot of nuance in the meaning of a word so then we can let uh for example the first I mean now all of these digits are the same but of course in practice in real life they will be different right because there these words are different but we could let the first uh position of this High dimensional Vector correspond to the species right so if they the same number in this slot they're from the same species and then we can let the third number represent age so if they're similar in this spot they're in a similar age in terms of what animal that it uh corresponds to okay so now we just kind of quickly concluded that we need to represent the meaning of words using High dimensional Vector because meaning of a word is very high dimensional it's nuanced but actually the meaning of a word is not the best perspective because it's not the meaning of the word itself that we want to focus on let's take the example of river bank versus Financial Bank so here bank is exactly the same word in the both context but they mean very very different thing depending on how they appear right together with other words so really like meaning of a word is very very contextual and it depends on the other words that that surrounds it so somehow we need to uh also take this into account to really learn you know to map uh the meaning of text into useful vectors we need to consider the whole text together somehow and of course like I mean when when people started working on this there was word embedding like people started focusing on just mapping words to to meaning it actually works quite well but you know when you want to do the next Lev you have to also consider the the context and the the whole sentence as like a contextual clue so we're going to also not learn to map you know a word to a vector we're going to map a any sequence of text to some meaning that's key here because we want to have that uh flexibility and that meaning is actually contextual depending on the sentence things change and how it you get depending on other words it appears together with okay so we now know what we want to accomplish and we have this High dimensional vectors as a tool how do we uh how do we learn contextual meaning well uh I we talked a little bit about this in the last course as well but the fact that we just already kind of understand the meaning or like the meaning in language is so contextual also gives us the cue or the the guide of how we can learn from it and and we're going to think about kind of cooccurrence of words and we're going to as talked about before represent and learn meaning by defining meaning as as defined by the company it keeps it's like self- referential in that way but that also that's very very powerful because we just going to learn the meaning of words depending on how they appear together with other words and vice versa but this actually sounds kind of like circular logic but it actually works really really well and and just kind of shows that mean like meaning of language is defined by its use there is no abstract dictionary really that will work it's just defined by its use and the meaning of language changes as the use of language changes we all know that right um okay so one thing we can do here that that works surprisingly well is to uh take a sentence or some text from the internet right there's endless of supply and then we just uh randomly mask or beep a word in the sequence and then we train a model to uh an AI model to predict what that masked word is right so it's very easy to build a script that downloads text from online randomly picks a position you know removes that word or put a mask you know there and then you know you hide it from the computer so the computer only sees I went to the beep bank for a swim and then it has from the you know it has to derive from the surrounding word what that missing word is so it has to learn how to correlate the meaning of a word and its surrounding words all right and what does it mean right like what does it mean if it's able to say that you know successfully learned to say that I went to the be bank for a swim if it's able to predict River it means that it knows how to interpret Bank in this context right the cue or clue here of swim gives it away and it also knows to then suddenly start to understand the meaning of text here because it's able to decipher this and similar similarly if it's is I went to the beep bank to deposit money if it's able to predict Financial again it also now know how to interpret this this sequence and this word of bank because it understands money and how they cor corresponds to kind of Financial Bank rather than the River Bank okay fantastic um so we have like our first iteration where we're now going to train this uh model on Mass language modeling so we're going to take a lot of text from online we're going to mask words and try to predict those Mass words based on the surrounding words and this actually leads to really really really good feature representation of text and language so we get this model we run it on uh some text we get a a feature that corresponds to the context and now we can fit it into these different tasks that we care about and this I mean this is was extremely revolutionary uh approach that change language bace forever it works extremely extremely well right so suddenly now you can optimize a uh model that learns language in general and now it's like a starting point and representation that you can you know build on top of right so then you would have uh this uh you know big model in the middle here that would represent and learn the meaning of language and you can you know you can use this across all different task but then you would train a small model on top of these features to just to correlate it to some kind of uh uh question answering or or translation or or sentiment sentiment analysis right and completion so you would use these kind of features for a downstream task and yes it's like not extremely clear how you get from this uh High dimensional vector and how you would define engineer all of the smaller computers and models to learn on top of it that's not clear and so it still involves some engineering and specific solutions for each task but just the just the ability that you can now leverage this big model understands language on some high you know from some uh for like from some uh representation learning you know approach gives you so much in terms of performance right so now you can start optimizing these modes get better and better and you can use them across a wide array of tasks which is extremely extremely powerful okay so now also you know when we trying to solve these problems people started talking about pre-training training versus Downstream stask Downstream task or fine tuning so now when we take this model and we use math language modeling right to train it to get some general understanding of language that's called pre-training so that's like how we train a model before we get to the specific downam Downstream task that we might actually care about in the end right so there like two separate steps you pre- this model in some setting and then you keep training it or you train on top of it as fine tuning for the specific test that you want so let's say you get you know you you you do this Mass language model on text and then you want to classify Amazon reviews as positive or negatives right then you use this big model to create a feature for their Amazon review and then you train a small model on top of that just to kind of learn if it's positive or negative on some label data that you have right but but just the fact you have this big model that understands language makes this fruitful and and useful but okay so but again so like you have the pre-training which is you know important but what you actually do care about is the collection of Downstream tasks so in research we make it a little more abstract because we're like well we don't know what Downstream task you might be interested in so we got to kind of Define a collection of them but in an applied setting you typically know what Downstream task you care about uh and so I mean but but there's definitely even in research is a set of Downstream tasks that are more popular because these are useful generally so what what the focus is as well then is to make the pre-training and the downstream tasks kind of as similar as possible right so they should they should somehow the pre-training should be as useful as possible for the down street test that you care about I mean let's say and let's say you want to you know learn how to drive a car for example and you know someh before you sit down in a car you want to have the best possible observations and the best possible kind of games or whatever you you know you do before you actually sit down in your car to learn as much as possible about driving a car before you sit down right that's exactly what we're focusing on when it comes to these things as well we want to make the pre-training as as useful as possible you know if you want to drive learn to drive a car so maybe you sit down with some instructor or something or like you know you just observe somebody you know you actually sit in the car together with your your dad or something when it drives right that's more useful than than seeing something from afar and uh this is also related to simulation and reality right we want somehow if you cannot afford and we don't know exactly about the specific reality or situation we're going to apply something we want our simulations to be as close as possible all right so now we've you know and this what we did so now we've learned to embed text into this High dimensional vectors that encode meaning so this this is great but you know language in itself conveys meaning so why go into into this High dimensional embedding space when we can just go from kind of text to text because language itself is very very flexible you know when we look up a word for example right we look up and we want to have a dictionary definition of it somehow to get more context around it so why not kind of start using language itself as a very flexible way of representing meaning or even uh expounding on the meaning of of text so uh that's a little bit The Next Step where people start doing and what we're going to try to do is we're going to try to remove these high dimensional vectors in the middle uh and turn them into text instead so here for example right someh put the some type of definition of a river bank a financial bank that takes the whole context together and then explains what it means with other words right it's self self-referential but meaning itself is self self-referential right so we might not be losing out much okay and this is also nice because as language is a very flexible you know way to encode things we can also encode all our tasks as just text to text like we don't need to encode you know positive or negative into zeros and ones right we can just encode as positive and negative as as words or as text so you know this is in the beginning people would uh encode all these different tasks and the data into form the computers want but now we're just going to encode that into text that they kind of make sense from a a human perspective and if we now have text to text models we not we not not said how we're going to arrive at them but if we have text to text models that digested text and output text at least then in terms of engineering these are the same formats right so if we if we take all our down strring tasks and we rewrite them as text to text right you can you can rewrite any task as text to text right you see here I mean I mean this is you know you can conven yourself it's very possible to take any TA task and then write in terms of text instead so you know instead of saying uh you know answer this question what is of your bank right you just give the the answer in text uh if it's multiple choice you can just say well it's option A or something and if you transl to German is just a text completing is also just text and classification positive negative right just output the actual description if it's positive or negative so now the nice thing this actually is text to text h and we can put all of our Downstream task whatever they might be in the same format so there might be additional training on some label data set around the specific task but there will be no Engineering in terms of going from a high dimensional Vector to some specific output because now all the input and output is just text to text and that's that's very nice because it also means it's going be easier for us to kind of find tuns on a lot of tasks at the same time and get the synergies across these tasks so this is what Google did with their model called T5 and it's also very very successful where basically this is what it did it's like okay we're going to Define all the different tasks that people might care about and the benchmarks that exist into just text to text and then also our model is trained to text to text so it's going to be a smaller Gap in terms of engineering and hopefully in terms of performance uh how they arrived this model is very very simple so on a high level they just basically now took some text they uh mased out multiple words in this text and then just try to predict start just predicting you know the specific word that was masked out they not predict and recreate the whole sentence right with those words filled in so instead of now being kind of a text to masked word prediction it's a mask text to the full text so it's a very kind of a small conceptual change but it requires some work to make it work in terms of engineering but now you have a text to text model and uh you can do this All right so um there's basically two different uh uh things that we've been talking about this basically one thing which is mass language modeling where we randomly take a uh I mean this is very similar to the the T5 approach of masking words right and predicting the complete sentence but Mass language modeling is when you mask a word any at any point in the sentence or text and you try to predict that based on the surrounding words Cal language modeling on the other hand is actually you only mask the very last word and you you only try to predict the last word based on the previous words and we're going to talk about this when we when we dive into the chtp and Transformer and how this is trained but how this is trained and how we can optimize this you know math language modeling versus uh Co language modeling is that in Mass language modeling it's going to be a little bit uh slower but the the what you get in return is that you know when you try to um predict the the mass word of like I went to the B bang for swim for Mass language modeling basically each word as it as it tries to embed itself or be useful is able to attend to all other words right both in front of it and behind it so that I'm not telling you why but that's how it works what in coastal language modeling you know Bank as it embeds itself and makes itself useful to the bigger bigger sentence it can only look behind it right and this is how the Transformers is Implement and I'm going to tell you why this works this way but uh this is a big difference and it terms out the coil will be faster to train because you can make some engineering tricks to make it uh more paralyzable but hopefully you know one of the should kind of make more sense to you right so if meaning is relational and contextual you should be able to do better if you're able to correlate and understand what's in front of you as well as behind you right somehow you have a larger context which means that you have more context which means you have a better understanding of meaning and it's pretty cool actually so Mass language modeling works better at almost everything besides one task right because how Cal language model is optimized and that's the only task that Cal language modeling works better at is predict in the next word based on previous word right generation because it's optimized for that way and it can be trained on more data using less compute but it turns out that that actually is a very very good task to be good at why well because now we can just say like well all of these different tasks are not text to text isn't that just text completion now if we take some input prompt and then we start generating a word a Pand dat and start generating next word like incrementally right because we have a model now that can generate the next word based on previous words we can just run this on its own output again and again and we can generate until some period or something until we're happy but just it just takes complet generation right so what what's the most logical thing to follow you know answer this question what is a river bank I mean the most logical completion of it if it's good at it is to actually give the answer right the same for translate you know translate to German my name is but the most logical completion of this in terms of generate the next word based on previous words is actually to generate the correct translation right and this is true for all of these different problems and this is exactly I mean it sounds perhaps crazy simple but this is the Breakthrough behind CHP just just taking this extremely simple concept of predicting the next token or word based on previous ones to a tremendous scale and it becomes a multitask solver because all you need to do is just predict the next word based on previous words right so you can ask all this things Tob solves it easily and for the people behind open ey actually like I mean it sounds you know in hindsight it sounds simple and like how could we see that but like that was a huge bet like who thought that if we take this simple language modeling predict in the next word based on previous words to a tremendous scale actually we get kind of human level intelligence that can solve any task there was a very like like you're crazy but like no we're gonna try this spend like hundred million dollars on Compu just train and train and see what it when at some point it starts kind of having these emergent abilities that are very humanik and that's exactly what happened okay awesome so to summarize a little bit our our language language uh uh stuff again meaning is defined what the company it keeps and so we cannot focus on a word by itself it has to be a sequence of words right because they affect each other but also that's the cue that we can learn meaning by this self self- referential right we just learn to predict a massed word based on a surrounding words so it's very very nice in language you can learn from just observing it h first we said we're going to use high dimensional embedding spaces to accurately or like sufficiently encode meaning because it's high dimensional nuanced but then we said like oh actually be nice if we can line the pre-training to the downstream task even more both in terms of engineering so we don't need to go from like T to some high edding uh space to some uh indicator variable or some number right now we can just go from text to text that's nice and then even we can just basically have the same objective in the pre and downstream task meaning just to predict the next word based on previous words it's even better right and for you know if you just take and train a model to predict the next word based on previous words if have enough data it should solve all the tasks that we care about if it's able to do it well we can almost start throwing away the downstream data we don't even care about it right we don't even know about the downstream test because as long as you can describe it in a prompt it will work for you and that's exactly what chtp is right it's just like you describe it with a prompt some input and it solves the task for you okay so let's now uh jump to other uh modalities like vision for example I mean this is also what the people behind CHP did try to do the same thing for uh images so you just take an image say like hey it's a sequence top to bottom left to right right a pixel at a time you just start predicting it uh and if it's able to uh you know accurately or realistically complete an image based on previous ones somehow need to understand the concepts involved right like hey am I completing a cat right now or is it uh a sidewalk or something right uh so this definitely kind of uh you see it works but that's not people use because it's not giving the stateof the art results and it's not computationally efficient something people uh use especially when it comes to vision is is this learning by contrasting or contrastive learning so we uh t a little bit about this last lecture but it works with this idea of positive uh Pairs and actually negative pairs but the whole idea here and I think it's actually something that's that's fundamental that we shouldn't take for granted but the whole idea here is that we say well people take photos of things and it's like natural images and we're like well you know we don't take random photos we take photos of things you know that are in the same environment right and maybe that we even put together like you know like a soccer ball and a soccer go or something so we're going to say that things that appear in the same image are on average more related to each other than things that appear in different images on average right you can find edge cases this is not true at all but when you look statistically at like billions of images that would be true that things just the fact if I take a photo here somehow you guys have more some have something more in common than you would have with people in a random image that I put from online right you're all I somehow you're interested in this stuff so that that makes sense uh statistically okay so also what we need to do because we need to be a little bit careful so let's say you know we just try to take so we take an image right you know you're have something in common because you're in the same image so we just crop this image we just have a computer make to Crea two different random crops sometimes they might you know be overlapping somebody they might capture wrong things but again statistically when you do this enough on enough data it works but you take an image you take two different crops and you push them together so like the representation now the high dimensional vectors this model creates of these two of these two crops will be pushed together so they're closer right when they're from the same image but of course there's a very simple solution to this because if we just take images we crop them and then we want the crops to be close together what if we map everything to just the same Vector then everything will be very very close right right so again it also kind of points out that everything is relative like similarity is relative for for the to be similarity needs to be dissimilarity for this to be kind of something to be close also need to be a sense of something being far away so what we need to do is actually to take this not to not allow it to collapse but to say like yes I want these you know two crops from the same image to be mapped to you know vectors High dimensional represention close to each other but far away from other images I'm going to push you away from other images so I cannot collapse so there also a sense of distance I need to be close and far away and and again right like you can get also at some type of uh more kind of abstract relationships like things that don't appear in this like things don't need to appear in the same image ever to be close right you might have two different species of dog that never appear on the same image ever but as long as they appear you with other shared Concepts right so if these two dogs appear with owners with leash if they appear to with frises then there will still be closed representation so it can kind of bridge that Gap okay so how's this done in practice well in practice we turn this into a classification task basically where we for each set of uh you know images that we in front of each other and crops we feed the computer uh the one crop and then we say you know hey we have uh three other crops here you need to classify or pick which one of these three ones is your positive pair right which one do you belong to and which one you know don't you belong to so we basically has to classify and retrieve the correct positive pair and here it's it's quite simple right but you can already see like somehow it needs to understand a little bit you know needs to understand this image of a owner walking with a leash and retrieve the dog I mean you can see the leash here but typically you know it's harder than this and say like you know a dog is something that's walked in a leash typically a cat is not so like and I can do this and then you know this is just for crops in total typically you would have you know 10,000 or something so it becomes even harder to pick out the right positive pair among a 10 you know among 10,000 candidates okay so this works really really well uh it's you know gives us really good representation of images that we can use a lot of different settings and so people use this a lot and also of course we can do this anywhere we're able to Define positive and negative pairs so it's very general uh right so if we take we can do the same thing basically for language modeling where we take a sentence we create the positive pairs by just kind of corrupting them or masking them and say that they should be mapped still that shouldn't destroy the meaning of the sentence so they should be ma close together and far away from this other random sentence that we might find from online or something so as long as we're able to have this kind of positive Pairs and negative pairs we can apply contrastive learning which makes it very general as well okay so now we're going to take A New Perspective and not think as much in terms of you know defining meaning by the company it keeps and these kind of philosophical buzzwords but we're going just going to say just going to play Let the computer play games and say that you know if the computer is able to solve this simple task we think that it has to understand the the concepts involved um and I mean this is also uh a perspective that's been very fruitful and it's good way as well to think about how these models learn you def find some simple task that you can uh or game and you let the computers play by interacting the data and you start learning learning about it um yeah so now meaning is not defined by the company keeps but how it helps you solve the task so I think one example of this is when you uh take a some text from online you just scramble all the words and you let the computer try to put them in the correct order again right like why would you do this well it doesn't make maybe a ton of sense but then you start thinking about it like well if you get a collection or a bag of word of this random words like how would you start thinking about putting in the right order right you would have to think about the grammar you have to think about the meaning what describes what and and how to build up a story that makes sense so let's say you know let's say you're able to take this and actually start putting in the right order it means that you're leveraging your know your understanding of language to do it so somehow this task if if if done well requires language understanding and ter it does so you can train this on scale on on tons of uh Text data and it starts doing a really really good job in fact it's able to put it in such good you know in the right order so often and so accurately that these computers don't even need to have language in the right order it can just get the uh all the words Mumble together and then it can implicitly put them in right order and use that right so it just kind of shows that the order of words are less important than the bag of words that appear somehow um similarly we can do the same for images we can just take an image we can uh create this kind of Jigsaw pule where we create the squares right we we uh Shuffle those squares and then we let the computer try to predict the correct ordering of those squares right uh and again yes if you want to be able to put the I in the right order and stuff somehow you need to start understanding how a face is typically uh outlined right and if you have other concept as well like you know what's the tail of a dog where the front stuff like that so if you want to do this scaling really well you'll start to understand you know how visual concepts appear and uh you know we need to be able to understand them we can also do something I call a game of den noising and like again also kind of a common denominator here is that we can get an image from online very cheaply right we can we can create squares and and shuffle them very cheaply like there's no human being need need to be evolved this a script only right similarly you can take an image from online and and very cheaply you can have billions of them trillions and then you can add noise like this this gy noise here a computer can create noise very very cheaply so you can take an image from online you can add some noise different levels right and then you can say like you know you can train the computer to try to remove the noise so the noising um so right somehow we can start seeing the Contours and we can kind of jump to conclusion of what this picture probably is and we can fill in the details right and this it ative step of denoising is called diffusion right so if you know the name stable diffusion this is where it's from right so and and probably when you think about it as you try yourself here to kind of complete this sequence you're leveraging your understanding of images and vision in general to do it so this model actually learns you know how visual objects and images are uh you know constituted and how they work because it's able to solve this task and also notice that you know if you incor incorporate something sometimes this like pure random noise on the very left where there's no image at all it can basically start conjecturing or making up images from Pure Noise so suddenly as well we have like a generator can generate images from nothing uh so here like we TR this on faces we can just generate faces and Sample any you know different set of faces that we can use for commercials or something I don't know um okay we also have something called a game of compression here we take a some data an image for example and we map it from a high dimensional object like you know the pixels and the colors and stuff to just a few numbers like let's say 32 numbers from like a few maybe even a million pixel values and stuff to just uh 32 numbers and then we try to have the models to recreate it afterwards so it has to push the data through this bottleneck and then recreate the image the original image accurately so it has to save only the most important uh numbers for it to be able to recreate the the original image so like does this work yeah if you if you have some good architecture it works really well and it's able to accurately kind of compress the image and you can use this compression to solve tasks right because this summarize the image in a very efficient way and these are called Auto encoders okay you can also play games against other people so here we have a uh model we can call the artist that tries to create realistic looking image uh faces all right so what it will do is that you're going to have a data set now of tons of faces right real faces you can get from some data sets online just download and then you're going to let the this model the artist to create a you know it's best attempt of creating a face then you're going to feed this it it's attempt to create a phas and an actual real phas to critic which is another AI model is trained to distinguish between these two right so then you know the critic is trained to distinguish between uh the real and the fake image and the artist is trained to fool the critic right because the artist wants the critic to be unsure he wants to make its images look so realistic in terms of faces so the critic critic is unsure about what actually is real or not and probably when you T yourself to draw you draw something and then you look at and be like oh actually does this look realistic or not not like what would Mom say and then you improve right and it's not the same thing but by itself and this is called uh genive adversarial training and networks it's uh quite unstable to train as you can imagine right I mean what if somebody gets very good at uh critiquing you know it's almost hard to make any progress then because they're almost just too far ahead right so there needs to be somehow a fairly evil Level Playing Field when you train this model so the input and output is on the on a similar level in terms of accuracy um all right okay so something that you know we've been talking about now in all these different approaches we went through quite quickly is both you know some approaches that focus on embedding the data or like representing the data some type of representation learning and also generating data okay so these are still two slightly different concept right we want to embed some data to a embedding space where we can use those features for certain task right we want to compress the data basically so we can use it for image classification or for you know text classification or what have you and also we have generation where we want to generate data based on some typically some sampling or some prompting right we want to say like hey you know we want to have a image looked at this and we want to be able to generate it and uh a lot of times actually a lot of the models that uh are really good at embedding things you know might not even focus and being trained on generating and like order and cers are trained on both embedding and generating might be much better at embedding than they are generating so it's also a lot of trade-offs where like you know you might only want to train and focus on the embedding aspects you don't want to have basically twice the number of parameters in compute you don't care about generation and some methods might might be better at one and and and worse at other Etc um so this is why this is the case right so you know something that we started to understand and and I'm not talking right now how you achieve this but something that we you know in a research base and and as well how these models are used that would be nice to be able to map this to the same embedding space like a sharing banding space mean that it Maps you know equivalent meaning to the same vectors uh and maybe it's less clear how it's nice if you just go from images to images but now if to add other modalities it's very nice if this embedding space is shared you can just take your favorite embedder and you can uh your favorite generator you can go in between so you can like you know plug in a embedder for uh images and you can like accurately embed uh a face and then you can you know plug in a generary in terms of language that understands the same embedding space and then it can generate the corresponding language right the corresponding meaning in language like caption right so like face face of a man so you can capture images and vice versa right you can out input an EMB better for language and text and then you can put in your generator for uh images and it can generate you know text image models so this of course very very useful and as we add more modalities like audio and stuff it becomes even more useful right and somehow you know us as human beings we're able to hear things you know we're able to read about them able to talk see them like we definitely have a shared embedding space in our own head you know we can see things you know from you know different modalities but there we still know how to relate the meanings of those things right like just because we hear a dog we don't see we don't that's not completely separate embedding space than if we would see it right we still understand it's a dog and we generalize about the dog you know depending like no matter what what kind of sense that we use right because the embedding space is shared it be very very inefficient if we have a separate embedding space for each each specific uh sense right we have to relearn what a dog is in terms of smell right and recre that some we want to leverage the synergies uh all right and it's not only useful when it comes to like different senses and modalities is also extremely ESS in business so let's say you have a model that's able to you know based on a uh consumers uh purchases in groceries is able to kind of create an embedding like a user profile but a high dimensional Vector of who you are based on your grocery habits but if I can embed that but then if if a generator dis able to go from that embedding to generate fashion you know items or or products that's useful because then I can say like well I know you know I know your grocery habits now I can recommend what you would buy in terms of fashion items right so I can go from knowing who you are and how grocery items corresponds to Fashion items I can cross sell and sell more things to you right and understand who you are at a deeper level so it's like that's useful okay so I I promise we would talk talk about the claim that language is an efficient medium uh for communicating and verifying meaning so like okay now we've still you know talked about these embeding spaces a lot as high dimensional vectors of numbers but again like maybe we can give language a uh uh privileged position that's just like okay we're going to use language itself to be our Universal embedding space right so when whatever it is this is audio if it's images right or if it's text we're going to put this in a embedding space of text and language right language itself uh and you know why is this useful maybe it's like it's good because we can peek peek behind the scenes and see what's going on a little bit as human beings because it generates text as some as as some intermediate representation like yes that's interesting useful but that's not actually why it's that you know why it's so uh useful and and and fundamental and I think it why it's useful is the same reason for why actually we think in terms of language even if we wouldn't be even if we wouldn't talk or communicate language is still a very very useful uh medium uh and that's because like our knowledge of language helps us to standardize knowledge and it helps us to learn new things faster and we can improve and we can ensure consistency in what we learn so let me give you example of this all right let's say we want to train a robot to solve tasks for us so we might want the robot you know to give we give some description of the task like make me a sandwich and then we want to teach the robot to do this so how we did this initially and you know how we would do this is that we would uh have a model an AI that map is prompt right and some input to some high dimensional embedding space in terms of numbers right and then it maps from that space to a set of actions right so this now this you know numbers that represents the task and the plans that it needs to do know we we don't know exactly what's going on but it represents in terms of digits and then it has to execute that plan right so somehow it implicitly encodes like the steps involved and what it needs to do Etc uh and this is what we did in the beginning for Robotics and people still do Etc but what if we don't map this to high dimensional vectors what if we just map this to more language right so we mapped is a prompt now of make me a sandwich to like some kind of plan and breakdown so it says I will go to the kitchen take bread from the pantry butter from the fridge and a knife from the drawer then use the knife to put butter on the bread lastly I will leave the kitchen with a sandwich to bring it back okay great this makes sense right we can we can look at it what it does and like okay this makes sense nice but why would we do this well we said that reinforcement learning like training these robots is very very hard if you have no understanding of how the World works so typically you won't do very well and it's very very hard for you to solve so let's say you're actually outputting something that doesn't make complete sense so let's say you're not doing a good job so now you know a bad model this outputs I will take butter from the fridge and a knife from the freezer like taking a knife from the freezer doesn't make sense to us okay then use the butter to put bread on the knife again like butter to put bread on the knife doesn't make sense I mean the right words are involved but not you know they're not interacting in the right ways here I will then go to the restroom for bread like who keeps bread in the restroom uh probably a few people lastly I will leave the kitchen with the sandwich to bring it back like okay you were just in a restroom but now you leaving the kitchen like this plan doesn't make sense like I don't even know need to know about the task just me knowing language it's like um like this looks like it's going to go wrong right so this is what's very very cool that we just by knowing language can verify and look at and say like well this is not going to end well you need to correct yourself right and if we're able to do that by just using your language right language models are able to do this so then they can look at this and be like well uh this is a great intered intermediate representation because I know language as a efficient medium for verifying ideas right and correcting them so I can look at this now and be like well robot you don't know how the world Works clearly because you don't know how things work I'm going to help you out and put this in better format right so we can just ask TP hey please correct this text doesn't know what tasks involved but it can make it much much better maybe not perfect here but just make it much much better put in a way where it makes much more sense right so I will take butter from the fridge a knife from the drawer then use the butter to spread the bread I will then go to the kitchen to make a sandwich lastly I will leave the kitchen with a sandwich to eat it right so not only then is CH p a revolution when it comes to just text task right it's also a revolution when it comes to Robotics and other tasks because understanding language in this sense extremely useful to to make improvements other areas right it allows to make kind of shortcuts in in our reasoning and thinking and also generalize other things that understand and know to a specific setting and do much better with much less data and in fact for like reinforcement learning you know you need to have this world model to even start making any progress whatsoever because you might want to know that like hey I'll put my knife into this human being right that's not a good plan if you know language like well no no no don't do that right you don't you don't you don't want to learn that by trial and error you want to know that already um all right okay so this is actually extremely similar to how autonomous agents what they leverage in terms of language right so now you have this large language models that are trained on the internet to just understand language tasks right now you can leverage this actually for planning and using Tools in a way that we didn't anticipate almost right it's almost very very humanik so a an autonomous agent is just basically a large language model that's applied iteratively right on its own output so it's allowed to you know correct itself improve on itself step by step typically you know your TTP model will just you know have one chance to look at your prompt and then give the output that you want but if it's able to you know look at your input Analyze That create an output and then look at his own output Analyze That and create some more output it can improve itself find its own mistake and do planning and saiz me it can be extremely powerful and when you allow it to use its own output as input you know iteratively it becomes much much smarter and better right it's like almost a little bit like you know you having to solve something on the spot by giving you some time to actually iterate and improve on your ini initial estimation or plan um right so it's allowed for retrospection um and improvements in this way right so for large language model analyzing generating is it's kind of related but you can look at some prompt analyze and generate some output then you Fe that output into the model again and we redo it and then we can throw in some some tools into this as well right so you basically describe some tools using language and then the Lang large language model can use these tools as well you know iteratively so uh for example let's say you write to your large language model that is autonomous agents now agent because it's allowed to rerun itself create me a website with like two pictures of dogs and descriptions right the first you can do is like okay great I'm going to make a plan so maybe the first output is just like five steps right and then it reading that and okay I'm going to start with step one so you can make this plan of like five steps start with step one then it's like okay an that I'm going to use internet to search internet retrieve two pictures of dogs right I get those pictures I put them a website I put those pictures pictures in right great now it uh looks at this it's output in the website code output it sees like great I've solved task one you know let me move on to task two and I'll add the the the text it adds the text again to verify that know then it runs itself on this output again to verify the website looks good maybe it's like okay actually I found a bug here this shouldn't be here I I incorporate that and then it runs itself on its output a last time to say like well now everything looks good let me now H tell my you know me or the owner right that I'm done and and I'm happy with it like this this ability to use tools and just like run on itself this way is what autonomous agent does and it adds a lot of kind of performance and abilities to these models and some examples right of the tools that these models use are internet right scrolling through internet retrieving information basically which is very very useful so you don't need to remember everything from yourself you can use and analyze information uh calculators are one example right like even for large language model is very very hard to uh compute the product of very very big numbers ET so why don't just use a calculator to fit it in and also what we're seeing now is other Foundation models right so like this kind of privileged position of language understanding and kind of Consciousness you know that's just one part of our brain and surprisingly I mean Consciousness and what we conscious of is a very small part of our brain and most of our brain are much more automatic and special some specific tasks so what I think we're seeing and what I think we're going to see more of is that like these launch language models are like a connector across different brains and stuff but it it feeds things into the specific regions of other Foundation mods are specialized in certain settings okay so still what are we seeing right we're seeing that in robotics that having a language model is extremely useful actually critical right and then of course you know so it helps you even in robotics and you want to add other abilties as well or you want to have Vision abilities and a all is trained on understanding images and vision as well because you don't want to retrain that for robot has to start with the starting point and it's it's useful right and and also this shows kind of that all of these uh different intelligence are very synergetic and I think that's also something that that took people by price is this new AI That's intelligent in a much deeper way also shows that intelligence is is very synergetic every time we try to split it apart it loses abilities and it becomes you know even harder to do right A lot of times now as well when you try to solve more it becomes easier and not only does it become better performance also but it also becomes easier uh so intelligence is very centralized you don't have a brain in each fingertip you have one single brain right but also probably there a limit to this because we don't have one single brain you know as Humanity right and our brain is constructed by a lot of sub brains so there's there's Nuance to this and I think I'm going to talk about this in sub subsequent lectures like well we're seeing different Foundation models emerging and it's going to be useful to know how will those Foundation models look like right but uh I think something that we saw early is that people felt that the market is going to be very fragmented like it's going to be a lot of room for specialized you know slightly specialized brains for specific tasks but it's it it isn't really right it's a very much like a winner takes all market in terms of AI because if you try to do more you'll do better H so I mean there won't be separate brains for uh language and vision and for for uh music right there's a lot of synergies in between so it needs be some way to connect them right and get those synergies for Optimal Performance right and similar when it comes to business right like the day of like when we attack recommendation separately with you know a separate intelligence and like recommendations with a isolated separate intelligence like that and a business intelligence is like a separate thing no these are related things it's about understanding your business right understanding your consumers in this case these are synergetic aspect and intelligence is centralized you should leverage data and know how from all of this all of these different disciplines right so you basically put in all this into one single huge brain I mean but but I mean also the internet bubble for example already SE starting to see this me everybody put a website on the internet for dog food whatever it was in the 1998 right but now it's basically only Amazon right somehow like we I mean I'm exaggerating but it's going to be the same thing in AI a lot where like there's not going to be a lot of room for for some some of these small hyp players right now H again like this of course is kind of a warning and scary because putting more and more power in few people's hands right at the end of the day it's like you know these are not conscious entities like it's people Behind These models their stakeholders Behind These models they'll have the power to decide and they'll they they'll benefit from this and they're getting like more centralized power and and now you know chps is taking a big chunk of the market and then suddenly people want CHP everywhere in all our products right and everybody's using it and if everybody on the whole earth is using the same large language model it's very very fragile because we're all you know doing the same things so if it's off it's going to be off systemically and we going to talk more about on a eics but also right it's also exciting because the intelligence that we're seeing is so tremendous so it can solve problems that we can't do ourselves and it's going to be able to uh benefit us greatly if we use it responsibly and smartly and I'm going to have the answer to that later in later lectures okay so next time we're GNA be even more Hands-On and next time I think we're going to talk about chtp in detail so we're going to through go through exactly how that works we're going to demystifies the the Transformer model which is actually quite simple and I think it's not that important and I'm going to tell you why so you can tell your friends that your cocktail parties the Transformer is not cool and then lecture after that we're going to dive into text to image model like stable diffusion uh and uh yeah and and then after that we're going to have lectures right on basically how what type of different Foundation models are emerging uh and how it's going to look commercially as well and then we're going to have a guest lecture uh two guest lectures so in uh after these two lectures we're going have two lectures with guest lectures right and then we have an Ethics as well okay so thank you so much [Applause]

MIT_6S087_Foundation_Models_Generative_AI_2024

MIT_6S087_Foundation_Models_Generative_AI_BIOLOGY.txt

goe all right uh well welcome to manolis he's a professor at MIT he's doing amazing stuff in computational genomics and Ai and biology he's going to talk about the AI Frontiers here super excited to have so give him a warm Applause thank awesome welcome everyone so um basically there's a lot going on in biology and there's a lot going on in Ai and my goal is to tell you a little bit about both and how the field is dramatically changing so um uh I'm going to tell you primarily about health and understanding biology and Medicine who wants to live forever here good good who wants to live at least live like until next year yeah so so there's this joke where like like oh who wants to live till 100 well the person who's 99 and uh yeah we never actually want to die we just don't necessarily want to live forever but anyway so the goal is how can we use AI to truly understand the mechanism through which human biology works and how we can use that to basically develop new Therapeutics that put an end to disease as we know it who's excited about that good so what's our goal our goal is to understand medicine and and medicine has truly come a long way so this is I was just giving a talk in Athens last week that's when I made this slide and uh this is how uh medicine used to work so you would have some type type of uh it's unclear whether here you're depicting a God or a physician but you know even nowadays the distinction is kind of blurred in many ways um and then they're doing this magic and the patient is sort of subjected to that and you know there's some peer review Comedia apparently um but this is how it used to be done and then eventually we started looking at things closer and closer and this is more than 100 years old the um structure of neurons inside our cortex and our uh different regions of the brain we can basically now see the the fact that we're based on smaller and smaller parts and the first diagnosis of Alzheimer's dates to a 100 years ago from Imaging where we could actually see the plaques and neurop fiary Tangles that are still today the definition of Alzheimer's but something dramatically changed in the last few years and this something is human genetics human genetics basically tells us that there's something playing a causal role here and what that allows us to do is start going Beyond just correlation to causation and the last part that changed is a lot of our own work in being able to gather massive massive amounts of data for integration so you can basically think of this as the Next Generation microscope where instead of gathering four cells at single cell resolution we are gathering 2 million cells at single cell resolution and instead of measuring whatever we can stain and we can stain about like 5 10 20 things at best we can measure the expression of 20,000 genes for every one of these dots okay so this is a 20,000 dimensional space with 2 million cells projected down into something that we humans can visualize in 2D okay so what are the Paradigm shifts that are happening the first paradigm shift is that we're going from hypothesis driven to data driven instead of just saying oh we have a very specific hypothesis like gather a bunch of data and then we're going to say yes no answer we now have just massive data we shoot first and we ask questions later so we basically have systematic data sets we're building resources massive data sharing and really comprehensive use of biology the second as I mentioned we're going from correlation to cation correlation means the countries that eat more chocolate also get more noble prices does the chocolate need a noble prices do they just buy uh I don't know more chocolate with prices is it correlation causation or reverse causation so that's what epidemiology has always been fuzzy about whereas with genetics we actually now understand mechanism we know that this if if there's a genetic difference then eventually you can establish causality and then the last step which is the most relevant to this class is we're going from classical data analysis where there was a different methodology for every problem we would just like come up with a question develop a statistical test and answering and and the humans did all the thinking to now and where the goal was to develop very few parameters and very targeted models to understand so that we're not overfitting the data whereas now we're basically saying billion parameters no problem bring it on and we're building this Foundation models that are very often multimodal where we're learning representations we're learning hierarchical deep representations and we're truly understanding concepts and yielding insights everybody with me so far so these are the major shifts what does that mean that means that we com we can combine now causality from genetics and big data to truly understand the mechanism of disease genetics that means that we're starting with causality because we know these regions have something to do with disease the problem is that we don't understand the mechanism and that's where massive data come in we can basically say this correlates with Alzheimer's let's go and find out what changes in the brain of people that have this genetic difference or the people that have Alzheimer's or people that have environmental exposure and then we can figure out the specific genes and proteins that are responsible and then use those to understand mechanism so we're gathering this massive data and that's where the Deep learning comes in where we can now go from sequence information to a model that understands the language of biology understands how mutations are acting understands how proteins are folding how chemicals are resulting into their functions and then eventually make predictions that we can validate experimentally and that's another amazing thing about biology in society it's very easy to say well maybe this causes that but intervention would cost like I don't know billions of people changing the way that they do stuff whereas with Biology we can take a cell and then change a gene and then see what happens so they're much more transparent those models everybody with me awesome so what we need to do is basically go from Simply there's something going on genetic here to here's the circuit here's the genetic variance the differences in the letters here are the motifs the sequence patterns that these letters perturb here are The Regulators that bind these motifs here are the control regions or enhancers and the cell types where they become active and here are the target genes that are controlled so effectively a circuitry and then using that my lab has basically worked on applying this type of methodology to a dozen plus disorders from cardiac disease obesity cancer Alzheimer's addiction neurod degeneration pathogenesis schizophrenia psychosis bipolar Down syndrome autism PTSD so every aspect of the human body and the human brain we can now start studying systematically across dozens of cell types across hundreds of uh tissues and millions of cells and hundreds of individuals we can now start asking how is the action of disease percolating through to give you an example I like to joke that we published a paper in the New England joural medicine about one bit of information in the human genome this is about changing a t into a c and what we showed is the mechanism through which we can translate a region of genetic Association the strongest association with obesity to a mechanism that basically tells us how is that variant acting what is the Upstream regulator the downstream Target genes the cell types of action and the mechanisms and we were able to by understanding this mechanism switch human cells from fat storing to fat burning just by flipping one letter or switch a whole animal into a fat burning machine so mice where we knock down one of these two genes are burning every calorie in their body they are there's no white fat in their body they're just completely red juicy healthy uh organs and then the um normal mice gain weight when you put them on a highi diet these mice are unable to gain weight you can basically feed them all you want they don't have to exercise more they don't have to eat less they just sleep it off they just burn it off so by manipulating the circuit we are now able to reverse disease circuit rink who's with me so far great okay so that's one example another example for Alzheimer's disease we're able to now reverse the circuitry of Alzheimer's and basically take individuals that have apoe4 this is the alil that increases your risk for Alzheimer's by factor of 10 so if your homozygous risk for apo4 you're almost guaranteed to have Alzheimer's disease at some point in your life pretty soon and um what we found is altered cholesterol biosynthesis we're able to trace it down to the transport of cholesterol to form myelin that protects neurons and by reversing this process by restoring cholesterol transport we were able to effectively restore myelination and restore cognition both in human cells and in mice second example from a region to a circuit to manipulation third example in cancer immunotherapy about 50% of the patients respond this used to be a death sentence now for 50% of the patients this is amazing they completely beat the cancer but for the other 50% the cancer comes back and what we were able to show is that by understanding the circuitry we were able to predict a regulator Upstream of the genes that turn back on shut them off and then suddenly not n of the cancers come back so again in very different applications obesity Alzheimer's cancer we're able to reverse this so how is this all possible I'm going to give you a number of vignettes for how we're now able to understand the language of biology and reverse it systematically okay so the first application is on regulatory genomics that basically means going from the DNA sequence to how that sequence functions how does Gene regulation work we basically have small patterns that we call regulatory motifs that are recognized by corresponding Regulators or proteins that bind the DNA at those motifs and then together they cooperate to to turn regulatory regions on or off and eventually genes on or off everybody with me here that's all you need to know basically we can measure The Binding of these Regulators across the genome by building assets that pull these regions down and then sequencing and then asking where did the sequence come from and then we can map all of the regions that are bound by different proteins and now comes the fun part we can basically put that all in a deep learning framework and say what is the sequence pattern that will predict the activity here and we can basically build these multi-layer neural networks these convolutional neural networks that learn to Novo these convolutional filters and these are the the sequence patterns that ultimately allow us to predict activ a and repression of different regions and I can show you here The observed experiment and then the prediction for every one of these regulators and you can see just this extraordinary ability to just capture the language of DNA and then the beauty of it is that it is mechanistically insightful basically tells you that this Motif contributes here and that Motif contributes there and so so forth who's with me so far awesome good so that's uh language of DNA then if you have a mutation that we've never seen before can we predict the impact of that mutation why for personalized genomic we want to be able to adapt Therapeutics adapt everything to your uh DNA so how do we learn the sequence of uh the genome and how do we learn the impact across individuals and across cells of genetic variation what we would like to know is for any genomic sequence if I change a single letter have a deep learn framework that allows us to predict how is this going to affect the Machinery that binds DNA the transcription factors the accessibility of the DNA and then the modifications of the packaging of the DNA so we can basically put this into again a deep neural network and then start predicting from thousands of nucleotides across multiple different layers of information the impact on expression and then start predicting the impact of these genetic variants and you can see that you know this this makes enormous impact The Next Step Beyond that is to understand how cells change in their activity patterns so we have about a trillion cells in our body and for these cells we have the uh extraordinary different functions between our neurons and our heart cells and our bones and our eyelids and our tears and all of these things have basically very different functionality and then the immune system itself has hundreds of cell types that are playing very distinct functions so can you understand both the dimensionality and the drivers of gene expression space at single cell resolution so we can now basically measure the expression of millions of cells and we can start building deep learning models that allow us to from the sequence information and the expression information compress that through an auto encoder into this bottleneck layer and then expand it back out why is that exciting because this is where the fun happens that's where the variation matters and in the same way that we can sty we can transfer styles for example sunglasses or men versus women or old versus young we can now start asking about the dimensionality of expression variation at single cell resolution and we can use this to start teasing apart the different components of variation through these variational autoencoders so we can basically say how much of that is core how much of that is label specific can we now decompose these parts and start varying the labels to turn the expression patterns of a patient into a healthy individual in vice versa or a young to an old a male to female and so so forth so everybody with me so far who feels that they're learning stuff good it's a very shallow overview but um you know the goal is to expose you to all of these different types of problems the next step is electronic health records so basically through massive amounts of phenotyping we can now gather huge diversity of individuals with genotype and with dozens of phenotypic variables so that basically means that for every individual I'm measuring six million common variants very often the whole three billion letters of that person of which only a tiny subset like for example 6 million and then I also measure the phenotype the disease the medical information for that person and what we're doing for that is that we can now use that to now start decomposing the electronic health record of a person into modules into components that are associated with distinct phenotypic combinations that appear together and we can now start uh mapping that to the distinct expression patterns that are changing in either the blood or in postmortem tissue samples from that individual in this particular case we looked at 430 individuals we looked at their brains postmortem and then we looked at the phenotypes that they had prior to death and we're able to correlate now these transcriptional Hallmarks of Alzheimer's disease to recurrent patterns of physiological differences between those individuals and we've coupled that with electronic health record uh information longitudinally across different individuals which we have also coupled with the uh large language model interpretation of these changes so we can now basically take these large language models and start interpreting what is actually changing in the uh activity of those individuals okay so so that basically means we can now start decomposing these patterns and we can start using large language models start interpreting the weird deviations from expectation for many of those so basically we can now look for patterns in the electronic health records and then use large language models to interpret these patterns automatically and basically come up with insights as to what is actually driving different aspects of the biological variability and we can now take these phenotypic components about how your electronic health record is changing and then combine that with genetic variation for every individual across millions of varant to look for the covariation between them so we can now start building up the building blocks of both genetic and phenotypic variation in uh the human genome okay so the next part is beyond just the quantitative variables such as lipids and all kinds of other aspects of the electronic health record you can basically now look at Imaging and pathology and if you take these slides you can now start using AI to automatically annotate what these slides actually mean and where is a tumor in those slides how by basically extracting features associated with The annotation of those images doing a massive curation for 1 million medical papers that you can download from the web and then annotate every panel based on the legend of these images and then for panel a b c d you have different annotations you can now use multimodal learning that you heard about already to basically do joint learning between the image and the text description of those images and start understanding the features of the image that are responsible for the different annotations so that basically means that across these 1.17 million image text pairs for histopathology you can basically now build a foundation model that allows you to look at a new image that you've never seen before maybe even for a tumor class that you've never seen before and start reasoning about the pixels is everybody excited about this good so that allows you to now do zero shot which means in a class that I've never seen before retrieval of ton of information and that allows you to now start learning the foundation of all of this the next Frontier is looking at chemistry being able to to go from Individual atoms to the function of a chemical how do you do that through graph neural networks that interpret the function of a whole molecule through the function of the parts how by building an embedding representation of every atom initially dependent only on the properties of that atom and then through the graph neural network we're propagating convolution operations that basically say in layer zero this only depends on myself but in layer one I depend on all of my neighbors and in Layer Two I depend on the layer one representations all of all of of all of my neighbors which themselves depend on all of their neighbors so I can use this to now start propagating a final prediction for the function of a chemical through the individual organization and layout of the chemical graph representing these molecules so we can use this type of operation to propagate information from the bottom up and eventually start predicting the function of a new molecule and even designing new molecules with particular characteristics you can now start synthesizing those using the techniques that you've seen already by building them from the ground up using either individual atoms or individual motifs in the chemical space and then designing molecules with a particular function in mind everybody with me so far yeah who has a right arm good who's with me so far good awesome nearly everyone with the right arm um so so that basically allows us to now start designing new chemistry and then as you heard in the news the next Frontier after that is understanding the function of proteins and the structure of proteins so on the function side you can basically look at Evolution and amino acid properties and this used to be done by Massive physics where you would basically simulate how a protein chain would eventually fold into three dimensions whereas now we can basically start looking at how this chain Maps into multiple species comparison Co Evolution analysis and also the profile of that sequence itself to start understanding the relationship between sequence and structure and the next Frontier after that is now looking at how that structure is encoded into function so basically here you have the prediction you have the modeling and um the next Frontier that we're working on currently in my team in collaboration with Mar kitnik over at the Harvard Medical School is building Foundation models for translating protein function and chemistry together so the goal is to basically be able to systematically intervene at the chemical space and at the protein biological space and together with Brad pent Luther here in chemistry we're basically developed this program that allows us to now drug all of biology and we are building this uh this Therapeutics llm that allows us to now take protein structure protein sequence and the textual description of biological function of that protein through the medical literature together with user queries similar to chashi PT where you can ask anything and we can now start predicting which proteins will have what kind of function from the building blocks of their amino acids and their structure okay so that allows us to now reason about the types of chemicals that I should use to intervene and to start reversing cardiovascular disease metastatic melanoma Alzheimer's and so on so forth so we want to take on disease systematically transform the way that we understand biology enable personalized medicine and bring all these uh together the last snippet of information I want to tell you about is how we can use AI to now start understanding not just biological function but also any idea that Humanity has ever had every paper that has ever been published every poem that has ever been written can map somewhere into the space of ideas and what we have built is a tool that allows you to directly navigate that space of ideas this morning I was actually speaking with the uh Development Bank of Greece to basically see how we can use this to understand all of the loans that they're giving out all of the startups all throughout Greece where we're working with the uh New York Times and the um Cathy Marin newspaper to basically parse all of their articles and understand their similarities here we're looking at 60,000 Papers written by MIT authors and then we can color those according to either the lab that wrote them or we can actually show them on the map of MIT as to which buildings are generating what types of ideas in this space of ideas here we're basically looking at every course that is being taught at MIT and the graph of pre prerequisites and the department from which these courses are happening and then the ultimate view is that there's an island for every concept and we can now use a Google Maps like navigation technique to look at side by side the paper that I'm reading where it sits on the map where are nearby papers and map all of knowledge uh this part in this particular way so again the major Paradigm shifts are that instead of basically trying to develop a new machine learning method for every new data set we are building these multimodal embeddings and this can happen in biological space but also in idea space and the goal is to gain insights that are previously inaccessible to human scientists using this way so again from hypothesis driven data driven from correlation to cation and instead of classical data analysis understanding Foundation models and multi-dimensional learning who feels that they've learned stuff today yeah awesome great uh should we take a couple of questions I I you know maybe you should speak on Thursday I'm I'm afraid we don't have too much time it is now uh 21 what time does the class end like 30 okay well let's question first okay quick questions yeah yeah is this last thing that you showed available this mentis um so so we have just built it we have applied it to as many data sets as we could find that are kind of cool and fun if you have massive data that you'd like to apply to Let's collaborate uh the goal is to eventually release it so that anyone can sort of use it but our initial mod super Randi is that we are working with people who have massive data sets and building one for them as well so that you can navigate your own data do you have specific data sets in mind good that's very nice yeah it's it's it's a lot of fun it's it's uh I mean I've wanted this forever but it's now finally possible I've been recording uh all of my meetings for the last 10 12 years and this is the embeddings of 10,000 meetings that we've had in my team and sort of how they cluster together this is 150,000 papers that have cited our work and how they cluster like I I want to understand everything that way every time I I read a New York paper a New York Times article I want to understand and what are all of the nearby articles and and when I'm reading a paragraph I want to see what is the nearest paragraph anywhere and then what other papers come there you know if I if I want to read five papers on the war of Ukraine you know I don't want to read the same intro every time I want to sort of blend these papers and weave together the distinct paragraphs and it's now at our fingertips we should be able to do this systematically so it's kind of distillation of a Knowledge Graph so yes it's basically taking the 74,000 DI ditional latent space of uh llms so we've trained our own llm so this is uh a buildon parameter model and it's uh trained using a slightly different objective of reconstruction of the sentence rather than next word prediction so that allows the LM to capture much more information about the context and the relationships and we're now using that to sort of see if we can you know do out Bings where you know from this paper plus that paper like start generating something new and actually relevant question for the encod you show so you had like ENC which kind of data do you use what's like an input for training there yeah so so a lot of that is self-supervised learning when you have Auto encoders you can just like predict figured it was more about genes right so it wasn't like language based right so so basically in the auto encoder that I showed earlier this was predicting the actual expression patterns so we were basically saying what is the um expression uh initially how can I predict it from you know a much condensed bottleneck layer so that's what allows us to now learn this uh you know so so basically the output is the input so we're trying to predict the original data but through this bottleneck layer but what so gene expression levels so basically the amount of expression the amount of mRNA for every protein yeah yeah yeah the everything show about and the way that we see the the patient data how how do you think will be the the new way of diagnos diagnose yeah a disease and uh do a personalized treatment yeah so so um multimodal is the way to go if you look at images for example you can trick an AI that learned only with images but it's much harder to trick an AI that learned with multimodal information so basically by having these multiple modalities humans are constantly integrating multiple modes of information and now with AI we can have a clinical view of a patient based on of course their clinical record but also their image how they're looking how they're feeling how they're talking their Cadence their gestures their eye contact the the way that their hand shakes or stays steady and so on so forth so basically um I expect that a lot of uh patients will be asking for AI augmented diagnosis going forward that right now it seems weird hey is the doctor cheating by looking at AI whereas in five years I think the patients will say um excuse me shouldn't have an AI assistant what happened is it broken today you know why why don't I get the best service I can get so basically I feel that right now we're diagnosing basically by training doctors to think like a machine over many many years of studies where they're basically saying oh I see this evidence I see that thing that basically you know they're following flowcharts in their heads of course they're multimodal creatures so they're understanding you know much more deeply they have intuitions Etc but it takes an enormous amount of training for the doctor not to be biased for the doctor to sort of think effectively just like AI is able to think now so basically my expectation is that moving forward a lot of the human aspects of doctors will be the rate limiting step the much more important aspect of empathy and understanding and sort of getting the the patient to speak to to express themselves to sort of you know think insightfully and that we'll be able to use AI to integrate all that information number one and then number two doctors currently don't want to look at genetic information they're like oh just want to look at you I don't want to look at your genetics whereas by having this multimodal information from the blood expression patterns from Imaging of your brain from lipid analysis Etc they'll be able to use AI will be able to use all of this information in the context of all of the information that you're Gathering each time to basically build a much more complete model of a person yeah with elect one of the big issues is's bias yeah yeah yeah so so there's something known as non-missing at random an m and when a doctor orders a test most of the time the test is abnormal the doctor will not order a test that they're expecting a normal result so if you look at the distribution of observed data versus unobserved data it's hugely skewed and many models would basically model the missing data as if it was you know the same as the observed data when in fact the missing data comes from a very different distribution because it was never ordered so there's many ways around this one is to build cohorts where we measure everything systematically so there's less bias in the specific test that we order um and then when you have that you can learn the distribution of covariates with other variables and then start predicting the state of a person prior to knowing whether that information is there or not or use the information about the presence absence of a variable to build an expectation for the distribution of that variable and then impute based on whether it was observed or not um as to the existing biases of humans humans are extremely biased every doctor is biased every human is biased AI in my view is the best hope for building unbiased models or you know anti-bias models or sort of correcting for these biases because through this variational to coders and other techniques we can untangle the different components of variation and look at sex age ancestry you know all kinds of aspects as orthogonal variables and then use counterfactual an is that basically says well if this person was a man instead of a woman you know how would the data look and you can actually predict that data and sort of make inferences about that so uh yes medicine is extremely biased current algorithms are extremely biased because they're trained based on the data that we have but there is hope for finally overcoming these human biases okay question so DNA seems to the language biology right it's very different from the language used you know in Civilization so probably we talked about this before probably requires you know separate Foundation mod DNA data and of course this huge amount of DNA data how far would you get by just training a huge you know infinitely big model on all the DNA that exists how far would you get in terms of understanding life um I like to say that human thought is much simpler than biology uh and and the reason is that biology has been added for three billion years uh and humans have only been thinking for I don't know maybe 100,000 years and uh like thinking deeply um that's one aspect the other aspect is uh just one second let me put Riley hi Riley I'll I'll come back to you soon um that's my next meeting stting um so so um the second thing is that human thought is constrained to be generated by this thing which itself is constrained to evolve which itself is constrained to develop at every generation from one cell like the zygote is one cell and it takes half the DNA from Mom half the DNA from Dad and divides and eventually sets up a neural network which then learns all of human knowledge from scratch at every generation so it's extraordinary that humans can think so well I mean you know frankly if I were to write an NIH Grant about developing such a machine I would get rejected like this will never work so I'm surprised that humans can even think and that can they can invent AI ET ET but AI doesn't have these constraints AI can basically use dramatically different architectures they don't need to fit in your skull they don't need to evolve they don't need to develop they don't need to be a local maximum so in a way it's almost surprising that we haven't found dramatically better architectures than human brain uh for AI I mean Transformers are kind of getting there but I wouldn't be surprised if humans are doing a very Transformer like uh you know hippocampus connectivity of engrams when we're thinking about Concepts Etc so the question is how much harder is biology than human language so I would argue human language is kind of simple and that there's like you know only so many islands of ideas if you wish and that uh biology by contrast is dramatically complex moreover D biology has no logic there's no like physics is beautiful there's like a few equations and that's it whereas I mean a few hundred equations whereas biology there's billions of tinkerings that happen at every generation you're just like changing one letter here and it kind of works and kind of adapts Etc so in a way the fact that we're even able to understand protein folding and DNA language and Gene regulation and pathology and all of that to me is mind muggling the fact that it even works so it suggests that somehow biology is still constrained and and the Saving Grace might be that we have only 20,000 genes and that's that's a rate limiting step in terms of complexity and all of this multimodality is sort of helping Channel this knowledge into much smaller space and the variations that we see are much smaller than all possible variations of 20,000 gen why because there's a regulatory circuitry and there's only so many Regulators there's only only so many states that a cell can enter so yes you know biology could be in theory infinitely complex perhaps chemistry is infinitely complex but the space of biology chemistry protein folding protein structure Etc that inhabits the this world might at least be at least 95% of it might be sort of within the bounds of our existing models awesome thank you so much beautiful thank you right oh

MIT_6S087_Foundation_Models_Generative_AI_2024

MIT_6S087_Foundation_Models_Generative_AI_INTRODUCTION.txt

all right well um sorry we're a little bit late but let's get started um so welcome to the first lecture on the lecture series called future of AI Foundation models and generative AI this is actually the second year we we hold this class and so I started working on this course before the recent hype and breakthroughs of CHP um and I really felt that we're starting to see kind of a new approach to AI in the community that was really going to change things for real H and I think we started to see that right now um and really what I want to accomplish in this lecture series is to give you an understanding of why this is happening right now what's the underlying kind of change in perspective and also going Beyond just kind of the tip of the iceberg which is chtp so I'm going to give you a deep but non-technical introduction to these subjects and and uh last year when I gave this course we were excited about uh t to video and text image models right guess we still are uh we were excited about uh superum robotics uh self-driving cars and AI applied now to other domains as well like genomics Etc and um of course A lot of these breakthroughs even if they're in very different domains they come back to this underlying technal technological achievement of foundation Ms generative AI um we going going to dive into last year uh we were excited about chat GPT right so we could ask it to write an engaging introduction to an a lecture if we ask the updated gp4 it does it produces more text but maybe does a better job as well uh last year we asked it to produce a engaging artwork of artificial intelligence uh and now we asking the newest version uh it also might be doing a better better job it looks more involved at least uh AR is subjective but uh I think it's better so if we were excited about these things uh in you know early 2023 what's happened right what H what's happened during this year well uh a lot of things of course there been a tremendous hype so there's been a lot of uh you know money pouring into these uh areas we've had companies that only you know few weeks or months old reaching A2 billion dollar valuation which is a team of five people um we've had excitement about autonomous agents we're going to talk about during this course as well like GPT engineer that's able to plan and and even act in a more humanlike way in terms of intelligence uh Nvidia that provides all the different dpus right that these models need have reached a a huge valuation like the $1 trillion club with we've seen uh sweeping uh regulatory uh kind of Acts and and uh uh initiatives right uh both from the White House and the European Union for example there's been a lot of drama in the a space right openi for example the CEO and the company behind CH CHP the CEO was outed and then came back in so maybe the transparency problem in AI doesn't only apply to the models but to the structures and companies behind them and also you know we're seeing kind of some hype and some winners u in terms of the this new AI Technologies but also now some companies are actually losing uh users and usage right stack Overflow for example people saying that kind of is being killed by an AI That's training its own data which is kind of ironic uh of course one of the big questions that remain is have you reached artificial general intelligence yet H some people say we have I think uh there's still quite a long way to go but we're going to try to also explore a little bit uh you know what what can we actually mean with a AGI and how could we potentially reach it given the technology that we have right now and can we give some kind of very uh order of magnitude estimation to when we'll get there so uh I'm a Richard I was uh born in the land of abania which is Sweden and uh before MIT I was at Stanford uh for seven years and I did research as well on AI and and this stuff also started a company that that does uh Foundation models inative AI in commercial settings I hope to bring in a little bit of that those perspectives as well H for the last four years almost now I think yeah almost four years I've been at MIT uh where I do research on on South press learning and and financial models and that good stuff okay so uh quickly on this SC course schedule so today we'll give an introduction and kind of a history of AI from a high level perspective as well what's going on and giving some kind of intuition and and then in the next lecture we'll dive much more into details about how these different algorithms work and how we arrive at these models that we use H after that we'll do an in-depth analysis of CHP like a case study then we'll do a similar case study on image generation and stable diffus then right and these four first lectures will be very similar to last year's offering but then we adding on Jan 23rd we going to talk about kind of emerging Foundation models right it's going to be a combination of them existing out there probably not one single model to rule them all so we're going to talk about that especially how that looks in uh industry and corporate setting uh so Professor man Kellis uh will come as well as you he's an expert on U biology and genomics and then also artm working who's uh an MBA from MIT he will talk about autonomous agents and then we'll have a fun kind of uh or it's going to be a fun hopefully fun lecture on AI and ethics which is of course perhaps a little bit more fussy but we also bring in regulations what kind of what's happening in terms of the institutions regulating AI H and after that we also have a panel with manolis and artm uh so should be fun all right so what will cover well we'll cover all the busws and new network supervised learning representation on supervised learning reinforcement learning genive AI Foundation model self superus learning and we'll try to put together with a lot of applications a lot of intuition uh because it really you know should be non-technical and I think as well I want to try to explain things in in simple but true you know deep ways and I think if you're not able to explain something in a simple way you're actually not doing a good job explaining it so that's what we're going to try to do at least today we're going to give you a short succinct answer to what is the secret Source behind Foundation malternative Ai and then when we've done that we're gonna ask how's the world structured because how we think the world is structures structured uh really influences how we learn in the world so we're going to kind of explore that from a more philosophical perspective and see how that actually leads us to uh Foundation models and generative AI and then we'll at the end we'll cover two applications of how we can use this in in in both research and in business again right we're going to try to use intuition examples and example from both sciences and business and hopefully you'll you'll understand why the hype actually is real I said it's last year but it is real and maybe we understand what's actually just hype and what's the the kind of more foundational aspect of it right what matters okay so I think in trying to uh understand and and U you know have a uh in understanding of AI it's good to use ourselves as a reference frame right we're human beings and I think one of the core questions that we can ask is well at some point right you're born you're a blank slate baby fairly useless you grow up you interact with the world and different stakeholders in the world and you acquire knowledge and you become a fairly useful knowledgeable adult right in terms of AI and learning right one of the key questions is what happens in between you being a blank slate fairly useless baby to you being a knowledgeable useful adult it's one of the key questions and we're going to use ourselves as a reference frame here so let's consider a few candidates that might be responsible for giving us most of the knowledge that we have about the world is it our parents they they give birth to us they raise us they give us a lot of our core values so they're definitely good candidates to consider is it perhaps our DNA and our genes they're responsible to giving us a lot of our physical characteristics also most some of our personal characteristics but maybe it's you know nurture versus nature is very flexible people can come very different based on where they grow up and also maybe DNA works on a more generational scale sometimes again very delayed feedback loop here not a lot of learning happening in a shortterm scale maybe it's uh Academia right maybe it's teachers and professors and and and the educational system it's supposed to be to educate us it makes us make a use make us useful learn different skills and also know how the learn how the world works right so that's also a good candidate and lastly maybe it's more kind of our immediate environment and we maybe what's more important is our goals that we want to be loved we want to be happy we want to be successful and by optimizing those objectives and goals we learn about the environment in the process you know we learn how the environment and different components and environments help or doesn't help us reach our goals and therefore we learn about it okay so we can speit this apart a little bit more and high level perspective see how they correspond to different disciplines within AI right so this parent uh child student teacher leadership that supervised learning is supervision so here there's an human expert like a teacher or a parent that puts the world in order for you it structur it it labels it and put it in a condensed format so you can learn from it and on the other hand this more delayed feedback like kind of evolutionary and you're interacting with the environment optimizing some goals and those goals are what we're mostly focusing on right that's reinforcement learning and as this delay gratification and you're optimizing is gold uh in an environment but it turns out actually that none of these approaches are responsible for giving you most of the knowledge that you have about the world in fact you can thank yourself for that because most things you know you learn by yourself so how is possible well um it's possible by defining meaning by the company keeps so and learning from observing the world so let's take the the example of a dog you actually don't learn what a dog is from your parents telling you or for your emotions guiding you you learn what a dog is by observing dogs in different context and correlating and contrasting dogs with other Concepts so a dog is something that's walked by an owner with a leash it's something that has an antagonistic relationship with cats it's something that chases free speace when those are PR and and this is this kind of relational context is what allows to understand what a dog is and as you learn what dogs are by correlating contrasting dogs with other Concepts like cats for example you intern learn what cats are right this is where you get this relational understanding of different concepts and and even language can be learned this way because the word or name dog would be uttered more in context where dogs appear and if you think about it if a you know a little child child points at a dog and asks his parents like hey what's that like all the parent is really doing is giving some label and the and the child having the ability to generalize that label to all dogs shows how has a very very robust understanding of an entity already a concept of a dog just not the name yet Perhaps but that entity concept is got is gotten by just observing dogs so for example let's take the uh cat and mouse here so I think intuitive for ourself when we hear cat and mouse there's a strong correlation and we can understand a cat in terms of mouse and vice versa right a cat is something that chases mice and M a mouse is something that's being chased typically by a cat but when it come to when it comes to a mouse and a dog I think that the relation is less strong and less obvious right what's the relationship between a dog and a mouse is not completely clear in our kind of Contex understanding of the world so let's take this observation and uh fit it into this Foundation generative AI models that also have this similar understanding of meaning right so a lot of this I mean basically all the the images you see in this lecture is produced by generative AI so let's ask uh thetive AI text to image to generate a cat playing with a mouse right this makes sense based on how it's trained you understands and generates a mouse you know that's or a cat playing with a mouse in some sense but if you asking instead to generate a dog playing with a mouse it gets confused because it doesn't really make as much sense in in a contextual relation understanding that this model has of the world so it generates a you know a mouse looking dog playing with a computer mouse so this shows that like when the context is not clear meaning is not clear and this is fundamental to how this model has start to learn about the world and it also corresponds to how we think about it probably when I said cat and mouse to you like you immediately collaps to a meaning space where there was obvious what a mouse was you didn't even think about a computer mouse when I said c mouse because it just contextually made meaning clear for you okay so basically more relations leads to a better understanding of meaning you understanding what love is for example helps understand what a dog is because an owner loves his dog so you know why not train a huge model with a ton of parameters that's able to compress a lot of these different relations on as much data as possible to learn as many relations as possible right so you get the most precise understanding of all of these Concepts involved and then you use this model basically anywhere where any of these Concepts appear right and that's exactly what a foundation model is rest upon and somehow also you know your brain is a prime example of some kind of foundation model you get a world you know model and knowledge model ofout the world from just observing it and building this relational uh structures and this is also what's you know what's behind this current AR Evolution that we're seeing this is the building block that is uh fueling all of the different breakthroughs that we're seeing right now okay so this might sound kind of intuitive and make sense but why did it take such a long time to get here well this is where I think is quite use useful to go on a little bit of a philosophical digression and and I mean provide a little bit of my uh uh fig picture on how this happened and also it's shared with other other researchers some of them so um I think this kind of rest a little bit on two kind of opposing perspectives about how the world is structured and how we learn in the world so I'm going to try to uh get a sense of by giving some examples and contrast some stuff so on one hand we have learning versus designing where designing is this uh you know like a Clockwork we know how every piece works we put them together every piece has like a a perfect role to play we know what's going on and it's built with a specific blueprint and purpose in mind right that's how we design things and on the other hand learning is something that we perhaps not fully understand and we're not completely conscious about it just happens we just get better at something as we're exposed to the phenomena more and more and more and there's really no you know end goal in mind before we start learning right we just adapt uh and it's very very flexible and intuitive um okay similarly we have Chaos versus order where order is kind of what exist on the scale of planets or atoms where things behave according to beautiful Simple Rules like math physics and beautiful Theory right what chaos is perhaps you know the more unpredictable reality that we find ourselves as human beings right the animal world the human world where things are unpredictable and and chaotic and one of the core questions here as well is that well you know the chaos that we experience in our everyday life is that like is there some simple order behind of all of that if you just find that order will all of our experience make sense or is there perhaps a limit to the order of the world and how much equations actually can explain and do we have to deal somehow with this chaotic World in another way similarly uh we have this perspective of bottom up and top down right in a uh top down organizations there is a boss or some you know top person that's able to come up with u a nice uh framework for our things should be done from the top just by analyzing data and then push that through throughout the organization to everybody involved somehow h on the other hand a bottom up organization really then it's really necessary to have a lot of people at the bottom the interact surpris with customers and products and and deals with all these different you know chaos that happens in all the particulars and there's no real simple top down decisions that can be made to make your business really work you need to account for all the particulars and it really matters how you engage with a customer on a you know personal level it's not enough for the boss to come up with some 10 simple rules for to Sol these things you need to deal with this chaos H and also a lot of the wisdom in a in a bottomup organization comes from really listening H to the people that are closest to the the end consumers okay uh so I think at least in the Western World We have had for quite a long time this uh designing ordered top down perspective and I think this is kind of due to the ancient Greeks right so Socrates for example had this allegory of de Cave where uh basically human beings are have very imperfect senses and a very kind of imperfect uh understanding of how the world really works and what's going on and they gave this this allegory of the cave where human beings are actually here on the left hand side look looking you know at the reflections of the cave so basically our experience of the world is so untrue and imperfect so we don't even get to experience the world firsthand we get to experience God's walking with depictions of real objects right and we don't get to see that those even we get to see the reflection of those depictions through you know a fire on the cave wall That's How distorted our view of reality is and for socr like well we to accept that we have very distorted imperfect experience of the world we have to always strive to understand the real true world the beautiful world of the Gods which exist you know outside of the world we get to experience so I think like for for Socrates a dog for example like there is a true perspective of a perfect dog in this Godly world that we should try to understand and all the variation that we're seeing in the real world are just some kind of imperfection from our sense and we should strive to understand the true dog in the in this perfect absolute world so also I think it makes sense because the Greeks this time they were discovering mathematics and in math for example there exist A Perfect Circle that obeys very very simple equations but every time you take a perfect circle and try to recreate it in the real world it's always off it's always imperfect so like this this kind of Correspondence somehow influence socr thinking and also make sense in this kind of mathematical perspective and I think this has been extremely uh fruitful for us is led to the kind of golden era of design we've had a a scientific revolution and Industrial Revolution right we've had modern math physics and modern medicine and we even went to the Moon with this design way you know top down order way of thinking so it's been extremely extremely good for us but you know assuming that this top down order design way of thinking is the be all of our existence how come we're not better at it right we've had billions of years of evolution here on this Earth how come we're not more like a computer or calculator right if if we can just try to find the simple mathematics and Order behind the world and we'll be able to perfectly exist in it and I think this is um kind of a strong indication that there actually is a limit to what order can explain because and we're not in like we're not intrinsically very good at logic and math because we don't live in a top down order world we live actually in a bottom up world of chaos where math is not that useful instead in a bottom up world of chaos what's useful is intuition flexibility and speed the things that we actually are good at right so in this order versus chaos perspective when it comes to our everyday interaction as human beings uh you know besides the scale of Panet or atoms actually the world is chaotic and we cannot Escape that fact we have to deal with it there won't be some just simple equations that explain everything for us that we can rely on we have to deal with all this chaos that is somehow unavoidable so what can we do right you just give up so if is there an instrument that can help us to contain and navigate all of this chaotic world that we find ourselves in well if we had billions of years of evolution and we didn't create a computer calculator or nature didn't then what did it did it create well it created a brain which is our best tool of navigating and making sense and learning in a world of chaos and then the new network in artificial intelligence is just our best attempt of replicating the brain inside of a computer right so it's very very flexible and adaptive it consists of a ton of kind of neurons or parameters H and it's very very simple computations but done in a hierar scale uh and also it's extremely slow to train but very very adaptable and flexible and fast to ex execute when actually learn something okay so we now have accepted the r as chaotic and we have a tool that's able to uh still function and navigate in a chaotic world how do we how do we use this well the thing is that these new networks still exist inside of a computer right and a computer only speaks the language of code and math H so still there is a divide now where we have the real chaotic world and we have a computer side of machine so we somehow have to go through the world of order and math to tell the brain inside of the computer what to to optimize for or what to focus on right to uses brain for we have to somwhat describe this in a more exact way for it to be useful so we still have we can't still completely ignore the world of order so how can we Define such objectives rigorously H and that's what we're going to try to go through right now so first thing we can do is to say that well we understand how the works Work World works right we understand how things work and we have a lot of knowledge about the world so why don't we just impart that knowledge onto computer why don't we just structure the world in a way that makes sense for computer we label all of it and then we can feed that information to a computer so you can start learning from that right that's that's our first attempt and that's supervised learning where we structure the world we label it and so computers can learn from us um and you know pretty immediately we run into some problems first off of course this scales with human labor human experts and labels that's why you know you have these Outsourcing centers people label data constantly and can everything even be labeled like do are we that kind of self-conscious about like are we that conscious about how things work like love for example can we label Concepts like love it's probably going to be quite hard and we maybe overestimate our understanding of how things work and how we can uh isolate Concepts and uh really you know maybe things again like love or other things are not very labelable and maybe things are not as categorical or like distinct maybe the world is actually more continuous in a sense and only in the limit of unlimited number of labels do you actually start to understand the world is really structured right maybe as you know maybe have set of labels and you learn from them but you will always find these uh points in between where labels don't really explain what's going on for example here right you know is this a is this a dog or a cat the left hand side I mean somehow at some points things get more and more close and we we reach a limit to how much we can label the world um and then we just need more and more labels actually make sense of it so because of this somehow uh we see that the supervised learning doesn't by itself generalize well enough it just doesn't work enough it's it's too expensive in terms of having human human people expert label data it doesn't generalize really well to uh you know the diverse setting of The Real World okay so let's say now that we try to rely on ourselves defining things structuring the world and we do almost the opposite direction so we say that as human beings we have goals we have desires that we want to optimize or you know that we do optimize and we hope that if you just impart those goals and desires onto computer it will learn about the world in the process right focus on the end like where you want to end up and the computer have to figure out how to get there so this is a reinforcement learning but I think you know if a good uh analogy for a blank slate computer it's like a blank slate baby I think it's very very hard for us to even understand what it means to be a complete Blank Slate I think a lot of our knowledge is so intuitive we just take it for granted so what does it actually mean for something to be a complete Blank Slate like having no understanding of how the world works at a starting point um so let's say you know you you you know you're this Blank Slate baby and no understanding how the world works and you want you want to optimize certain goals like maybe you want to optimize successor or uh becoming rich right first of there's a huge delay in the feedback right you do something you won't immediately know if it's actually helping you or not you have to wait a long time before you get a signal right so it's very very you know difficult to know what's working and not you need to keep added for a long time before you you you see that it makes a difference right but even if you take something that's perhaps a little bit more immediate like trying to become less hungry it's still some delay in terms of maybe minutes or something but let's say you you try to reduce your hunger right but you have no understanding of how the world works you just randomly pick a concept that you're observing and you start to explore how that concept affects your ability to become less hungry right to become full maybe that's you just start focusing on the moon and how the moon and the characters around the Moon affects your ability to become uh less hungry right you're going to spend so much time uh exploring complete nonsense that has basically no relations to your ability to become less hungry and you're gonna die out of hunger way before you make any progress whatsoever so I think that's kind of that's kind of hard to even understand let's take the the example of a car that's a complete black slate you want you know you want the car to learn how to drive to your home using reinforcement learning I mean again maybe it starts exploring how the you know Moon affects the ability to reach home and it will make no progress but even if it starts focusing on things that are actually relevant but just share coincidence and it starts focing on other drivers or other human beings in traffic right still we cannot afford the car to hit like a million human beings and and crash a million times before it actually reaches home get some signal and starts making some progress right this is too expensive and too risky in real life uh if you take you know imagine putting a real baby in the driver seat of a car I mean how many billions of years if this baby would live forever how many billions of years would it take for this baby to just by coincident reach home I mean it's going to take forever and then when it does that you say like hey good baby awesome work you know here's your signal now do it again I mean this is somehow how reinforcement learn works when there's just a delayed feedback and this A blank slate and this I mean this works better in in chess or something where there's an idolized set of rules and the state space is much smaller but this kind of just blows up in in real life so what we need is a basic model of how the world works that we get from just observing the world because it's the only thing we can really afford right reinforcement Lear as we just said is too dangerous you'll die before you make any understanding of how the world works and so it's too risky and too expensive and too slow and supervised learning is also too expensive because it relies on human experts to label the world and at the end of it it doesn't work because label you cannot label the world okay again so what comes to rescue is this you know breakthrough behind Foundation models called self-supervised learning where you learn from self-supervision by just observing the world and we Define meaning by the company keeps which allows us to learn from just observing okay so quick recap uh we talked about Chaos versus order and and we conclude that actually the world is chaotic we cannot ignore that we have to deal with that uh the brain is the best tool that we have to learn and compress and navigate in a chaotic world but we still have to uh Define how this brain should interact with the world supervised learning didn't work because it's too expensive and doesn't generalize like labeling the world reinforcement learning is too dangerous and too slow so that's why we end up with uh learning from observation and self superv learning okay so let's dive into some specific use cases of this so uh here's a kind of a collection of different uh self supervised learning algorithms and how they learn from data from just observing the data we'll cover all of them in in subsequent lectures uh but what they all rely on is learning from data itself right there's no need for human experts in the loop so it scales extremely well to UNL amount of data and then they use these very you know broad capabilities in a wide range of of different tasks and in the in this lecture we'll just talk quickly about predicting the future and positive pairs okay so this idea of learning by predicting the future based on the past relies on this idea that in order to predict the future like we need to understand the past so let's take a example of language model or learning you know how learning from Text data so this is good because we have an unlimited basically amount of text Data from the internet so we can just download a sequence of text and then we can remove the last word and try to predict the last word based on previous words that is extremely simple to Define and then we can let the huge huge brain of a new network start doing this task and let's now assume that this new network or computer has become really really good at predicting the next word based on previous words what is does that actually imply does it does what does it learn does it learn grammar well in order to generate and predict grammar like you know grammar grammar grammatically correct sentences it has to understand grammar of course I mean does it have to understand the meaning of words I mean if the sentence that we give it is you know the dog is needs to understand what adjectives that describ to dog so also need to understand the the meaning of words does have to understand a difference between a informal social media post or a formal news article well you know if it's right now being fed a a informal Facebook post and he wants to complete that accurately need to understand what kind of language that corresponds to and vice versa right and similarly like if people in the if the way people write changes based on their political beliefs they also start to need to pick up their political beliefs based on how they write things right which only gets kind of scary but a really really powerful model can start picking up those things that are so implicit because it it's optimized to do this this fairly simple task and again if I ask it you know what's the capital of stock on question what's the capital of Sweden question mark right I give that sentence to the uh model and it has to complete it accurately and optimally it has to give the correct answer which is Stockholm so it also becomes very knowledge about about the world in a bigger sense and this is the core you know approach behind chtp and why it works well and why it's so flexible and capabilities right and how such a simple objective can lead to extremely you know broad sense of intelligence and uh again you know similar things uh applies to real life example like frames in real life or in a mve or something right if a model is able to say that okay it sees a human being with a leash and a dog and in the next frame it sees a frisbey if it's able to prod predict that and say well probably the human being will throw the frisbey and the dog will run to to catch it right being able to do that you know prediction to combining these object objects show that this model understands how these objects relate to each other so it's extremely extremely flexible and Powerful okay another approach that's very popular in vision is it's called positive par contrastive learning here we learn uh know here again we can just say we say that we think that objects that appear in the same image are more related than objects that appear in different images so then we can just download a ton of images from online and we can just kind of randomly crop the images and push the crops from the same image close together and far away from crops mother images and if we do this we're going to see like well okay so a human being this on the left hand side right the crop here with the human being a leash a dog is going to push a dog kind of close meaning to a human being with a leash and a dog and a frisbey being pushed together somewh because they're more related well also the really really cool thing here is that it kind of rest on this assumption as well that that you don't even have to appear in the same context to be understood it's like people that have similar friends or similar people so here again like okay the frisbey and the human being in the leash don't appear in the same image but they they appear together with similar OBS like a dog so someone as well it start it will start to capture the relationship ship between these which can be kind of more abstract and and more isol like more far away in some sense but it still captures that in a in a very robust sense okay so let's apply this so we're going to talk from one example in science and another in business okay so let's say we want to apply this new paradigm to dyamics it's a good setting because we have a abundance of DNA base pair like this basically this Tech sequence that we get from people from SE sequencing people's genomes so what we've done and what we could do is that we could use our own human intelligence and look through this data and see like well there seems to be this reoccurring uh sequences like genes for example that rear and we can look at these evolutionary threes how these things are being passed on Etc to build up a structure and start building features around parts of the genome and then use that for for example protein structure prediction like start understand how the genes work or we can just kind of rely on on self super learning and just try to predict the next DNA based parall letter based on previous ones so let's say we train a huge model on a ton of DNA data to just predict the next DNA base pair based on previous ones like what does it learn in this process implicitly so if it's able to do this really really well like again does it learn the meaning of genes and genes yes so genes are just reoccurring sequences so if it's if it's wants to able to like complete this uh the genetic sequence really really well need to understand like identify hey I'm inside of this Gene right now and that's why I need to generate these new letters right does it need to understand as well implicitly if it's a if it's kind of completing the Genome of a dog versus a cat yes if they differ it needs to because need is it going to change how it how it thinks about predicting the next uh uh base pair uh and it turns out there actually creates really really good features that maybe it takes this awesome time for human beings to should understand exactly what they encode but if you then use these features and and you trade another R to predict protein structure right what kind of protein structures DNA will lead to it works really really well okay another example uh in the business case that I was involved in in my startup here um let's say you come to a retail company and they want to understand uh their consumers and maybe do you know have a better assortment and give recommendations Etc so typically what they do is that they uh come in takeing some Consultants to start trying to Define some user profiles right if it's uh young single people or or families with kids and they try to build that up and kind of label people and and consumers and they do a lot of questiones and stuff where they ask customers like hey who are you you know what do you prefer ET to build up some kind of knowledge uh structure um so first off right there's a problem because every every human being is unique so every time you you enforce some kind of user profiles you typically lose a lot of actual understanding and performance of your models because they uh rely on very very coarse uh information and and and structure and also it's actually very very hard to ask people what they want and what and how they work because they don't are not completely self-aware of what makes them make a certain decision and what they would prefer and and it takes also a lot of lot of work like a lot of manual work to map up all these people ask them questions and questionnaires to start to build up this understanding of your consumers so what we can do instead is start to kind of look at the behavior of the customers and let the customers do the work for you by just acting your channels right and and tracking some of their interaction points so uh here for example right we have a on the top somebody buying a wine bottle a cheese and some chocolate and if the mod is kind of good at understanding predicting for example the next step of a consumer it can say like and you know in the bottom we have a soda and a candy it can perhaps start predicted like okay the top person is an adult or something that has that wants to relax on a Friday night and the the bottom is a a a child equivalent version and then if he looks at some behavior of a of a family coming into the store it can kind of recommend both maybe because if it's a family you know it has Som both of these features um and this like the capabilities of these models when you train them enough data becomes extremely sophisticated and it's very very cheap to track data rather than building up all yourself from from Human you know human work um and also right some of what this eventually gets to is a more deep understanding of your consumers like your products and the customers right how to interact and if you're in retail that's basically all of your business it's understanding your products and your customers and making the best combination so then is start building up this understanding it can use it in a in a broader sense across the company for business intelligence Etc so so by building a more deep and general intelligence make it more applicable uh broadly in the company instead of instead of uh approaching every problem in isolation we doesn't scale that well okay so summarize we uh we started with a short uh answer to like what's the core fundamental change in perspective behind Foundation mulative AI right learning from observation we uh start to ask like how's the world structured Chaos versus order top down versus bottom up design versus learning and we said there is a limit to order and design and new networks are a way to compressing and dealing with this chaos but as we Define these different objectives we still end up with Sol super learning and learning from observation as somehow the at least we know now a viable option forward we have to learn from unlabeled data directly and learn from observing rather than interacting because that can be too dangerous and expensive uh and then we also cover two applications uh one in science and one in business okay and I want to leave you with this picture I think um right the meaning is relational and also maybe ask yourself like do you have implicit self- supervised learning algorithms going on in your own head right this is how you learn if I would take this clicker for example and I would just drop it and it would float in there I mean you probably be upset or even you know surprised and upset this is happening like almost subconsciously because you probably have this mental model in your head that's always trying to predict what's going to happen next based on previous action right and by doing that and slightly adjusting When You observe the world you implicit learn about the whole world in the process right you have these algorithms going on in your onhe head okay next lecture we will uh make this much much more concrete we'll go through different algorithms and that should be very very exciting we also have a a website future of a.m mit.edu where you can get all the the updates and uh other good stuff so thank you super much and if you have any questions feel free to [Applause] ask well the inition on the first two mod that you said but in the case like model didn't really get inition behind it you got the intuition of the two first uh prict in the future and uh POS par did you understand the intuition there a little bit but you didn't understand self supervised learning that's exact South Suess learning these are those are two examples of self Suess learning so I mean actually the first time we offered this course it was called Foundation models and self supervised learning so self supervised learning is how you train uh Foundation models and generative AI right foundation model generative AI is the output so those those two examples are self super learning so self super learning is a family of algorithms that give you generative Ai and Foundation models does that make sense so actually I think maybe yeah pointing it out is important because self supervis learning is the approach and how we train these models right and when we train them we call them Foundation models so CHP is trained using self-supervised learning so it get a little bit technical I mean if you had like Yan leun for example he said like oh just call it self superv learning why call it Foundation model generative AI but then it's like oh let's call it Foundation model because that's more sounds better than what it is and there's some definition from Stanford that they use but then the media is like well dtive AI sounds cooler stuff like that so there terms as well that are different based on based on the context but all those those two examples you understand are self supervised learning that's examples of it thank you

MIT_6S087_Foundation_Models_Generative_AI_2024

MIT_6S087_Foundation_Models_Generative_AI_ETHICS.txt

all right let's get started so uh today should be very fun because we're going to talk about ethics and regulations so first I'll provide a kind of a very high level lecture cover a lot and then we're going to have a panel uh where manolis comes back uh and you get a chance to ask a lot of questions we can discuss this uh more in detail right so I mean there's there's little doubt that Foundation models and generative AI becoming uh very impactful as a technology right so then the question is how do we develop this safely and responsibly and what kind of air do we want and what can we affect and also should we regulated and how can we balance Innovation and regulation is what we kind of try to uh cover today so I'm actually going to open up and asking you a question who would you blame if an AI hurt you I think this is this is a key question and so there's a lot of different Alternatives um maybe you want to right let's say tatb does something should you blame the company that developed it open AI or is it perhaps in institutions like government institutions agencies that supposed to regulate things and protect us is it maybe you know the AI itself is an entity or a person is responsible like we're you know we see AI as as an individual with its own aims and and desires is it the data that's been trained on is that the problem or is it maybe you know the best kind of tool that we have is the people behind it right the stakeholders behind this AI um and I think like anthrop anthropomorphizing sorry anthropomorphizing AI is very dangerous because it makes us feel that AI has an agency and responsibility and it can allow stakeholders to hide behind Ai and relieve themselves of responsibilities uh but at the end of the day still right even this type of AI that we're seeing right now that's very powerful there's always people behind the AI That's benefiting from it right and that's the ones that that somehow uh probably should be kept responsible I think that's a very it's a useful perspective because they're they're motivated and they have G something to gain or lose so you can try to understand what AI is doing but looking at the stakeholders behind it so right if you ask CHP itself to gbt forhead to create an of yourself you get this you know personlike entity that media also likes to portray which I think is a dangerous and false representation that allows uh the stakeholders behind them to hide behind the problem of the AI rather than the problem of the stakeholders right that wants to build it and benefit from it okay so what are like let's take TP for example in open eye what are the stakeholders well you know there the people behind the company uh uh the leader the management Etc right we have my picture here then we also know that Microsoft has bought a 50% stake in openai so then suddenly they also kind of Stak holders and have a lot of motivation here uh and then on top of that right suddenly the CEO of openi is pushed out and then he's he returns right so this this a you know when we talk about transparency responsibility in terms of AI models there's a huge lack of transparency in the stakeholders Behind These AI models and I think it's it's kind of a you know great challenge to kind of hypocritical that these people are supposed to build transparent Ai and that's what we care about when the organization is completely you know non-transparent and they can't keep their things together right and that should make us worry this is nothing that's not related to AI at all but the problem already starts there right when you try to establish who's the stakeholders okay so um right so we're going to start off covering a lot of different uh topics BAS basically the first part will be you know jumping between topics of AI and ethics and then we're going to say we're going we're going to say like well these are complicated issues and then we're going to see how government and institutions try to regulate and address this issu and then we're going to talk about potential future directions and I think also I don't want to like pretend there's a mainstream perspective on these issues right these are very nuance and complicated I want to hide behind some some uh you know mainstream or say that these are you know solved solved problems and there's clear answers so just to be completely you know transparent uh this is you know I'm sharing these uh thoughts and perspectives H and really it's just to get you guys to think about these problems also just make you ask more questions really uh so I'm going to cover a lot of different things and jump back and forth not going to cover everything just going to try to make you guys think and and be excited and try to make up your own opinion right I don't want to lect you and tell what's good or bad and moral that's for up for you I guess to decide I want to provide you with certain aspect and problems that we're seeing in AI okay so before we jump in I just want to emphasize certain perspectives so one perspective I think is important is that is AI categorically different from other Technologies or other problem areas that we've been seeing in our history right maybe AI is just part of a type of development or Evolution I've been seeing for a long time maybe AI is just you know part of a Continuum rather than something that's very different and maybe AI is not the problem in this trajectory maybe we went wrong somewhere earlier maybe when we started using electricity maybe that's where we really went off right and did something bad so also like is AI really that different from what we seen before I think it's an important uh perspective and and actually is this not where a real challenge it's not AI but something else okay another important point I think is that ethics is a lot about how things should be how the world should work right philosophy and ethics love to talk about the shoulds I think it's a big difference between how the world should work and how the work how the world actually works right so I think this for me I think it's less utility in building up a Utopia of how we would like the world to work I think it's more interesting talk about how we think the world will work uh and you know realize there's a difference here okay and last well I mean so these are you know real problems that are uh high stake problems right and now there's definitely you know using these Technologies uh to boost their nation's military security and it's becoming an arms race and I think it's also you know that makes the stakes higher and certain things become like a necessity to protect your nation and typically when there's an arms race it's a lot of you know there much less should but it's a lot of competition uh and things move very very quickly um yeah so I think that's also you good to keep in mind that these things are not just affecting ourselves but they're affecting this you know bigger picture uh right everything is is fair in Love and War and I think that's uh historically quite true okay so we're going to start off walking through some different threats that we see from Ai and well there's a lot of them right we have misinformation and manipulation we have deep fakes we have privacy bias and discrimination lack of transparency and accountability centralization job displacement unnatural living conditions and the unintended unknown so let's get started and and of course some of these are quite related okay so misinformation manipulation there's a you know fairly famous example of Cambridge analytica right that uh maybe was more big data than AI but they definitely use the Technologies to do something that they called uh behavioral microt targeting so they're able to go into a nation for example and just look at a lot of Facebook data or data on inhabitants get to know them pretty quickly and you know it turns out that people might be fairly easily influential in influence influenced influenced easily influenced if you have some data on them you can do some targeted you know targeted uh attacks or actions um I mean how effectful this was I mean it's hard to tell but there's certain indications on that quite effectful in actually making one election go one way another right which sums that seems that our democracy seems to be quite fragile to these kind of things but of course right again this is a new threat there a very famous example of you know the late uh 19th century where there was this uh American warship called Maine that blew up uh close to Cuba right and there was a in war Civil War going on in in Cuba and there was before America entered the war a warship B blew up okay and then a lot of like newspapers in America was fairly centralized so there's a lot of vested interest in these newspapers actually for these people that control the newspaper to go and enter the war so they made up this story saying that it was the enemy who blew up this ship right it was the Spanish that blew up the ship but a military report showed this was an accident right it was a mistake that made the ship blow up but the newspaper said it's like no there was the enemy and the general public went with the N narrative and America did enter the war right so I think that maybe then it was more of a you know Monopoly on information in terms of newp controlling it but this a as symmetry of information you were very very vulnerable to before as well uh so maybe it's not really a new threat in that sense um okay impersonation so okay this is something that come came up to my Instagram it's not super good but oops see if I can play I mean you probably maybe recognize the people that it's not super good but it's I mean it's pretty good it's getting cheaper to produce I start seeing this stuff in a lot of places right I was fooled for a little bit um and of course I mean this you can see this can be used for for much more uh bad stuff than this right but I mean at the same time right actors are scared of losing their jobs even writers as well are scared of AI coming and impacting and stealing their their livelihoods so it's it's also a real problem there and of course maybe as well we starting to see this when you start impersonating uh public officials and stuff it can really fool people people in a sense that we're vulnerable too which also very very scary um you could probably also impersonate me uh so here's let's see if you can hear this so here's just a website I uploaded a little bit of my lecture audio and this is what it produced hey so welcome to my lecture in artificial intelligence um your instructor Richard I mean it's pretty good this is so completely a generator right I just write it down and it starts producing this it even gets my Swedish accents you know fairly well which is impressive and then imagine you know coupling this with my Facebook data which says a lot of things about me you know can grab and then have a large language model and then this is basically real time I mean if it calls up my grandma to ask for money it could fool my grandma quite easily right so it's it's uh impressive you see this actually becoming a real problem now because the these abilities uh okay right so in terms of impersonation and scams that right the scale now is completely different because AI at a scale can really get to know you on individual level impersonate you uh and create like a lot of different personas as well online it seems really realistic and it can create a sense of of public opinion or an opinion of the people that you care or the groups you care about you know that just are fake right and we seen this in terms of comments on online Etc that and is becoming a real problem and I think it's kind of you know something new that we haven't seen before um okay and I think you know with this as well we're getting almost a little bit of a informational reset uh where before you know several hundred years ago when we didn't have newspaper even or Internet Etc information was kind of fussy right if somebody came to your village said they heard something you know it was secondhand information you always doubted probably a little bit like hey I don't know if it's true or not I think we start to see the same thing now we almost have too much information it's too noising we don't know what's real or we can trust so kind of reciting information somehow that like whatever we see even with our own eyes we cannot trust right so everything has to be critical we have to like somehow use our own uh judgment more I me that's maybe that's good right but uh yeah I mean back in the day somebody might say like I heard that Biden had an affair right and you would have to like be like that secondhand or third hand information you would have to judge that and be like Oh I'm skeptical but now you might be like well I saw Biden actually kiss a woman but it could have been a fake uh so I think that's that's quite interesting okay so manipulation right and I think AI is really really good to learn H from data like human weaknesses and be able to exploit them and it it can do this at scale by addressing every person tailored right in the whole world basically which is which is quite scary and of course how is able to do this well it's able to do this because also because we share so much of our own privacy and our own data online that really there's so much data on each person here that a smart intelligence can really impersonate you and get to know you and manipulate you on a very targeted level um and uh I think this is you know compared to what we've seen before in terms of monopolies of newspaper one directed you know one directed communication channels this is really a new type of threat that this AI Works to scale and really personalize his Target right and at the level they can get to know us and have almost a complete master of right how this Al algorthm just keep us consuming things and it starts to almost get to know us better than ourselves so it's almost like a drug that is very hard to get rid of when you start consuming it because it's so good to keep us involved you know engaged and manipulate us um okay bias and discrimination right so this is of course extremely uh difficult subject and I think all I want to do here is just you know give some some perspective on why this is very diff difficult and present some challenges right so here's a few uh key features to summarize me okay so I'm Swedish I'm from a small town I'm a non-smoker I'm a history nerd I sit a lot atat a lot of chocolate I stress a lot and I'm a PhD right so one thing that you might want to do when you look at these features if you want to let's say Implement like Insurance pricing right H you might say that okay certain features here are protected you don't want to use these features right you don't want to discriminate discriminate on Bas certain features let's say that you know you want you don't want to discriminate based on Swedish features right let's say that we there's an insurance company that's active in both Denmark right our neighbors and in Sweden and they want to set some pricing but they don't want to discriminate based on where you're from if Sweden or Denmark right so one way we do this you look at all these other features that we that you have and you try to predict Sweden right because like you don't want to you know you don't want to ignore Sweden and then other features correlate very much with Sweden so what you do is that you try to predict my nationality based on other features okay you train an AI M to do that and see like what features are predictive in ter terms of my nationality and then you remove these and say you know great and kind of now we've removed the ability to make decisions based on the features because we're not able to predict uh your nationality based on these features uh of course like you know that's good maybe but also it's quite true that this big difference between Sweden and Denmark is that people in Denmark love to they love to smoke and they love you know they're a little bit less healthy so if there's a health insurance something like why should they be benefiting right somehow and we we BR you know we're getting a higher price even though we're healthier in some sense like this kind of non smoker smoker might correlate very very you know very a lot with your nationality and then somehow they getting a free card by having the same premium insurance right so it might seem like it's quite unfair in that sense um and then right you might s say like well you don't want to discriminate uh based on nationality but also not on your profession education level so you try to predict you know PhD from other features and you remove those too and then more you try to you know the more protective features you're trying to remove Etc the less information the model has to to you know make its decision so it can be quite hard as well and you know there's like nothing left to decide on um so like there is a quite hard in a a very robust or structured way to somehow say What's the difference even between bias and knowledge there needs to be certain bias to able to make decisions you know based on your features and and then of course like yeah what groupings should matter it's quite subjective like what should matter and what should be fair to make decisions based on and what is not fair to make decisions based on and then yeah sometimes when you're you know trying to equalize things you're helping uh someone and hurting somebody else one way you know that we tried in in research to make these things uh more robust or like more mathematical so you can think about fairness for example one way to define fairness in AI is to think about something called a lip shet condition and this means that if you change the input to your AI Model A little bit the change in output should be very little right so you're not able to single out people you are not able to single out people positively or negatively because if the input is similar the output is similar so if they are similar they treat it similarly okay okay um but also like this also kind of show the fairness then is a Continuum it's like a slider because you can make this constant how much it's allowed to change you know up and down and if we to zero then of course you know any change in the input leads to no change in the output and you have a constant function just like treats everybody the same but then then again the AI is kind of useless so this is like very very very difficult to solve and address mathematically into these models in a lot of senses I also feel like these big companies they love to throw around like nice math and make you feel that you you know you're safe we have addressed this with these different approaches that don't really work well at all but it's some kind of impeachement just to make you like oh they're actually addressing these things but they don't really care it's not really working that well so I think I should also be very you know skeptical of some of these approaches uh they might sound good but and the question is how well they're working and how much these big companies actually care about uh solving this or adusting this problem s adequately and also I think maybe right all data is always going to be potentially biased in some way whatever you do for example if you put data online just by the fact that you put it online means that it's biased right there's a difference between things you publish and put online and things you don't so for for example in research is a big difference between things we decide to publish and we don't publish okay so the internet and certain information is extremely skewed to post results ET like of this kind Hallucination is typically just visual thinking because the internet is a very different place than than the real world so maybe the problem is more to diagnose the bias that data has than completely removing it right make it more transparent than trying to solve the problem completely okay transparency and accountability so I mean people say there's a problem with AI is not transparent but I don't know if that's the real problem right so we rely on human decision- making humans are not very transparent we don't know how we how we work ourselves we don't know what how the brain works and what really motivates us so like maybe the transparency probably is not the real problem because humans suffer from from a similar problem you know potentially uh maybe the problem is really that there's a lack of skin in the game for AI a human being we kind of understand a little bit what they're seeking to optimize and and they might be selfish that we can try to understand you know what's involved and what they have to gain or to lose but for AI it's much harder right what are that actually optimizing their own interest so this lack of the SK like lack in terms of skin of the game I think is uh is uh difficult again that's why I think also going back to the stakeholders behind the AI is very important because they do have skin in the game so maybe that's the one we can actually regulate and focus on because the AI maybe shouldn't be seen as an it's own entity also I think another you know aspect this might be problematic is that it's not the lack of transparent but the scale of AI you know human beings are not very transparent but typically their decision making is more localized I mean typically we don't like dictators making decisions for everybody right but an AI is kind of a dictator in Disguise typically it's a model that's used by a lot of people at the same time a single you know person or like a single intelligence is used by everybody at the same time um so I think I mean okay for some small claim stuff or in court even like small things I think I would trust gp4 to do a fairly objective you know compared to like a random human being I think that gp4 might be fairly you know an average decision if I would ask it to like okay we have this dispute help us solve it I would trust gp4 gp4 JP to do pretty good job right but the problem is I think is that if we start using gp4 to you know augment judge or be us in every courtroom is that then suddenly we using the same same AI model everywhere which makes it very very you know fragile because this scale like it's like you know that's like putting the same judge in every courtroom right you're just taking one human being and the same human being every courtroom and that's bad because if he's off or is if he's biased he's going to be systemically biased in all of these instances a human being is typically very inadequate but it's inadequate maybe more in his own ways and that's going to be localized but if you put this everywhere it's going to be biased in the same way you know Sy systemically and I think that's that's really uh dangerous and right and this judge will appear to be personable right and local but it is really one single huge AI brain on an open AI server right and that's that's might be uh quite bad okay Industrial Revolution on drugs so what are we seeing well we're seeing a lot of change happening quickly probably there's going to be a lot of you know change in terms of how we spend our time what kind of jobs we do Etc now that we see AI transforming Society so what are some of the potential consequences right and potentially unintended consequences they're not really relate to AI itself but just the change happens really quickly right so job displacement again is something that's likely to happen we've seen it happen before Industrial Revolution on natural living conditions right now we spend so much time in weird cities and also on our phones and on you know the VB Etc just changes how we interact Etc and maybe people are more lonely Etc there's a lot of things going on here and it's completely changing how we uh used to live before and I think you know maybe also gives us more free time Etc but typically you know a lot of change happening quickly creates a lot of unrest among people people don't take change happen quickly typically historically I believe very well so that can be quite bad and maybe when people get more time and they feel more you know insecure insecure they don't start writing poetry and and paint they might start blaming each other and they start might might start taking out on each other right so what happened when we had the Industrial Revolution well I would say that quite quickly after industrial revolution we had some you know major really terrible world wars and it's kind of unintended right I think it's also something that that's um quite uh possible here as well that we develop these Technologies a lot of change happens quickly and it creates a lot of unrest there an arm race and and it becomes a war or something that's not really you know AI to blame is something the change happen quickly and there's basically unintended consequences um and I think these unintended consequences can be really really impactful because if you're not able to right I mean you're typically unintended because they're unpredictable so you Wen able to predict them predict them so you w not able to address them so they happen had a huge impact and it had a huge impact just because you could not anticipate them right something that happens you're not able to anticipate will typically have a big impact because you're not prepared for it right and this is part of this you know broader theme that we're going to talk a little bit about in terms of un unintended consequences and things have been unpredictable and this from uh a think called Talib in terms of black ons right events that we don't we didn't see coming and see coming and they're extremely extremely important we need to somehow be able to function as a society and as a system without being able to predict everything's going to happen right we have to still be able to survive and prosper when things are unpredictable um okay so a big problem is that when you have this centralization that we're seeing in technology especially in AI right and couple with things being unpredictable what it's going to do what's going to happen it creates a very very fragile system I think intuitively we can understand like well if we have a single AI to rule the mo that everybody uses it creates a lot of you know if that AI goes Rog or goes wrong then we're all screwed right so centralization in a way where something's too big to fail is very very fragile because most things do fail because we cannot predict all potential outcomes address it and make it completely robust most things fail and when they do right we somehow have to survive so I think we can all agree that this kind of seems like a bad and bad idea and maybe we should be inspired more by nature for example where things are allowed to you know fail or things are very unpredictable in nature but it's it's still able to thrive and prosper right so it's all it's it's almost uses the unpredictability and the change to its advantage and something I think there something that we also need to uh be able to do right it's very very decentralized nature nature okay so this comes right from this Nim talb and it's this concept of fragile robust antifragile so fragile is something that basically is everything from loose from something being unpredictable or something unpredictable happen right just everything to lose is very very fragile and it's fragile to change right just wants things to be constant so if you have a glass something just wants everything to remain the same you if you change things something unpredictable happens is very fragile and everything to lose from from something changing and then actually the opposite of fragile most people say like well it's robust right but it's not really right so uh freder is something that has so much to lose from change but antifragile is the opposite of fragile which means actually have something to gain from change you can use it your advantage and then in the middle we have this idea of robustness mean that we try to create something like you know steel metal that is it's robust to change mean that like it can handle it and it won't change right what we're going to see now like but it's really really hard to think about to build truly robust systems because building robust systems that we going to design it and build it and say well we predict what's going to happen in the future we're going to address this in our design and how we build this uh system or whatever so it can withstand these things right and not suffer too much but it's very very difficult because it rest on this assumption we can build something or we can predict the future and and then build something based on those predictions right Fukushima ET that was really really robust and people made predictions of what's going to happen in terms of earthquakes so they tried to build a really robust system could withheld or survive earthquakes right but they just were off in the predictions a much bigger earthquake than ever happened before happened and they were completely screwed because robustness is very very hard right so do you think then like and too big to fail bank that you try to make really robust is also very hard right do you think it's is it easier to build a bank and think about all the potential things that can happen so it can so it never fails you know do you think it's easy to build a bank that will never fail or to build a system where banks are allowed to fail and the system survives right and I think it's it's just impossible to build a bank that's too big to fail because everything fails at some point so you need to have a system actually that allows for things to fail for banks to fail right this is a really good kind of allegory terms of how we should think about AI think because we don't want AI That's too big to fail it's a very very fragile system right so again this relies on the fact that we cannot predict the future we actually if you look about if you look at people trying to break the future our own historical record of trying to break the future we suck at it it's just basically random guesses and it doesn't work so uh we shouldn't we shouldn't you know rely on our own predictions we should be able to be fine even if we're wrong so uh yeah we have to you know say like something is going to happen we don't know what but when it does we're able to handle it you know quickly and well and one of the best ways right to uh actually then if you cannot predict and anticipate what might happen so we build sa like you know that's not how we can judge what's safe and and actually works long term uh the best thing actually we can do then if we don't want to rely on our own assumptions and predictions is to use a test of time right so we just put something we we try it out in small scale where it can fail and we see how well it works we actually use this a test of time to see if something works or not which is typically the most robust way of seeing some if something safe or not H and I think here again right there's a real big difference between how nature built intelligence over a long period of time and like we as human beings are very very anti fragile right and robust because we've been developed over roughly 3.5 billion years so there a lot of test of time here has happened over in Nature's version of intelligence but for us for example right I mean order of magnitude the latest AI has developed over the last 3.5 years so it's like a factor of a billion off here which like we're moving so quickly not testing anything we're just deploying it and suddenly everybody has their own hand and it's been no test of time whatsoever so we basically have no idea what's going to happen if we're safe or not and that's I think it's a you know very very scary and and and dangerous path right so this is kind of a nice image of what we want to accomplish for an antifragile system where you have multiple you know deployments and systems in place uh everybody you know every every each one of them subject to a test of time and each one of them allowed to fail and the system prosper and is able to function you know even when those failure happens right I think we have to really learn from time and change right that's what our options are not to be a victim of this change and unpredictability okay so those were a lot of different aspects and now we're going to see okay well these are clearly difficult so how did our institutions and our governments solve this and protect us right what's the regulations that people now you know put in place to try to address these problems okay so uh I'm just going to give a little bit my own opinions quickly before I jump in and cover the the original sources so it's definitely like you know I went through the European and the American and the Chinese one right and the Chinese one I use translations from Stanford and carneg insute I have references later but this right what I don't like is this tons of forecasting again like predictions did look like hey we predict what's going to happen next 10 20 years I'm like how the hell do you know what's going to happen AI the next 20 years like I'm a researcher I have no idea so I think that's like that that's I just think is fragile and bad we rely so much on people's prediction this you know in this in these different regulations I just think it's very very hard okay there's a lot of bureaucracy meaning that like okay you know all this basically all these acts are like you should report everything and anything to us as as institutions and we'll keep record everything it's like yeah sure but what are you going to use that for us that useful and it sounds like create a huge bureaucracy which maybe we be counterproductive there's a lot of fussy words I think to make you feel good it sounds safe and and like hey we care about your kids in your future but when you read about it like well what does it really mean it feels like it's it's very very uh you know there's nothing behind these words and just promises that we're going to take these things into account in the future create a lot of auditing and Benchmark around these things we want to accomplish but there's no actual action plans or actual you know well defined procedures here okay uh another thing I think is bad is that there's a lot of talk in all this regulation about preventing AI basically from thinking bad thoughts cover this I think it's also very hard to say like Hey we're going to we're GNA somehow be able to restrict how the AI thinks how it thinks right if it thinks in immoral or moral ways I think it's very very difficult but they seem like they seem like that's something they believe is very doable uh red team testing so America especially has somehow found this term red team testing it sounds cool uh it's about kind of ethic hackers attacking the AI and trying to find vulnerability vulnerabilities h i mean it's yes it's useful but it just seems like they think this is going to solve our problems they keep saying like hey they're going to use red team testing which is I don't think it's going to solve our problems okay good part well it's a consens that we should disclose AI gener content sure like great let's Watermark AI generate content something and make that into law I think it's great it's very actionable at least right so we can do that Al I think is interesting as well because they're starting to be forced to provide more definitions of like what is AI what our foundation mulative Ai and also that's interesting I think useful to push for more exactness in terms of these definitions need for talent yeah there's a huge push for Need for talent right so they're gonna help people out that want to do Ai and which I think is good because we need more talent and people addressing these things I'm European uh I think the European one is quite tired and bureaucratic and wants to have more regulation you know other ones the US one is kind of in between terms of Regulation they still care about control but they're like well this is important is kind of an armor R going on we don't want to lose so we're going to be a little bit looser regulation okay China seems to be like okay we're going to be the least regul regulation uh it's a little bit more unclear sometimes uh but they seem to be very kind of aware that there's an armed race going on and they see like they can have a chance here to became uh the Supreme kind of AI uh Nation okay so let's jump in and compare so we have Europe and we have America and China right so in terms of the aims that they want to accomplish so I think it's interesting in Europe we don't like to say like hey European values or something so we use nice fancy words instead so we want to make AI systems used in Europe to be safe transparent traceable non-discriminatory and environmentally friendly and overseen by people are an automation right it sounds great what does it really mean I don't know but that's how they put it up right starting America is a little bit more like okay great we want to have civil rights Democratic Values but also foundational American principles right what does it mean I don't know but uh for Americans it probably sounds good at least and China as well right to go the route of kind of uh you know soci core values right this that's that's uh uh at least has some kind of name right like foundational American principles and versus socialist core values those care a lot about uh having indigenous Innovations like okay we're going to push for doing great things in China and they want to upset the economic order or social order right so these are some of the different aims and that's what they say at least right maybe they they're quite similar still um okay so what kind of definitions do you use to define this different type of AI so Europe seems to be focusing more on the impact of the different models right and also name call so they you know they they say chtp and gp4 right so high impact general purpose am also might poose systemic risk such as the more advanced am gp4 so they name calling OPI open AI which is interesting American company and again like general purpose AI mod in my post systemic risk like okay what does that mean well this actually turns out this very very difficult to you know very precise or even Define that precisely or closely to something precise it's very very hard so becomes quite arbitrary but actually poses systemic risk that's hard uh okay so uh the American ones are a little bit similar but they try to go to the next level right so the focus they say like well could be easily modified to exhibit high levels of performance Tas that post a serious risk of security National economy Security National Public Health or safety or any combination of those matters and it said it also can can can permit the evasion of human control or oversight through means of deception or obfuscation this I think is very interesting because I think it's almost a pretty good definition what AI is like AI might be what we get when we give up our ability to understand and control right it's like a human being it's it's very powerful and creative because we cannot know exactly what's going on it's not very compressible okay but then you also go and talk about numbers which I like so they say that you know this dangerous AI has computing power greater than 10 to 26 integer of point to point operations at least it's like that's exact it's kind of nice I feel like because then we can using this and we start iterating is it an adequate adequate definition we don't know but we'll try and it said that if has a training on something train on a data center networking with data center networking of over 100 gigabytes per second and having a theoretical Max maximum compute capacity of 10 to 20 integer floating points operation per second maybe doesn't mean a lot to you guys but it does mean something to Like Us in computer science this is something that we can at least understand and can make exact I like that they're forced to try to make it more exact okay so the summ of the Chinese one is that they've done a little bit of a different path right so Europe focused more on impact America focused a little bit on the compute and and that needed to create the AI and China focused a little more on a specific algorithm right they have an algorithmic registry where you where you register your algorithms and they try to make decisions based something is fine or not based on a specific algorithm you know a little bit independent of how big it is right or how it's used the focus on the algorithm so a large language model you know train using self super staring with a transform model might be an algorithm that they might register and then they would regulate that in the same way if it's big or small or however it's used okay um intervention so what how do you want to solve these things um okay so the European uh Parliament wants to disclose that content was generated by AI That's True for all of them I think it's good design the mod to prent it from generating illegal content it's like yeah but how can you do that how can you design like a a trillion parameter new network TR on internet not to generate legal content I think it's just very very hard and it seems like yeah publishing summaries of copyright data used for training okay great sure let's do it and again undergo thorough evaluations of and any serious Insight would have reported to the European commission it's like report report and some bureaucracy uh okay America they create initi initiative to create guidance and Benchmark for evalu in auditing AI capabilities it's like yeah but what is it mean how is it helpful we don't know again red AI red team testing shows up multiple times H they seem like it's a great path forward I don't know how it's going to solve everything but maybe H and companies again that you know use this Foundation mod Etc should on Ono ongoing basis ongoing basis report and report and Report uh China um is a little bit kind of I think that like nobody's safe right because whoever is using the AI if you're creating the AI if you're using it if you're using somebody's else AI if you have an API whatever you do if you're interacting with in AI you can be responsible if anything happens um yeah uh and then they seem to be a little bit more vertical and iterative meaning they go look at they're a little bit more flexible and and look specific cases and try to allow for a little bit more quicker Innovation less sweeping okay so what are some uh uh feedback on these regulations well one uh pretty big uh uh criticism of the eu1 is that they take a lot of pride in creating the kind of first super huge sweeping regulation of AI but they basically create no AI H so I think also as a European is quite it's a quite uh strange situation where you try to regulate big American AI companies H right that's what happened Italy Italy banned chtp open a is like well then we'll just not be active in your market and then Italy is like well we want your AI still so I think it's quite uh difficult for example the EU to regulate AI without being the ones actually producing AI so I think we talk about I think you want if you want to be able to regulate you should also be able to produce it and uh I think that's that's uh the more uh trable uh procedure forward okay another thing that we've been seeing is that uh open source seems to be fairly discouraged both in the European and American ones right American ones especially seems to say like well openi is bad it's dangerous you know we shouldn't allow people to have access to these things H the EU act as well but it's this potential you know there's some rumors the EU will allow a little bit more for open source models right again the only the thing that Europe does have is the biggest open source AI like mraw and hugging face so like maybe you know they're pushing and they are pushing for changing the eu's regulation America has the all the biggest companies in Ai and they're all closed Source basically right so no surprise open source is not allowed um okay then we also have this thing about you know not thinking bad thoughts right so where I think it's interesting like where do we draw a difference right you thinking something that's bad or immoral you drawing something on a piece of paper that's bad or immoral right all these things are illegal illegal right or you know an AI model thinking or generating something it's illegal based on your prompt or imore based on your prompt right I think it's like where do you draw the line is it fine to prompt an a model as a human being to generate things that might be immoral right so it's it's it's okay for human being to think it to draw it but an AI it's not so clearly it's a distinction because all the regulations are basically saying like yeah you can think bad things you can draw bad things but an AI can't so that's where they draw a line I think it's quite interesting and also I think it's very very difficult right I think uh it creates deception in these models because maybe they're not going to Output literally bad things for us but they might think them and use them in their competitions but just keeping them away from from us so it creates dishonest thinkers if you will and also I think that maybe actually the ability to think bad thoughts and consider all the different cases might necessary to be able to be a good person and do good things maybe to be able to think bad thoughts to be able to do good things right maybe it's necessary uh right so if you remove that maybe it's just really really bad okay again need for talents especially now is you're in your America as well they want you to stick around in AI so take advantage of that um again so for the Chinese one I used the carneg UL for natural peace and stand for University for translation Etc okay so again right who knows what AI is going to happen like what's going to happen AI for the next 20 years do the Parliament and our like politicians know this well you know I don't think so you can actually look at the track record right so if you if you follow these and you know kind of backtrack how they've done historically there's no correlation between their their estimates and what actually happens so if you just look at this a regulation there's already problems so the big EU regulation took years to finish our thousands of pages okay and now like okay so they did a lot of work on it on 2020 what do you say 2021 did a lot of work on this regulation this is pre CHP prenative Ai and not like well actually we just released the AI act but it's missing a lot of things because we made made the most progress in 2021 before all this AI so now we have to like restart because we were off even before we were you know we were off in our expectation in before we were done and same you know the Chinese one were unlucky because they released their deep synthis regulation just 5 days before CHP was released so they kind of completely missed the the what was going on right and was like oh five days Pro relevant and then something happened um okay and I think again a lot of this wording in these regulations are like we're going to protect our citizens it's for you it's for the you know the little people the Common People okay I think that's might be very deceptive and dangerous because a lot of times the big companies want to be regulated right we see open eye in America people oh please regulate regulate us and historically there been a lot of instances where regulators and big big government goes hand in hand with big big business right because the big players a lot of times big players established players benefit from regulation because it keeps the keep the status quo you know and keep small players out so if you look at like how the military industrial complex was created in America they both both wanted you know they both wanted regulation going into the war like well we should only like go you know we should only allow big companies to produce War material right and then when they went When The War stopped stopped and they wanted to go from a war eony into regular economy also the big uh companies want to have regulation because like well we don't want to lose out now that we're switching our economy to small Players let's regulate everything so I think also like regulation might not be in your best interest uh maybe actually it's placing the hands of big business right okay can we have it all again I think that someh as well we need to be fine with certain tradeoffs I think AI is a lot of times when we get when we give up our you know desire and ability to explain understand and control things so we can we cannot have the cookie needed to so I have to give up something I believe and we have to decide what we're willing to give up um right and and we we do give up a lot of things when we use other human beings to control things and take decisions for us and we need similar kind of checks in place for AI I think if we're going to be able to trust them because we're not going to able to understand exactly how they work and make them completely predictable Etc I think what we need to again to do is that we need to say like yeah things that we don't know and bad things and good things are going to happen let's not try to predict those things and rely on that let's try to make systems that are fine when it happens and be reactive and quick to to react when it does happen I think that's that's the viable path forward and also I I'm a European I've been in Europe a lot and I'm a little bit I mean a little bit tired of Europeans and the like the bureaucrat in Europe just want to regulate so much and make all these long plans when they're not when we're not really building Ai and I think really you know for Europe for example I think needs to have its own AI company so they don't you just control and regulate America ones and plead to these big American companies that should need to build their own AI companies to be able to be secure and be able to dictate how things are going to be used and also think typically actually the creators and the innovators in a technology are more important to set the standards how things are done Than The Regulators right so we need good people and moreal people to build this stuff because they decide what's going to be used or not so show lead the way right I think that's key okay we're almost done uh right so this is also like these are very difficult problem like ethics and morality right philosophers and human beings have been thinking about these issues forever and I don't know if we made a ton of progress the suddenly is well like now we're like okay we're going to use AI in all these different places and make decisions for us we just have to solve a we just have to solve ethics and morality in a few months and it will be fine it's like well I don't think we can solve ethics and ethics and morality in just a few months like these are maybe unsolvable questions and we have to somehow you know be fine with things being unclear or uncertain right maybe that's how the world is that's maybe how value things you know value system morality subjective and everybody thinks different things and maybe actually we should be scared of Institutions or companies saying like no we have the answer for you this is our you know 10-step approach to moral Ai and it solves all the issues like well it gives you a full sense of security and actually I think that the world is much more complicated and nuanced so so I think we should be you know not ask and expect institutions and and AI companies to tell us what's good or bad right we have to like judge for ourself and understand that there's no clear answers so I take more responsibility as well as Citizens actually read up and make your own make your own decisions on detailed questions because there's no simple rules and I think also it's it's very interesting because now right we have to take all these different value systems that we have and we need to code them into computer right so we have to translate our different values that we might might want to incorporate into actual code and math and and put it in it's very interesting because also shows how little we actually know because we can't Define this exactly and a lot of times when we try to Define you know fairness in one way and then fairness in other way they actually contradict each other so like they're not consistent with each other these are like very very difficult things to solve right there's a lot of things that we haven't solved as a society in terms of ethics and morality that AI is like you know how are we supposed to solve AI when we haven't done able to solve the problems that we already have we we in in in with companies in a financial world we have tons of companies that are too big to fail right and the financial system itself is EXT extremely non-transparent right who knows what's going on uh and then like equal equal access to Opportunities right and it com equal access to compute like yeah we haven't solved that it's super unfair and it's going on and nobody's doing anything about it and so I think you know we're not doing a great job also again the arms race right have you ever seen an AI play a game against a human being like a real great AI like it completely destroys it like imagine that as well now you have like an AI who can who can fly an airplane doesn't feel the gForce and his training can just be super fast I mean it just can completely out compete human beings maybe just a few airplanes can have a complete dominance of airspace and in Modern Warfare like Warfare if you have control of the airspace you basically control the war so like of course this too juicy for the American Military to give up for example so they're already having AI Pilots driving these planes so like ethics whatever who cares this is just too good like you know they need this so again like it doesn't really matter as much maybe what we want to happen this is happening okay so what where are we heading well I think we're heading towards a very you know centralized system one single AI model to rule them all they will rely on and if it fails right one single talk if it fails we're all screwed right I think that's how we build system and that's what we're heading towards right what do we want to do well we want actually to be antifragile we want to be you know inspired by nature and allow things to be able to fail and we're fine and we're going to be thriving in that environment and we want to allow the test of time to test our systems to make sure that they're safe right but I don't think we're doing that in fact I think that every hundred years we're doubling our risk for human extinction Okay so ending on a very pessimistic note because I also know that we a visitor here today that's going to be much more optimistic and we can all now debate this so I think this is a great place to got I can take some quick questions but we're going to set up the panel in the meantime thank [Applause] you okay oh yeah yeah I I think I think that I think the closed Source I mean I think closed source is going to wing in win in the shortterm uh perspective because it just like it just makes more sense shortterm and we love to optimize shortterm as Humanity right we just don't really think in terms of long term I think open open source is what we want but the market is going to decide with closed Source do that makes sense yeah I I would I'm totally down for open source but look at open AI open AI was supposed to be open Source nonprofit and now it's like the most closed Source you know system we have or company we have in AI so I think that hiring Sam and rehiring him was the best thing that happened to closing yeah maybe all right any other questions before we go into the panel

MIT_6S087_Foundation_Models_Generative_AI_2024

MIT_6S087_Foundation_Models_Generative_AI_IMAGE_GENERATION.txt

okay welcome to the fourth lecture uh on the course called fation Model intive AI uh today we will do a very brief uh managing of data uh data is one of the key components in this new type of AI and really deserves its own course but we're going to talk a little bit about it and then we're going to cover stable diffusion which is a uh text to image uh generation ai dtive ai um okay so what do we have left on this course so next week we are going to have a lecture on emerging Foundation models and and their applications so this is foundation models in the wild in the in the market and in commercial settings especially we'll have two uh guest speakers as well uh so manoles will talk about AI in genomics and Applied Biology and art time will talk about autonomous agents um and then the last lecture will be on AI ethics and Regulation and then we'll have a panel at the end of it then discuss uh the ethical aspects of AI okay so to summarize a little bit right first lecture was an introduction a very quick intuitive answer to what is uh Foundation models and generative AI we went on a little bit of a philosophical digression and and the history of AI the second lecture went through all the different algorithms uh from a high level perspective and then we went into depth the last lecture on chpt and right so what do we do in in uh what's the key behind Foundation models it's be able to learn from observations you don't need human beings in a loop and you can scale up as as much as you want and that you get from this is a very kind of contextual relational understanding of meaning that mean is defined by the company it keeps it's self referential right so dog is something that's walk by owner with a leash it's something that has anistic relationship with cats it's something with that chases F chases fris PES when those are thrown this is how we understand what a dog is right not actually your parent labeling it or really you optimizing certain goals like reinforcement learning it's you observing dogs in different context and correlating dogs with other Concepts that's how you understand what a dog is that's your know that's the main trick uh again uh uh something that I think is you know true when it comes to AI is that you know there a very most understanding is intuitive and is relies on all the different relational uh edges so you don't never really fully understand something you can always become better and it's a lot about just familiar familiarizing yourself so if you don't understand everything in lecture that's fine just try to get some intuition and familiarize familiarize yourself with it and then kind of keep going okay so uh data why is it so important well I think a kind of uh complimentary perspective on all the breakthroughs in AI that we've been seeing is thinking about in terms of data and actually how the new AI just looks at data the old type of data but looks at data in a new type of way so it becomes more powerful and can use more of it and that's basically a very big part of what's happening right now and the data is is really really key um and in some some ways I think data is the you know if you want to apply AI in actual settings you might care about looking at the data understanding data is going to be key for you and so that's a very interdependent concept like Ai and data go goes hand to hand it's very very hard to develop better models if you don't understand data and vice versa so if you start applying AI to your own settings understanding how the data uh you know what data you have and also how AI how AI leverag data and what kind of data he wants is going to be very very important for you and if we take this picture we had before about kind of you look at the this new AI development as some kind of Iceberg right the tip of the iceberg are uh chat GPT and stable diffusion for example right hype stuff that that uh people are talking about and then course below that it's about understanding soft Su press learning the training methodology to get the these AIS right foundation models and generative Ai and then a really big you know significant chunk below that the people don't talk about as much is the data right that's what feeds this whole Revolution and right and I think so let's look at open AI for example the build chipt right maybe it's like 10 to you know 10 engineers working for a year or six months actually developing the chbt version that we're using right now right that's not a lot I mean that's why you have a lot of startups now that try to replicate CHP chbt just for a lot of money in computer train on on data with a small team and they can reproduce it so that's like in itself I mean it's of course impressive but it kind of pales in comparison to how the internet was created right so they they're able to leverage all of internet download it and and train on it but the inter was you know we spent 20 years you know billions of people putting a lot of data online that that's a huge effort that you Cann replicate right nobody can replicate that uh process of creating the internet from scratch no company can it would be too too expensive and so somehow the internet makes this all this available and is basically the greatest data collection effort in human history and that's I think is even more vital for CHP than any of the technology right right so data is really really key and again um I mean if I had to choose between and for you as well if you had to choose between having chtp or having the internet you would rather have the data and the internet because you can retrain CHP you can create better versions so the data is really key and having access to it as well it's also going to be problematic now with copyright Etc where people want to say like well stack Overflow but we have this data now you you kind of out competing us by using our data and people are going to Value data more and more now as well and it becomes very interesting how it's going to affect the development of AI because the internet has been so easy to use and and people haven't really cared and again when you work for a company whatever you do and you have your own problems start looking at the data the data has all the secrets so it's a it's a very very important to think about and not Overlook um all right so the small piece of philosophizing in this class will be about data and so I mean equally that that for chtp the interesting thing is not the technology right it's not the brain of CHP but it's the internet maybe it's a similar perspective of also human beings like maybe we're not that impressive as intelligence but actually the data that we've created and the whole like get biosphere has created that's what really matters so maybe we're just more data creators for the gene for example if you know about this theory about the selfish Gene trying to reproduce itself like maybe we just collecting data for for another purpose basically so right let's say that an alien came to uh Earth and discover earth and we be like oh they probably want to kidnap us and look at our brains and dissect us understand us maybe they'll be like well no we don't care about you guys we want your data we just want like all the history of the earth and what's going on here collect that and then can they can derive their own human beings improved versions so maybe it's like the data somehow is is really really key and we try to focus mean sometimes we focus on the AI itself but uh the data behind this these creations are extremely important all right um okay so conditional image generation text to image generation that's what we're going to talk about today and so two of the first most popular models here was called do and and uh stable diffusion so this is a year ago and you can see the two different versions compared uh on the left hand side you have do and on the right hand side you have stable diffusion and this says a Hightech solar Punky toop on rainforest so it does a good job uh looks realistic H and it's it adheres to the prompt and and it has some kind of imagination as well to be able to create this and we can just check quickly how it's progressed so this is dpt4 now asking the same thing I would say it looks uh even better um so that's good that's been some progress h of course like I mean almost all of the images that's in this lecture slides are from these text to image models and this is how I created the this strange image so prompting can be quite hard I think prompting has become easier and easier but you can also sell prompts online to get you know to for people can get exactly the image that they want so that's interesting I think that people are very hyped about prompt Engineering in the beginning but now it seems to be you know a little bit less hype and it's be as the model become better it's also easier to create the prompts and the images that you want and again this also a year ago where where Facebook I think or Google released this imag uh so they went from takes to videos which was something that we seen improved subst finally and and hopefully this year we'll get something that works extremely well so we can start creating videos from scratch and of course the underline approach to technology is very very similar but you have to scale up now and and think about frames right which is just a collection of images okay so to be able to successfully create images and have a text to image conditional generation there's a few things we want to accomplish so we want to have a High real image quality we want our imag to look real realistic uh right that's important we also want a high correspondence between language and the image that's being created so if we write something we want something that looks like we what we're writing about right we don't want to write something and then get get something completely different it's going to be a strong correspondence and also we want to somehow capture a variety of images around a concept so let's say we want to generate a dog we want to be able to generate all possible dog somehow that's how that's that's going to show that the AI understands what a dog is is more you know what a dog is in a more nuon sense because it can create all the different versions and it's going to be useful as well because you want a sample and it shows some kind of performance uh uh criteria and we want it to be fast to train and to use so how do we accomplish this well we're going to use engineering and smart tricks to accomplish everything besides High language and imil correspondence which we'll have to rely again on data so it's going to be you can say it's label data so it's correspondence between images and text h you can get it up from you can get that from different scripts and scraping online but you need that correspondences to get a really good performance and you can reduce the need for data but you going to you need to have that data in how you build these models so far okay so let's just talk about how we can go about this how can we generate learn to generate images and maybe we can first talk about uh drawing faces as a starting point uh so let's say we want to learn uh to draw faces want to train our AI model to learn to draw faces um and we we arrive at something that's able to generate the same phase over and over again right do we trust this model somehow does a good job and it's able to understand what a you know entails a face or what a what's included in a facee probably not because just you know it collapsed a single instance maybe just memorizing something so we want to have some variety right also for it to be more useful we want to be able to say like generate 10 faces I can use in my slides right and you want different faces you don't want the same phase it's less utility is decreased and so you want to be able to generate different things so the first thing you notice when you think about this that okay well a these deep learning models are still still deterministic functions so if you give it the same input you get the same output so there's no Randomness here for the new network to leverage to create different samples instances you know if you give it the same input it it's going to produce the same output and Randomness is very very hard for new network to kind of create by itself so you need to feed that Randomness as a input to it like it's as part of its environment let's say um and you know maybe it's same for ourselves as well that if you would be able to very very carefully recreate exactly the same environment you would also act in very similar ways if you find yourself in exactly same environment and maybe that when we find ourselves in new environments actually that's conducive to us being creative so maybe we can learn something from this as well but uh it's very important for networks at least that we give them some Randomness that they can leverage to explore and generate uh different versions okay so we're going to as part of the input in new network we're going to feed some Randomness right so it's going to every time it's run it has some different type of input so you can leverage that to say like hey I'm going to now create a face that looks like this or like a face looks like that you can leverage this Randomness to create a distribution of faces so now each time when it gets a slightly different input it can generate a different type of uh pH and now I think we feel like okay now it's we believe more than I start to understand what drawing a phase means and it's actually pretty good at it because it's able to do a lot of different faces and it's also becoming more useful to us um and also what's important is that we want to have Randomness in a way that's very easy for us to sample from a computer right so we can create this Randomness very cheaply from a computer fit it in and then run the model right so we don't have to create Randomness but it's created by the computer itself and computers are very good at creating this pseudo random uh input that we can use okay so um let's say we want to give this uh random uh input to the model and you want to generate some facei how can we go about doing this well uh we could try to just draw the face in a single goal right just from having the randoms and removing all Randomness completely at one single goal for the model get one try or we can say you can remove some Randomness iteratively and it's going to be very very useful because it's going to be somehow uh easier for the model to do right it doesn't have to be do everything in a single step it can remove some Randomness and then basically feed it you know run run itself on its own output and try to remove some more Randomness and I think you know as well as human beings we probably when we learn to draw we draw things we don't just draw everything in a single goal we kind of sketch it out and kind of itally improve on our drawing and we're going to try to uh allow for the same uh abilities for this new network um and okay so why is this useful as well well it's useful because now also when you think about you know different levels of Randomness and you can remove uh only part of it right then most have some kind of different levels of diff difficulties so maybe if you want to go from uh a complete random input some noise to just removing some noise and putting some features there that's kind of maybe that's harder right because it requires more of imagination to get going but at the very end here when we just have to finish the cheek maybe that's easier so suddenly as well we're going to have different kind of levels of Randomness in the input and we have different level of difficulties so this is both useful like both useful for the model when it's being used use because it can iteratively try to improve on it random like improve on itself right you just have to generate anything everything in one single goal and can generate a little bit and then kind of add to that you know it it runs itself on its own output all right but also if it's it's if it sees different different levels of Randomness during training so it has to go from left to right at these different levels it's also good because it can start making some progress at a certain level of difficulty you know and then it can start improving and you know it's learned something there and then when it's learned something there reach the next level it can make more use of the more difficult examples so I know you guys rememberers but when we talked about uh generative adversarial networks it was very difficult because then you have a h kind of a Critic and a artist and they have to be very synced so the critic needs to be able to give feedback on the level at which the artist is right now right it skill level but if one gets ahead of of the other one it breaks down that's actually what happens during training but here somehow we give a you know different levels of difficulties during training if you can create these examples so the model is able to do make use of some examples and start improving and then we can make make use of all examples um okay so in order to be able to create a a VAR like variet variet sample like samples of different outputs right H we need to be able to sample noise and computers are good at creating certain type of noise we're going to use that uh creating everything in one single goal is is very difficult so we're going to allow the computer to iteratively improve on itself so it's going to be able to start from some noise add some part of an image you know but not remove all noise in a single go and then keep going iteratively and and if you want to do that right somehow the noise needs to be part of both the input and output so if it doesn't have to remove all the noise it can still be part of the output that it generates and it can run itself on its own output and okay uh and handling different noise levels are good both for training because it gets training example at different difficulty levels and it's also good when we actually use it because it can iteratively improve on itself again we like uh in Gans there's the artist only have a single attempt to create a image and then a Critic criticizes it U and here the artist and a Critic has to be on a similar level because the artist is supposed to fool the critic to feel like this is a good creation of a face for example right so it's gonna the the artist creates a image of a face uh the critic gets a real face from a distribution of images and then it also looks at this fake face that artist created and it learns to critique and say it's supposed to learn to say like well which one is the real image which one is the fake and the artist is supposed to fool the the critic so the critics the critic feels or believes that the artist is actually creating uh relooking faces but it's very very difficult because the feedback needs like the the artist and cre needs to be on a similar skill level for them to kind of co-develop and colearn together okay um so if we actually look at these uh you know what drawing professionals do when they draw something is that they start with some outline and then they add details so it seems to be a a good uh idea also from how human beings do this uh uh professionally and that's what we're going to try to uh imitate and also as you can see here they don't go from having some random noise and then removing it bottom to bot like top to bottom they start from some outline and they make add more details right not just removing the the top of the face middle face to low face they go from some kind of outline to details and it's also what we want to do and and I think this is why this is also very very powerful is because when you want to learn something it's much easier to point in the right right direction and make some progress then find the exact location immediately so right if somebody asks you where uh usbekistan is right it's probably you wouldn't know the exact coordinates or location but you can point in right direction so maybe you point if you're America you point towards Europe and you make some progress and then you can keep going kind of but it's easy to learn that way H and making some progress that way when you have to get kind of directional feedback okay so how we're going to try to recreate this is that we're going to have this type of uh uh setting where uh we start with some um noisy image like a complete noise gy noise that the computer can generate and then the the model will start adding some features right adding some beginnings of an image so basically remove some noise and start going towards some type of some type of image in this case a face and then when it's done this one step it can run itself on its uh output and keep going and then it can make even more improvements and even more decisions around how this face will look like right it starts somehow from Pure Noise and adding some outlines of the face and then it adds more and more detail at at subsequent steps and then we can run this until you know we're pretty happy with output and deliver it and this is called you know again one single step of this is called the noise so you remove noise and add details of an image and if you do the noising S step it's called diffusion so that's where stable diffusion comes in it does multiple step of den noising okay and we also of course want to uh add some uh abilities to go from text to images so we have this uh correspondences images with text U descriptions we also feed this a condition the model on this so in its uh creation process it can guide Generations by having access to this uh text so it can kind of collapse into a space that's uh uh useful based on this prompt and um we don't need to start trying to understand language from scratch we can leverage a pre-trained language model to get some good starting point of understanding language right that's helpful but still we need this correspond to actually train the model okay so that's like a very very high level of uh how the diffusion stable diffusion model Works we're going to go a little bit more detail and talk about how you actually create the training data around this how you train the model and then how it's being used and then we'll end the lecture talking about some more integrated details so of course it would be very very nice if we had uh tons of these different outlines you know so we would take an image uh from the internet give it to some human being and the human being could uh create some type of outline to details uh examples that would be extremely extremely expensive to give a billion images to millions of human beings to kind try to create different levels of abstractions and so we're going to try to recreate it by using on the computer and we do this by uh adding taking an image from the internet which is cheap creating noise which is cheap and then we just add these two together basically and then we're going to add different noise levels so we get an image which is the image that we actually want to recreate we add some noise that we can generate a computer can generate and then we get the input on this left side and then we have the target which is just the image that we we had from the beginning so we want to learn to go from this noisy version of the image to the Target so this is very very cheap to create because noise is very cheap to create a computer can do it and image is it's easy to download that's why there this needs to be no human being in this Loop and and of course we want to uh uh have different types of uh noise levels because we want to be able to go from any type of noise level to less noise because when we actually when we train it we want to go for any noise level to less noise because when we deploy it it's going to be deployed you know iteratively running on itself so it might come across a lot of different type of noise levels when actually being deployed H so we're going to just have different types of no noise level to create different types of pairs for tons of different images and For Tons tons of different noise levels and also we're going to throw in uh Pure Noise sometimes so then it has then basically it has no idea about the image whatsoever it has to just hallucinate to make up an image from Pure Noise that's also important because when we actually deploy this we want to start we want to be able to start from Pure Noise right because we want to be able to generate from thin air something H and of course we want to to make this even more useful useful in order to sample randomly we're going to add text that corresponds to the image to be able to guide that generation so we can tell we can give a prompt to do or stable diffusion it generates the image that we want to see okay okay um right and then when we train this model we now have a uh input which is a noisy uh you know image plus some noise level and noise the model gets this input and it gets the uh The Prompt that we feed through a language model just to kind of make it more useful features or or latent features for of it we feed this to the model and then it's supposed to generate a guess of how the how it believes that the the real image looks like right and make a step towards the real image that we that we know that we have um and then uh we can look we can look or we can have some loss function looks at the difference between the guess and the actual Target that we want to reach and we can give some directional kind of feedback saying like hey you go in the right direction but uh this is you know how you can improve on yourself so that's how we're going to be able to train this and then when it's trained again uh we're going to feed when when it when it's been trained on all these different types of images and noise levels we're going to actually start them all off using pure noise and some uh text prompt and then it tries to take steps towards a real image like an actual image and then we can run this as many times as we want right step one step two three we just rerun the model and we get a better and better image and we stop at some point and say hey we're happy now and um but of course it's very important here because now that it's being deployed there's no need to download any images from online or anything because it can generate an image from Pure Noise uh which we can just uh create by by the computer can create by itself okay so um all right so that's that's kind of the little bit more detailed high level perspective of how stable diffusion Works uh and image generation works works we're going to now going to jump in a little bit more into how stable diffusion makes very interesting choices of how to improve this pipeline uh and they connect a lot of different concept that we've been talking about throughout the class so first thing that they um mention or realize is that details in images are very repetitive and maybe you don't want to spend too much effort in your training and in your model in just generating detail so if you look at this if you zoom in on this face like these are probably not that hard for AI model to add they're pretty pretty repetitive all these pores and dets and stuff you can just you know it's not going to be very very hard for the mod that's what they believe so they shouldn't put too much too much work on on creating the details it's hard to create a the outline or the you know the important features of a face and a detail is going to be uh less demanding May it's an empirical observation but that's what they uh believe and also because these diffusion models like work on the pixel space so it takes some pixel image right so maybe it's ,24 time 10244 adds noise of the same Dimension and then that's a pretty big input of like you know a million pixels and then it removes some noise and creates the same resolution again right with less noise and it keeps running on it over and over and refines itself and that's expensive then because resolution is quite expensive so it's a kind of a then the the resolution that you're using determines the cost of running this this training and and in deployment so be very very nice if we could magically reduce resolution of this image and run on run diffusion on the smaller images instead right so if you're able to make uh the resolution eight times smaller I think right uh the actual number of pixels are you know much much smaller quadratic right um okay so what they what they do to accomplish this is that they're going to do this thing that we talked about right game of compression and order encoders so they're going to try to push an image right uh and compress it to a smaller space um and instead of using just a random uh Vector small Vector they actually going to try to create a smaller image and then be able to kind of upscale and down scale without losing any significant information uh so they're going to learn to compress to a uh lower resolution image that basically is a very uh good compression and Summer of the image without all the details that might not be might not be that difficult to re recreate right so it's actually when you look at this when it's being trained the smaller image is actually like a smaller image you know so it just learns to compress this in a smart way and because this order quod that it creates is really good at creating this details so it's very good at upscaling the image and increase in resolution um and we're going to split this apart into the order encoder into an encoder and a decoder so the encoder is that one that takes a big image compresses it to a smaller image right that's the encoder and then also we're going to have a decoder that goes from a small image to a bigger image H and for now we're just going to assume that these we have this they work really really well at down scaling upscaling the image and we're basically losing no you know no information about the image in the process because these details are not something that we really care about okay so let's say now we have a perfect encoder and decoder how can we utilize this well we can now take uh our image from the internet we can encode it to a smaller image we can create noise in that smaller space and then H do you know the whole thing of diffusion the noise in this smaller space instead and then when we're done uh we can recreate the original image okay so let's talk about this actually works in detail but now you know in the factor we can just we do everything that we did before but we can just do it in a smaller space which saves a lot of Compu in terms of training and also it's much faster when it when it's being deployed okay so how does this work well we take an image from online that's big we run it through our perfect encoder we we get a smaller image right so that's our new Target now we add the smaller noise as well you know in resolutions we get an input and uh we now feed this input to our uh model with this prompt and we try to create remove some of this the noise right we try the noise remove some noise and then we can compare ourselves to our Target and give feedback and improve ourself right so we just do this training now but in this smaller space and of course we can it's easy for us to generate noise know different types of noise at different levels at in this smaller space and so we're just redoing this whole thing and this turns out to work really really well right and so what we do is that we instead of taking image from on line we take an image from on line we run it through our encoder to make it smaller and then we do the training there instead and when we use this right so now notice we only use the encoder during training but now when we want to deploy this we're going to have some user typing on his computer a prompt we uh when we receive a prompt we the computer creates Pure Noise right so the full noise level maximum possible uh the we have some you know model that encodes language feeded to our the noising model it gets the Pure Noise and starts making a step towards a real image and then we can run uh you know keep running this on the output as many times many reg that we want to and when we're you know happy and depending on how much compute want to make available right how many steps this some thres like okay you know get a kind of a diminishing return but you run it for number of iterations until uh it's good enough and then you fit it to your decoder and you upscale to a image of resolution that the user is interested in and you deliver that and as long as St code and decoder works really really well H you're not losing out anything okay so hopefully uh this makes some sense and it's a nice trick to be able to get the lower resolution images uh that makes training faster and deployment faster because this step of the noising we do multiple iteration maybe we do it 100 times a thousand times so it's going to be the most the number of iterations is going to be the most expensing part of this training and deployment that's why it's nice because the we only decode once during deployment and we only encode once during training so it's not very expensive okay um so we now said we had this uh perfect uh encoder and decoder in an order encoder and now we're going to talk about how we get them right so and why uh what kind of uh tricks we use to to make them work well okay so in a order encoder setup right we want to uh take some image from the internet any any image in data set we want to run through our encoder to a smaller image a compressed version and then we want to run it through the decoder and upscale it and we want to get back original image right we want to be able to do this this uh uh loop without losing any information and actually this is I mean there's a lot of nuances here that are harder than than you might at first think so first off you got to decide like how do you how do you measure if you get your original image back what's a good loss function or perspective on how close these images are because you want to have some loss function as you want to minimize the dis the similarity between these images after being run through this order encoder but how can you do that so one of the first things you can try you know what people did is that you can just say we're just going to compare pixel to pixel just going to take the pixels values in the input image the pixel values in the output image do a negative something or like do the negative and make sure that the difference is very small right so what's good about that the good thing about that if if if the order encoder recreates the original image perfectly then the loss is zero so the Optima is good right when you do a perfect job the loss is uh perfectly optimized um but of course uh there's clear kind of drawbacks in how it's actually able to uh measure similarity in a way that we think is sensible so let's say you take an image now and you just basically shift image a little bit right these are still very very similar images but because they shift a little bit the pixel correspondences when you do the negative between two images they're going to be very very different like here for example these images are kind of shift a little bit so comparing an eyebrow with an eye and you going say these images are very very different though they're clearly quite similar so the just doing the negative with the minus subtracting the two Imes and looking at that is not sophisticated enough to capture a similarity that that we as human beings perceive okay so what can we do well the first thing that stable diffusion decides to do is that uh they actually use now introduce Gans and they say well Gans have been used for quite a long time empirically we've seen that they're good at capturing details so uh they're going to leverage this in this work and since gandas are good at the detail level rather than the big picture comparisons they're going to take small patches of each image and then you're going to feed this patch right so have a patch now from the original image and the recreation you're going to f you're going to feed this two patches to a Critic right and this critic is trained like this CRI is trained to say which one is from the original image right the true actual patch from a real image and which one is from the recreation right so it it makes a guess and then we train it by saying okay this is actually uh this one is the original this one is the fake so next time you see something similar you do better right but then of course the encoder decoder is trained to fool the critic so the encoder and decod is trained to make the patches look very similar from the crit perspective and this turns out just to work really really well at capturing all the different details so you know when we when you generate an image of a uh phas like you won't go and zoom in on the pores too much and and care too much as long as it looks quite similar you're going to be happy with it and that's what they leverage here as well and of course they do this on all the different patches and free to this this scan okay so this gets this gets now solving kind of the generating the details of an image um so we can down scale and upscale again without losing uh those abilities right because we want to when we as doble F gener images for us we want to look realistic we want the details to be be there we don't want to have a phas with no pores or something like that right we want look realistic and now I'm able to do that but this like one thing that's missing um in the setup is that we just have this local loss but we still also want the overall image to look nicely you know similar in from from a bigger perspective as well right we want the face to be put together like a face not just the patches you know being skined but they want the eyes to be in the right place Etc so we're going to try to also include our loss now or or compare this images from a more Global Perspective not just patch level but more Global and we're going to be we're going to try because this is going to this is going to be used by human beings they're going to judge the the quality of this output we're going to try to imitate uh how human beings judge images or how they think images are images are similar or not so we're going to try to compare these images in a way that humans would yes groups how do you add oh yeah so H so I think in this word they actually they take they compare pairs of patches and it take the same patch on both places uh and is that a problem now like okay is it a problem now that these patches don't uh aren't perfectly aligned well if you think about it right because we're not using the the just this we don't we don't we don't subtract now and get a value this model literally looks at both of these and says which one could be for an actual real image right like which one is from the original image and which one is from the fake one uh and if you look at this yourself like this would be hard for you to tell because they look like they both come from real images from a face so then like them being off in terms of not being the same patch is uh now kind of Fairly irrelevant because you as a Critic will just judge looking at that patch and saying does this actually look like it comes from a real image or not which makes this laws not now so much more informative because it's it's it's it's not as as simple just like doing the negative it actually has has to understand more how patches look like the CRI the critic is another model yes so it's another model that's trained adversarially right so after after after we've done what I train the CRI the CRI is not used so that's just we're going to throw away the critic when we're done okay good question and all right okay so we have the comparing the patch level done so we are able to handle details but we want to be able to handle the complete image right so we want to have some more Global loss to compare the images and we want to have something that you know says this images look similar from a human perspective because human beings are going to use this eventually and okay so what they do actually is they say that there's been a lot of work so we talked about contrastive learning this class before so this is again um self super learning on how you learn a good image features for images right so when we what we do well in this case we we take images from online we say that Concepts that appear in the same image are more related and similar in meaning then constant appear in in different images on average right when you look at a lot of images so you just in con learning you just take an image uh you create uh two random crops so patches and then a you know you train this uh you train a model to H say like well if you feed a if you feed a one crop to the model the model supposed to be able to say which of all possible crops in my data set is from the same image and which one is from other images so it needs to be able to identify what patches or crops come from the same images right and so here somehow you know it would have to understand that if it sees a image or a crop or a patch of a human being being with a leash it's probably walking a dog right so some understand these two things are related and it's more likely to be a dog that is walking than a cat for example right another crop so it has to understand these different things how they relate H and when you train this it creates uh an output that you use like a feature a Laten based feature that summarizes the image so basically in this process it creates a compressed uh Vector that summarizes like a feature repres representation of this image if you will and it turns out that there's been a lot of work done where you look at the features and the representation that these models learn and you look at the distance so you take two images you run it through this uh embedder that we got from conal learning and then you get representation vectors features and then look at a simple Al like a simple you know negative subtraction distance in this embeding space Laten space that distance corresponds very closely to how human beings H judge similarity or dissimilarity in images so just they basically take uh how they T they take one image and then they might have uh uh or let's say maybe I don't know how to do exactly but maybe they take like five images and then let that human beings group them together of like okay which ones are closest or they take like one image and then four targets and say like rank these images in terms of how close they are to this image and then these models will do very similarly to how human beings rank this so this is kind of empirical which also suggests actually we are also learning in similar ways okay but why why is this nice well it's nice now because we have this embed that we can we don't need to train it now it's trained so we can freeze it but we have an EMB better that can create you know Global features of our images and then we can look at the distance in this latent space because the distance here when distance is when the distance is small in this representation or latent space that and they're very similar according to a human being so now we can optimize and minimize this distance in this lat in space to get a global similarity somehow between these images okay so now we just combine these uh so we have both a Critic you know that we tried to fool and we have this EMB better that we try to make happy but but aligning the representation in this uh latent space so we get it both details locally and then globally kind of aligned uh and turns out to work well and and converges nicely you you could imagine for example that these two losses would be contradictory and work against each other but empirically this works well and they converge nicely okay so um to summarize so uh the noise thing right when we take some uh noise and we try to refine it by adding details and taking steps towards uh Imes can actually create uh from thin air or Pure Noise if you train it on different levels of of noise and diffusion is when we able to do this iteratively right so we have a the noising model is able to take some noisy version of an image and taking steps in right direction to make it more look look more like a real image when we do this iteratively right improve an image in in several steps it's called diffusion uh if we condition on some language input so we have language and image correspondences we can make this generation not just random but actually uh you know make you happy based on the prompt so it adheres to the prompt that you gave it it's a correspondence between the prompt and the actual generated image and so diffusion works on the pixel level so it's expensive so if you have high resolution H you will have a very expensive model to train and to deploy so if you can do it in a compressed small resolution space you gain a lot so that's why stable diffusion uses an order encoder to compress the image to a smaller image and does all the training there and and when we create these order encoders so encoder and decoder we have to be careful about how we think about similarity to get something that is actually useful and also we have to take somehow inspiration from how human beings judge similarity because at end of the day our models are deployed to human beings and we want to make them happy with our product and I think also it's pretty cool because right in this uh all these different approaches we used a language model let's train on on let's say just next token or next word prediction so that's self super learning we use the noising diffusion that we talked about before we used order encoders that were trained using Gans like so generative adial networks and then we also even used a you know contrastive learning from images to get a useful representation that we can use for for training our order coder as well so it's like really cool how all these different self-supervised learning Concepts came together to create stable diffusion and how they able to kind of you know Stitch it together and this is possible of course because they'll rely on no label data right so they can be trained at scale and and um yeah so I think that's also like a very cool thing of how you can combine all these different things uh okay and and as well a little bit I mean I think also it's pretty nice that we can use a little bit of our human intuition to engineer and use AI that maybe we can say like how do we think about when we draw images this is useful to be able to improve on yourself and and uh also when we look for example at how do human beings judge similarity it kind of corresponds to how these uh self supervised learning contrasted learning models judge similarity that's also very encouraging that we're somehow also getting a little bit closer perhaps to understanding how we work ourselves and our brains learn because we starting to see these similarities and again also I think is quite interesting is that like that creativity from from new networks comes from noise a lot like if you give the same input expect the same output and maybe as well like I think maybe a lot of us we have you know the same routines if you want to be creative in your work and your research you might have to get out of your comfort zones once in a while and put yourself in in weird positions maybe even if you just go to like a new setting and work you'll be more creative because your brain is not good at making up Randomness it's good at leveraging Randomness in your environment which is also pretty cool okay we also talked about the tip of the iceberg and we made a little bit more we kind of fill in some bottom part here that of course at the top we have CHP and stable fusion and this cool stuff that people are talking about but this of course come from South supervised Learning Foundation motive AI that's been around for for some time and the research I've been excited about right but then also of course the data the data took a long time to create the internet you know we were not able to reproduce easily so the data is so key to making these AI models work and Al something that I think a lot of people that uh work with AI underestimate in research a lot of times our data sets are curated cleaned up and structured for us H in a way that the real world isn't um so the data is really really important if you want to go from Theory to practice okay so uh that's for today um yeah you can for if you have questions I can take them now yes so that is that oh that's a good question uh all right I mean we didn't even cover variational order encoders in this class right uh I don't know which one they used uh I don't they use the varation codor I think use regular C but yeah it's a little bit we can go talk about offline what was the functioned the diffusion model I know theyed yeah the diffusion loss is score kind of it's called score matching um yeah that's a I mean it's uh it's not super mathematically involved but uh it's a little bit uh mathematically involved how to define it so I think the high level intuition is that uh you're trying to you're trying to go from you trying to go from like a distribution that's random Gauss and noise which we computer love to create that's like one distribution is Gauss and noise and then you have a distribution of images all right you want to be able to sample from your gaus noise and then project yourself or transport yourself to this space of images okay and yes it's very hard if you just take random noise and you should take you know one giant step to the space of real images okay so I say the images leave over here and the noise SL is very it's very hard to just be like bam jump but if you're smart about it you can create training samples so during training you don't have to go all the way like this you can create samples all the way in between so mean you can take an image add some noise and end up in the middle between the full GA and noise and they complete images so then during training you can start to learn and and I lost is basically directional mean that like if you're here and you get a training sample you take a small step towards you know the real Imes you get like a directional feeding feedback saying like Okay are you taking a steps in the right direction that's the kind of feedback you get which is very nice it turns out it's like that's nice feedback to get that's easy to learn from and then it's nice because now during training you have seen all the different combinations of like being in between the full gaussing noise and real images and you've seen everything in between right during training so now that when you're being deployed and you only said this thing you can actually take steps like this along this from noise to this space and actually end up at a realistic image so you can find your way by removing noise iteratively like diffusion and find yourself in the space where you want to as long as you take steps in right direction so basically right now since this model has been trained to get from America to any place in Europe we can say you know and it learned by taking the right steps so then we can say hey let's you know go to usbekistan you can be like w I'm going to start taking a step towards Europe then a step towards there step step step and then eventually will find itself in the right direct right place basically does that make sense yeah yeah all right thank you guys

MIT_6S087_Foundation_Models_Generative_AI_2024

MIT_6S087_Foundation_Models_Generative_AI_AUTONOMY.txt

but uh tin flew in from silic Valley he's going to be here in th as well for B part but he's going to talk about agents so please en Jo yep hi everyone so I just going to quickly jump into it right away imagine you want to research a topic of the future of AI you want to understand what's going to happen soon in terms of AI space and so on and you don't want to see it on on a 1 hour and 30 minutes lecture um you don't want to click on a thousands of links you don't want to research and spend long time doing that you want to have a concise report right away right there this is possible now with autonomous agents it's also possible now to basically send a promt to chpt and get your favorite pizza delivered from the favorite restaurant um furthermore it is possible now to execute on any almost any online task like say for instance doing um a California driving test online and you can look at the Sinister face of this guy you can trust he did it himself so yeah uh my name is Art I'm uh originally I'm software engineer I was born and raised in Ukraine I have been developing AI products for the past more than a decade I guess as every second Ukrainian software engineer right now because of the geopolitical situation I'm also an ethical hacker and work a lot in cyber security space um and I'm a Serial entrepreneur I came to MIT to do my MBA degree but to many people ask me why software engineer do an MBA degree and I dropped out well not exactly because of that I started my own company which is called Kraken AGI and we are building agents with the mission to drastically Elevate Global digital reliability and security we basically build in autonomous AI agents for cyber security and software development space now today I want to quickly touch on the topic of um terminology in this field it's extremely confusion it's extremely confusing and convoluted unfortunately right now as in any new field I also want to explore autonomous AI Agents from the perspective of specifically AGI artificial general intelligence and I want to expose you to some um techniques and mechanics that are being used in the industry to build to build such an agents right now so that you maybe can go offline and research more on these Topics in terms of terminology GPT is a model gpt1 gpt2 gpt3 GPT 4 chat GPT is a SAS product gpts are now agents or co-pilots or assistants so the the terminology in the industry is extremely extremely confusing different things mean the same um although named differently different things that are named the same mean different stuff so it's it's like super confusing right right now and this is fine you need to understand that this is fine when you search something on the topic of autonomous a agents you will see well in some paper neuros symbolic linking in another retrieval augmented generation Google will tell you this is called grounding some other folks would use emotional grounding concept so this is fine but let's touch on a couple of key terms here so the agents have been developing for quite some time um since the Advent of AI basically in during this old um AI deep learning um hype um this architecture emerged so it it it was quite some time ago and basically the main components of it is we have a system that can autonomously perform it has sensors it it has actuators it interacts with an environment post generative AI hype so like a couple of years ago we got a more simplified architecture and a more simplified approach now actuators and sensors are kind of called tools now in this system we have either an llm as a core or a family of AI models basically um providing reasoning capability for the whole agent itself we give out a task to an agent um that it needs to achieve on a goal another thing I want to Define here today as well is what is Agi what is artificial general intelligence and I think um Google has taken an approach to that and and actually had very good definition AGI is basically something that can accomplish any task that human can accomplish or can do or kind of can interact and react in any environment human can interact and react now what's today's AI is missing to to to be AI well except of this lame joke about letter G um today's AI is like a monk in a cave meditating so it like the llm or Char GPD when you interact with it it's a snapshot of time and space and knowledge it's very wise it has a lot of general knowledge but it exists outside of time outside of the environment and basically you come up to char GPT you bring a letter with the task written on it CHP gets this letter reads it as a monk writes down the result gives you back and keeps meditating it also has quite a lot of constraints specific scalability constraints one of the scalability constraints is the fact that well if you bring like a bunch of books to this mon um it will just truncate half of the information that has been provided and we only work with information that allows that its context window allows another thing to look on the current AI to kind of put in an additional analogy what rard mentioned on today's lecture about the foundation models that the um current llms Foundation models are components of the brain but not the whole brain our whole brains are much more complicated it's a family of AI intertwined together and they're not the whole body the body has the sensors interfaces to act to feel to see to have vision and so on current llm models on only have limited interfaces to do that now still considering this can we achieve AGI like capabilities today and the answer is we kind of can or we at least can push to a capabilities and there are a couple of approaches to that one of the approach is what Yan leun he's a chief scientist at mattera proposes it's basically he basically says Hey llms are dumb they can't execute on a bunch of tasks like say planning they still hallucinate so we need to build a completely new architecture basically inherit our own knowledge and build a completely new approach completely new AI model probably based on Transformers or something else and um or give the new AI new architecture of this new ability but I'm software engineer I like love to problem solve and in software engineering industry we say composability over inheritance so uh we potentially still can try and use something that's already exist in the field and try to apply try to work around limitation that exist there try to uh create new techniques around it and actually still try to push um the existing llms towards agla capabilities only using what we have and that's another approach that currently the industry and most of the Silicon volley startups and companies are are taking right now how to do that specifically how can we build how can we push current llms those monks sitting in the caves to be more like AGI first we can give it we can give the llms um ability to self-reflect we as hum humans we are thinking iteratively we don't have like when we produce an idea our next thought is being fed of the S thought that happened a second ago so we iteratively think about something until we come up with the result we don't just go ahead right away and produce the idea or or suggestion or something this technique of like thinking associatively in iterations is called The Chain of Thought and it's being applied in prompt engineering you can apply it on your own even now when you go to char GPD and ask it to think about some concept um in a couple of iterations Bas in your uh new iteration on the result of the previous one you basically will be implementing a Chain of Thought technique and it is extremely powerful so much powerful that that Google actually decided to use it for their marketing purposes when they released Gemini Ultra which is a new model by them they basically claim that their model is much more powerful than gp4 but it actually is much more powerful than GPT for on the Chain of Thought at 32 iterations so again it can drastically increase the performance of the model through simple prompting technique another technique is critique prompts it's basically you sit um look at yourself in the M mirror after your well maybe successful maybe a fail date and you're like thinking what you did wrong or what you did right so basically you can implement the same approach in the prompt for the current llms and Foundation models now if we so also as humans we not only like self-reflect and think iteratively in time we also continuously learn from our experiences we continuously kind of adjust our weights and biases in our brain it's not that we are a snapshot that we are static and only trained on some particular knowledge no we continuously improve ourselves this Improvement can be achieved with a strategy that is called reinforcement learning with human feedback when you open up chat GPT you see like dislike that's basically open AI getting data about how well Char GPT produce the result and basically at some point in time they will fine-tune the model that they have on this likes and dislikes producing a better more uh performant fine tuned model but now ai itself is very smart and can we ask AI to do this like and dislike can we ask AI to remember do this critique prompts if we ask a to do this critic prompts this technique will be called uh reinforcement learning with a ey feedback it's quite a recent one and but it already is extensively used in the industry for me this concept of err if is like sleep like a concept of sleep for humans when we sleep we adjust our weights and biases again we remember new information we learn new things that we we kind of like engrave the things that we learned during the day now still it might not give a capability of for the current llm to produce information and produce uh value um working with the tasks related to recent events but can we just give this monk that's sitting in a cave and and computer an access to new source or can we give them a vocabulary or some kind of a book so that a computer an llm I'm sorry an llm can search um this new source to get the most relevant information for the task that is also the most recent from May maybe today from maybe a couple of seconds ago even or even can we give them um a notepad to remember things that happened recently to write down jot down some notes in terms of the task that have been um asked just a couple of prompts before and for that we in the industry currently use so-called retrieval augmented generation Google also calls it grounding this approach ground but most of the people call it rug um again coming back to the terminology aspect so this is extremely high field right now retrieval augmented generation is extremely extremely um active um I mean myself I work up in wake up in the morning um we have an internal tool developed in the company that basically a funeral so the new sources um around the um AI topic that is relevant to all the academic papers and things like that it basically filters out only the relevant items and gives it back it gives it to the team to read and every week I read something new on retrieval augmented generation every week a new technique is coming up every week a new approach is there so it's very active fi and actually it produces quite quite valuable results for those of you who who are kind of familiar with software engineering be want to play around with something check out l index framework um they are doing extremely amazing job of gathering these techniques and implementing them as they are published in academic papers now I want to do a quick giveaway um if you um solve the next task on the next slide in five seconds and um if you if you can do that I'll just buy two burgers back home when I'm in California um if you can do that um you will buy three the prices for Burgas here in Boston um but anyway um basically on the next slide I will give you a task you have five seconds to solve it ready 5 4 3 2 1 you lost okay two burgers for me when I'm in California why are we so bad at symbolic calculations for calculator is just is just a microsc to solve the task for calculator it it's extremely computationally efficient for this we are extremely inefficient for symbolic stuff for working with structured data our brains are not wired for that same problem with llms unfortunately the llm architecture is not wired to work with symbolic calculations structured tools and and so on but rug the retrieval augmented generation that I mentioned before also gives the capability for llms to work with these tools and synergize with this tools llms now can use calculators they can use some interfaces to act and interact in the environment and they can basically um be much more efficient for the task where structured and symbolic computation needs to be have to happen now if we teach llm to self-reflect we do continuous learning we um give access for llm to tools to ured knowledge to new sources and so on the only thing that's left for us to do is just to teach it how to plan um planning is a separate quite large subject thousands of papers there if you want to play around with planning techniques similar to what I mentioned about Lama index check out L chain there are just actually a couple of Frameworks for planning um at our company we build our own framework but L chain will get you started with just basic stuff that you can check out um baby AGI llm planner and and a thousand of other papers basically proposes different approaches and planning techniques but llms in it in itself they're still very bad planners and that's what I was showing on this Yan Leon's uh Twitter post they it has been shown in in research and in papers that the planning capability of llms are are quite limited um here on you can see on the screenshot of this chpt that's not my screenshot you can check out the Twitter post lately but basically it it fails to plan for simple instructions for a simple task is it is it critical though is it a critical issue is it a critical problem well I believe it's not because when we combine planning with acting we can actually solve for this problem of planning long term meaning that um in simple terms go do your startup go I don't know build your crafting item you always wanted to build and actually see which issues you will get actually uh try do an action see the result of this action and the effect of it work with this effect perceive it and then repan again to get to your goal action is the most efficient environment Al computation and we don't need llms a to have a capability to plan plan too far away planning too far away increases informational entropy we don't need that we can just plan we can basically we basically need the first action to be planned well other subsequent action in the process might not be planned well we will still replan everything after we did this most efficient environmental computation in time and space Sorry for for such a complicated term here but basically what I'm saying here action speaks louder than planning action speaks louder than prediction if we have an llm if we open up open it up and give it an ability to act and perceive the result perceive the effect of action we don't need um planning too far in advance too far in the future we only need the first steps to um to be planned well to be executed well now with all of this components we pretty much can build a functional AI agent um that can actually achieve on quite a lot of task and will be quite AGI like so it will be quite close to artificial general intelligence the problem it won't be practical at all if you want to put it outside or work for some particular goal work for some particular task it will be extremely expensive very inefficient and this is now my speculation in terms of what's coming next for this industry I feel like this industry Now is working a lot on optimizing and optimization of different um approaches and techniques and that's actually let me let me show you the move I I I learned recently so I'm not like thinking about the next motion I I learned it through imitation learning after watching some YouTube video and I'm like I'm not having any inefficient reasoning proc in my brain I don't think about it I just perform the move I have a muscle memory already for now ai agents they think about the next section they think how to put the hand where to put its butt and so on so they actually do some quite an efficient computation there is no muscle memory technique there that optimizes that kind of caches the process and for this I believe action and behavior based models can be that component of a brain that helps with this muscle memory I also believe that imitation learning motion diffusion planning um is going to be there for AI agents as well um there is something going on in the industry around it but it's not quite there yet in terms of implementing this approaches as a part of autonomous agents but I think that's a part of the future another part of the future is um video processing and efficient video processing right now llms are not processing video information well they are not prated on the videos in any way um at MIT at cell the liquid neural Nets have been developed with sparse wiring um that's basically a technique for autonomous vehicles to better navigate with the information that they receive from from their computer vision sensors um as well as for say drones for cars for autonomous for everything that exists in a real environment and that can perceive the video information it it is this extremely efficient model very small one that can still provide this navigation capabilities I feel like this can be can be um aligned with something similar to instincts that human has and for the last part that I want to touch um when we put AI agent in the environment when we optimize it when it's efficient Maybe not today maybe sometime in the future um we potentially can have this agent socialize and work together and there is already there are a couple of um projects that are exploring this like's say my favorite one is chadev it's basically a company made of AI agents that do software 24/7 each AI agent is specialized one does coding another one does designing the third one does testing the fourth one does development and you bring them a task they all try to solve this task together as a as a collective intelligence and another thing for you to Google or chpd after this lecture is Microsoft's autogen framework where they explore specific interfaces and interactions between agents how to make this happen and how to implement it at last again when I come back to the subject about AGI and artificial general intelligence and ask you questions are there yet I personally believe where we have all the component that we need the problem now is we need to glue them all together another problem now that the gluing process is not going to happen like this year next year it's going to take us probably a decade well we can we can specifically predict how much or forecast how much but I pretty much sure that from the theoretical perspective with llms we have solved one of the um key critical problem which is reasoning we did didn't have this capability before now we have this capability and bringing this interfaces bringing this tools putting this reasoning um model in time and environment is just a matter of engineering and combining these components in the right way until we get there until we get to this AGI part until we get to this point we still need to consider how can we make this AI agents practical and this is critical right now we cannot trust at Max the a AI agent that we built so we need to understand how we put the human in the loop the huge topic of user experience and user interface for AI agent right now is in Silicon Valley also many companies are talking about that we have hackathons about that where we explore how to specifically put human in the loop to approve what AI agents want to do and I feel like till we get to till we gain all the trust we need um this is going to be probably one of the key developments um in the industry and we probably never would would be at the point when we have all the trust always humans will be accountable for something we will just reduce some stuff or delegate some stuff that we are comfortable delegating accountability for but most of the time the decision will still be driven by humans so we need to learn now how to effectively integrate human with an agent so that they can collaborate well so today after today or maybe after this course um overall check out this concept I think the slides will be uh published um Char GPD about them Google them again don't worry about the terminology it might be a bit effed up um it might be a bit confusing but yeah that's how the industry is structured today unfortunately um all this techniques and all of these approaches are extremely powerful and and bring the autonomous AGI like agents to life even today and I'm always happy to discuss this topic with you that's my LinkedIn um on the QR code that's my tw Twitter although I don't use it that much um welcome to reach out to me and chat about autonomous agents overall thank you very much okay guys you spent like two hours listening about so many Concept at switching back and forth between between different topics I guess like AG huh like autonomous agent yeah but anyway questions yeah yep um so the first part is like are these current like models like the way you're thinking about it it's kind of like inspired by like human intelligence right like when you said like there's different types of parts of their brain and like the model like would also like work this way but do you believe that like this is like ultimate goal or there's like another way of arranging that's much better than like hum and we should use like humans like inspiration you know extremely yeah extremely great question I feel like because also I'm I don't have an answer to that I feel like um from what I see well there are some subject like what rard discussed today right we still can use foundational models with the modalities that are unusual that we're Unthinkable of to be used before where humans are I mean we don't see in biology in in in our environment around us something that also uses this data this information in the way um that say foundational models are being built right now right like foundational models for behavior and that are been used for retail for instance so it's kind of like an intuitive so still there is something that Evolution hasn't been able to achieve but I feel like Evolution was still a extremely powerful process right and it it has built extremely efficient machine in our brain and in our head so I still feel like we will draw and there will be a lot of a lot to learn about it and we will draw a lot of lessons some historical artifacts right will be probably eliminated since we are constructing it at artificially right but I'm pretty sure that's like vast amount of information U provided for by nature to us that we haven't explored fully yet and I feel like we um we pretty much will lean towards how the human brain is being developed and and and how we interact and act so yeah make sense yeah yeah thank you okay awesome um thank you very much everyone all right guys thank you

MIT_6S087_Foundation_Models_Generative_AI_2024

MIT_6S087_Foundation_Models_Generative_AI_PANEL.txt

um I'm going to ask you each one um kind of a more targeted question at first and you guys also think about what what you want to ask our poist today so um Professor first question to you um rard touch on this topic and you are an expert in computational biology probably exposed a lot to Evolution and and mechanics how it worked I want to um get your opinion your perspective on the Dilemma that exists right now um in terms of centralization versus decentralization in terms of alignment versus um more risk and diversity so let me pass this to you in terms of perfect yeah in terms of alignment versus more risks and diversity specifically meaning that well we as humans are very diverse uh we have diverse cultures you know um lived in in Greece and France rard lived in Sweden I lived in Ukraine um we are BR in different environments and the evolution might have been pushed by our differences um but now we have a very defragmented um defragmented AI defragmented organization defragmented Society in terms of who's pushing AI forward they are implementing their own AI alignment systems um they're reducing the diversity but potentially also reducing biases and stereotypes that have already existed in society so we kind of have a dilemma between high risks um but more opportunity for Innovation or lower risks and lower opportunity for Innovation what's your perspective on that coming from biology Evolution and things around that beautiful uh fantastic question extremely rich extremely uh deep broad reaching Etc so um let me start with Biology a little bit so basically uh as you mentioned sort of humans are forced to be diverse we don't have a choice we basically have genetic variation that modifies every aspects of our brain and of our body and of our behavior and of our inclinations and so so forth I have three children uh you know they are completely different from each other and and and they were completely different when when they first came out and they're still completely different now and um as much as we would love to as parents think that nurture matters a lot it's only about 50% and another 50% is just like nature and and there's very little you can do about that and um that's I think part of the beauty of humanity the fact that whether we like it or not we're all programmed to actually think differently to interpret things differently to uh Etc and that that's just the nurture component the nature component Al sorry that's just the nature component the nurture component also gives us extraordinary diversity in sort of where we grew up the things that we saw as cultural references at different points in our lives as you mentioned different cultures different families even in the same sort of street block you can have kids growing up with completely different perspectives on life and I think that's what makes MIT work that's what makes any team work the fact that we think differently and we can bounce ideas with each other with mutual respect but also uh completely different perspectives and that shapes the ideas very interestingly so I think one way to achieve that with AI even with a single underlying large language models is to instill different personalities in a set of agents that are interacting together in the same system so that forces the agents to actually process ideas in different ways so if you want to have the most Creative Solutions you don't want a single AI That's going to give some average you want a lot of different AI that are going to be bouncing off each other each with own personality and you can encode that you can give them personalities you can basically say you know you are a professor who grew up in Iran and who has you know these kinds of backgrounds you are a waiter who grew up in I don't know Scandinavia and has this background Etc and then based on these personalities you can sort of build a life story and a set of attributes for each of the agents and then push them towards uh more creativity um in terms of bias we all worry so much that AI will be biased but I have to say that humans are you know have a terrible track record on bias we are horrible when it comes to bias and icii as a hope for being able to not just debias but anti-bias uh our thoughts to be able to sort of artificially tag on different biases with different attributes and push us off our comfort zone in terms of expectations and have the AI push itself off its comfort zone so you can basically create create again these personalities with very different stereotypes and with mismatch of these stereotypes and sort of have the AI interact with those and actually learn how to uncouple uh those biases so that's on the bias a little bit on the diversity and in terms of the centralization I you know again I think the scenario of Skynet in Terminator is exactly one of centralization it's basically US versus the AI and I think that the for of the market are such that as centralization happens in One Direction you will have forces pushing against it in the other direction and there is uh there are laws against Monopoly there are antitrust laws that are sort of go going to kick in if we see that in fact centralization is pushing too far and I think that's healthy I think that the forces of the market are are healthy I think that the best way to combat the Skynet scenario of the AI apocalypse is not to pause AI it's to double down and to sort of you know expand out and to democratize and to sort of you know provide opportunities for many others to build on the same architectures on the same Hardware on the same software and sometimes even on open AI to basically create diverse agents on top of it and that's what we saw a few months ago with the Chachi pts the fact that everyone can program their own Ai and even if there's an underlying architecture you can still have diversity in the utilizations and in the outcomes okay so thank you that's very interesting I do think that saying that you have one single big big AI that you incorporate different personalities into it sounds like if you take the biology and evolution similarity like well all of humankind would share a single brain you know that would be prompted differently and that sounds like w why don't Humanity have a single brain because it's very fragile if it screws up we're all screwed so I I you see I mean I think it's and also I think if you have that such a big thing it's gonna even if it's just less biased it's going to be biased systemically in exactly the same way for all of those users right well a human being exactly is very biased but differently so which is I think much more in line with nature and evolution which I think is great guiding Stars uh so like since you since you work with this I feel also the last thing you point as well that let's just push through right but like what's and what's the what can we learn in terms of innovation and change from nature well most of change is bad and how you know we understand that is by the passing of time like if you push things very very quickly what systems like Evolution will understand what's bad Innovation is going to kill us and what's not if you don't give it enough time to see the effects does that make sense uh no absolutely these are a great idea so so basically on the first comment of the single brand many many personalities even if you have a single giant llm it has 5 billion parameters if you look at the human brain the way that thought happens it's not a single thought like all of our neurons rarely fire together you basically have communities within our brain you have engrams that are firing together you have a central sort of memory module if you wish which is the hippocampus but even it basically connects to a bunch of different parts of the brain and a bunch of different sort of memories are sort of co- ing at once and the way that we're interpreting the world is also multimodal you basically have visual cues auditory cues emotional cues cognitive cues you know written verbal auditory like there's a huge diversity of inputs and a huge diversity of references that even a single brain has and even within chat PT if I'm interacting with a single chat with a single user with a single personality if you look at the underlying architecture a lot of researchers are basically suggesting that in fact it's a a community of models that is cobbled together that basically even the answers that you get from seemingly a single chat a single interaction are in fact a plurality of these interaction sort of diversity of interactions so basically I don't expect a single mode of thinking and if you look at even Chach PT even if you had a single mode of thinking there's the temperature parameter that basically tells you how close are you to the best solution versus how much are you lowering that probabilistically to basically Give A diversity perspectives and the best models are running at 08 not 1.0 and and basically that tells us that inherently if I ask exactly the same questions I'm going to get slightly different answers every time so diversity is almost built in even in a single system but do you have any what's the Sim most similar system in nature to this that you propose like is there any like can you be like well it's safe because nature also has an inspiration for us here you see what I mean so I would say the worldwide web is the closest thing where basically all humans have different cognitive modes and yet we have a a shared exchange Marketplace for ideas and basically you can think of all of the billions of web pages as the neurons of Chachi PT in a way and basically all of us are interacting with some common knowledge frame but every one of us is basically making connections with with the subset the second comment that I wanted to get at is uh change being bad and um I like to say that if evolution was Flawless if human Engineers had designed a Flawless replicating machine we would still be to this day perfectly replicating bacteria so basically what makes evolution work is breaking things it's like you know climate change it's asteroids I mean we here in this room would not be around if it wasn't for the chicku you know asteroid that basically wiped out the dinosaurs 60 million years ago mammals had been around for about 30 million years and had just never escaped from their holes because they would just make such a lovely breakfast every time they stuck their head out to these giant dinosaurs and and the Dinosaurs Ruled the Earth for 175 million years we have been around since their Extinction for only 60 million years and we diverse from chimps only 5 million years ago basically the the primates have only been around for 5 million years and the human lineage are only 1 million years so I just want to give you a little bit of perspective there's no such thing as oh you know evolution is so wonderful and perfect the only reason cognition is ruling in this earth is because of an asteroid but I mean so the din dinosaurs died out are you saying that well I mean I'm not saying change is bad I think change in inevitable I mean we're not changing as quickly as other you know like virus or something but like yeah the dinosaurs were pretty fragile they died Humanity might die like is that good thing then right because like you're taking the dinos as an example and like yeah I think that you know at commat hitting Earth is a very rapid change that we're not used to and all of dinosaurs died I think like yes AI can be a similar thing if change happens too quickly and we're not robust to it we're not antifragile because change is in inevitable right so it's like yeah sure it's not bad but if we die out as humanity and some other more antifragile life from Earth takes over I think that's the outcome we don't want that sense like I'm like yeah but we don't want to be the dinosaurs we want to be a little bit more smart in how we build our system let me be completely blond basically AI has no will to take over in fact I could have stopped that sentence that AI has no will period we evolved with a set of evolutionary constraints of scarse resources competition cooperation for our kin uh Gene level selection basically selfish Gene selection where a gene for altruism will be selected if it is beneficial for the species that carries it and so so forth so basically all of these things that have made Humanity have wars but also save you know children and I don't know stop for trues even when the bosses are saying kill each other the soldiers on the field will basically sort of make a truce I think there's there's so many different complex aspects of humanity but the urges that pushed Natural Evolution to scarcity competition for food competition for mating for progeny for Speed of replication Etc they do not apply to Ai and if you look at the path again of all of at least the lineage near us you basically see this move to higher and higher levels of abstraction basically very early on in life you basically have chemotaxis which basically means turning to where the chemical gradient is to find the density uh the higher density of food molecules and therefore the source of the food chemotaxis eventually got replaced with I mean that that was smell it eventually got replaced with vision being able to see stuff and be better better catch it then eventually a central nervous system that integrates the information from multiple sources and eventually this appendage like between your ears that basically grew larger and larger and larger for integrating more and more information at the source however we started from food replication AI doesn't have any of that it started with a brain and it started with solving problems and it doesn't have will and it doesn't have agency and it doesn't have any desire to take over if there is such a weaponization of AI it is the humans behind it it is not the AI yeah I mean I mean I'm not saying that a has a will at all I don't I don't think that's not the this is a lot of change happening quickly and human beings I get like yeah we can definitely Kill oursel by using AI in the wrong ways uh so it's not that AI has a will or anything and I would say like AI just part of Technology development would you say that that humanity is more robust now to exting than it was 2,000 years ago do you think it's more robust today than it was then if you look at the diversity of human populations you can basically measure the density of population over time by taking any pair of humans from any population you can basically see across the entire genome what is the time to the most recent ancestor of coalesence of the alals in any one chromosome that basically tells you if a lot of them coales here and then there's a gap and a lot of them coales here that basically tells you about low diversity High diversity and so so forth in the human population itself so you can actually trace the size of the human population over time and we almost went extinct several times so Humanity has almost disappeared several times there we were down to a population of a few hundred and and somehow resurfaced so yes we have been very fragile the other thing I want to say is when people say oh AI can be dangerous let's wait six months it sounds like oh this fire thing that we just invented is very dangerous maybe we should wait a few hundred thousand years it's very similar in my view and yes fire is is extremely dangerous it can burn your houses it can basically wipe out your Colony it can burn your Fields it can basically lead to famine fire is extremely dangerous but fire is also so what allows us to cook our foods to change our diet to basically have the necessary uh energetic resources to grow our brains and to basically uh develop into modern humans and in my view AI is a tool it's a it's just a fancy typewriter it's the next generation of computers it's the next generation of the book the book led to extraordinary changes in Civilization extraordinary by spreading knowledge the internet accelerated all that exponentially by sharing knowledge way faster uh you know bio archive and archive and and Wikipedia and and YouTube and all of these different tools for learning have dramatically Accel the spread of knowledge and what AI is right now these chat Bots they are the integration of that knowledge so you don't have to go and like read a thousand documents you can just like get some kind of average answer right away and I think this is of course a great accelerator of course it will lead to climate change it will lead to more energy utilization it will lead to more CO2 I mean don't get me wrong we are almost killing our planet we are basically in a very fragile state for the world as we know it that does not mean humans will go extinct that just means that a lot of other species will go extinct that means that you know we will change the way that we understand life around us Etc and there's like enormous conservation efforts to combat this horrible thing that's happening to our planet and to the other species Etc but that just comes with the increasing entropy of progress and and this has been happening for thousands of years this is not a new thing this exponential that we're writing seems steeper because we are on it and it always seems steeper exactly where you are then a little bit ago just because of how exponentials are but if you zoom out all the way every single time it looks like the same EXP itial so I don't think much has changed right now I don't think we are um insurmountably doomed in terms of climate I think that it's it's very dire I completely agree and we need to do massive amounts for it and and we should and we will and AI will be our partner there I think being able to find better materials better biofuels better like all of these things require a 100 times more scientists than we have now 100 times more Engineers than we we have now we are not about to lose our jobs I think that we are about to hit a period of productivity that's unprecedented in the history of humanity and that's the only way to overcome the crisis that we have in education where children don't have enough time with teachers where patients don't have enough time with doctors where the elderly don't have enough time with home carees where parents don't have enough time with their children or their parents or basically we are stretched so thin the only saving grace of this current instability in my view is massive changes in productivity and AI is part of that solution all right yeah we have I love how differently we think and I'm enjoying this debate and if you took the other position I'm sure I would take yours yeah and um yeah we have amazing speakers today an amazing topic I guess um this fuels the conversation and F fuels the discussion but I think we are ready to take questions from the audience take it away be be provocative be challenging while while okay go ahead or what we're not even thinking so along the lines of what we're discussing I I do think that with the Advent of data science and computer power we are with analysis of data we are en covering a lot of correlations and causations and therefore we're automating a lot that before we could not automate um and maybe maybe it's us how do we adapt to the new realities of the automation right do we let the auton car heal uh the baby or the young um the old lady if it's inevitable right those those are the things that we need to question but one of the things I do worry is that because this is so powerful there massive produ gains um you can have a change in the world order and what would you described before is we have China we have Europe we have United States and I think that's the real threat and I wanted to to see what's your view position and or prediction in that case are fantastic question I can I can start with the Dilemma of do you want to kill the grandma or the child and I maybe I'm a horrible parent but I asked this to my children I have an 11-year-old a 9-year-old and a seven-year-old and I said okay great you know let's talk about ethics let's talk about these dilemmas you have a self-driving car what should he do and at first they're like oh you should do this or you should do that I'm like okay that means that you're inherently assigning a value of worthiness to different humans is that what you and they're like no you know life is priceless you can't compare and then they're like do nothing and in fact that's what most humans eventually come to basically there's this Paradox where if you're steering the train taking an action to save five people and to kill one Grandma basically is an action that most humans are just not able to ever cope with and then what they will choose is in action and possibly killing five you know people instead of you know killing one so humans just have no ability to sort of choose to kill in order to save this is not within the way that we're wired and uh apparently this was demonstrated in my own kids you know within like a few minutes of this conversation um the second one about the world order I agree completely we are like basically AI is putting everything back to the drawing table I think that the traditional areas of competition are superseded by the competition in this realm and you know geopolitics are shaping Ai and AI is shaping geopolitics so I you know I think you cannot think about innovation abstractly without the context of the geopolitical area and spectrum that that you live in that is shaping The Innovation the way you're thinking the types of financing the types of applications Etc and um uh I don't know if I have a right answer except that you know yes I I agree completely that this is you know shaped not just in One Direction but in both directions all right um more questions any more questions okay yeah you dodged who's going to win that was good um so clicking on like the geopolitical climate uh tomorrow afternoon the slone embas are going to have a debate on uh the impact of AI on unemployment globally in Aggregate and how that's going to play out for different different countries and different economics situations um interested in some of your thoughts on that so again I'm an optimist I basically see AI as liberating millions of humans from the state that they're in children who are too challenged by school who cannot follow can ask their own personal AI tutor which is available at the push of a button uh for help to get unstuck to ask the quote unquote dumb questions children who are at the top of their class with their brain oozing out because they're so bored can ask their own personal AI tutor to basically push them to the limits of their own capabilities so IAI as lifting up the human race across all countries of the world um I see also nearly all current jobs being extinct in 5 to 10 years every sing Le job we have now will have to be dramatically rethought the like I would say 80% of the way you spend your time will be different your job might still be called that but 80% of your time will be doing something completely different and a lot of it will be working with your AI agents to basically uh you know execute probably a lot more than you were able to do before um there will be new jobs created completely so I'm not as worried about unemployment at least in the short term of 5 to 10 years I expect many many more jobs will be created but what I am worried about is exactly what you mentioned that these jobs will be created in very different places than the places where they're getting lost so call centers will disappear in India will they be you know will that also be the place where the the new I don't know prompt engineering or you know just to give the the typical example jobs will be created it's unclear so basically there might be dramatic shift in capital across the world but My overall View and hope is dramatic increases in productivity and a raise of the global standard of of living of well-being of cognition of thought of Engagement of motivation and so so forth I think at least the short term will be extraordinarily productive the long term there will be existential questions of do I you know am I needed in society anymore but if you look at many periods of the Renaissance and and sort of ancient Greece Etc theater in ancient Greece was free for the citizens it was like you know provided because they had sufficient wealth that they could support the Arts I'm hoping that the spirit of productivity will come also with a flourishing of innovation of creativity of the Arts of the not necessarily necess not necessarily needed for my survival and and for my kids but more like places of interest and hobbies longm we get to that sorry I I share your optimism but I thought about it differently that long term we'll get to that positive place and we'd have a short-term challenge where folks would need they be out of their job and would need to get retrained and it wouldn't be equal because folks that were just like nearing retirement it'd be harder for them to get retrained thought workers who are used to learning new skills would pick up on things but folks that more labor intensive manufacturing might be more difficult three tool I'm with you there on the short term but I don't see dramatic changes of jobs in this immediate short term I think that AI still has a lot way to go before yeah and also to reflect on that that as a person working on AI agents I see this right now in the industry this transform the shift the talks about human in the loop the talks about giving out accountability or having control ourselves so yeah this is very good uh topic I want to since we are touching more on the regulation side rard I wanted to ask you the question now as a kind of a researcher um or even like just a person um considering your lecture and and what you discussed uh about regulations so what do you think no regulations or some regulations or a lot of regul regulations what I mean again dilemma right no regulations um give us freedom freedom for Innovation but increases risks some regulations might even not perfect one um reduce our capability to innovate but on the other hand it protects us from again from say cyber warfare happening and and and things around that deep fakes um utilization of uh our private information and so on what do you think about that uh this a I mean so of course the answer has to be probably some regulation uh I I do think I mean I think maybe regulation is not the right way to think about it right I mean building a system where regulations aren't needed maybe that's what we should aspire to uh and that's like just flipping the perspective a little bit because typically when we think about regulation it's like well we have these giant players we're going to regulate them it's like well maybe if we have small localized player we don't need that type of regulations because it's going to self-regulate and I think self-regulation is great it's very antifragile and very you know something that actually I think if we look historically it's been very very successful when we self-regulate locally instead of having this global system we try to kind of regulate on a global scale I don't know if that makes sense but yeah I think that's that's what I would say what I'm concerned about is regulatory capture basically the uh the concept that the big players will be able to cope with regulation because they have the resources to do so and if we ask AI to be interpretable to be traceable to be fair to never be biased to never be racist Etc then there's like three or four players at most that will be able to cope with that regulation and all of the little guys will will die so I'm a little concerned about that basically my um how do we how do we achieve this without the um without stiffling Innovation and my take on this is the human using it is ultimately responsible um I don't want a car that will prevent me from you know going on the sidewalk or avoiding a collision or you name it because it stiffles my uh roles with it basically I I want rules that prevent people from killing others with their cars basically the car is absolutely a weapon but changing the car so that it cannot be used as a weapon basically means that the 99.9999 % of users will be not able to be as safe anymore not be able to sort of you know be as fast or as sort of dance on the highway and so on so forth so um I I think that if you want to make an AI that will just never say anything racist you will just stiffle all the little players If instead you punish the user who basically now spews out racist rhetoric by using AI then AI goes back to being the tool rather than the agent so I feel that right now we're kind of delegating responsibility for the action to the AI the action should stay with the human and then the regulation becomes much easier because the the user is ultimately responsible and it's not about o can I jailbreak the system so that it says nasty things no just like if you say nasty things you're responsible I'm sorry it's not the AI I think we have time for the last question from the audience if anybody um have something okay some exercise so like artificial intelligence is inherently tied to the data we use to trade it on right so like large language models there's been lots of controversy on where do we get that data from who owns the data like how is it related especially in like biology too when you're using like know people's genes how how do you see this like how do we continue to use and like innovate in AI when like D I guess the data kind of question is going to always be there of like who's it is how do we regulate it how do we distribute it effectively like I just want to see your opinions on where that's going to go in the future as that gets more and more of a challenging question it's a question about privacy Mo mostly yeah what was the last one you made privacy yeah so let me answer that at first basically um yes absolutely the New York Times has been taken advantage of by open AI no doubt about that um that said um a book I'm reading a book I'm now writing something that's inspired by that book that's what we call um you know Academia that's what we call you know uh research that's what we call academic work basically um if you steal from One Source it's plagiarism if you steal from a 100 sources it's research so so in my view open AI yes absolutely stole from The New York Times but not uniquely from The New York Times and and the question is at what point will a lawsuit that prevent open AI from training on the New York Times will suddenly prevent me from doing research because everything is copyrighted the other thing is um plagiarism has an enor gradient and I can take thousand articles written on a given topic and basically align them that's what we do in genomes all the time we do genome alignment and and we have tools for basically aligning every New York Times article to an article that precedes it that is almost identical so basically if you open that kind of worms then where is human Innovation because everything is a gradient there's very little that I can write or that anyone can write or that an AI can write that hasn't been kind of said before and I think there's a song about that woohoooo um um and um I I I feel that we need to rethink what Innovation means we need to rethink what attribution is we need to rethink what novelty and what plagiarism is and what research and integration is and in my view AI is really about integration of knowledge it's not about copying you really have to go out of your way to get CH GPT to spit out a whole New York Times article it's very very hard they may have manage to but in very weird sort of esoteric scenarios most of the time it's going to be some kind of integration and the New York Times will not be the only place where that sentence appeared and if you want to start suing people then everybody's in trouble um there was an experiment with music and Melodies that was done recently where BAS basically a group I think at the media lab basically created just every possible Melody systematically and then just put it all in the public domain basically saying listen stop suing each other Melodies are finite and and you know we've we've kind of created them all in some way human thought might be finite and I'm hoping that the boundaries will be pushed by exp exposing the mundane and then sort of recognizing the more Innovative so my My Hope again as an optimist is that we will push the frontiers of human creativity and that uh this will reach a new level that we don't we don't have to worry about the pettiness of similarity of small TX but really about the leaps that we can make going forward yes have a okay so I don't know much time we left but I mean I I think I'm also very optimistic about the future in AI I I really am but I think I mean what I really do think and the pessimist part is that like I don't trust our own thinking and beautiful ideas and theories I'm much more Prat pragmatic and I think that's uh important moving forward as Technologies just being like well let's not assume that we're going to be able to figure everything out or that research is going to be able to you know decide all the questions about what's going to happen in the future I think we have to be pragmatic and and try things and uh yeah also my comments I'm very optimistic about the future of of AI but not that optimistic about the future of humans but on the other hand maybe that's a good thing maybe we push the evolution further down the road I mean essentially philosophically we don't know whether this is good or bad but anyway can I make oh yeah sure absolutely so so genetically humans are not going to change any anytime soon just like let's be blunt the scale at which and the speed at which things are changing is like basically if you look at the curve of how humans have gotten smarter and we have is very very slow if you look at the speed that how AI has gotten Smart in that scale it's basically a vertical line just and no matter how much you zoom it continues to be a vertical line um we're not going to outpace AI by genetic changes however if you look at the human brain there's tiny numbers of changes that made that made us so much smarter compared to chimps in terms of the speed of neuron replication the number of Divisions the growth The Waiting before you start fixing basically if a kid develops too fast then and sort of fixates their neurons too fast they they don't have the ability to grow as much and and sort of we have delayed this growth so there's a small number of regulatory changes that have sort of made chimps into humans if you look at the brain um if if you now look at how much remains with nurture how much remains with a small number of pills that you could give to sort of you know expand the number of neurons that get created or the speed of connectivity or you name it basically we are starting to understand a lot of the knobs that you can turn to sort of really make human brain like in biological cognition dramatically uh more capable even with today's understanding and tools you know the ethics of course are enormously complex but I'm just saying that we're not going to be seeing genetic changes in humans but we can't see biotechnological and techn like basically biotech changes that sort of push humans to the next level and that's just humans if you now add nurture and sort of the way that we train that's enormously higher and if you add the human machine interfaces that's enormously high as well so basically I see human capabilities as dramatically expanded without any changes to our heart without any change of of the ones that I just mentioned just by using AI agents and sort of pushing further and allowing kids to be more creative in mathematics in you know physics in all of these other areas because the rate limiting step will no longer be the teachers I feel that High School teachers have their own limitations they're humans but by if you replace the extraordinary empathy of the teacher and you Argent it sorry if you arent the empathy of the teacher with the knowledge and capabilities of AI and the whole world then suddenly teachers don't have to be constrained by the stuff that they understand deeply no they can enable their their students to understand stuff way more deeply than they do and I think if we if we approach this with humility we can let children dramatically increase the pace of learning so I think that the moment AI is in the picture and we want to push humans to new heights we haven't even begun to tap that and I think AI will be our partner there sorry do want the comment we are quite over time but yeah I mean I think that I mean this this just a I mean of course I think it sounds great and I think you're under I mean I agree with you a lot I just think that there is a lot of uncertainty in a lot of the outcomes right it's going to be a lot of great things there's going to be a lot of bad things so let's just you know take both possib or like I mean every possibility in you know in our consideration and also the things don't even think about right the unknown the unknown unknowns so like there's a lot of things we don't have we have very little idea what's going to happen what's going to read outcomes here so let's you know accept accept that and don't assume that we're going to know everything okay so we're sure that teachers and waitresses maybe um those who have social skills Will Survive we are not sure about everyone else but anyway I think that's a good closure uh for today's topic um thank you very much for Professor Manu skis for participating in this panel and Ricard BR gabrielson thank [Applause] you I'm going to pass back to scale of one to five how much did you guys enjoy this conversation yeah awesome great

MIT_805_Quantum_Physics_II_Fall_2013

20_Multiparticle_States_and_Tensor_Products_continued_and_Angular_Momentum.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: I'm going to get started. And after a couple of minutes there will be an announcement. So today's lecture will begin by looking at the singlet state for a few minutes, a couple of its properties. And then we go into this Einstein-Podolsky-Rosen argument. We'll talk about what they said, what they wanted to criticize quantum mechanics for. And then we'll go through the answer given by Bell that actually demonstrated that Einstein, Podolsky, and Rosen were wrong. And after we'll do those Bell inequalities, we'll close this chapter on quantum states of spins and begin our treatment of angular momentum. So I want to remind you of a couple of things. We've been discussing this so-called singlet state. Plus, minus. For the second particle, minus minus, plus. First and second particle. This state. A few things we've discovered about this state--you've been working with it. We calculated its z-component of total angular momentum, the angular momentum of the first particle and the second particle. And it was 0. The x-component was 0. This y-component is 0. In fact it doesn't have any total angular momentum. So this state is rotationally invariant, we say, because it doesn't have angular momentum. You did very fine in the homework that the state is in fact rotationally invariant. Now this state is a very interesting state. It's one of those entangled states that we discussed last time when we were talking about teleportation and Bell states. And apart from that, it's a state that is not hard to realize physically. In fact, it typically takes place, for example, in reactions in decays of certain particles. For example, pi 0 can decay into two photons. Then the two photos can be in the state of total angular momentum 0. But more precisely, since we're talking spin-1/2 particles, if you have a meson called an eta 0, it's a interacting particle of strong interactions. A meson decays rather quickly into a mu-plus plus a mu-minus. Actually it decays into other things as well. So it decays into two spin-1/2 particles. And this particle has 0 angular momentum. It's a scalar. It's not spinning. And therefore, if these two particles go into a state of 0 orbital angular momentum, conservation of angular momentum implies that these two particles are in the state of this form that has 0 spin angular momentum, total spin angular momentum. So the realization of an entangled state like that is fairly common and fairly easy. So you have a decay of this form and you get particles that are entangled this way. You showed that this state actually could be a written as one over square root of 2, n-plus, n-minus, 1, 2, minus n-minus, n-plus, 1, 2. Precisely because this invariant and the rotations, you could use instead of the plus-minus basis, any basis n. For any direction n, you have this state. Now we'll talk about the probability that is of interest to us. We'll write to this symbol, probability to get a plus, say, b plus. And this means that probability to get particle-- let me see-- to find the first particle-- particle with spin along a so that this first particle to be in the state a plus. And the second particle-- particle will spin along-- along b. So in this state, b plus. Now that looks-- this calculation of this probability-- So you're going to do some measurement. And you ask, what is the probability I find the first particle in this direction, second particle pointing in this direction? It may look like a somewhat non-trivial calculation. And it is. But if you use the fact this the state over there-- because we're asking for this probability on this state. We're going to be talking about this state. So if you put the state and you write the state in the form, you pick one of the two vectors, say, a. Well, the state is a-plus, a-minus, 1, 2, minus a-minus, a-plus. Because I could choose n to be anything. So might this well choose one of the two vectors to be a. And then you ask what is the probability to find first particle in this state and second particle in this state? Well, when somebody tells you, what is the probability to find a particle in a state, you put that state into-- you sandwich your psi with that state. And this overlap, which is a number, you square it. So we're going to do the same thing here. So what is this probability? Probability to find a plus, b plus, would be the absolute value. And then we put here a plus in the first state. Tensor product, we could say. Well, b plus in the second state. And we should put-- because this is what we want to find, the state on the tensor product that we're looking for. And we put the psi here. So we must put the 1 over square root of 2 times the a-plus 1, a-minus 2, minus a-minus 1, a-plus 2. So [? overlap ?] time, well, we go 1 with 1, 2 with 2. So well, a-plus with a-plus will give me 1. b-plus with a-minus, I don't know. Second term, a-plus with a-minus gives me 0. So this term is irrelevant. I just need this. So the a-plus with the a-plus gives me 1. I have the 1 over square root of 2 here. And I must close this and square. I forgot to write that. So what is this probability? A-plus, b-plus is equal to-- the 1 over square root becomes 1/2. And then all we have left is b-plus with a-minus. In the second state space, in the particle 2 state space, the label doesn't matter at the end of the day now that we've disentangled the 1 and 2. So it's b-plus a-minus. I don't have to write that it's 2, 2. Or you can write it. And it's this squared. So it's simplified a lot but not quite yet the answer. What we need here is the overlap between these two spin states. And I remind you that when you had any arbitrary spin states with n and n prime, long, long ago, homework three or four, something like that, you calculated the overlap between these two spin states. And the answer was that you would take cosine of 1/2 of the angle-- if there is an angle gamma. The overlap between the spin states squared was cosine squared of 1/2 the angle. So here we have the vector a, see, the vector b. Here is the vector a-minus, a-minus direction. And if we call this theta ab, this is pi minus theta ab. So this should be 1/2 cos squared 1/2 of pi minus theta ab. 1/2 cosine square of half of the angle between the two relevant vectors. So this is cosine of pi over 2 minus theta over 2. That's sine squared of theta ab over 2. So here it is. It's our calculation of this thing. It's a neat formula that we're going to need later today. So one more comment before a little stop, if b is equal to minus a. Now in that case you should be in luck because precisely what's happening here is that if one spin is along the plus direction the other has to be along the minus direction, whichever you choose. So in order to have that a be in plus and b be in plus, well, this first term would do it. a is in plus, the first particle. Here, no. And b, which is minus a, would be in plus. So the probably should be 1/2. So if b is minus a, the angle theta ab is equal to pi. And the probability of ab is 1/2, correctly. And it's 1/2 because half of the cases a is in plus. The other cases, a is in minus. So the other case, for example, that could be interesting is, what is the probability that the first particle is in z-plus and the second particle is in x-plus? Well, these two vectors form 90 degrees. So you should have 1/2 of the sine of half that. So that should give you 1/2 of the sine squared of pi over 4. And it's 1/2, then it's 1 over square root of 2, that's another 1/2. So it's one quarter, for example. OK, I went long enough for a moment. There's an announcement that [? Preshanth ?] wants to make so please listen to him. [? PRESHANTH: Hi ?] I'm back for another announcement. So tonight, the MIT SPS is going to be holding its fall UROP lightning lectures. So if you don't have a UROP it's a great opportunity for you to come and see what other research your classmates are doing in physics. If you do have a UROP, I would encourage you to come as well because you can actually come and share your stories about the technical content of what you're doing in your UROP. It's 7:30 this evening in the PCR 8329. There will be free food, so please join us then. And see you all then. Thanks. PROFESSOR: Thank you. All right. So, this was the introduction to what we really need to do today. So before we get started, are there any questions on what we've done so far? On these properties of this entangled state? So this is a measurement of an entangled state. These two particles could have flown away a big distance. Two observers, one tries to see what is the probability. The first observer sees the spine pointing in some direction. The other observer sees the spin pointing in another direction. It's a the natural question which can be done experimentally. And we've calculated that answer. Now so let's begin with this EPR story. Now, you've seen some of EPR last semester in 804. The only complication with that is that you really needed to have these mathematics to appreciate it completely. So this second look at EPR should be fairly complete in that we won't leave almost anything out of the story. There are many ways of doing EPR and essentially these Bell inequalities, which is the really non-trivial thing that comes after that. So some are in the homework, some elaborations. And probably in recitation later in the course we'll see a little more. But it all begins with a strange thing, the kind of thing that you wouldn't expect people in physics to discuss. And it's the point of this so-called-- Einstein, Polosky, and Rosen wrote a paper. And they talked about local realism. Now, that sounds like philosophy. And for awhile people thought, well, this is interesting, but undecidable. Can't really do anything with it. So what is local realism? Now, again, not being exactly physics, it's not all that easy to say what it is. And people discussed that. But some notion of it is fairly clear. The notion is that this reflects something-- it's basically two assumptions about measurement results. So you measure something and obtain a number. And the first assumption, one, is that these measurement results correspond to some aspects of reality. Just said like that it seems a little funny. That you measure something, if get some numbers, because that was something real about this object, it had this property. And so measurement corresponds to some aspect of reality. So measurements-- assumptions about measurement results. So measurements. m, correspond to some aspect of reality. Two, the measurements that you do in you lab are not affected whatsoever by the measurements that somebody else is doing at the same time at the moon. There's no time for the information of what that result in the moon has given to reach you. So at that instant of time, what they are doing at the moon doesn't affect the result of your experiment. So it's measurement is independent of actions performed at a distant location at the same time. Now to Einstein and Polosky and Rosen-- but Einstein was very vocal-- Physics must satisfy that. It's kind of sad I think, actually. The person that managed to see through and discover how nature works at so many deep levels-- the photoelectric effect, special relatively, general relativity-- somehow became convinced that this had to be true. And unfortunately, he was wrong. Or fortunately, I guess. It's not worth trying to qualify that. But these two things that seem just so reasonable are just not true. This one, measurements correspond to some aspect of reality-- you see, you have a Stern-Gerlach apparatus, you throw a spin, it goes up. You say it ended up with spin up. Well, Einstein would say it always had spin up. It was a reality about that object, at it had spin up. You just didn't know. You did the experiment to you discovered it. So the thing that people try to do in order to understand this concept, that it corresponds to something having to do with the reality, is that you admit that half of the particles go up and half go down. But you say, actually, there's something about these particles you don't know. And if you knew that, you would just be able to tell. This is a particle that has spin up. And it will go up. But in quantum mechanics, we have abandoned that. We've said these particles here are a superposition of a state up and a state down. And there's nothing definitely up about this particle or definitely down. So the way people do this, to correspond to this aspect of reality that you don't know, is by introducing what is called hidden variables. Some things that allow you-- there's something hidden about this spin particle that you don't know. But if you knew it you would know exactly how it's going to come out through this Stern-Gerlach experiment. And you say, well, that sounds fairly untestable. But the fact is that it's not. Now, so this is implemented, this assumption is-- when people try to modify quantum mechanics, they use what is called hidden variables. Some things that you don't know about the particle, but if you knew, you would see that in fact this particle has spin up. This second is in some ways even more disturbing because we got accustomed to the idea that, locally, simultaneous things that cannot be reached-- events that cannot talk to each other via the exchange of light cannot effect each other. So simultaneous things that's happened far away can't effect each other. So this also sounds very reasonable. But that's also wrong. And there's the obvious question, so if this is wrong, can you send information faster than the speed of light? And people looked at it in many, many ways. And it's very interesting. And we could discuss that. But it would take us long, so I will leave it to, maybe, the recitations, maybe other forums. But here, actually, there's no contradiction, no way of finding real information going faster than the speed of light, even though things far away at the same time can affect you. So two very interesting things that seemed very dangerous to discard but turned out to be wrong. So this is what EPR did. And they made some thought experiments that we're going to review to some degree and see if we can discard these assumptions. So that's what we're going to do now. We're going to try to understand that. Now, if you're interested in what hidden variables are, my discussion will not use hidden variables. Although they are kind of implicit, you will see, as I state some things. I'm basically going to be explicit on the fact that things, [? the ?] [? real ?] [? facts, ?] as Einstein would like you to think. And we'll try to see if those real qualities about particles get us in trouble. So let's begin and try to discuss this first experiment. You see observer one, Alice and Bob again, if you want. Alice is going to measure about the z-axis. Bob is going to measure about the z-axis. And therefore, if Alice finds spin up, Bob finds spin down. If Alice finds spin down, Bob finds spin up. And that's a correlation. And it's very interesting. But Einstein, Bell, and Polosky would say, look, it's not all that interesting. There's nothing all that mysterious happening here. EPR would says, the pairs that you've built, the so-called entangled pairs, are pairs of particles with definite spins-- spin directions, spin vectors. So what you have built, EPR would say, if you have here particle 1 and particle 2-- I'm going to list the properties. Suppose you have a particle 1, you've created. Einstein would say some particle 1s with spin up in z and particle 2 with spin down in z. Or you've created particles 1 with spin down in z and particle 2 with spin up in z. And you've created [INAUDIBLE] 50% of the particles are of this type, of the pairs. And 50% of the pairs are of this kind. So don't tell me all this superposition hocus pocus. Half of your pairs, one particle has spin up, one particle has spin down. The other 50% of your pairs, particle one is down, the other particle is up. No wonder they're correlated. You get plus, gets minus. Get minus, get plus. What is the probability that you get plus? 50%. What is the probability you get minus? 50%. Everything is reproduced. Mystery over. No quantum superpositions. OK? There's no mistake here. No mathematical mistake. You're saying that the particle has a definite spin. You maybe not know it. But we say, look, your particles in fact have a definite spin, z-plus and z-minus and those things. And this is where hidden variables would come along. You would say, well, if you have the particle 1, it's spin is a function of some hidden variable that you don't know. But if you knew it, you would know what the spin is because it has a definite spin. You don't know what is the hidden variable. But as a function of the hidden variable the spin is known, definite, not a superposition, nothing like that. It's a very aggressive attack on quantum mechanics and something that troubled people. And in fact fascinates people even up to now because the idea, these kind of things are really absolutely wrong. It's very shocking. Perhaps the second even more shocking because, by now may be you're accustomed to all kinds of variables that are a little more detached from reality. You have electromagnetic fields. And they forced you to learn about potentials that seem to be a little more abstract. And similarly here. So no problem at all. So people say, OK, this is a simple situation. But it may be that we're going to do more measurements. And we're going to consider two directions that are different. Maybe Alice has two Stern-Gerlach machines, one that measures about z and one the measures about x. And Bob has also to Stern-Gerlach machines, one of the measures about z and one that measures about x. And they are going to ask different questions. Because we know the spin, how it transforms. So getting those results right with two directions is going to be a little more interesting. So we're going to try measuring in two possible directions. Both A and B, Alice and Bob, can measure in two directions, z and x. And Einstein would say, look, you can measure in z and in x. To avoid confusion, let's not talk about one measurement after another. Particle comes, you measure in z or you measure in x. And realism says that you don't know what you will get because maybe there are hidden variables. But each particle has a definite answer if you ask what is the z direction of the spin and has a definite answer if you ask what is the x-component of the spin. Definite answers, real answers, realism again. So I'm going to label the particles. For example, this is a label for a particle z-plus, x-minus. This particle labeled like that would be such that if you measure its spin in the z direction, it's plus. If you measure the spin in the x direction, it's minus. So measured in z, it is up. Measured in x, it is down. These are the kind of particles that EPR would say exist. It's not that you got a particle out and it's in some strange entangled state. These particles are flying away. They're not talking to each other anymore. And this particle, since you can measure in two directions, there's some reality, and the measurements correspond to reality, so there are attributes. And this particle is classified by having these attributes. If you measure z, plus. If you measure x, minus. So how about this situation. Well, let me make a list now. Particles 1 and particle 2. And this would be the list that EPR would do for you. EPR comes along and says, look, here's what you're doing. Particle 1, suppose it's a z-plus, x-plus. Well, in your beams, actually, when particle 1 is that, your other particles are z-minus, x-minus. So if particle 1 is of this type, particle 2 is of this type. That way, EPR protect themselves because they say, look, if you measure z-plus and you measure z-minus, you get correlation. If you get plus and plus, you get correlation as well. Now there can be particles in z-plus, x-minus. This will go with a z-minus, x-plus. There could be a particle z-minus, x-plus, and this as z-plus, x-minus. And there could be particle z-minus, x-minus, z-plus, x-plus. So there are four cases, four types of particles they say have been produced. And 25% of pairs are of this form. 25% of pairs are of this, 25 of this, and 25 of this. So you could ask some questions to EPR. What is the probability that you get z-plus for 1 and z-minus on 2. Well, z-plus on 1, it's these two cases. And z-minus on 2, well, those two cases. So this is 50% of the times. So it's 1/2 that probability. It's correct. That's what we would predict from an entangled state viewpoint. But let's ask something mixed. Let's see if we get in trouble. P of z-plus on 1, and let's say x-plus on 2. Well, z-plus on 1 and x-plus on 2, that's not it. This case. z-plus on one, x-plus on 2. 25% of the times, one 1/4. This was the probability we calculated. So you tried with one direction. You don't need quantum mechanics to produce a result. You try with two directions, you don't need quantum mechanics to get the result. So people got stuck and they said, well, maybe it's undecidable. Maybe this is philosophy. Maybe this is something. And people had to wait for Bell. He said I'm going to try three directions. Now, three directions makes a big difference. This is the first time it really goes wrong. So it's kind of surprising, perhaps, at some level. But this is subtle stuff. So it takes a while before you find something wrong. You're talking about showing Einstein it's wrong, so that's not so easy. So three directions. So particles are going to be of types-- EPR would say, look, I'm going to use the same strategy. I'm going to say that particles have three attributes now. They're all physical. They correspond to reality. Because if you measure in either of three directions, they have to have an answer for that. So here's a label for a particle. And for example, a-plus, b-minus, c-plus. So if you measure, it would give spin in a direction plus h bar over 2. Spin in the b direction minus-- well, in the b direction would give you minus h bar over 2. Spin in the c direction would give you plus h bar over 2 . So we're not measuring simultaneously. We're just asking, well, you take a particle, do a measurement, and see what you get. And we're always going to be asking for probabilities of this kind, probably that the first particle is doing this and the second is doing that. Well, EPR would start now with particles again, particle 1, particle 2. And populations. So let's list quickly the particles. a-plus, b-plus, c-plus. Then you will go a-plus, b-plus, c-minus. Then you've done the c-plus, c-minus here. You go for two more. A-plus, b-minus, c-plus. a-plus, b-minus, c-minus. I was supposed to fit four more there. Can I? Well, I will try. a-minus. Now you've done all the four a-pluses so you need four a-minuses. Then you have b-plus, b-plus, b-minus, b-minus, c-plus, c-minus, c-plus, c-minus. I got all, I think. And particle two of course is correlated. You would say, well, I don't need to write it. But it helps seeing what's going on so I'll write it. a-minus, b-minus, c-minus, a-minus, b-minus, c-plus, a-minus-- let's use bars here-- a-minus, b-plus, c-minus, a-minus, b-plus, c-plus, a-plus, b-minus, c-minus, a-plus, b-minus, c-plus, a-plus, b-plus, c-minus, a-plus, b-minus, c-- this one is minus so it's plus here. We're done. Lots of labels. And you could say, well, maybe you want to put 1/8 in each of them. But actually the argument is more interesting. It doesn't need you have to try to put fractions. So let's consider that there's a total number of particles N, which is N1 plus up to N8. And here are N1 of this, N2 of this, N3 of this, N4 of this, N5, N6, N7, and N8. Let's see how-- All right, so we have that. Well, it takes imagination to see how you're going to run into some contradiction. So what is the basis for the contradiction? Somehow this formula, which is really quantum mechanical must eventually go wrong with all these attempts to deny that the world is quantum mechanical. So we could split again those particles into equal fractions. But there's no need to do that. And it's clearer if you don't. So you try to combine the three directions into one equation. So one way to do that would be to say, OK, what is the probability that you get a-plus and b-plus? So this is for the particle number 1. And this is particle number 2. So then you must look at the table and say which one's do that. Well, you need a-plus in the first so it's one of the first four rows. And b-plus in the second. So actually, it's these two cases, N3 and N4, over N. We want to involve three directions. So let's go for another one. P of a-plus with c-plus. Let's see how much is that. Again, this is for the first particle. This is for the second particle. So I must look at the first four rows. And see that you have an a-plus. First part is in a-plus for the first four rows. But the second should be in c-plus and that is cases N2 and N4. So we get N2 plus N4 here over N. Well, we've involved this a with b, a with c. How about involving b with c? So I'll put c-plus, b-plus, for example. OK c-plus, b-plus. I must look at c-pluses, and b-pluses, no. C-plus and b-plus, yes. N3 is there, which is good because it already was there. And which else? c-plus and b-plus. c-plus, no. c-plus here b-plus, yes. N7. Now N3 plus N4 is less than N3 plus N7 plus N4 plus N2. You see you have N3 and N4. And now I add whatever N7 and N2 are. And that's then an inequality because it's going to be more cases. Now I divide by N. So you obtain an inequality that P of a-plus, b-plus is less than or equal than N3 plus N7 and N4 plus N2. I'll right the second first, P of a-plus, c-plus plus P of c-plus, b-plus. So I didn't put specific populations 1/3, 1/4. But in general whatever populations you choose, this inequality must hold. So that's the more clever strategy. Because suppose you choose some populations and you don't get in trouble, well, maybe with some other populations you would get in trouble. Maybe it's not so easy to get the relative factors. So here is something that must be true whatever populations you choose. And now that is Bell's inequality. So the achievement of Bell is to somehow translate this assumption of realism into an inequality. And now quantum mechanics has a formula for these things, for this probability. So we can test whether this is true. So let's do that. This is the so-called Bell inequality. So if quantum mechanics is true, the following should hold. Let's see that. If QM is true, well, there should be a problem with this inequality. So let's see what happens. Let's see if it's true, this inequality. Is it really true? Well, the left hand side would be, given the formula that we had, 1/2 of the sine squared of theta ab over 2. So let me just emphasize, this was derived using local realism. Local realism gives that. So you do the experiment, get these probabilities. And if realism is true, this should hold. Let's see what quantum mechanics has to say. Let's plug-in the values that you get from quantum mechanics. Now we calculated this probability. We put the first term. Here is sine squared of theta ac over 2 and 1/2 of sine squared theta bc over 2. So does that work? Does that always work? Can I orient this axis in such a way to disprove EPR? And in fact, it turns out to be quite easy to do that. So you choose three vectors like this. ac, bc, so c here, a here, and b here, I believe. Yep. So put an angle theta here. An angle theta here. And then what do you have? Theta ab would be 2 theta. Theta ac would be equal to theta bc equal to theta. That's a pretty nice simple choice of angles. If you choose these angles now, let's see what happens with our inequality. So you get 1/2 sine squared theta ab over 2 would be theta. Is it less than or equal to this? Well, these two become the same. So you get sine squared of theta over 2. Violated or not violated? Is it true or false, this, for all theta? What do you say? Yes? STUDENT: If theta is less than pi, that's not true. PROFESSOR: Close. It's not true for a small theta. So if you're this, and you're desperate to know, the thing you have to do is assume theta is very, very small. See if you get in trouble. How much is this? 1/2 theta squared. Let's see. Half of this one? Yeah. And how much is this? This is theta squared over 4. Sorry, I was not seeing it. Sine theta for small theta is roughly theta. So here is theta over 2 squared. But here is 1/2 theta squared. And it's false. The 1/2 theta squared is not smaller than one quarter theta squared. And in fact, for theta equal pi over 2, I think this is an equality. Because theta equal pi over 2, you get on the left hand side 1/2 less than or equal than sine of 45 degrees, which is correct. So it's an equality of pi over 2. Fails below. So it was a shock. That if you could do an experiment in quantum mechanics and experiment with correlated, entangled particles, that you could to measure these probabilities, these correlations. And you would obtain a result that actually contradicts for certain alignments of your experiments the assumptions of local realism. So it was a great result of Bell to show that quantum mechanics is in contradiction with local realism. There's no way to keep the ideas of quantum mechanics and put hidden variables and assume that there's real values for things and that there's no effect at a distance. It would all be contradicted by an experiment. That was done later and the definite version of the experiments around 1980 or '82 by Alan Aspect and others. Very clever experiments worth reading and understanding. But confirmed that the quantum mechanical result is really true by measuring correlated pairs. And this inequality is violated. Yes? STUDENT: So what about David Bohm's theory of hidden variables quantum mechanics? So-- PROFESSOR: David Bohm's theory of what? STUDENT: His hidden variable quantum mechanics theory allegedly reproduces the same results as quantum mechanics but it's still a hidden variables theory. PROFESSOR: I don't think there's any hidden variable theory that works. David Bohm, I think, actually was credited by rewriting EPR, who essentially talked about position and momenta to talk about spins. And he might have been the first one that began to try to do hidden variable theory. But no hidden variable theory works at this moment. And this shows it. So people say that actually this assumes that there's local hidden variable theories and there's non-local hidden variable theories and all kinds of strange things. But it's more and more unnatural. So it doesn't seem to do something very interesting. Yes, there are many questions. Aaron, maybe you want to say something. STUDENT: Let's see, I think Bohm has a non-local hidden variable theory solution. It's kind of awful looking. But I guess violates the second principle rather than the first one. It also doesn't extend to-- this doesn't really work for everything. We don't know how to make it work for a spin-1/2 particle to find [? a dimension. ?] So [INTERPOSING VOICES] really believe to be a true theory of [? equivalence. ?] PROFESSOR: Thank you. More questions. Steve? STUDENT: In the case with EPR, is it a problem that we can have scenarios where the spin is greater than a spin-1/2 particle would have? If we had the state a-plus, b-plus, c-plus, we would have 3 h bar over 2? PROFESSOR: No. The statement that is done here is not that-- well, this is a label for a particle. EPR just assumed that if you measured a you would be able to get this. If you measure along b, you would get this. And if you measured along c, you would get this. So this is a single particle. Any measurement gives some results this side of the list. There's no sense in which these are added. STUDENT: But even when you measure one or the other, the other values still exist for those measurements. PROFESSOR: Well, I believe there's no need to discuss that. So they don't talk about the statement of doing subsequent measurements within this statement. You just take this particle and you decide. You measure a or measure b or measure c. You don't try to measure simultaneously. You don't try to measure one after another. You just do one measurement. And that already, which is the minimum you can do, gets you in trouble. So I'm not sure how EPR would phrase subsequent measurements after they've done the first measurement or things like that. But they're not necessary for this stage. OK, so look, it's a very interesting thing. There's lots to discuss here, but it's best if you read it. James had a question. Just let's take one quick question. STUDENT: I was just wondering if there was any other extension beyond three directions for Bell's inequalities. Is there a N-direction of Bell's inequalities or some type of form of it? Or is there [INAUDIBLE] PROFESSOR: There are other forms of Bell inequalities. I'm not sure if it's popular with four directions or anything. But certainly Weinberg, for example, discuss other ways. There are alternative ways to phrase it. I've talked about here probabilities to observe results. There is a more, perhaps, [? common ?] way talking about expectation values or correlation functions. This is something you'll do in the homework. The sort of game that is done in the homework that was a suggestion by Aaron to put it in the homework-- this game in which with quantum strategy and entangled pair you beat the system-- is yet another formulation of the Bell inequalities as well. So lot to do. But I think it's better now that we stop and talked about angular momentum from now until the end of the semester. This was about spins, but we now have to put together angular momentum which is orbital and spin and all the various kinds. So we'll begin with angular momentum. So there are notes on the web on that that I wrote and modified a little this time. And lots of little exercises. So what I want to do now is guide you through the things that happen there so that you get a view of what we're going to do. We're going to work with this thing in a elegant way using vector notation for operators is going to help us understand things better. So we've seen angular momentum before. And let me summarize simple things that we know about it. If you have angular momentum, and let's begin with orbital. This is what we have when we take Lx to be y Pz minus z Py. Ly equals z Px minus x Pz. And Lz to be equal to x Py minus y Pz. These are the angular momentum, orbital angular momentum operators. Now, it's better for many things to use labels like x, y, and z, those operators. Call them x1 hat, x2 hat, x3 hat. And Px, Py, Pz, P1, P2, P3. In that way you can write commutation relations like xi, Pj equal i h bar delta ij. xi with xj equal pi with pj equal 0. So it allows you to write things more quickly. In fact, the angular momentum operators become also a little simpler. It's sort of x2 P3 minus x3 P2. And you'll have the x, y, z labels and all that. So we want to use vector notation. Now, vector notation, you can do it in two ways. You can talk about triplets. Those are vectors. Or you can form the vectors themselves. Now if you form the vectors, you get objects that sometimes are a little disconcerting. But we try that they not be so. So here is the r vector operator. You could think of it as the triplet of x, y, and z. But let's call it like this, x thing times the first basis vector plus y operator times the second basis vector plus z operator times the third basis vector. And we've done things like that. And we understand that these basis vectors really don't talk with these operators. They can be moved across each other. The basis vectors are things that help you write expressions, rather, at talking about triplets. Same thing for momentum. Let's do it this way. P1 e1 plus P2 e2 plus P3 e3. And finally, well, angular momentum, the vector operator-- you've done a lot of the angular momentum vector operator for spin. So here you would put Lx or L1 e1 plus L2 e2 plus L3 e3. So those are ways to write equations that carry all the operators and treat them as vectors, even though they're operators. So they're unusual vectors. They're vectors whose components are operators, are not numbers. So the obvious question is, what changes then? So we're going to define dot product and cross products as we had before. But we have to be a little aware that when we write these things we could make a mistake unless we're not careful. So here are two vector operators. What is the dot product of these two vector operators? Well, you know it's supposed to be the first component of this and the first component of that, second second, third third. So it should be ai bi, summed. Repeated in this is our [? summed. ?] Now, I should not write bi ai with a reverse order because this thing, the components, are now operators. And maybe they don't commute. So I've defined this once and for all to be this. And a cross b, the i-th component of this thing is going to be defined once and for all to be epsilon ijk aj bk. Definition. The a to the left of the b. And with this, we can check our usual rules of manipulation of operators. So one more definition. a squared is going to be aa, and it's going to be ai ai. Simplest calculation is a dot b equals to b dot a. Yes or not? No, they're operators. So let's calculate the difference. Let's get a little practice calculating differences. So I write a dot b is equal to ai bi, its sum. So then I say, that's ai bi plus bi ai. You see the commutator is ai bi minus bi ai. And I add it back. But this thing, bi ai is b dot a. So here I've got a formula. a dot b is equal, actually, to b dot a plus this commutator, ai bi. And now you've got a new formula for operator vector analysis. a dot b and b dot a are not the same but they differ by this thing. And a very important corollary, very famous corollary, what is r dot p? Is it the same as p dot r? If you're working quantum mechanics you maybe tempted to, oh r dot p and p dot r are the same, but r and p don't commute, so what is the difference? R dot p is equal to p dot r plus xi pi. And how much is that? [INAUDIBLE] Sorry? STUDENT: i h bar. PROFESSOR: i h bar? STUDENT: 3. PROFESSOR: 3 i h bar. Yes, don't forget the sum. This is supposed to be summed. So it says x1 commutative, p1, x2 going to 3 i h bar. So here's a famous formula. rp differs from pr plus three h bar i-- i h bar, people write. i h bar. Another formula that you would be curious to know. Well, the dot product was supposed to be symmetric. It's not. The cross product is supposed to be antisymmetric. Is it or not? a cross b sub i is equal to epsilon ijk aj bk. What do I do next? I want to move the a and b's around. So I'm going to replace this by a commutator plus the other ordering, ijk. And I put here-- well, this would be a parentheses-- aj bk commutator plus bk aj. Now, what do we get? Well, you have to look at the second term. Let's put the first term here because that's-- pretty much, we're not going to be able to do much with it. aj bk. And the first term, I would write it like this. Minus epsilon, flip these two, ikj bk aj. If you do it like that, then it sort of fits nicely with the definition of the cross product. Because in the cross product, the first label those here, the second label goes with the last labeled of the epsilon. So this thing is minus b cross a. And that was a cross b. Plus epsilon ijk aj bk. So that's your formula for of how the cross product. Now fails to be antisymmetric. It's not necessarily antisymmetric unless you're lucky. So here is a property that you should try to think about. How about r cross r? Is it 0 or not 0? r cross r. You could say, what can I do here? Well, r cross r minus r cross r-- two r across r's should be equal to this. But both components are x's. So r cross r is 0. p cross p, that's also 0. And therefore, say, L cross L, is it 0? Maybe. Is that right? No. L cross L, we'll see it a little later, but it's not 0 because one L with another L don't commute. Lights, high. OK. So L cross L is actually not 0 because this thing is not 0. We'll talk about it a little later. But actually this is a very famous one. L cross L is proportional to L with an h bar. It's a lovely formula. So another interesting thing. Well, what is r cross p? r cross p, from this formula we would minus p cross r. And how about the other term? Is it 0 or not 0? Is r cross p equal to minus p cross r, or does that fail? Well, r and p don't commute. Actually this formula-- I should have-- somebody should have complained. This should be the i-th component, the i-th component, and here is the i-th component. So i-th component the i-th component. But then let's look at here. Epsilon ijk xj pk. But xj and pk is i h bar delta jk. Now, this delta jk is symmetric. This is antisymmetric. You should get accustomed to the idea that that is 0. If this is antisymmetric and this is symmetric, this is 0. Let me do it one more time. Maybe this is not completely obvious. But epsilon ijk delta jk. The intuition is also obvious. Epsilon must have the three numbers different and this forces them to be the same. But it's more general. If this is antisymmetric and this is symmetric, they should be 0. And the way you do that is you say relabel j and k. Whatever was called j call it k. Whatever was called k call it j. So this is epsilon ikj delta kj. And then we use the symmetry properties. So when you exchange this back you get a minus sign when you exchange this back you don't get a sign. So minus ijk delta jk. So you have shown that this thing is equal to minus itself. And therefore it's 0. Something that's equal to minus itself is 0. So this is 0. And you've got that r cross p is really minus p cross r. And that's what we call the angular momentum, r cross p. But now you know that it's also equal to minus p cross r. Let's see, one other one, for example, that is classically true. Let's see if it's quantum mechanically true. r dot L. The angular momentum L is supposed to be perpendicular to r and perpendicular to p because this is the cross product. Is that true quantum mechanically or not true? Well, maybe I didn't say-- well, we said L is r cross p. So this is r dot r cross p. So this would be ri times epsilon ijk-- no, x-- xj pk. So what is this? It's epsilon ijk xi xj pk. Well, this is 0 because these two x's are operators but they commute. Therefore this object is symmetrical in i and j. This is antisymmetric. This is 0. So r dot L is actually 0. How about p dot L. Well, there's two ways of doing that, one that looks pretty obvious, one that looks a little harder. Let's do the one that looks a little harder. This would be pi epsilon ijk xj pk. So here is the temptation. Must be 0. Because there is a pi pk. Symmetric in i and k. And here's antisymmetric in i and k. But it's this wrong. These are not obviously symmetric unless you can move them across. And there's an x in the middle laughing at you and saying, beware. This could go wrong. So you have to be a little more careful. So let's be careful. ijk and you put pi xj pk, all our operators. But pi with xj the commutator would be a delta ij and that vanishes. So this p actually can be moved across the x and therefore show that the p is on the other side. Because the epsilon is there. So it's a lucky thing. So pi xj pk is not symmetrical in i and k. But if there's an epsilon, you're in better shape. So here the p can be moved. ijk xj pi pk. And now nobody can stop you from saying this is symmetric. They're next to each other, they can be moved across each other. It's really symmetric in i and k. And therefore i and k antisymmetric, this is 0. p dot L is equal to 0 as well. So this was a little bit hard way. If you had to used that L is actually equal to p cross r, then the p would have been next to the p from the beginning. And you would have saved a little time. The same time that it took you to realize that this equality is true. So those are true equalities. p dot L is 0. And it's also equal to L dot p. It's not obvious that L and p commute. But in this case L dot p is 0. If you use this formula for L, it's obvious. r dot L is 0. It's also equal to L dot r. Doesn't make a difference. It's also true as well. So that's roughly what goes on. Look, I would like you to just read those pages. It's a continuation of this. It's about eight pages. I leave exercises to be done. That's in the homework. They're of this type, playing with it. And this is a good thing that you get accustomed to it. And the material that you need just for the last problem of the homework will be covered on Monday. And we can talk about it in recitation tomorrow. So that's it for today.

MIT_805_Quantum_Physics_II_Fall_2013

12_Quantum_Dynamics.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality, educational resources for free. To make a donation or to view additional materials, from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. So, this homework that is due on Friday contains some questions on the harmonic oscillator. And the harmonic oscillator is awfully important. I gave you notes on that. And I want to use about half of the lecture, perhaps a little less, to go over some of those points in the notes concerning the harmonic oscillator. After that, we're going to begin, essentially, our study of dynamics. And we will give the revision, today, of the Schrodinger equation. It's the way Dirac, in his textbook on quantum mechanics, presents the Schrodinger equation. I think it's actually, extremely insightful. It's probably not the way you should see it the first time in your life. But it's a good way to think about it. And it will give you a nice feeling that this Schrodinger equation is something so fundamental and so basic that it would be very hard to change or do anything to it and tinker with it. It's a rather complete theory and quite beautiful [? idea. ?] So we begin with the harmonic oscillator. And this will be a bit quick. I won't go over every detail. You have the notes. I think that's pretty much all you need to know. So we'll leave it at that. So the harmonic oscillator is a quantum system. And as quantum systems go, they're inspired by classical systems. And the classical system is very famous here. It's the system in which, for example, you have a mass and a spring. And it does an oscillation for which the energy is written as p squared over 2m plus 1/2 m, omega squared, x squared. And m omega squared is sometimes called k squared, the spring constant. And you are supposed to do quantum mechanics with this. So nobody can tell you this is what the harmonic oscillators in quantum mechanics. You have to define it. But since there's only one logical way to define the quantum system, everybody agrees on what the harmonic oscillator quantum system is. Basically, you use the inspiration of the classical system and declare, well, energy will be the Hamiltonian operator. p will be the momentum operator. And x will be the position operator. And given that these are operators, will have a basic commutation relation between x and p being equal to i h-bar. And that's it. This is your quantum system. Hamiltonian is-- the set of operators that are relevant for this are the x the p, and the energy operator that will control the dynamics. You know also you should specify a vector space, the vector space where this acts. And this will be complex functions on the real line. So this will act in wave functions that define the vector space, sometimes called Hilbert space. It will be the set of integrable functions on the real line, so complex functions on the real line. These are your wave functions, a set of states of the theory. All these complex functions on the real line work. I won't try to be more precise. You could say they're square integrable. That for sure is necessary. And we'll leave it at that. Now you have to solve this problem. And in 804, we discussed this by using the differential equation and then through the creation annihilation operators. And we're going do it, today, just through creation and annihilation operators. But we want to emphasize something about this Hamiltonian and something very general, which is that you can right the Hamiltonian as say 1/2m, omega squared, x squared. And then you have plus p squared, m squared, omega squared. And a great solution to the problem of solving the Hamiltonian-- and it's the best you could ever hope-- is what is called the factorization of the Hamiltonian, in which you would manage to write this Hamiltonian as some operator times the dagger operator. So this is the ideal situation. It's just wonderful, as you will see, if you can manage to do that. If you could manage to do this factorization, you would know immediately what is the ground state energy, how low can it go, something about the Hamiltonian. You're way on your way of solving the problem. If you could just factorize it. Yes? AUDIENCE: [INAUDIBLE] if you could just factorize it in terms of v and v instead of v dagger and v? PROFESSOR: You want to factorize in which way instead of that? AUDIENCE: Would it be helpful, if it were possible, to factor it in terms of v times v instead of v dagger? PROFESSOR: No, no, I want, really, v dagger. I don't want v v. That that's not so good. I want that this factorization has a v dagger there. It will make things much, much better. So how can you achieve that? Well, it almost looks possible. If you have something like this, like a squared plus b squared, you write it as a minus ib times a plus ib. And that works out. So you try here, 1/2 m, omega squared, x minus ip over m omega, x plus ip over m omega. And beware that's not quite right. Because here, you have cross terms that cancel. You have aib b and minus iba. And they would only cancel if a and b commute. And here they don't commute. So it's almost perfect. But if you expand this out, you get the x squared for sure. You get this term. But then you get an extra term coming from the cross terms. And please calculate it. Happily, it's just a number, because the commutator of x and b is just a number. So the answer for this thing is that you get, here, x squared plus this is equal to this, plus h-bar over m omega, times the unit operator. So here is what you could call v dagger. And this is what we'd call v. So what is your Hamiltonian? Your Hamiltonian has become 1/2 m, omega squared, v dagger v, plus, if you multiply out, H omega times the identity. So we basically succeeded. And it's as good as what we could hope or want, actually. I multiply this out, so h-bar omega was the only thing that was left. And there's your Hamiltonian. Now, in order to see what this tells you, just sandwich it between any two states. Well, this is 1/2 m, omega squared, psi, v dagger, v, psi, plus 1/2 half h, omega. And assume it's a normalized state, so it just gives you that. So this thing is the norm of the state, v psi. You'd think it's dagger and it's this. So this is the norm squared of v psi. And therefore that's positive. So H, between any normalized state, is greater than or equal to 1/2 h-bar omega. In particular, if psi is an energy eigenstate, so that H psi is equal to E psi. If psi is an energy eigenstate, then you have this. And back here, you get that the energy must be greater than or equal to 1/2 h omega, because H and psi gives you an E. The E goes out. And you're left with psi, psi, which is 1. So you already know that the energy is at least greater than or equal to 1/2 h omega. So this factorization has been very powerful. It has taught you something extremely nontrivial about the spectrum of the Hamiltonian. All energy eigenstates must be greater than or equal to 1/2 h omega. In fact, this is so good that people try to do this for almost any problem. Any Hamiltonian, probably the first thing you can try is to establish a factorization of this kind. For the hydrogen atom, that factorization is also possible. There will be some homework sometime later on. It's less well known and doesn't lead to useful creation and annihilation operators. But you can get the ground state energy in a proof that you kind of go below that energy very quickly. So a few things are done now to clean up this system. And basically, here I have the definition of v and v dagger. Then you define a to be square root of m omega over 2 h-bar, v. And a dagger must be m omega over 2 h-bar v dagger. And I have not written for you the commutator of v and v dagger. We might as well do the commutator of a and a dagger. And that commutator turns out to be extremely simple. a with a dagger is just equal to 1. Now things that are useful, relations that are useful is-- just write what v is in here so that you have a formula for a and a dagger in terms of x and p. So I will not bother writing it. But it's here already. Maybe I'll do the first one. m omega over 2 h-bar. v is here would be x, plus ip over m omega. And you can write the other one there. So you have an expression for a and a dagger in terms of x and p. And that can be inverted as well. And it's pretty useful. And it's an example of formulas that you don't need to know by heart. And they would be in any formula sheet. And the units and all those constants make it hard to remember. But here they are. So you should know that x is a plus a dagger up to a constant. And p is a dagger minus a. Now p is Hermitian, that's why there is an i here. So that this, this anti-Hermitian, the i becomes a Hermitian operator. x is manifestly Hermitian, because a plus a dagger is. Finally, you want to write the Hamiltonian. And the Hamiltonian is given by the following formula. You know you just have to put the v and v dagger, what they are in terms of the creation, annihilation operators. So v dagger, you substitute a dagger. v, you go back here and just calculate it. And these calculations really should be done. It's something that is good practice and make sure you don't make silly mistakes. So this operator is so important it has been given a name. It's called the number operator, N. And its eigenvalues are numbers, 0, 1, 2, 3, all these things. And the good thing about it is that, once you are with a's and a daggers, all this m omega, h-bar are all gone. This is all that is happening here. The basic energy is h-bar omega. Ground state energies, what we'll see is 1/2 h-bar omega. And this is the number operator. So this is written as h-bar omega, number operator-- probably with a hat-- like that. So when you're talking about eigenvalues, as we will talk soon, or states for which these thing's are numbers, saying that you have a state that is an eigenstate of the Hamiltonian is exactly the same thing as saying that it's an eigenstate of the number operator. Because that's the only thing that is an operator here. There's this plus this number. So this number causes no problem. Any state multiplied by a number is proportional to itself. But it's not true that every state multiplied by a dagger a is proportional to itself. So being an eigenstate of N means that acting on a state, N, gives you a number. But then H is just N times the number. So H is also an eigenstate. So eigenstates of N or eigenstates of H are exactly the same thing. Now there's a couple more properties that maybe need to be mentioned. So I wanted to talk in terms of eigenvalues. I would just simply write the energy eigenvalue is therefore equal h-bar omega, the number eigenvalue-- so the operator is with a hat-- plus 1/2. So in terms of eigenvalues, you have that. From here, the energy is greater than 1/2 h omega. So the number must be greater or equal than 0 on any state. And that's also clear from the definition of this operator. On any state, the expectation value of this operator has to be positive. And therefore, you have this. So two more properties that are crucial here are that the Hamiltonian commuted with a is equal to minus h omega a and that the Hamiltonian committed with a dagger is plus h omega a dagger. Now there is a reasonably precise way of going through the whole spectrum of the harmonic oscillator without solving differential equations, almost to any degree, and trying to be just very logical about it. It's possible to deduce the properties of the spectrum. So I will do that right now. And we begin with the following statement. We assume there is some energy eigenstate. So assume there is a state E such that the Hamiltonian-- for some reason in the notes apparently I put hats on the Hamiltonian, so I'll start putting hats here-- so that the states are labeled by the energy. And this begins a tiny bit of confusion about the notation. Many times you want to label the states by the energy. We'll end up labeling them with the number operator. And then, I said, it will turn out, when the number operator is 0, we'll put a 0 in here. And that doesn't mean 0 energy. It means energy equal 1/2 h-bar omega. So if you assume there is an energy eigenstate, that's the first step in the construction. You assume there is one. And what does that mean? It means that this is a good state. So it may be normalized. It may not be normalized. In any case, it should be positive. I put first the equal, but I shouldn't put the equal. Because we know in a complex vector space, if a state has 0 norm, it's 0. And I want to say that there's really some state that is non-0, that has this energy. If the state would be 0, this would become a triviality. So this state is good. It's all good. Now with this state, you can define, now, two other states, acting with the creation, annihilation operators. I didn't mention that name. But a dagger is going to be called the creation operator. And this is the destruction or annihilation operator. And we built two states, E plus is a dagger acting on E. And E minus is a acting on E. Now you could fairly ask a this moment and say, well, how do you know these states are good? How do you know they even exist? How do you know that if you act with this, don't you get an inconsistent state? How do you know this makes sense? And these are perfectly good questions. And in fact, this is exactly what you have to understand. This procedure can give some funny things. And we want to discuss algebraically why some things are safe and why some things may not quite be safe. And adding an a dagger, we will see it's safe. While adding a's to the state could be fairly unsafe. So what can be bad about the state? It could be a 0 state, or it could be an inconsistent state. And what this an inconsistent state? Well, all our states are represented by wave functions. And they should be normalizable. And therefore they have norms that are positive, norms squared that are positive. Well you may find, here, that you have states that have norms that are negative, norm squareds that are negative. So this thing that should be positive, algebraically you may show that actually you can get into trouble. And trouble, of course, is very interesting. So I want to skip this calculation and state something that you probably checked in 804, several times, that this state has more energy than E and, in fact, has as much energy as E plus h-bar omega. Because a dagger, the creation operator, adds energy, h-bar omega. And this subtracts energy, h-bar omega. This state has an energy, E plus, which is equal to E plus h-bar omega. And E minus is equal to E minus h-bar omega. Now how do you check that? You're supposed to act with a Hamiltonian on this, use the commutation relation that we wrote up there, and prove that those are the energy eigenvalues. So at this moment, you can do the following. So these states have energies, they have number operators, they have number eigenvalues. So we can test, if these states are good, by computing their norms. So let's compute the norm, a dagger on E, a dagger on E for the first one. And we'll compute a E, a E. We'll do this computation. We just want to see what this is. Now remember how you do this. An operator acting here goes with a dagger into the other side. So this is equal to E a, a dagger, E. Now a, a dagger is not quite perfect. It differs from the one that we know is an eigenvalue for this state, which is the number operator. So what is a, a dagger in terms of N? Well, a, a dagger-- it's something you will use many, many times-- is equal to a commutator with a dagger plus a dagger a. So that's 1 plus the number operator. So this thing is E 1 plus the number operator acting on the state E. Well, the 1 is clear what it is. And the number operate is clear. If this has some energy E, well, I can now what is the eigenvalue of the number operator because the energy on the number eigenvalues are related that way. So I will simply call it the number of E and leave it at that. Times EE. So in here, the computation is easier because it's just E a dagger a E. That's the number, so that's just NE times EE. OK, so these are the key equations we're going to be using to understand the spectrum quickly. And let me say a couple of things about them. So I'll repeat what we have there, a dagger E a dagger E is equal to 1 plus NE EE. On the other hand, 888 aE aE is equal to NE EE. OK, so here it goes. Here is the main thing that you have to think about. Suppose this state was good, which means this state has a good norm here. And moreover, we've already learned that the energy is greater than some value. So the number operator of this state could be 0-- could take eigenvalue 0. But it could be bigger than 0, so that's all good. Now, at this stage, we have that-- for example, this state, a dagger E has number one higher than this one, than the state E because it has an extra factor of the a dagger which adds an energy of h omega. Which means that it adds number of 1, So if this state has some number, this state has a number which is bigger. So suppose you keep adding. Now, look at the norm of this state. The norm of this state is pretty good because this is positive and this is positive. If you keep adding a daggers here, you always have that this state, the state with two a daggers, you could use that to find its norm. You could use this formula, put in the states with one a dagger here. But the states with one a dagger already has a good norm. So this state with two a daggers would have also good norm. So you can go on step by step using this equation to show that as long as you keep adding a daggers, all these states will have positive norms. And they have positive norms because their number eigenvalue is bigger and bigger. And therefore, the recursion says that when you add one a dagger, you don't change the sign of this norm because this is positive and this is positive, and this keeps happening. On the other hand, this is an equation that's a lot more dangerous. Because this says that in this equation, a lowers the number. So if this has some number, NE, this has NE minus 1. And if you added another a here, you would use this equation again and try to find, what is the norm of things with two a's here? And put in the one with one a here and the number of that state. But eventually, the number can turn into a negative number. And as soon as the number turns negative, you run into trouble. So this is the equation that is problematic and the equation that you need to understand. So let me do it in two stages. Here are the numbers. And here is 5 4, 3, 2, 1, 0. Possibly minus 1, minus 2, and all these numbers. Now, suppose you start with a number that is an integer. Well, you go with this equation. This has number 4. Well, you put an a. Now it's a state with number 3, but its norm is given 4 times that. So it's good. Now you go down another 1, you have a state with number 3, with number 2, with number 1, with number 0. And then if you keep lowering, you will get minus 1, which is not so good. We'll see what happens. Well, here you go on and you start producing the states-- the state with number 4, state with number 3, state with number 2, state with number 1. And state here, let's call it has an energy E prime. And it has number equal 0. Number of E prime equals 0. So you look at this equation and it says aE prime times aE prime is equal N E prime times E prime E prime. Well, you obtain this state at E prime, and it was a good state because it came from a state that was good before. And therefore, when you did the last step, you had the state at 1 here, with n equals to 1, and then that was the norm of this state. So this E E prime is a fine number positive. But the number E prime is 0. So this equation says that aE prime aE prime is equal to 0. And if that's equal to 0, the state aE prime must be equal to 0. And 0 doesn't mean the vacuum state or anything. It's just not there. There's no such state. You can't create it. You see, aE prime would be a state here with number minus 1. And everything suggests to us that that's not possible. It's an inconsistent state. The number must be less than 1. And we avoided the inconsistency because this procedure said that as you go ahead and do these things, you eventually run into this state E prime at 0 number. But then, you get that the next state is 0 and there's no inconsistency. Now, that's one possibility. The other possibility that could happen is that there are energy eigenstates that have numbers which are not-- well, I'll put it here. That are not integer. So maybe you have a state here with some number E which is not an integer. It doesn't belong to the integers. OK, so what happens now? Well, this number is positive. So you can lower it and you can put another state with number 1 less. Also, not integer and it has good norm. And this thing has number 2.5, say. Well, if I use the equation again, I put the 2.5 state with its number 2.5 and now I get the state with number 1.5 and it still has positive norm. Do it again, you find the state with 0.5 number and still positive norm. And looking at this, you start with a state with 0.5, with 0.5 here. And oops, you get a state that minus 0.5. And it seems to be good, positive norm. But then, if this is possible, you could also build another state acting with another a. And this state is now very bad because the N for this state was minus 1/2. And therefore, if you put that state, that state at the minus 1/2, you get the norm of the next one that has one less. And this state now is inconsistent. So you run into a difficulty. So what are the ways in which this difficulty could be avoided? What are the escape hatches? There are two possibilities. Well, the simplest one would be that the assumption is bad. There's no state with fractional number because it leads to inconsistent states. You can build them and they should be good, but they're bad. The other possibility is that just like this one sort of terminated, and when you hit 0-- boom, the state became 0. Maybe this one with a fractional one, before you run into trouble you hit a 0 and the state becomes 0. So basically, what you really need to know now on the algebraic method cannot tell you is how many states are killed by a. If maybe the state of 1/2 is also killed by a, then we would have trouble. Now, as we will see now, that's a simple problem. And it's the only place where it's interesting to solve some equation. So the equation that we want to solve is the equation a on some state is equal to 0. Now, that equation already says that this possibility is not going to happen. Why? Because from this equation, you can put an a dagger on this. And therefore, you get that NE is equal to 0. This is the number operator, so the eigenvalue of the number operator, we call it NE. So in order to be killed by a, you have to have NE equals 0. So in the fractional case, no state will be killed and you would arrive to an inconsistency. So the only possibility is that there's no fractional states. So it's still interesting to figure out this differential equation, what it gives you. And why do we call it a differential equation? Because a is this operator over there. It has x and ip. So the equation is x a E equals 0, which is square root of m omega over 2 h bar x x plus ip over m omega on E equals 0. And you've translated these kind of things. The first term is an x multiplying the wave function. We can call it psi E of x. The next term, the coefficient in front is something you don't have to worry, of course. It's just multiplying everything, so it's just irrelevant. So have i over m omega. And p, as you remember, is h bar over i d dx of psi E of x zero. So it's so simple differential equation, x plus h bar over m omega d dx on psi E of x is equal to 0. Just one solution up to a constant is the Gaussian that you know represents a simple harmonic oscillator. So that's pretty much the end of it. This ground state wave function is a number times the exponential of minus m omega over 2 h bar x squared. And that's that. This is called the ground state. It has N equals 0 represented as a state. We say this number is N equals 0. So this state is the thing that represents this psi E. In other words, psi E of x is x with 0. And that 0 is a little confusing. Some people think it's the 0 vector. That's not good. This is not the 0 vector. The 0 vector is not a state. It's not in the Hilbert space. This is the ground state. Then, the worst confusion is to think it's the 0 vector. The next confusion is to think it's 0 energy. That's not 0 energy, it's number equals 0. The energy is, therefore, 1/2 h bar omega. And now, given our discussion, we can start building states with more oscillators. So we build a state with number equal 1, which is constructed by an a dagger on the vacuum. This has energy 1 h bar omega more. It has number equal to 1. And that's sometimes useful to just make sure you understand why N on a dagger on the vacuum is a dagger a a dagger on the vacuum. Now, a kills the vacuum, so this can be replaced by the commutator, which is 1. And therefore, you're left with a dagger on the vacuum. And that means that the eigenvalue of n hat is 1 for this state. Moreover, this state is where normalized 1 with 1 actually gives you a good normalization if 0 is well-normalized. So we'll take 0 with 0 to be 1, the number 1. And that requires fixing that N0 over here. Now, these are things that you've mostly seen, so I don't want to say much more about them. I'd rather go through the Schrodinger thing that we have later. So let me conclude by just listing the general states, and then leaving for you to read what is left there in the notes so that you can just get an appreciation of how you use it. And with the practice problems, you'll be done. So here it is. Here is the answer. The n state is given by 1 over square root of n factorial a dagger to the n acting on the vacuum. And these n states are such that m with n is delta mn. So here we're using all kinds of things. First, you should check this is well normalized, or read it and do the calculations. And these are, in fact, orthogonal unless they have the same number of creation operators are the same number. Now, that had to be expected. These are eigenstates of a Hermitian operator. The N operator is Hermitian. Eigenstates of a Hermitian operator with different eigenvalues are always orthogonal to each other. If you have eigenstates of a Hermitian operator with the same eigenvalue, if you have a degeneracy, you can always arrange them to make them orthogonal. But if the eigenvalues are different, they are orthogonal. And there's no degeneracies in this spectrum whatsoever. You will, in fact, argue that because there's no degeneracy in the ground state, there cannot be degeneracy anywhere else. So this result, this orthonormality is really a consequence of all the theorems we've proven. And you could check it by doing the algebra and you would start moving a and a daggers. And you would be left with either some a's or some a daggers. If you're left with some a's, they would kill the thing on the right. If you're left with some a daggers, it would kill the thing on the left. So this can be proven. But this is just a consequence that these are eigenstates of the Hermitian operator n that have different eigenvalues. And therefore, you've succeeded in constructing a full decomposition of the state space of the harmonic oscillator. We spoke about the Hilbert space. Are now very precisely, see we can say this is u0 plus u1 plus u2 where uk is the states of the form alpha k, where N on k-- maybe I should put n here. It looks nicer. n. Where N n equal n n. So every one-dimensional subspace is spanned by that state of number n. So you have the states of number 0, states of number 1, states of number 2. These are all orthogonal subspaces. They add up to form everything. It's a nice description. So the general state in this system is a complex number times the state with number 0 plus the complex number states of number 1, complex number, and that. Things couldn't have been easier in a sense. The other thing that you already know from 804 is that if you try to compute expectation values, most of the times you want to use a's and a daggers. So the typical thing that one wants to compete is on the state n, what is the uncertainty in x on the state n? How much is it? What is the uncertainty of momentum on the energy eigenstate of number n? These are relatively straightforward calculations. If you have to do the integrals, each one-- by the time you organize all your constants-- half an hour, maybe 20 minutes. If you do it with a and a daggers, this computation should be five minutes, or something like that. We'll see that done on the notes. You can also do them yourselves. You probably have played with them a bit. So this was a brief review and discussion of them spectrum. It was a little detailed. We had to argue things carefully to make sure we don't assume things. And this is the way we'll do also with angular momentum in a few weeks from now. But now I want to leave that, so I'm going to take questions. If there are any questions on this logic, please ask. Yes. AUDIENCE: [INAUDIBLE] for how you got a dagger, a, a dagger, 0, 2 dagger, 0? PROFESSOR: Yes, that calculation. So let me do at the step that I did in words. So at this place-- so the question was, how did I do this computation? Here I just copied what N is. So I just copied that. Then, the next step was to say, since a kills this, this is equal to a dagger times a a dagger minus a dagger a. Because a kills it. And I can add this, it doesn't cost me anything. Now, I added something that is convenient, so that this is a dagger commutator of a with a dagger on 0. This is 1, so you get that. It's a little more interesting when you have, for example, the state 2, which is 1 over square root of 2 a dagger a dagger on 0. I advise you to try to calculate n on that. And in general, convince yourselves that n is a number operator, which means counts the number of a daggers. You'll have to use that property if you have N with AB. It's N with A B and then A N with B. The derivative property of the bracket has to be used all the time. So Schrodinger dynamics, let's spend the last 20 minutes of our lecture on this. So basically, it's a postulate of how evolution occurs in quantum mechanics. So we'll state it as follows. What is time in quantum mechanics? Well, you have a state space. And you see the state space, you've seen it in the harmonic oscillator is this sum of vectors. And these vectors were wave functions, if you wish. There's no time anywhere there. There's no time on this vector space. This vector space is an abstract vector space of functions or states, but time comes because you have clocks. And then you can ask, where is my state? And that's that vector on that state space. And you ask the question a littler later and the state has moved. It's another vector. So these are vectors and the vectors change in time. And that's all the dynamics is in quantum mechanics. The time is sort of auxiliary to all this. So we must have a picture of that. And the way we do this is to imagine that we have a vector space H. And here is a vector. And that H is for Hilbert space. We used to call it in our math part of the course V, the complex vector space. And this state is the state of the system. And we sometimes put the time here to indicate that's what it is. At time t0, that's it. Well, at time t, some arbitrary later time, it could be here. And the state moves. But one thing is clear. If it's a state of a system, if we normalize it, it should be of unit length. And we can think of a sphere in which this unit sphere is the set of all the tips of the vectors that have unit norm. And this vector will move here in time, trace a trajectory, and reach this one. And it should do it preserving the length of the vector. And in fact, if you don't use a normalized vector, it has a norm of 3. Well, it should preserve that 3 because you'd normalize the state once and forever. So we proved in our math part of the subject that an operator that always preserves the length of all vectors is a unitary operator. So this is the fundamental thing that we want. And the idea of quantum mechanics is that psi at time t is obtained by the action of a unitary operator from the state psi at time t0. And this is for all t and t0. And this being unitary. Now, I want to make sure this is clear. It can be misinterpreted, this equation. Here, psi at t0 is an arbitrary state. If you had another state, psi prime of t0, it would also evolve with this formula. And this U is the same. So the postulate of unitary time evolution is that there is this magical U operator that can evolve any state. Any state that you give me at time equal 0, any possible state in the Hilbert space, you plug it in here. And by acting with this unitary operator, you get the state at the later time. Now, you've slipped an extraordinary amount of physics into that statement. If you've bought it, you've bought the Schrodinger equation already. That is going to come out by just doing a little calculation from this. So the Schrodinger equation is really fundamentally, at the end of the day, the statement that this unitary time evolution, which is to mean there's a unitary operator that evolves any physical state. So let's try to discuss this. Are there any questions? Yes. AUDIENCE: So you mentioned at first that in the current formulation [INAUDIBLE]? PROFESSOR: A little louder. We do what in our current formulation? AUDIENCE: So if you don't include time [INAUDIBLE]. PROFESSOR: That's right. There's no start of the vector space. AUDIENCE: Right. So is it possible to consider a vector space with time? PROFESSOR: Unclear. I don't think so. It's just nowhere there. What would it mean, even, to add time to the vector space? I think you would have a hard time even imagining what it means. Now, people try to change quantum mechanics in all kinds of ways. Nobody has succeeded in changing quantum mechanics. That should not be a deterrent for you to try, but should give you a little caution that is not likely to be easy. So we'll not try to do that. Now, let me follow on this and see what it gives us. Well, a few things. This operator is unique. If it exists, it's unique. If there's another operator that evolves states the same way, it must be the same as that one. Easy to prove. Two operators that attack the same way on every state are the same, so that's it. Unitary, what does it mean that u t, t0 dagger times u t, t0 is equal to 1? Now, here these parentheses are a little cumbersome. This is very clear, you take this operator and you dagger it. But it's cumbersome, so we write it like this. This means the dagger of the whole operator. So this is just the same thing. OK, what else? u of t0, t0, it's the unit operator. If the times are the same, you get the unit operator for all t0 because you're getting psi of t0 here and psi of t0 here. And the only operator that leaves all states the same is the unit operator. So this unitary operator must become the unit operator, in fact, for the two arguments being equal. Composition. If you have psi t2, that can be obtained as U of t2, t1 times the psi of t1. And it can be obtained as u of t2, t1, u of t1, t0, psi of t0. So what do we learn from here? That this state itself is u of t2, t0 on the original state. So u of t2, t0 is u of t2, t1 times u of t1, t0. It's like time composition is like matrix multiplication. You go from t0 to t1, then from t1 to t2. It's like the second index of this matrix. In the first index of this matrix, you are multiplying them and you get this thing. So that's composition. And then, you have inverses as well. And here are the inverses. In that equation, you take t2 equal to t0. So the left-hand side becomes 1. And t1 equal to t, so you get u of t0, t be times u of t, t0 is equal to 1, which makes sense. You propagate from t0 to t. And then from t to t0, you get nothing. Or if it's to say that the inverse of an operator-- the inverse of this operator is this one. So to take the inverse of a u, you flip the arguments. So I'll write it like that, the inverse minus 1 of t, t0. You just flip the arguments. It's u of t0, t. And since the operator is Hermitian, the dagger is equal to the inverse. So the inverse of an operator is equal to the dagger. so t, t0 as well. So this one we got here. And Hermiticity says that the dagger is equal to the inverse. Inverse and dagger are the same. So basically, you can delete the word "inverse" by flipping the order of the arguments. And since dagger is the same as inverse, you can delete the dagger by flipping the order of the arguments. All right, so let's try to find the Schrodinger equation. So how c we c the Schrodinger equation? Well, we try obtaining the differential equation using that time evolution over there. So the time evolution is over there. Let's try to find what is d dt of psi t. So d dt of psi of t is just the d dt of this operator u of t, t0 psi of t0. And I should only differentiate that operate. Now, I want an equation for psi of t. So I have here psi of t0. So I can write this as du of t, t0 dt. And now put a psi at t. And then, I could put a u from t to t0. Now, this u of t and t0 just brings it back to time t0. And this is all good now, I have this complicated operator here. But there's nothing too complicated about it. Especially if I reverse the order here, I'll have du dt of t, t0 and u dagger of t, t0. And I reverse the order there in order that this operator is the same as that, the one that is being [INAUDIBLE] that has the same order of arguments, t and t0. So I've got something now. And I'll call this lambda of t and t0. So what have I learned? That d dt of psi and t is equal to lambda of t, t0 psi of t. Questions? I don't want to loose you in their derivation. Look at it. Anything-- you got lost, notation, anything. It's a good time to ask. Yes. AUDIENCE: Just to make sure when you differentiated the state by t, the reason that you don't put that in the derivative because it doesn't have a time [INAUDIBLE] necessarily, or because-- oh, because you're using the value at t0. PROFESSOR: Right. Here I looked at that equation and the only part that has anything to do with time t is the operator, not the state. Any other comments or questions? OK, so what have we learned? We want to know some important things about this operator lambda because somehow, it's almost looking like a Schrodinger equation. So we want to see a couple of things about it. So the first thing that I will show to you is that lambda is, in fact, anti-Hermitian. Here is lambda. I could figure out, what is lambda dagger? Well, lambda dagger is you take the dagger of this. You have to think when you take the dagger of this thing. It looks a little worrisome, but this is an operator. This is another operator, which is a time derivative. So you take the dagger by doing the reverse operators and daggers. So the first factor is clearly u of t, t0. And then the dagger of this. Now, dagger doesn't interfere at all with time derivatives. Think of the time derivative-- operator at one time, operator at another slightly different time. Subtract it. You take the dagger and the dagger goes through the derivative. So this is d u dagger t, t0 dt. So I wrote here what lambda dagger is. You have here what lambda is. And the claim is that one is minus the other one. It doesn't look obvious because it's supposed to be anti-Hermitian. But you can show it is true by doing the following-- u of t, t0 u dagger of t, t0 is a unitary operator. So this is 1. And now you differentiate with respect to t. If you differentiate with respect to t, you get du dt of t, t0 u dagger of t, t0 plus u of t, t0 du dagger of t, t0 equals 0 because the right-hand side is 1. And this term is lambda. And the second term is lambda dagger. And they add up to 0, so lambda dagger is minus lambda. Lambda is, therefore, anti-Hermitian as claimed. Now, look. This is starting to look pretty good. This lambda depends on t and t0. That's a little nasty though. Why? Here is t. What is t0 doing here? It better not be there. So what I want to show to you is that even though this looks like it has a t0 in there, there's no t0. So we want to show this operator is actually independent of t0. So I will show that if you have lambda of t, t0, it's actually equal to lambda of t, t1 for any t1. We'll show that. Sorry. [LAUGHTER] PROFESSOR: So this will show that you could take t1 to be t0 plus epsilon. And take the limit and say the derivative of this with respect of t0 is 0. Or take this to mean that it's just absolutely independent of t0 and t0 is really not there. So if you take t1 equal t dot plus epsilon, you could just conclude from these that this lambda with respect to t0 is 0. No dependence on t0. So how do we do that? Let's go a little quick. This is du t, t0 dt times u dagger of t, t0. Complete set of states said add something. We want to put the t1 here. So let's add something that will help us do that. So let's add t, t0 and put here a u of t0, t1 and a u dagger of t0, t1. This thing is 1, and I've put the u dagger of t, t0 here. OK, look at this. T0 and t1 here and t dot t1 there like that. So actually, we'll do it the following way. Think of this whole thing, this d dt is acting just on this factor. But since it's time, it might as well be acting on all of this factor because this has no time. So this is d dt on u t, t0 u t0, t1. And this thing is u of t1m t0. The dagger can be compensated by this. And this dagger is u of t0, t. This at a t and that's a comma. t0, t. Yes. OK, so should I go there? Yes. We're almost there. You see that the first derivative is already d dt of u of t, t1. And the second operator by compensation is u of t1, t, which is the same as u dagger of t, t1. And then, du of t, t1 u dagger of t, t1 is lambda of t, t1. So it's a little sneaky, the proof, but it's totally rigorous. And I don't think there's any step you should be worried there. They're all very logical and reasonable. So we have two things. First of all, that this quantity, even though it looks like it depends on t0, we finally realized that it does not depend on t0. So I will rewrite this equation as lambda of t. And lambda of t is anti-Hermitian, so we will multiply by an i to make it Hermitian. And in fact, lambda has units of 1 over time. Unitary operators have no units. They're like numbers, like 1 or e to the i phi, or something like that-- have no units. So this has units of 1 over time. So if I take i h bar lambda of t, this goes from lambda being anti-Hermitian-- this operator is now Hermitian. This goes from lambda having units of 1 over time to this thing having units of energy. So this is a Hermitian operator with units of energy. Well, I guess not much more needs to be said. If that's a Hermitian operator with units of energy, we will give it a name called H, or Hamiltonian. i h bar lambda of t. Take this equation and multiply by i h bar to get i h bar d dt of psi is equal to this i h bar lambda, which is h of t psi of t. Schrodinger equation. So we really got it. That's the Schrodinger equation. That's the question that must be satisfied by any system governed by unitary time evolution. There's not more information in the Schrodinger equation than unitary time evolution. But it allows you to turn the problem around. You see, when you went to invent a quantum system, you don't quite know how to find this operator u. If you knew u, you know how to evolve anything. And you don't have any more questions. All your questions in life have been answered by that. You know how to find the future. You can invest in the stock market. You can do anything now. Anyway, but the unitary operator then gives you the Hamiltonian. So if somebody tells you, here's my unitary operator. And they ask you, what is the Hamiltonian? You go here and calculate I h bar lambda, where lambda is this derivative. And that's the Hamiltonian. And we conversely, if you are lucky-- and that's what we're going to do next time. If you have a Hamiltonian, you try to find the unitary time evolution. That's all you want to know. But that's a harder problem because you have a differential equation. You have h, which is here , and you are to find u. So it's a first-order matrix differential equation. So it's not a simple problem. But why do we like Hamiltonians? Because Hamiltonians have to do with energy. And we can get inspired and write quantum systems because we know the energy functional of systems. So we invent a Hamiltonian and typically try to find the unitary time operator. But logically speaking, there's not more and no less in the Schrodinger equation than the postulate of unitary time evolution. All right, we'll see you next week. In fact-- [APPLAUSE] Thank you.

MIT_805_Quantum_Physics_II_Fall_2013

15_Quantum_Dynamics_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. BARTON ZWIEBACH: Let's begin. So today's lecture will deal with the subject of squeeze states and photon states. And it all builds up from the ideas of coherent states that we were talking about last time. So let me begin by reminding you about the few facts that we had about coherent states. So a coherent state was born by taking the ground state of the harmonic oscillator and displacing it with a translation operator some distance, x0. And then we let it go, and we saw that this sort of wave function would just move from the left and to the right coherently, without spreading out, without changing shape. It would move in a nice way. Now, this was obtained with a translation operator, which was an exponential that had on the exponent the momentum operator. But we realized eventually that the reason it all works out is because its exponential of something that depends on creation or annihilation operators, and we could do something more general, which was to use a complex number, alpha, and define a displacement operator, a more general one, that is some linear combination of a and a dagger with alpha and the complex conjugate of alpha. So this is only proportional to the momentum if alpha is real, but if alpha is not real, that operator in the exponent is not quite the momentum. It's something that has a bit of position as well. So this is a more general operator, but on the other hand, it's clear that it's anti-Hermitian, because if you take the dagger of this thing, this term becomes that and that term becomes this one, each one with a change of sign. So you're really with an anti-Hermitian operator. Therefore, the whole operator is unitary and you're acting with a unitary operator on the vacuum. And therefore, this state is also well normalized and represents a state with some expectation value of the position. Just like a coherent state, we moved it to the right and it had some expectation value of the position. But this one also has some expectation value of the momentum. So in fact, we realize that the real part of alpha in this axis was related to the expectation value of the position divided by to the 0. So if you produce a coherent state with this value of alpha in the complex alpha plane, well, you go down and that's the expectation value of the position. You go horizontally, well, that's the expectation value of the momentum scaled because this alpha is a pure number, has no units. So this x over square root of 2 d0 and p d0 over h bar have no units, and that's how it should be. So we learned also that the annihilation operator acting on this coherent state was alpha times the coherent state. So it's a very simple property. That number, alpha, is the eigenvalue of the destruction operator. Now, that's a one line computation based on this, or a two line computation maybe. But it should be a computation that is easy for you to do. So make sure you know how to get this very quickly from this definition. So that's a coherent state. And then the thing we finished the lecture with was with the time evolution of this coherent state. And the time evolution was that as the state, alpha, in time becomes the state alpha at time t, it remains a coherent state but the value of alpha is changed. In fact, the value of alpha is changed in such a way that you can just imagine this thing rotating, and rotating with angular velocity, omega. So this thing was the coherent state, e to the minus i omega t alpha. So this whole complex number, instead of being alpha, is just this. There's no comma t. This is the time development of the state. And there was a phase here, e to the minus i omega t over 2, that was not very relevant. But that's what the state was doing in time. So basically, that's where we got last time. Before I push on, do you have any questions? I did post the notes associated to coherent states about half an hour ago, so you have them. You do have two problems on coherent states in this homework, so the notes should help you. But any questions about this picture? OK. So I want to develop it a little more before starting with squeeze states. So here's what I want to tell you. And this is an intuition that people have about these states. Alpha is a complex number. And you know, it's a well-defined complex number. But you know, this is a coherent state, so it's not a position eigenstate. It's not a momentum eigenstate. It's not an energy eigenstate. It has all kinds of uncertainties. It has uncertainties in position, in momentum, and in energy. Yes? AUDIENCE: On that complex plane where you have the x and the p, are those expectation values of the position of momentum? BARTON ZWIEBACH: Yes. I think I wrote them with expectation values of the position of momentum last time. Yes. So given alpha, that number you get is the expectation value of position or expectation value of momentum. Correct. So actually, this is the expectation value of the position, but the position is a little bit rounded. Yeah, this is the expectation value of the position. It's a number. But intuitively, this is spread out a little. The position is not just one point. This is a coherent state. It looks like a Gaussian wave function. The momentum is also spread out a little. So in some sense, many people draw this as a little blob. And that blob represents your intuition that yes, the expectation value is this and the expectation value is this, but the position is, well, somewhere around this thing and somewhere around that stuff. You can complain, this is very hand wavy, but it's useful. It's good to have that physical picture that the state really is some sort of blob here, not that the expectation values are not well defined, but rather that it's something like this. And I want to relate it to an idea that comes along with waves, and it's important for what we're going to be doing later today. If you have a wave with energy e, and suppose your wave is a light wave. Light wave with energy e. And it's described by, say, A cosine omega t. That's some component of the electric field or the magnetic field for this wave. It is like that. Well, we've been talking about energy time uncertainty, and we know that unless we make things very precise. It's easy to get things wrong. So I will first do something fairly imprecise to give you a feeling of what people talk about, and then we'll use this picture to do it more precisely. So if you have this wave, the phase of this wave-- we'll call it phi-- is omega t. And if we are naive there, the error in the phase is divided by omega is the error in time. Now, this wave has energy E. It has some number of photons. So the energy, E, is the number of photons times h bar omega. N is equal to number of photons. And again, we could say delta E is delta N h bar omega, and then substitute these two relations into this to see what we get. Well, delta E is delta N h bar omega. Delta t is delta phi over omega. This should be h bar over 2. The omegas cancel, the h bar cancels, and you get delta N delta phi is about 1, or 1 over 2 or 1 over square root of 2. And that's, in fact, the relation that people do take somewhat seriously, if you have a wave in quantum optics and say, well, the uncertainty in the number of photons and the uncertainty in the coherence of these photons, the phases, if they're out of phase, they're not coherent, there's a relation of this kind. And this derivation is certainly pretty bad. It's just not precise because even we started with this that is not precise unless you really explain what you mean by delta t. So let's see if we can make some sense of this picture here. So here we go. I want to do a small calculation first. So let's see what we have. In this coherent state, what is the expectation value of the number operator? Expectation value of the number operator in alpha. Well, the number operator in alpha, you would do this a dagger a in alpha. These are easy to do because a on alpha is alpha times alpha, and if you dagger that equation, a dagger on alpha is alpha star. So you get-- I'll go slowly-- alpha alpha star here, and then you get alpha alpha. And alpha has unit norm. These are numbers, so this is equal to length of alpha squared. So if this is a harmonic oscillator, the expectation value of the number operator, in fact, is the length squared of this vector. Now, how about N squared? N squared is a little more work because you have alpha a dagger a a dagger a alpha. This one gives me the factor you know, this one gives me the factor you know. And therefore, we already have alpha squared times alpha a a dagger alpha. And here, the a's and the a daggers are kind of in the wrong order because I know what a is on a ket, but a is now on the bra. And I know what a dagger is on the bra, but now a dagger is on the ket. But the answer is simple. You replace this by the commutator plus the reverse order. So this is equal to the commutator, which is 1, plus the thing in the reverse order. And this is 1 plus alpha squared. So you have alpha squared times 1 plus alpha squared, and that's the expectation value of N squared. All that, because we're actually interested in what is delta N, the uncertainty in N in the coherent state. And that would be this, square root of this, which is alpha to the fourth plus alpha squared minus the square of the expectation value, which is minus alpha to the fourth. And this is length of alpha. So the uncertainty in N is just length of alpha. It happens to be the square root of the expectation value of N. So in fact, if you think of this picture, you're tempted to say, oh, this represents the number of excited states that you have. This length represents the expectation value of N. No. The expectation value of N is this length squared. This length represents delta N in the picture. So what else can we say? Well, this picture is useful because now, I can be a little more precise here. This thing is rotating. That is time evolution of your coherent state. Now, this thing this rotating, but I can ask now how wide this is. So what is the uncertainty in x in a coherent state? Well, the uncertainty in x in a coherent state is the same as the uncertainty of the ground state because you just moved it. Uncertainty doesn't change. So the uncertainty, delta x, is in fact this quantity that we call d0 over square root of 2. That's the uncertainty delta x, and the uncertainty delta p is h bar over d0 square root of 2. These are not hard to remember. d0 is the length scale of the harmonic oscillator, so that's typically what the uncertainty should be. The square root of 2, yes, it's hard to remember. But delta p is this one. And then the other thing that you know is that the product should be h bar over 2, so that is correct. Now, look at this. How big is this thing? If the uncertainty in x is d0 over square root of 2, this width is about how much, roughly? Nobody? This is the uncertainty in x, d0 over square root of 2 in these units, if you move the expectation value of x plus the uncertainty of x over 2 and the other uncertainty of x roughly. This thing is d0 over square root of 2, so it represents basically 1/2, because you change the expectation value of x by this amount, and then this thing moves 1/2. The size of this is 1/2, roughly. Could be 1/4 or could be 2, but it's roughly 1/2. And the vertical one corresponds to the uncertainty in momentum. So intuitively, this is h over square root v0, so if you plug it in there, this amount, p plus delta p, you'll get 1/2 as well. So in this plot-- yes? AUDIENCE: Wouldn't the width be 1 because the uncertainty is the width in one direction [INAUDIBLE]? BARTON ZWIEBACH: Well, the uncertainty is neither the width in one direction or not. It's a Gaussian, so I don't know where it stops. This picture is not very precise when I talk about this, so let me leave it with 1/2 or something like that. I don't think we can do better. Now, there's also 1/2 here. So finally, we get to something that is kind of interesting. If really the state in some sense, in terms of x and p, is spread here, and this is moving around, the phase is a little ambiguous. Because you would say, well, the phase is this one, but well, you could go the whole uncertainty that you go here. The uncertainty in where the coherent state is, we could call the phase here delta phi in this picture. We don't know where this state is because it's a little blob. We know the expectation values where they are, but the state itself is a little imprecise. So there's an angle here in this diagram that represents the phase because this is going with frequency omega t. So this is the phase as this goes around, so this angle, delta phi, is how much. Well, if this is 1/2 and this is 1/2, I'm going to assume that this is 1/2 as well, or 1, or something like that. So it's 1 over this length. That's the uncertainty. But delta N, we calculated. This is roughly. And delta N we calculated, and it's exactly alpha. So delta phi delta N is about 1 correctly. And here, there is at least a picture of what the phase uncertainty is and why it originates. Yes? AUDIENCE: Can you tell me again how the Gaussian relates to the uncertainty? BARTON ZWIEBACH: One second. Let me see. I've got one question first. AUDIENCE: Yes. Can you explain one more time where the 1/2's come from? [INAUDIBLE] the graph. I'm not sure why. BARTON ZWIEBACH: Yes. AUDIENCE: Are you saying that the width is 1/2, or is that how high it is? BARTON ZWIEBACH: The width of this little ball. AUDIENCE: So how does that follow from the graph? BARTON ZWIEBACH: It's a little hand wavy, but I'll say it like this. I expect the position, if measured, to be between expectation value of x plus minus delta x. So if I'm going to measure the position of something in a state, the most likely measurement that I will get is the expectation value of x, statistically, after I repeat this many times. But if I just measure, I'm probably going to get some number between this and this. So if you think of this diagram not as the expectation value of x in here, but whatever you got for x as you measured, if you do 1,000 measurements, you're going to get points all over here in some region because you measure x in one case, then you measure the momentum, you get a plot of data, and you measure them all. And then suppose you're doing it with x first. You measure x and you say, well, I get all kinds of values. I don't know what the momentum is, but I get all kinds of values. They're going to run all over here between these two positions. So when I add to the expectation value of x this thing, when I want to see in this graph, what it is, I must divide by square root of 2d. So I divide by 1 over square root of 2 d0 to see how it goes and how I plot it in this graph because these are the units in this graph. So if delta x is d0 over square root of 2, I'm going to get some values that go from the expectation value of x up to 1/2 more and 1/2 less. AUDIENCE: So it's not actually 1/2. It's actually 1/2 times whatever that set amount. BARTON ZWIEBACH: Well, if I say this, that you obtain between this and that, then I should say it's 1. Maybe I had in mind that you sort of get most things between 1/2 of delta x. It's not terribly precise, but it's roughly this is the picture. You measure the position, you're going to get that. Similarly, you decide to measure momentum. You don't measure position, measure momentum, and you're going to get roughly the expectation value, but you're going to get a little plus minus uncertainty. So you're going to all points here in your measurement. So this dashed thing is your histogram after doing lots of experiments. You have lots of dots in here. And roughly, this is how it comes about. It's not terribly precise because I cannot put a point either here, because if I say the measurement was this, I'm suggesting that I also measured the momentum on that state, or I could only measure the position. But it's a rough idea, rough picture, of how big the spread is here. There is a mathematical theory to do this more precisely, although physically not much clearer, which are called Wigner distributions. I don't think it helps too much to understand it, but the rough picture is relatively clear. So if you divide by 1 over square root of 2, this quantity that was equal to d over square root of 2, you get, in this scale plus 1/2 and minus 1/2, so 1, 1, 1, and this value there. There was a question there. Yes? AUDIENCE: Can you explain again how the Gaussian relates to the uncertainty in p and x? BARTON ZWIEBACH: So I don't know how the Gaussian relates to uncertainty in x. So basically, we computed the uncertainty in x for the ground state, and I claimed that for a coherent state, the uncertainty in x cannot change because you just took the state and you moved it away. And the uncertainty of x doesn't talk about what the expectation value of x is. That changes when you move a state. But just how much it's spread and how much the state is spread is not changed by a translation. So this is the old result for the ground state uncertainty, ground state uncertainty, and neither is changed. Let's go now into our squeezed states. So what are going to be squeezed states? They're going to be pretty useful states. They have lots of applications nowadays. They've been constructed over the last 10 years more and more often, and people are now able to construct what is called squeezed states experimentally. And the way we're going to motivate it is by imagining that you have a harmonic oscillator, a particle n, and some spring, k, or an omega. And there's a Hamiltonian. H is equal to p squared over 2 m plus 1/2 m omega x squared. But this Hamiltonian is going to be the Hamiltonian of a particle that has mass m1, and the oscillator has frequency w1, and that's what we're going to call the first Hamiltonian. After a little while, you observe this Hamiltonian. I will erase this thing. We don't need them anymore. We observe this thing and this has an uncertainty, delta x, proportional to-- we know this v over square root of 2, so h bar over 2 m1 omega 1. And an uncertainty in p, which is delta p equals square root of h bar m1 omega 1 over 2. And again, they saturate the bound if you have your ground state. So ground state is here. These two uncertainties, the bound is saturated, all is good. Nevertheless, suddenly, at time equals 0, this state, this particle, is in the ground state. The Hamiltonian changes. There's an abrupt change in the physics. Maybe the temperature was changed and the spring constant changed, or the particle, a drop was added to it and its mass changed, but the Hamiltonian has changed all of a sudden. So this Hamiltonian, H1, is valid for t less than 0 and a particle in the ground state. So the particle's in the ground state, the Hamiltonian is fine there, but suddenly, the Hamiltonian changes. The particle identity has not changed. The particle is there, but it is the Hamilton that changes. So there's an H2, p squared over 2 m2 plus 1/2 m squared w squared 2 x squared. The picture is physically clean. The particle is sitting there in the ground state, and suddenly, the parameters of the system change. So this particle was having a good time, it was at the ground state, relaxed. Then suddenly, the wave function didn't change at time equals 0. It was spread over some distance. No measurement was done, nothing. And suddenly, this particle finds itself with some wave function but in another Hamiltonian. From now on, its time evolution is going to be governed by the second Hamiltonian. Now, since the second Hamiltonian is different from the first Hamiltonian, this particle is not going to be any more in the ground state. Even though it was in the ground state of the first Hamiltonian, it's not anymore in the ground state of the second Hamiltonian as soon as the thing gets turned on. So for t greater than 0, this Hamilton is there. So actually, the wave function does not change, so let me write delta x, and I'll write it the following way, h bar over 2 m2 omega 1. But you say, no, delta x didn't change. Correct. So I'll put the factor m2 w2 over m1 w1, and now it's the same delta x. Similarly, for delta p, I will write that this is square root of h bar m2 omega 2 over 2, and put the factor m1 omega 1 over m2 omega 2 in front in such a way that it is the same delta x and the same delta p. Now, delta x times delta p multiply to be h bar over 2, and they still multiply to that number because I didn't change them. But this is equal-- I'll call this number e to the minus gamma. I'll go to another blackboard. Delta x is e to the minus gamma times square root of h bar over 2 m2 omega 2. And delta p is e to the gamma, because it's the inverse factor on the one that we call gamma, square root of h bar m2 omega 2 over 2, where e to the gamma is the square root of m1 omega 1 over m2 omega 2. Look, we've done very simple things. We haven't done really much. But already, we start to see what's happening. From the viewpoint of the second Hamiltonian, these uncertainties are not right. They are not the uncertainties of the ground state, because from the viewpoint of the second Hamiltonian, the ground state uncertainty is this and the ground state uncertainty is this. And indeed, this particle was in the ground state, it had some Gaussian, but that's not the right Gaussian for the second Hamiltonian. It's the right Gaussian for the first Hamiltonian. So it's not in the ground state of the second Hamiltonian, but it's in a particular state in which, if gamma is positive, the uncertainty in x is squeezed from the lowest uncertainty that you get in an energy eigenstate. And the uncertainty and the momentum will be stretched in that direction. So you see, in the ground state of the harmonic oscillator, you get that uncertainty, and that's a canonical uncertainty. But this uncertainty is squeezed because it's different from what it should be, and this is squeezed. So from the viewpoint of the second Hamiltonian, the ground state of the first Hamiltonian is a squeezed state. It's a state whose uncertainties have been squeezed. And those states exist, and the purpose of what we're going to do now is try to determine them, find them, see what they are, how they behave. Any questions? Yes, Nicholas? AUDIENCE: I'm a little confused why we can say that delta x [INAUDIBLE] these new ones are just related to the old ones by this factor. BARTON ZWIEBACH: OK. You see, what I assumed is that before time equals 0, you had a Gaussian. That was the original Gaussian. That was the original wave function, and you had some delta x and some delta p that were given by this one [INAUDIBLE]. Now, I didn't do anything except rewrite the same quantities here, because what I said next was that even though at time equals 0, the Hamiltonian changes, at time equals 0, the wave function doesn't change. The wave function remains the same. After that time, it's going to start changing because the new Hamiltonian kicks in. But this delta x's are the same as that I wrote, and here are the same. But here, you see clearly that this delta x with respect to the second Hamiltonian is not the one that it would be if it would be a ground state, nor the delta p. Yes? AUDIENCE: Just at the instant you change the Hamiltonian, because they might evolve and, the uncertainties will change. BARTON ZWIEBACH: Sorry? AUDIENCE: Is this just at the instant where we change the Hamiltonian, because after some time, the wave function might change [INAUDIBLE].. BARTON ZWIEBACH: That's right. This is just after I change the Hamiltonian. The time evolution of this state is something that we have to figure out later. But after I've changed the Hamiltonian, the state looks squeezed. So how can we calculate and understand these things? So the way to think of this is the following. You see, you have this system of two Hamiltonians. There's an x and a p operator, and the second Hamiltonian has an x and a p operator. These are the properties of the particles. Therefore, what that I'm going to think of is that the x and the p operators are the operators that describe the particle. They are unchanged because we're talking about this same object, same particle. So if I have the x operator, which is equal to this formula, h bar over 2 m1 omega 1 a1 dagger plus a1 like this. From the first Hamiltonian, the x's are related to a1's and a1 daggers, but this is the same x describing the same position as you would do in the second Hamiltonian. So m2 w2 a2 hat plus a2 hat dagger. It's a very strong physical assumption I'm making here. It's an assumption that's so strong that in many ways, you could almost say, well, I'll buy it, but we'll see if it gives something reasonable. I'm saying the x operator is really the same thing, and you could view it as constructed from ingredients of the first Hamiltonian or the second Hamiltonian. So is the p operator. p, which is-- well, I have a formula here-- minus i m1 omega 1 h bar over 2 a1 minus a1 dagger-- should be the same as minus i m2 omega 2 h bar over 2 a2 minus a2 dagger. So x and p are not changing. We're not talking about two particles that have an x1 and a p1, and the second particle, an x2 and a p2. It's just one particle has an x and a p is what you observe when you measure position and you observe when you measure momentum. Nevertheless, x and p are related in this way to the creation and annihilation operators. So we're going to find from this some very strange relation between the creation operators, the annihilation operators of the first system and the second system. So what do we get, in fact? Well, the constants disappear from the first equation roughly, and you get a1 dagger is equal to-- you get the ratio of m1 omega 1 over m2 omega 2, so you get e to the gamma a1 a2 plus a2 dagger. From the bottom one, a1 minus a1 dagger is equal to e to the minus gamma a2 minus a2 dagger. These two equations give you that. It should be clear. You just cancel the constants and remember the definition of e to the gamma. And now we can solve for a1 and a1 dagger in terms of a2 and a2 dagger. And what do we find? a1 is equal to a2 cosh gamma plus a2 dagger sinh gamma, and the dagger is what you would imagine. So a1 dagger is equal to a2 dagger cosh gamma plus a2 sinh gamma. The second equation that you can calculate is the dagger of the first. It should be that. And now you've found the scrambling of the creation, annihilation operators. The old annihilation operator is a mixture of the new annihilation operator and a creation operator. They're mixed. It's a very strange thing that has happened, a mixture between creation and annihilation operators. This is so famous in physics, it has a name. It's called the Bogoliubov transformation. It appears in the analysis of black hole radiation. There's a Bogoliubov transformation between the fields far away of the black hole and the fields near the black hole. It appears everywhere. And here it has appeared, so we're going to try to understand what it does for us. Similarly, you can find what a2 is in terms of a1's by the symmetry of these equations. This corresponds to actually letting gamma go to minus gamma, because if you pass these gammas to the other side, the equations are of the same form. By letting 1 become 2, 2 becomes 1 and gamma goes to minus gamma. So we don't need it right now, but in case you want to find the other ones, the 2's in terms of the 1's, you would just change the sign of gamma and it would work out. So this relation is the key to allow you to calculate things. So what do we want to calculate? Well, here is what I would like to calculate. The ground state of the first oscillator is this thing we had. It's the thing that has the wave function. But I want to express it as a superposition of states of the second oscillator because the second oscillator is what gives you the new Hamiltonian and what's going to tell you how the state is going to evolve later. So presumably, this state is some number times the ground state of the second oscillator, plus maybe some creation operator on the second vacuum as well with a constant. Now, this wave function of the ground state is even, and I would expect that it's a superposition of even eigenstates of the second oscillator as well. And even eigenstates are things that have even occupation numbers. Those are the even Hermite polynomials. So presumably, it goes like this and things with four oscillators and things like that. So what that after is this sort of expression of the original state in terms of energy eigenstates in terms of anything of the second oscillator. So how can we do that? Well, one thing we know about this state is that a1 on it is equal to 0. It's killed by a1, but that a1 is an interesting thing. It's a2 cosh gamma plus a2 dagger sinh gamma, and that thing must kill that state. So I could at least, if I had infinite time, put a few terms and try to calculate more or less what kind of state is killed by this strange combination of creation and annihilation operators. You see, we know a ground state is killed by the normal annihilation operator. That's what this is. But this operator, now we know it's given by this formula over there, and then it must kill all that. So we're faced with a problem that is in principle fairly difficult, and you could not hope for an exact solution unless there's something very nice going on. Happily, squeezed states are still very nice and tractable states, so let's see what we can do. Well, what I'm going to do is to put an ansatz for this state based on this expansion that I had there. I would say, look, there's going to be a normalization constant, but at the end of the day, we have things acting on the vacuum, so there's going to be something very messy acting on the vacuum of 2. And what is that going to be? Well, we've learned about coherent states that are exponentials of oscillators, exponentials of a's and a daggers added. So here, we're going to attempt something a little more general. I'll put an exponential minus 1/2, and what should I put? Well, let's try to be simple minded still. It seems to go in even power, so if we're very lucky, maybe we can put just an a2 dagger a2 dagger here, an exponential something quadratic in oscillators. And I don't know what the coefficient is in front, and it may depend on gamma because I have to solve an equation with gamma. So I'll put minus 1/2 f of gamma times that. And we'll see if we can solve this. So what does it mean to solve it? Well, it means that it must be annihilated by this operator. So our computations with the creation and annihilation operators are becoming more and more complicated. They look more and more complicated. They're really not harder. Let's see what happens. So I need now that a2 cosh gamma plus a2 dagger sinh gamma kill this state. So the N is going to go outside. It's a number. So acting on e to the minus 1/2 f of gamma a2 dagger a2 dagger on the vacuum sub 2, that must be 0. How does one solve this? Well, let's see what we have. Let's see this term. a2 dagger, good. a2 dagger commutes with a2 dagger, so I can bring the a2 dagger all the way to the right and it doesn't kill the vacuum, so I don't gain anything. Can be to the right or to the left because it commutes with this whole thing, so I haven't gained anything if I move it, so false start. I don't want to move that one. This one, I want to leave it here, and this one somehow must produce something that cancels this one. Now, a2, on the other hand, is the kind of thing that always should be dealt with because this is an annihilator and that does kill that. So as it moves along, it encounters obstacles, but obstacles are opportunities because an obstacle means we're going to get something that maybe cancels that. So if it also went through and killed the vacuum, we're finished. This doesn't kill the vacuum. Happily, it gets stuck here. Now the thing that we have to hope is that we can disentangle that commutator. Now, here is a universal thing. How do I want to write this? I'm going to write it like this. I have an a2, a number, I don't care about the number, and a complicated thing, and a vacuum. Whenever you have an a, any operator, and a vacuum, this is equal to a commutator with the operator on the vacuum. That should be second nature because this is even given to that minus oa, but oa, the a is near to the vacuum and it kills it. So whenever you have an a o vacuum, you can put the commutator, so I'll do that here. So I put a2, the cosh gamma, I take it out. I put this whole thing minus 1/2 f a dagger a dagger 2. This whole thing and the vacuum. That's the first term. And the second term, I have to just copy it. sinh gamma a2 dagger e to the minus 1/2 fa squared dagger on the vacuum. All that should be 0. So what do we get? Is that commutator doable or undoable? It's happily a simple commutator, even if it doesn't look like it, because whenever you see a commutator like that, you think A to the B, and then you know if you're in luck, this is just AB e to the B, and this is true if AB commutes with B. So that's what you must think whenever you see these things. Am I in this lucky situation? Yes, you are, because with this commutator, one a will kill an a dagger, so you will be left with an dagger. But a dagger commutes with b, which is a dagger a dagger. So AB, A with B is just add an a dagger up to a function or a number, and then a dagger commutes with B so you are in good shape. This is true. So what do we get here? We get cosh gamma, and then we just get the commutator of a2 with minus 1/2 f a2 dagger a2 dagger times the whole exponential-- I won't write it-- times the vacuum plus sinh times a2 dagger times the whole exponential times the vacuum. We have to do this commutator, but the f doesn't matter. It's a constant. It's a function. No operator in there. a2 with a2 daggers are 1. There are two of them, so you get a 2, and the 1/2 cancels this, so you get minus cosh gamma f a2 dagger times the exponential plus, from the other term, sinh gamma a2 dagger times the exponential on the vacuum equals 0. And, as promised, we were good. We get an a2 dagger, a2 dagger. These two terms cancel if f is equal to tan hyperbolic of gamma, which is sine over cosine so that these two things cancel. I can write this, of course, as minus cosh gamma f plus sinh gamma a2 dagger, the exponential, and the vacuum, equals 0. So it's just a simple relation, but there we go. Tanh gamma is the thing. Tanh gamma gives you the answer, and let me write this state so that you enjoy it. Let's see. The state is just a fairly interesting thing, this 01 expressed in the new Hilbert space of the second oscillator is some n of gamma times the exponential of minus 1/2 tangent hyperbolic of gamma a2 dagger, a2 dagger on the vacuum sub 2. And you need the normalization, n of gamma, and it will be done. Now, the normalization, you may say well, look, normalizations are good things. Sometimes, you work without normalizations and you're OK, but it turns out that these normalizations are pretty useful, and unless you get them, some calculations are kind of undoable. So it's a little bit of a challenge to get that normalization. You can try in several ways. The most naive way is to say, well, this must have unit norms, so n squared, and then I take the bra of this and the ket of that, so it would be a vacuum, an exponential of minus 1/2 tangent a a, and an exponential of minus 1/2 tangent a dagger, a dagger. Must be 1. n squared times that. The problem is that I've never been able to compute this. At least it takes a long time and you get it by indirect methods, but getting a number out of this is painful. So there's one way of getting the normalization here that is not so bad. It's a little surprising what you do. You do the following. You declare, I'm going to compute the overlap of 2, the vacuum of 2, with the vacuum of 1. And now, what is this, n gamma vacuum of 2 here, e to the minus 1/2 tanh gamma a2 dagger a2 dagger vacuum of 2. How difficult is it to compute this inner product? AUDIENCE: [INAUDIBLE]. BARTON ZWIEBACH: Sorry? AUDIENCE: Not difficult. BARTON ZWIEBACH: Not difficult. What is it? AUDIENCE: [INAUDIBLE]? BARTON ZWIEBACH: Yeah, that thing. AUDIENCE: e to the negative 1/2 tanh gamma. It's 1. BARTON ZWIEBACH: Sorry? AUDIENCE: I mean, you multiply the a2 dagger right across to the left hand side of the ket. BARTON ZWIEBACH: Yeah, you're saying it, indeed. Look, this thing is as simple as can be. This is just 1. Why is that so? You expand the exponential, and you have 1 plus things, but all the things have a daggers. Now, a daggers don't kill this 1, but they killed the other 1 on the left, and there's nothing obstructing them from reaching the left, so this is 1. It's completely different from this one because if you expand this one, the a daggers kill the thing but there's lots of a's to the left. And the a's want to get here, but there's lots of a daggers to the right, so this is hard, but this is easy. So n of gamma is 0 2 0 1. But what is that? If you introduce a complete set of position states, zx, This is 0 2 x x 0 1. This one is the ground state wave function of the first Hamiltonian, and this is the star of the ground state wave function of the second Hamiltonian. And those you know because you know m, omega. You know the ground state wave functions, so this integral can be done. So this whole normalization is given by this integral, and this integral gives you 1 over square root of cosh gamma. That interval takes a few lines to make, but the end result is there. So you got your coherent states. You got now the squeezed state completely normalized, so let's write it out. 0 1 is equal to 1 over square root of cosh gamma exponential of minus 1/2 tanh gamma a2 dagger a2 dagger on the vacuum sub 2. Wow. That's it. That's a squeeze state that has been squeezed in such a way that the squeezing parameter appears here in the exponential. Now, this is the way we got to it, but now I wanted to just think of it independently, just from the beginning. If you had a Hamiltonian, this is an interesting state all in itself because it is a squeezed state. It's a Gaussian, but of the wrong shape for this system. This is a Gaussian of the right shape for system two. But once you put all these oscillators, it's not anymore a Gaussian of the right type. It's a squeezed Gaussian. So if we forget about this system one, let me write this thing from the beginning and say like this. We have a Hamiltonian, we have a ground state, we have m and omega, and we have a and a dagger. Let's just define what we call the squeezed vacuum, vacuum sub gamma, to be precisely this thing. 1 over square root of cosh gamma exponential of minus 1/2 tanh gamma a dagger a dagger, not 2 anymore because we have just a single system. A single system, the ground state, and now we've defined this state, which is what we had there before, but we don't think of it anymore as, oh, it came from some other Hamiltonian, but rather, this is a state on its own. It's a squeezed vacuum state. And from the computations that we did here, the delta x for this state would be e to the minus gamma h bar m omega. and m omega over here. So these are these, and you don't need to know what gamma is. That's a number that somebody chose for you. Any number that you want is gamma, and therefore, you use it to squeeze the state. And that's what you've achieved. So you have a Hamiltonian of a harmonic oscillator. You can construct the vacuum. You know how to construct coherent states by acting on the vacuum. Now you know how to construct squeezed states, states in which the expectation values do those things. We had a very nice formula where we began the lecture today in which the coherent state was just a unitary operator acting on the vacuum. Now, we made sure to normalize this, so we did check in this calculation that o gamma 0 gamma is equal to 1. So this thing must come from the action of some unitary operator acting on the vacuum. Which is that unitary operator that acts on the vacuum and gives you that? Not so easy to find. All the computations here are a little challenging, as you've seen. But here's the answer. Cosh gamma e to the exponential of minus 1/2 tanh gamma a dagger a dagger should be something like an e to the what? Should be something like e to the a dagger a dagger minus aa acting on the vacuum. Why? Because certainly, the aa's are going to disappear, and you're going to get products of this one squared. And this is anti-Hermitian, so that operator is unitary, but I now must put the gamma somewhere there. So what should I put here in order to get that to work? Well, it's maybe something you can try by assuming gamma is very small and expanding both sides, or finding a differential equation, or doing things, but the answer is incredibly simple. It's e to the minus just gamma over 2. That's it. Gamma appears here, and by the time you reorder this quadratic form-- you see, what you have to do here is expand, and then you have powers of these, and then you have to bring all the annihilators to the right and kill them. And then you have a power series in squares of this thing. That will reassemble into this exponential. It's almost a miracle that something like that could happen, but it does happen. And it's a very interesting calculation, actually, to do that. We don't do it in the course. I may post some pages that I did once this computation. And that is a nice operator. We call it the squeezing operator. So s of gamma is a unitary operator, s of gamma. The squeezed state of 0 gamma is equal to s of gamma on the vacuum where s of gamma is equal to e to the minus gamma over 2 a dagger a dagger minus aa, that operator. It's a unitary operator and it does the squeezing. Actually, once you have squeezed states, you can do more things, and you can squeeze and then translate. Those are the most general states that people use in quantum optics. So you take a vacuum, you squeeze it with s of gamma, and then you translate it with v of alpha. And this is the state, alpha gamma, squeeze factor, translation factor. One picture of that is in our alpha plane. You take the vacuum that is some spherical ball here in the x expectation value, p expectation value. You squeeze it. You might decide, I don't want to have too much delta x, so you squeeze it and you produce something like this. That's a squeezed vacuum by the time you apply this. And then you do the alpha, and you translate it out, and this state is now going to start rotating and doing all kinds of motion. It's pretty practical stuff. Actually, some of you are taking junior lab, and the person that works a lot there in junior lab is Nergis Mavalvala, and she does gravity wave detection, and squeezed states has been exactly what she's been working. In order to minimize displacements in the gravity wave detectors, they have a squeeze vacuum state injected into the detector to make the harmonic oscillator that represents the mirror stabilize its uncertainty in position to the maximum possible. There's a whole fabulous technique that people use with the squeezed states. Now, the squeezed states allow you to construct some states that seemed to us that they were pretty strange and that we never had good formulas for them. So that's how I want to conclude the lecture. I will leave photon states for next time, but I want to discuss one more application of the squeezed states, and this comes from limits. So here is your squeezed state, e to the minus gamma. So let's squeeze the state to the end. Take gamma to go to infinity. What happens to the squeezed state? So you're narrowing out the ground state in position space to the maximum possible. What happens to the state? Well, it goes a little singular, but not terribly singular. Gamma is going to infinity, so cosh is going to infinity as well. So the state is going kind of to 0, but 0 sub infinity. It's proportional, but the exponential is good. Exponential of minus 1/2 tangent of gamma as gamma goes to infinity is just 1. And this is a dagger a dagger on the vacuum. This state is in almost terrible danger to be infinite. If you try to find its wave function, you're not going to be able to normalize it. You've reached the end of the road kind of thing because of this. Gamma goes to infinity. This is going to be infinite here because this state, if you compute its overlap with itself, is blowing up. And here, you see the niceness of this. It also suggests that gamma can go from plus infinity to minus infinity, and that's a natural thing. Nevertheless, here, it goes from plus 1 to minus 1. If you had a number 3 here, this is a state that blows much worse than the worst delta function or derivative or square that you've ever had. It's just unbelievably divergent because it just can't exist, this state. You're going beyond infinity here to go behind this thing. So it's just pretty much impossible. So the limit is states are reasonable as long as this quadratic form goes from minus 1 to 1. And when you go to 1, you get this, and what should this be? This should be the wave function I associated to a delta function. This would be the position state, x equals 0. Roughly, it's a delta function. And indeed, if you act with x on it, x, remember, is a plus a dagger. Act on this exponential. Now, do you remember how to do that? This a dagger doesn't do anything but the a goes here, and it's a trivial commutator. You get minus a dagger. So it actually kills this and gives you 0. So in fact, the exposition operator acting on here gives you 0. It looks like it is really the state x equals 0. If you go the other way around, and you take gamma to be minus infinity, the only thing that changes here is the sign. So this is like the delta of x, or the x equals 0 state. And if you take 0 minus infinity, goes like x minus 1/2 plus 1/2 a dagger a dagger on the vacuum. And this state is a delta function in momentum. It's the momentum state p equals 0. Why? Because gamma is going to minus infinity. The uncertainty of momentum is going to 0. And therefore, indeed, if you act with the momentum operator on this state, it's like acting with a minus a dagger, and you've changed the sign of this, but you've changed the sign here, so it also kills this state. So it looks like we can really construct position and momentum eigenstates now with squeezed states, and that's what they are supposed to be. A squeezed state is something that has been squeezed enough that you can get a delta function. So how do you finish that construction? Here is the claim. Square root of 2 m omega over h bar x a dagger minus 1/2 a dagger a dagger acting on the vacuum. This is the claim, that this is the x position state. So basically, you have to squeeze first and then translate this thing to the x position. So how do you check this? Well, you should check that the x operator, which is something times a plus a dagger acting on this thing gives you little x. So you should have that x operator on this thing gives you x x. And that is going to work out because the a dagger is going to sit here and it's going to get canceled with the a with this, but the a with this part is just going to bring down an x with the right factor. So this state, which is a squeezed state and a little bit of a coherent state as well, is producing the position eigenstate. In the harmonic oscillator, you can really construct the position eigenstate and you can calculate the normalization. The normalization comes out to be a rather simple thing. So at the end of the day, the position eigenstate is m omega over pi h bar to the 1/4 e to the minus m omega x squared over 2h bar. And this whole exponential, square root of 2 m omega over h bar, x a dagger minus 1/2 a dagger a dagger acting on the vacuum. So your basis of creation and annihilation operators on the harmonic oscillator is flexible enough to allow for a concrete description of your position eigenstates, and a tractable one as well. And that's the extreme limit of squeezing, together with some little bit of coherent displacement. Next time, we'll do our photon states and we'll illustrate the ideas of both coherent and squeezed states at the same time.

MIT_805_Quantum_Physics_II_Fall_2013

10_Uncertainty_Principle_and_Compatible_Observables.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, last time we were talking about uncertainty. We gave a picture for uncertainty-- it was a neat picture, I think of the uncertainty, refer to the uncertainty measuring an operator A that was a Hermitian operator. And that uncertainty depended on the state that you were measuring. If the state was an eigenstate of A, there would be no uncertainty. If the state is not an eigenstate of A, there was an uncertainty. And this uncertainty was defined as the norm of A minus the expectation value of A acting on psi. So that was our definition of uncertainty. And it had nice properties. In fact, it was zero if and only if the state was an eigenstate of the operator. We proved a couple of things as well-- that, in particular, one that is kind of practical is that delta A of psi squared is the expectation value of A squared on the state psi minus the expectation value of A on the state psi squared. So that was also proven, which, since this number is greater than or equal to 0, this is greater than or equal to 0. And in particular, the expectation value of A squared is bigger than the expectation of A squared. So let's do a trivial example for a computation. Suppose somebody tells you in an example that the spin is in an eigenstate of Sz. So the state psi it's what we called the plus state, or the z plus state. And you want to know what is uncertainty delta of Sx. So you know if you're in an eigenstate of z, you are not in an eigenstate of x-- in fact, you're in a superposition of two eigenstates of Sx. Therefore, there should be some uncertainty here. And the question is, what is the quickest way in which you compute this uncertainty, and how much is it? So many times, the simplest way is to just use this formula. So let's do that. So what is the expectation value of Sx in that state? So it's Sx expectation value would be given by Sx on this thing. Now, actually, it's relatively clear to see that this expectation value is going to be 0, because Sx really in the state plus is equal amplitude to be Sx equal plus h bar over 2, or minus h bar over 2. But suppose you don't remember that. In order to compute this, it may come handy to recall the matrix presentation of Sx, which you don't need to know by heart. So this state plus is the first state, and the basis state is the state 1 0. And then we have Sx on plus is equal to h bar over 2 0 1 1 0, acting on 1 0. Zero and that's equal to h bar over 2. The first thing gives you 0, and the second one gives you 1. So that's, in fact, equal to h bar over 2, the state of minus. So here you go to h bar over 2 plus minus, and you know plus and minus are orthogonal, so 0 is expected. Well, are we going to get zero uncertainty? No, because Sx squared, however, does have some expectation value. So what is the expectation value of Sx squared? Well, there's an advantage here. You may remember that this Sx squared is a funny matrix. It's a multiple of the identity, because if you square this matrix, you get the multiple of the identity. So Sx squared is h over 2 squared times the identity matrix-- the two by two identity matrix. So the expectation value of Sx squared is h bar over 2 squared times expectation value of the identity. And on any state, the expectation value on any normalized state, the expectation value of the identity will be equal to 1. So this is just h squared over 2 squared. So back to our uncertainty, delta Sx squared would be equal to the expectation value of Sx squared minus the expectation value of Sx squared. This was 0. This thing was equal to h bar over 2 squared, and therefore, delta Sx is equal to h bar over 2. So just I wanted to make you familiar with that. You can compute these things-- these norms and all these equations are pretty practical, and easy to use. So today what we have to do is the following-- we're going to establish the uncertainty principle. We're going to just prove it. And then, once we have the uncertainty principle, we'll try to find some applications for it. So before doing an application, we will discuss the case of the energy time uncertainty principle, which is slightly more subtle and has interestingly connotations that we will develop today. And finally, we'll use the uncertainty principle to learn how to find bounds for energies of ground states. So we might make a rigorous application of the uncertainty principle. So the uncertainty principle talks about two operators that are both Hermitian, and states the following-- so given the theorem, or uncertainty principle, given two Hermitian operators A and B, and a state psi normalized, then the following inequality holds. And we're going to write it in one way, then in another way. Delta A psi squared times delta B-- sometimes people in order to avoid cluttering don't put the psi. I don't know whether to put it or not. It does look a little more messy with the psi there, but it's something you have to keep in mind. Each time you have an uncertainty, you are talking about some specific state that should not be forgotten. So maybe I'll erase it to make it look a little nicer. Delta B squared-- now it's an inequality. So not just equality, but inequality. That product of uncertainties must exceed a number-- a computable number-- which is given by the following thing. OK, so here it is. This is a number, is the expectation value of this strange operator in the state psi squared. So even such a statement is somewhat quite confusing, because you wish to know what kind of number is this. Could this be a complex number? If it were a complex number, why am I squaring? That doesn't make any sense. Inequalities-- these are real numbers. Deltas are defined to be real numbers. They're the norms. So this is real positive. This would make no sense if this would be a complex number. So this number better be real. And the way it's written, it seems to be particularly confusing, because there seems to be an i here. So at first sight, you might say, well, can it be real? But the thing that you should really focus here is this whole thing. This is some operator. And against all first impressions, this operator formed by taking the commutator of A and B-- this is the commutator A B minus B A-- is Hermitian, because, in fact, if you have two operators, and you take the commutator, if the two of them are Hermitian, the answer is not Hermitian. And that you know already-- x with p is equal to i h bar. These are Hermitian operators, and suddenly the commutator is not a Hermitian operator. You have the unit here. A Hermitian operator with a number here would have to be a real things. So there's an extra i, that's your first hint that this i is important. So the fact is that this operator as defind here is Hermitian, because if you take 1 over i A B-- and we're going to try to take its Hermitian conjugate-- we have 1 over i A B minus B A. And we're taking the Hermitian conjugate. Now, the i is going to get complex conjugated, so you're going to get 1 over minus i. The Hermitian conjugate of a product is the Hermitian conjugate in opposite order. So it would be B dagger A dagger minus A dagger B dagger. And of course, these operators are Hermitian, so 1 over minus i is minus 1 over i. And here I get B A minus A B. So with a minus sign, this is 1 over i A B again. So the operator is equal to its dagger-- its adjoint. And therefore, this operator is Hermitian. And as we proved, the expectation value of any Hermitian operator is real. And we're in good shape. We have a real number. This could be negative. And a number, when you square it, is going to be a positive number. So this makes sense. We're writing something that at least makes sense. Another way, of course, to write this equation, if you prefer-- this inequality, I mean-- is to take the square root. So you could write it delta A times delta B. Since this is a real number, I can take the square root and write just this as absolute value of psi, 1 over 2i i A B psi. And these bars here are absolute value. They're not norm of a vector. They are not norm of a complex number. They are just absolute value, because the thing inside is a real thing. So if you prefer, whatever you like better, you've got here the statement of the uncertainty principle. So the good thing about this uncertainty principle formulated this way is that it's completely precise, because you've defined uncertainties precisely. Many times, when you first study the uncertainty principle, you don't define uncertainties precisely, and the uncertainty principle is something that goes with [? sim ?] is approximately equal to this. And you make statements that are intuitively interesting, but are not thoroughly precise. Yes, question, yes. AUDIENCE: Should that be greater or equal? PROFESSOR: Greater than or equal to, yes-- no miracles here. Other question? Other question? So we have to prove this. And why do you have to prove this? This is a case, actually, in which many interesting questions are based on the proof. Why would that be the case? Well, a question that is always of great interest is reducing uncertainties. Now, if two operators commute, this right-hand side is 0 and it just says that the uncertainty could be made perhaps equal to 0. It doesn't mean that the uncertainty is 0. It may depend on the state, even if the operators commute. This is just telling you it's bigger than 0, and perhaps by being clever, you can make it equal to 0. Similarly, when you have two operators that just don't commute, it is of great importance to try to figure out if there is some states for which the uncertainty relation is saturated. So this is the question that, in fact, you could not answer if you just know this theorem written like this, because there's no statement here of what are the conditions for which this inequality is saturated. So as we'll do the proof, we'll find those conditions. And in fact, they go a little beyond what the Schwarz inequality would say. I mentioned last time that this is a classic example of something that looks like the Schwarz inequality, and indeed, that will be the central part of the demonstration. But there's one extra step there that we will have to do. And therefore, if you want to understand when this is saturated, when do you have minimum uncertainty states, then you need to know the proof. So before we do, of course, even the proof, there's an example-- the classic illustration that should be mentioned-- A equal x and B equals p, xp equal i h bar. That's the identity. So delta x squared delta p squared is greater or equal than psi 1 over 2i-- the commutator-- i h bar 1 psi squared. And what do we get here? We get the i's cancel, the h bar over 2 goes out, gets squared, and everything else is equal to 1, because h is normalized. So the precise version of the uncertainty principle is this one for x and p. And we will, of course, try to figure out when we can saturate this. What kind of wave functions saturate them? You know the ones that are just sort of strange-- if x is totally localized, the uncertainty of momentum must be infinite, because if delta x is 0, well, to make this something that at least doesn't contradict the identity, delta p better be infinite. Similarly, if you have an eigenstate of p, which is a wave, is totally delocalized, and you have infinite here and 0 here. Well, they're interesting states that have both, and we're going to try to find the ones of minimum uncertainty. So OK, we've stated the principle. We've given an example. We've calculated an uncertainty. Let us prove the theorem. So as we mentioned before, this idea that the uncertainty is a norm, is a good one. So let's define two auxilliary variables-- f, a state f, which is going to be A minus the expectation value of A on psi. And we can put the ket here. And g, which is going to be B minus the expectation value of B, psi. Now what do we know about this? Well the uncertainties are the norms of these states, so the norm squared of these states are the uncertainty squared. So delta A squared is f f, the norm squared. And delta B squared is g g. And Schwarz' inequality says that the norm of f times the normal of g is greater than or equal than the absolute value of the inner product of f with g. So squaring this thing, which is convenient perhaps at this moment, we have f f-- norm squared of f-- norm squared of g must be greater than or equal than f g squared, absolute value squared. So this is Schwarz. And this is going to just make a note-- here we know when this is saturated. It will be saturated if f is parallel to g. If these two vectors are parallel to each other, the Schwarz inequality is saturated. So that's something to keep in mind. We'll use it soon enough. But at this moment, we can simply rewrite this as delta A squared times delta B squared-- after all, those were definitions-- are greater than or equal-- and this is going to be a complex number in general, so f g in Schwarz' inequality is just a complex number. So this is real of f g squared, plus the imaginary part of f g squared. So that's what we have-- real and imaginary part. So let's try to get what f g is. So what is f g? Let's compute it. Well we must take the bra corresponding to this, so this is psi. Since the operator is Hermitian, you have A minus expectation value of A, and here you have B minus expectation value of B psi. Now we can expand this, and it will be useful to expand. But at the same time, I will invent a little notation here. I'll call this A check, and this B check. And for reference, I'll put that this is psi A check B check psi. On the other hand, let's just compute what we get. So what do we get? Well, let's expand this. Well, the first term is A times B on psi psi, and we're not going to be able to do much about that-- A B psi. And then we start getting funny terms-- A cross with B, and that's-- if you think about it a second, this is just going to be equal to the expectation value of A times the expectation of B, because the expectation value of B is a number, and then A is sandwich between two psi. So from this cross product, you get expectation value of A, expectation value of B, with a minus sign. From this cross product, you get the expectation value of A and expectation value of B-- another one with a minus sign. And then one with a plus sign. So the end result is a single one with a minus sign. So expectation value of A, expectation value of B. Now, if I change f and g, I would like to compute not only fg inner product, but gf inner product. And you may say why? Well, I want it because I need the real part and the imaginary parts, and gf is the complex conjugate of f g, so might as well compute it. So what is gf? Now you don't have to do the calculation again, because basically you change g to f or f to g by exchanging A and B. So I can just say that this is psi B A psi minus A B. And if I write it this way, I say it's just psi B check A check psi. OK so we've done some work, and the reason we've done this work is because we actually need to write the right-hand side of the inequality. And let's, therefore, explore what these ones are. So for example, the imaginary part of f g is 1 over 2i f g minus its complex conjugate-- gf. Imaginary part of a complex number is z minus z star divided by 2i. now, fg minus gf is actually simple, because this product of expectation values cancel, and this gives me the commutator of A with B. So this is 1 over 2i, and you have psi expectation value of A B commutator. So actually, that looks exactly like what we want. And we're not going to be able to simplify it more. We can put the 1 over 2i inside. That fine. It's sort of in the operator. It can go out, but we're not going to do better than that. You already recognize, in some sense, the inequality we want to prove, because if this is that, you could ignore this and say, well, it's anyway greater than this thing. And that's this term. But let's write the other one, at least for a little while. Real of fg would be 1/2 of fg plus gf. And now it is your choice how you write this. There's nothing great that you can do. The sum of these two things have AB plus BA and then twice of this expectation value, so it's not nothing particularly inspiring. So you put these two terms and just write it like this-- 1/2 of psi anti-commutator off A check with B check. Anti-commutator, remember, is this combination of operators in which you take the product in one way, and add the product in the other way. So I've used this formula to write this, and you could write it as an anti-commutator of A and B minus 2 times the expectation values, or whichever way you want it. But at the end of the day, that's what it is. And you cannot simplify it much. So your uncertainty principle has become delta A squared delta B squared greater than or equal to expectation value of psi 1 over 2i A B psi squared plus expectation value of psi 1 over 2 A check B check psi squared. And some people call this the generalized uncertainty principle. You may find some textbooks that tell you "Prove the generalized uncertainty principle," because that's really what you get if you follow the rules and Schwarz' inequality. So it is of some interest. It is conceivable that sometimes you may want to use this. But the fact is that this is a real number. This is a Hermitian operator as well. This is a real number. This is a positive number. So if you ignore it, you still have the inequality holding. And many times-- and that's the interesting thing-- you really are justified to ignore it. In fact, I don't know of a single example-- perhaps somebody can tell me-- in which that second term is useful. So what you say at this moment is go ahead, drop that term, and get an inequality. So it follows directly from that, from this inequality, that delta A squared delta B squared is greater than or equal-- you might say, well, how do you know it's equal? Maybe that thing cannot be 0. Well, it can be 0 in some examples. So it's still greater than or equal to psi 1 over 2i A B psi squared. And that's by ignoring the positive quantity. So that is really the proof of the uncertainty principle. But now we can ask what are the things that have to happen for the uncertainty principle to be saturated? That you really have delta A delta B equal to this quantity, so when can we saturate? OK, what do we need? First we need Schwarz inequality saturation. So f and g must be states that are proportional to each other. So we need one, that Schwarz is saturated. Which means that g is some number times f, where beta is a complex number. This is complex vector space, so parallel means multiply by a complex number. That's still a parallel vector. So this is the saturation of Schwarz. Now, what else do we need? Well, we need that this quantity be 0 as well, that the real part of this thing is equal to 0. Otherwise, you really cannot reach it. The true inequality is this, so if you have Schwarz, you've saturated. This thing is equal to this thing. The left-hand side is equal to this whole right-hand side. Schwarz buys you that. But now we want this to be just equal to that. So this thing must be 0, so the real part of f overlap g-- of fg must be 0. What does that mean? It means that fg plus gf has to be 0. But now we know what g is, so we can plug it here. So g is beta times f. Beta goes out, and you get beta f f. Now when you form the bra g, beta becomes beta star. So you get beta star f f equals 0. And since f need not have zero norm, because there is some uncertainty presumably, you have that beta plus beta star is equal to 0, or real of beta is equal to 0. So that said, it's not that bad. You need two things-- that the f and g vectors be parallel with a complex constant, but actually, that constant must be purely imaginary. So beta is purely imaginary-- that this beta is equal to i lambda, with lambda real. And we then are in shape. So for saturation, we need just g to be that, and g to be beta f. So let me write that equation over here. So g-- what was g? It's B, B minus absolute value of B on psi, which is g, must be equal to beta, which is i lambda A minus absolute value of A on psi. Condition-- so this is the final condition for saturation. now, that's a strange-looking equation. It's not all that obvious how you're even supposed to begin solving it. Why is that? Well, you're trying to look for a psi, and you have a constraint on the psi. The psi must satisfy this. I actually will tell both Arum and Will to discuss some of these things in recitation-- how to calculate minimum uncertainty wave packets based on this equation, and what it means. But in principle, what do you have to do? You have some kind of differential equation, because you have, say, x and p, and you want to saturate. So this is x, and this is p. Since p, you want to use a coordinate representation, this will be a derivative, and this will be a multiplication, so you'll get a differential equation on the wave function. So you write an answer for the wave function. You must calculate the expectation value of B. You must calculate the expectation value of A, and then plug into this equation, and try to see if your answer allows a solution-- and a solution with some number here, lambda. At least one thing I can tell you before you try this too hard-- this lambda is essentially fixed, because we can take the norm of this equation. And that's an interesting fact-- take the norm. And what is the norm of this? This is delta B, the norm of this state. And the norm of i lambda--, well norm of i is 1. Norm of lambda is absolute value of lambda, because lambda was real. And you have delta A here of psi, of course. So lambda can be either plus or minus delta B of psi over delta A of psi. So that's not an arbitrary constant. It's fixed by the equation already, in terms of things that you know. And therefore, this will be a subject of problems in a little bit of your recitation, in which you, hopefully, discuss how to find minimum uncertainty packets. All right, so that's it for the proof of the uncertainty principle. And as I told you, the proof is useful in particular to find those special states of saturated uncertainty. We'll have a lot to say about them for the harmonic oscillator later on, and in fact throughout the course. So are there any questions? Yes. AUDIENCE: So if we have one of the states and an eigenstate, we know that [INAUDIBLE] is 0 and we then mandate that the uncertainty of the other variable must be infinite. But is it even possible to talk about the uncertainty? And if so, are we still guaranteed-- we know that it's infinite, but it's possible for 0 and an infinite number to multiply [INAUDIBLE] PROFESSOR: Right, so you're in a somewhat uncomfortable position if you have zero uncertainty. Then you need the other one to be infinite. So the way, presumably, you should think of that, is that you should take limits of sequences of wave functions in which the uncertainty in x is going to 0, and you will find that as you take the limit, and delta x is going to 0, and delta p is going to infinity, you can still have that. Other questions? Well, having done this, let's try the more subtle case of the uncertainty principle for energy and time. So that is a pretty interesting subject, actually. And should I erase here? Yes, I think so. Actually, [? Griffith ?] says that it's usually badly misunderstood, this energy-time uncertainty principle, but seldom your misunderstanding leads to a serious mistake. So you're saved. It's used in a hand-wavy way, and it's roughly correct, although people say all kinds of funny things that are not exactly right. So energy time uncertainty-- so let me give a small motivation-- a hand-wavy motivation, so it doesn't get us very far, but at least it gives you a picture of what's going on. And these uncertainty relations, in some sense, have a basis on some simple statements that are totally classical, and maybe a little imprecise, but incontrovertible, about looking at waveforms, and trying to figure out what's going on. So for example, suppose in time you detect a fluctuation that as time progresses, just suddenly turns on. Some wave that just dies off after a little while. And you have a good understanding of when it started, and when it ended. And there's a time T. So whenever you have a situation like that, you can try to count the number of waves-- full waves that you see here. So the number of waves would be equal to-- or periods, number of full waves-- would be the total time divided by the period of this wave. So sometimes T is called the period. But here, T is the total time here, and the period is 2 pi over omega. So we say this is omega t over 2 pi. Now, the problem with these waves that begin and end, is that you can't quite see or make sure that you've got the full wave here. So in the hand-wavy way, we say that even as we looked at the perfectly well-defined, and you know the shape exactly-- it's been measured-- you can't quite tell whether you've got the full wave here or a quarter of a wave more, so there's an uncertainty in delta n which is of order 1. You miss half on one side, and half on the other side. So if you have an uncertainty here of order 1, and you have no uncertainty in T, you would claim that you have, actually, in some sense, an uncertainty in what omega is. Omega might be well measured here, but somehow towards the end you can't quite see. T we said was precise, so over 2 pi is equal to 1. I just took a delta of here, and I said P is precise, so it's delta omega. So this is a classical statement. An electrical engineer would not need to know any quantum mechanics to say that's about right, and you can make it more or less precise. But that's a classical statement. In quantum mechanics, all that happens is that something has become quantum, and the idea that you have something like this, we can associate it with a particle, a photon, and in which case, the uncertainty in omega is uncertainty in energy. So for a photon, the uncertainty is equal to h bar omega, so delta omega times h bar is equal to the uncertainty in energy. So if you plug it in here, you multiply it by h bar here, and you would get delta E times T is equal to 2 pi h bar. And then you have to add words. What is T? Well, this T is the time it takes the photon to go through your detector. You've been seeing it. You saw a wave. You recorded it, and took a time T-- began, ended. And it so it's the time it took you to have the pulse go through. And that time is related to an uncertainty in the energy of the photon. And that's sort of the beginning of a time energy uncertainty relationship. This is quantum, because the idea that photons carry energies and they're quantized-- this is a single photon-- and this connection with energy is quantum mechanics. So this is good and reasonable intuition, perhaps. And it can be the basis of all kinds of things. But it points out the fact that the more delicate part here is T. How could I speak of a time uncertainty? And the fact is that you can't speak of a time uncertainty really precisely. And the reason is, because there's no Hermitian operator for which we could say, OK the eigenstates of this Hermitian operator are times, and then you have a norm, and it's an uncertainty. So you can't do it. So you have to do something different this time. And happily, there's something you can do that is precise and makes sense. So we'll do it. So what we have to do is just try to use the uncertainty principle that we have, and at least one operator. We can use something that is good for us. We want uncertainty in energy, and we have the Hamiltonian. It's an operator. So for that one, we can use it, and that's the clue. So you'll take A to be the Hamiltonian, and B to be some operator Q that may depend on some things-- for example, x and p, or whatever you want. But the one thing I want to ask from this operator is that Q has no explicit time dependence-- no explicit time dependence whatsoever. So let's see what this gives us as an uncertainty relationship. Well, it would give us that delta H squared-- that's delta Q squared-- would be greater than or equal to the square of psi 1 over 2i H with Q psi. OK, that's it. Well, but in order to get some intuition from here, we better be able to interpret this. This doesn't seem to have anything to do with energy and time. So is there something to do with time here? That is, in fact, a very well-known result in quantum mechanics-- that somehow commutators with the Hamiltonian test the time derivative of operators. So whenever you see an H with Q commutator, you think ah, that's roughly dQ dt. And we'll see what happens with that. And say, oh, dQ dt, but it doesn't depend on T-- you said 0. No it's not 0. There's no explicit dependence, but we'll see what happens. So at this moment, you really have to stop for one second and derive a familiar result-- that may or may not be that familiar to you from 804. I don't think it was all that emphasized. Consider expectation value of Q. And then the expectation of Q-- let me write it as psi Q psi, like this. Now let's try to take the time derivative of this thing. So what is the time derivative of the expectation value of q? And the idea being that look, the operator depends on some things, and it can have time-dependent expectation value, because the state is changing in time. So operators can have time-dependent expectation values even though the operators don't depend on time. So for example, this depends on x and p, and the x and p in a harmonic oscillator are time dependent. They're moving around, and this could have time dependence. So what do we get from here? Well, if I have to take the time derivative of this, I have d psi dt here, Q psi, plus psi Q d psi dt. And in doing this, and not differentiating Q itself, I've used the fact that this is an operator and there's no time anywhere there. I didn't have to differentiate Q. So how do we evaluate this? Well, you remember the Schrodinger equation. Here the Schrodinger equation comes in, because you have time derivatives of your state. So i d psi dt, i H bar d psi dt is equal to H psi. That's a full time-dependent Schrodinger equation. So here, maybe, I should write this like that-- this is all time-dependent stuff. At this moment, I don't ignore the time dependence. The states are not stationary states. If they would be stationary states, there would be no energy uncertainty. So I have this, and therefore, I plug this in here, and what do we get? i H bar h psi Q psi plus psi Q i H bar H psi. Now, I got the i H in the wrong place-- sorry-- 1 over i H bar, and 1 over i H bar. Now the first term-- this thing comes out as its complex conjugate-- 1 minus i H bar, because it's on the first input. H is Hermitian, so I can send it to the other side, so psi, HQ psi. Second term-- the 1 over i H just goes out, and I don't have to move anybody. QH is there, psi. So actually, this is i over H bar, because minus i down goes up with i. And I have here psi HQ, and this is minus i over H bar, so I get HQ minus QH psi. So this is your final result-- the expectation value d dt of the expectation value of Q is equal to i over H bar, expectation value of the commutator of H with Q. So this is neat, and it should always stick in your mind. This is true. We will see the Heisenberg way of writing this equation in a little while-- not today, but in a couple of weeks. But maybe even write it even more briefly as i over H bar expectation value of HQ. So what do we get from here? Well, we can go back to our uncertainty principle, and rewrite it, having learned that we have time derivative. So time finally showed up, and that's good news. So we're maybe not too far from a clear interpretation of the uncertainty principle. So we're going back to that top equation, so that what we have now is delta H squared delta Q squared is that thing over there, the expectation value of 1 over 2i. There's some signs there, so what do we have-- equals 1 over 2i H bar over i d dt of Q. So what I did here was to say that this expectation value was H bar over i d dt of Q, and I plugged it in there. So you square this thing, so there's not too much really to be done. The i don't matter at the end of the day. It's a minus 1 that gets squared. So the H bar over 2-- I'm sorry-- the H bar over 2 does remain here, squared. And you have dQ dt squared. Q is a Hermitian operator. B was supposed to be Hermitian. The expectation value is real. The time derivative is real. It could be going up or down. So at the end of the day, you have delta H delta Q is greater than or equal to H bar over 2, the absolute value of dQ over dt. There we go. This is, in a sense, the best you can do. Let's try to interpret what we've got. Well, we've got something that still doesn't quite look like a time uncertainty relationship, but there's time in there. But it's a matter of a definition now. You see, if you have delta Q, and you divide it by dQ dt, first it is some sort of time. It has the units of time. And we can define it, if you wish, to be sub delta t. And what physically, does this delta t represent? Well, it's roughly-- you see, things change in time. The rate of change of the expectation value of Q may not be uniform. It make change fast, or it may change slowly. But suppose it's changing. Roughly, this ratio, of this would be constant, is the time it takes the expectation value of Q to change by delta Q. It is like a distance divided by a velocity. So this is roughly the time needed for the expectation value of Q to change by delta Q, by the uncertainty. So it's a measure of the time needed for a significant change, if the expectation value, if the uncertainty of Q is significant, and is comparable to Q. Well, this is the time needed for significant change. Now this is pretty much all you can do, except that of course, once you write it like that, you pull this down, and you go up now, delta H delta t is greater or equal than H bar over 2. And this is the best you can do with this kind of approach. Yes? AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, I simply define this, which is a time that has some meaning if you know what the uncertainty of the operator is and how fast it's changing-- is the time needed for a change. Once I defined this, I simply brought this factor down here, so that delta Q over this derivative is delta t, and the equation just became this equation. So we'll try to figure out a little more of what this means right away, but you can make a few criticisms about this thing. You can say, look, this delta time uncertainty is not universal. It depends which operator Q you took. True enough. I cannot prove that it's independent of the operator Q, and many times I cannot even tell you which operator Q is the best operator to think about. But you can try. And it does give you-- first, it's a mathematical statement about how fast things can change. And that contains physics, and it contains a very precise fact as well. Actually, there's a version of the uncertainty principle that you will explore in the homework that is, maybe, an alternative picture of this, and asks the following thing-- if you have a state and a stationary state, nothing changes in the state. But if it's a stationary state, the energy uncertainty is 0, because the energy is an eigenstate of the energy. So nothing changes. So you have to wait infinite time for there to be a change, and this makes sense. Now you can ask the following question-- suppose I have a state that is not an eigenstate of energy. So therefore, for example, the simplest thing would be a superposition of two eigenstates of different energies. You can ask, well, there will be time evolution and this state will change in time. So how can I get a constraint on changes? How can I approach changes? And people discovered the following interesting fact-- that if you have a state, it has unit norm, and if it evolves, it may happen that at some stage, it becomes orthogonal to itself-- to the original one. And that is a big change. You become orthogonal to what you used to be. That's as big a change as can happen. And then you can ask, is there a minimum time for which this can happen? What is the minimum time in which a state can change so much that it becomes orthogonal to itself? And there is such an uncertainty principle. It's derived a little differently from that. And it says that if you take delta t to be the time it takes psi of x and t to become orthogonal to psi of x0, then this delta t times delta E-- the uncertainty of the energies is the uncertainty in h-- is greater than or equal to h bar over 4. Now a state may never become orthogonal to itself, but that's OK. Then it's a big number on the left-hand side. But the quickest it can do it is that. And that's an interesting thing. And it's a version of the uncertainty principle. I want to make a couple more remarks, because this thing is mysterious enough that it requires thinking. So let's make some precise claims about energy uncertainties and then give an example of what's happening in the physical situation. Was there a question? Yes. AUDIENCE: [INAUDIBLE] PROFESSOR: You're going to explore that in the homework. Actually, I don't think you're going to show it, but-- AUDIENCE: [INAUDIBLE] H bar [INAUDIBLE] it's even less than the uncertainty [INAUDIBLE] PROFESSOR: It's a different statement. It's a very precise way of measuring, creating a time. It's a precise definition of time, and therefore, there's no reason why it would have been the same. So here is a statement that is interesting-- is that the uncertainty delta E in an isolated system is constant-- doesn't change. And by an isolated system, a system in which there's no influences on it, a system in which you have actually time independent Hamiltonians. So H is a time independent Hamiltonian. Now that, of course, doesn't mean the physics is boring. Time- independent Hamiltonians are quite interesting, but you have a whole system. Let's take it to be isolated. There's no time dependent things acting on it, and H should be a time independent Hamiltonian. So I want to use this statement to say the following-- if I take Q equals H in that theorem over there, I get that d dt of the expectation value of H would be what? It would be i over H bar. Since H is time independent-- the condition here was that Q had no time dependence. But then I get H commutator with H. So I get here H commutator with H. And that commutator is 0. However complicated an operator is, it commutes with itself. So the expectation value of the energy doesn't change. We call that energy conservation. But still, if you take Q now equal to H squared, the time derivative of the expectation value of H squared, you get i over H bar. You're supposed to be H commutator with Q, which is H squared, now. And that's also 0. So no power of the expectation value of H vanishes. And therefore, we have that the time derivative of the uncertainty of H squared-- which is the time derivative of the expectation value of H squared minus the expectation value of H squared-- well, we've shown each one of the things on the right-hand side are 0, so this is 0. So delta H is constant. So the uncertainty-- delta E or delta H of the system is constant. So what do we do with that? Well it helps us think a little about time dependent processes. And the example we must have in mind is perhaps the one of a decay that leads to a radiation of a photon, so a transition that leads to a photon radiation. So let's consider that example. So we have an atom in some excited state, decays to the ground state and shoots out the photon. Then it's an unstable state, because if it would be stable, it wouldn't change in time. And the excited state of an atom is an unstable state, decays into-- goes into the ground state. And it makes a photon. Now this idea of the conservation of energy uncertainty at least helps you in this situation that you would typically do it with a lot of hand-waving, organize your thoughts. So what happens in such decay? There's a lifetime, which is a typical time you have to wait for that excited state to decay. And these lifetime is called tau. And certainly as the lifetime goes through, and the decay happens, some observable changes a lot. Some observable Q must change a lot. Maybe a position of the electron in an orbit, or the angular momentum of it, or some squared of the momentum-- some observable that we could do an atomic calculation in more detail must change a lot. So there will be associated with some observable that changes a lot during the lifetime, because it takes that long for this thing to change. There will be an energy uncertainty associated to a lifetime. So how does the energy uncertainty reflect itself? Well, you have a ground state. And you have this excited state. But generally, when you have an excited state due to some interactions that produce instability, you actually have a lot of states here that are part of the excited state. So you have an excited state, but you do have, typically, a lot of uncertainty-- but not a lot-- some uncertainty of the energy here. The state is not a particular one. If it would be a particular one, it would be a stationary state-- would stay there forever. Nevertheless, it's a combination of some things, so it's not quite a stationary state. It couldn't be a stationary state, because it would be eternal. So somehow, the dynamics of this atom must be such that there's interactions between, say, the electron and the nucleus, or possibly a radiation field that makes the state of this electron unstable, and associated to it an uncertainty in the energy. So there's an uncertainty here, and this particle-- this electron goes eventually to the ground state, and it meets a photon. So there is, associated to this lifetime, an uncertainty delta E times tau, and I will put similar to H bar over 2. And this would be the delta E here, because your state must be a superposition of some states over there. And then what happens later? Well, this particle goes to the ground state-- no uncertainty any more about what its energy is. So the only possibility at this moment consistent with the conservation of uncertainty in the system is that the photon carries the uncertainty. So that photon must have an uncertainty as well. So delta energy of the photon will be equal to h bar delta omega, or h delta nu. So the end result is that in a physical decay process, there are uncertainties. And the uncertainty gets carried out, and it's always there-- the delta E here and the photon having some uncertainty. Now one of the most famous applications of this thing is related to the hyperfine transition of hydrogen. And we're very lucky in physics. Physicists are very lucky. This is a great break for astronomy and cosmology, and it's all based on this uncertainty principle. You have the hyperfine transition of hydrogen. So we will study later in this course that because of the proton and electron spins in the hydrogen atom, there's a splitting of energies having to do with the hyperfine interaction. It's a magnetic dipole interaction between the proton and the electron. And there's going to be a splitting. And there's a transition associated with this splitting. So there's a hyperfine splitting-- the ground state of the hyperfine splitting of some states. And it's the top state and the bottom state. And as the system decays, it emits a photon. This photon is approximately a 21 centimeter wavelength-- is the famous 21 centimeter line of hydrogen. And it corresponds to about 1420 megahertz. So how about so far so good. There's an energy splitting here, 21 centimeters wavelength, 5.9 times 10 to the minus 6 eV in here. But that's not the energy difference that matters for the uncertainty, just like this is not the energy difference that matters for the uncertainty. What matters for the uncertainty is how broad this state is, due to interactions that will produce the decay. It's a very funny, magnetic transition. And how long is the lifetime of this state? Anybody know? A second, a millisecond, a day? Nobody? Ten million years-- a long time-- 10 million years-- lifetime tau. A year is about pi times 10 to the 7 seconds is pretty accurate. Anyway, 10 million years is a lot of time. It's such a large time that it corresponds to an energy uncertainty that is so extraordinarily small, that the wavelength uncertainty, or the frequency uncertainty, is so small that corresponding to this 1420, it's I think, the uncertainty in lambda-- and lambda is of the order of 10 to the minus 8. The line is extremely sharp, so it's not a fussy line that it's hard to measure. It's the sharpest possible line. And it's so sharp because of this 10 million years lifetime, and the energy time uncertainty relationship. That's it for today.

MIT_805_Quantum_Physics_II_Fall_2013

17_Two_State_Systems_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: What we have to do today is study a very important example of a two- state system. That will be the ammonia molecule, and will lead us to understand how masers work. Masers are what for microwaves, the same thing as lasers are for light-- it's just a different frequency. Masers is microwaves and lasers is for light. It's the same thing. So it's a very nice application of two-state systems. And then we'll discuss over the last part of the lecture some aspects of nuclear magnetic resonance. I don't think I'll get to the end of it, because it's quite a bit of material. But we'll try to see what we can do. So let me remind you of the last thing we were doing last time, that this is going to be in the backdrop of what we do today. We spoke about Hamiltonians for a two-state system that were the most general two by two Hermitian matrix specified by four real numbers-- g0 and the three components of the vector g multiplied by the Pauli matrices. This is Hermitian. This can be written in this way, in which we've identified Hamiltonians for spins, in the sense that g dot sigma-- really, sigma is proportional to S, so this is equal to omega dot S, where omega-- Larmor-- is 2g over h bar. And we explained last time that if you have a term omega l dot S, spins will rotate with angular velocity omega l vector, which means they rotate around the axis defined by the vector omega l, with an angular velocity equal to the magnitude of the vector omega l. So that's Larmor precession. This Larmor precession in the case of a magnetic field is given by minus lambda times the magnetic field, gamma times the magnetic field, where gamma is that constant that relates the magnetic moment of the particle to the spin angular momentum of the particle. Then we got, moreover, that the energy levels of this Hamiltonian-- this is a two-state systems, so it's a two-dimensional vector space that can be at most two energy eigenstates. That's the simple thing about two-state systems. These two energy eigenstates have the energies equal to g0 plus/minus g. And the plus corresponds to the spin state n plus, and the minus corresponds to the spin state n minus. And you don't have to talk spin states when you write this spin states over here. The plus, you should think of spinning in the plus direction, but the thing that we call plus is the first basis vector. And the thing that we call minus is the second basis vector of this state space. Therefore, if you've given a matrix, Hamiltonian has nothing to do with spins. You still have the notion that the first basis vector, whatever it is-- an iron moving in this direction-- is the mathematical analog of a spin up. And the second basis vector-- whatever else it may be-- is the analog of the spin down. So this will be important for what we do now, as we begin the study of the ammonia molecule, and its states. So having reviewed the key ideas from the last part of last lecture, are there any questions? So let me begin with this ammonia molecule-- double M, M-O-N-I-A, is NH3. It's used as a fertilizer. It's a gas, has strong odor, no color, fertilizers, cleaning products, pharmaceuticals, all kinds of things. It has the shape of a flattened tetrahedron with a nitrogen atom at one corner, say, and the base the three hydrogen atoms. If it would not be a flattened tetrahedron, this angle over here-- if it would be an equilateral, regular tetrahedron, this angle over there would be 60 degrees, because this every face would be an equilateral triangle. But if it's a flattened tetrahedron, if it will be totally flat-- then n would be at the base. The angle in between these two edges would be 120 degrees, because they have to add up to 360. Well, this has 108 degrees, so it's pretty flat, this tetrahedron. And you can imagine, actually, this is a two-state system, because just as the nitrogen can be like this-- can be up with respect to the base of the hydrogens, it could also be down. And so you can imagine this molecule rotating, and suddenly the up nitrogen goes down, and this keeps rotating-- a possible transition of the system, in which, well, I don't know if I'm drawing it well. I don't think so. But roughly, it could be like this-- the nitrogen is down. And this, in principle, would be like two possible configurations of this system. There's a barrier. This is in equilibrium, so if you try to push it, it's not easy. But if you did manage to push it, it would be stable in the other direction. It is as if you had a potential for the nitrogen-- a V of z-- the direction is z-- in which it can be here, up or down. And it's stable in either one, but there's a big barrier in between. So that's the story of this nitrogen atom. And we're going to try to describe this as a two-state system. So I need some notation. Well, I'm going to have the first basis state to be called up for N up, and the second basis state is going to be called down for nitrogen down. And now, I'm going to try to write the Hamiltonian for this system. Well, you know what sort of happens here. Your intuition in quantum mechanics with wave functions should be similar. Look-- this is not the two-state system, because there may be many energy eigenstates, but you know that the ground state looks like a wave function, just like this. And the first excited state could look like a wave function that is like-- oops-- this. Pretty much the same thing, but you flip one, and if the barrier is sufficiently high, these two energy levels are not that different. So the question is, how do we model this? There may be an energy E0 for the system, a ground state energy maybe, and a little bit of a higher energy. So we're going to write the Hamiltonian. And I'm going to put E0 for the moment. And my first basis state is 1 and up. This would be the 1 0, and here would be the 0 1, the second basis state. And this is saying that 1 0, the N up, is an energy eigenstate of energy E0. And down is an energy eigenstate of energy in up as well. But that can't be the story. There cannot be two degenerate energy eigenstates. Your intuition tells you that this is impossible. One dimensional potential wouldn't say that. So there must be something else happening. This cannot be the whole Hamiltonian that describes the physics of the problem. So what we're going to do is try to tinker with this Hamiltonian, a simple tinkering that is going to give us the physics that we expect. So I'm going to put a constant delta here. This should be Hermitian, so I should put the delta as well, another constant there. For convenience, however, I'd rather put the minus sign there. I will define delta to be positive for definiteness, and for convenience, however, I will put here minus delta. Now, you could say look, you say that, but maybe it's not for convenience. Maybe it changes the physics. Well, it cannot change the physics, because these things are the matrix elements of the Hamiltonian-- the 1 2, and the 2 1 matrix elements. And I could decide to change what I call the first basis vector, to call it minus the first basis vector. This would change the sign of this, change the sign of that, without changing those signs. So this sign is a matter of a basis. So we certainly have not made any assumption by putting that to be a minus sign over there. Now, once you have this Hamiltonian, this delta is going to be some energy. And that's going to be what mimics the physics here, because these states are not going to be any more energy eigenstates. The matrix is not diagonal anymore. So the 1 0 vector, and the 0 1 vector are not any more energy eigenstates. Moreover, it's interesting to try to figure out what it has to do with our previous system. So this is E0 times 1 minus delta times sigma 1. And that's a good thing to know. So in this case, comparing to this g is the vector in the x direction, because its g multiplying by sigma 1. And it has magnitude delta. So we notice-- and we're going to make a picture later-- is that g, in this case so far, is equal to delta times the unit vector in the x direction, minus delta times the unit vector in the x direction. So g is equal to delta. So, OK, we've written those. Let's then figure out what are the ground states and the excited states. And this is a two by two matrix, and a simple one, at that. So you could just do it, or better to figure out what we're doing. We'll use our formulas before. Yes, George. AUDIENCE: So why is it that we mandate that delta has to be real? I mean, that's not the most general form. PROFESSOR: That's right, it's not the most general form. So at this moment, we're trying to do what any reasonable physicist does. Without delta, it doesn't match the physics. So let's try the simplest thing that can work, and a delta real-- we'll see if it works. And if it works, we'll worry later about different things. So we'll put the simplest thing at this moment, but indeed, we could put more complicated things. So given this, in fact, we know what it the energy eigenstates should be, or we more or less can guess what the energy eigenstates should be. Let me tell you the energies are g0 plus/minus g, so you're going to get E0 plus delta and e0 minus delta as the energies-- E excited, and E ground. And the gap between these two energies-- the gap between these two energy levels is 2 delta. So there we go. We've already produced something good. We have two energy eigenstates. There should be a small energy difference, and that gap is 2 delta. Now, what are those states? Well, it's not too hard to see that the eigenstate that has this energy is the excited state, is 1 over square root of 2, 1 minus 1. If you add with this matrix on this, that's the energy eigenstate. And the energy eigenstate for this one is 1 over square root of 2, 1 1. Let's write them. These are the eigenvectors. Let's write them as 1 over square root of 2, nitrogen up minus nitrogen down, and 1 over square root of 2 nitrogen up plus nitrogen down. So I want to, even though it's not complicated to do this, and we have called these states that way, so it's all clear. I want you to see how that comes from our spin way of thinking. So you know there's this molecule, and for this molecule, only one direction matters. We could have called it x, if we wanted. In fact, maybe x would have been a better name. On the other hand, for spin states, there are three dimensions-- x, y, and z. So we have to think in an abstract way. So where is this vector G? We said G is minus delta Ex. So this is the x-axis. This is the y-axis. This is the z-axis. G is here. The vector G goes back over here, is minus delta x hat vector. Now, what if you have g in that way? You know that the excited state is one of these states over here. Let's see-- this N plus is the excited state, and N minus is the lower state. So the excited state should point in the direction of g vector, because N corresponds to the direction of the g vector. G is positive here, this little g is positive. So g is in there. N is in there as well, because g and n are parallel. And the excited state should correspond to N, a vector in the plus n direction. So the excited state should be here. It's a spin state in that direction. That's what that formula says. And the ground state should be a spin state in the minus N direction, so this must be the ground state. So this I call the excited state, and this the ground state. And indeed, remember now that what is your translation. 1 and 2-- the 1 and 2 states are like the plus and minus of spins. So in terms of spin language, this excited state is the plus minus the minus. And this is the plus plus the minus, because the up is plus, the down is minus. So indeed, this state-- you probably remember it. This is a spin along the x direction. So the ground state must be like a spin along the x direction. That's here. The excited state is a spin, the orthogonal one in the minus x direction, so it must be a state orthogonal to this one, as to this, and it points in the other direction. So those are our spins. And we had that the gap delta-- the gap is 2 omega. It's an energy, so it's what we called h omega naught a photon-- the transition energy. I could give this energy in eVs, but I actually don't have it. I have the wavelength of the frequency of the associated photon. So this corresponds to a frequency nu of 23.87 gigahertz, and a lambda of about 1.26 centimeters-- more or less half an inch. So that is the transition difference between these two levels. So this is something people knew-- there's two levels above this molecule, and they correspond to the result of this perturbation that splits them. So in a sense, we've got a nice model-- perfectly reasonable model, without introducing much complexity-- of what this thing is doing. So let's do a little exercise. Let's see how does the N up state evolve in time? So we have psi at time equals 0 be in the state up. What is it later? There are many ways to do it-- many, many ways to do it. The quickest, in principle, is to think about spins, even if just a little painful. But let's think about spins. Omega l is going to be around the direction of g. So think of the state of the spin. The N up state, the up state, is here. And then it's going to precess with angular frequency vector in the direction of g. So it's going to precess in the direction of g. So you can imagine now this vector precessing. And it's going to go-- since it's essentially the minus x direction-- precession in time is going to flip it to the y-axis, and then make it rotate in the z-y axis. That's all what it's going to do. So you have a picture of what it's going to do. We might as well calculate a little, although the picture is complete, and the frequency is known, and everything. But what you do here, of course, is you try to write it in terms of energy eigenstates. And the up state is the 1 over square root times the sum of e plus g. And you know the energies of those two states, so you know how they evolve in time. It will be in the notes. You can do this. After you now evolve, with e to the minus i ht over h bar, you then go back from e to g, to up and down, because that's sort of the intuition that we want to Have. So it's not difficult. None of these steps are difficult. e and g are written in terms of up and down. So what does one get? One gets psi of t is equal e to the minus i, et over h bar times cosine of t delta over h bar, times the state up plus i sine of t delta over h bar, state down. This is the time evolution, so the probabilities, for example, to be up is the square of this one-- cosine squared of t delta over h bar. And the probability to be down is sine squared of the same thing. So this poor nitrogen molecule, if it happens to have the nitrogen up, is going to start rotating like crazy, even if you don't do anything. It's just sitting there, and it's rotating up and down, with a speed doing this thing 23 billion times a second. Molecule's up and down, because it's not in a stationary eigenstate. Now, here, actually, you may think that something is a little funny, because you would say, well, the frequency of rotation is like delta over h bar, but the Larmor frequency is supposed to be 2g over h bar, so it would correspond to a Larmor frequency of 2 delta over h bar, which is exactly the frequency of the photons. But there's no contradiction here. This is, in fact, rotating at that speed, at twice that speed. Because if you remember, for a spin state, this was the cosine of theta over 2. Therefore, as it changes, that's the way theta over 2 is changing. But theta, which is the angle of this physical rotation, changes twice as fast. So it's, again, those 1/2s of spin states that are very confusing sometimes. But there's no contradiction. The sort of Larmor frequency of the equivalent spin problem is exactly the same as the frequency of the original problem. So now we want to make this into something more practical. And for that, what we explore is the fact that this molecule has an electric dipole moment. So the molecule as we pictured it there, as it happens, the electrons of the hydrogen sort of cluster near the nitrogen. So this up region is kind of negative. The bottom region is kind of positive, and there is an electric dipole moment pointing down. So this is a pretty important property of this molecule, this dipole moment. And electric dipoles we usually call p. But for some reason-- maybe I should change the notes, at some stage or maybe this is discussed very nicely in Feynman's lectures on physics. He uses mu for this, like for magnetic dipole. So I will actually use mu as well, now. So this thing has an electric dipole, and therefore the energy is the electric dipole dotted with the electric field. And that electric field is an external electric field. You have this little dipole, which is this molecule, and you put it inside an electric field, and there's a contribution to the energy, just because the dipole is sitting on an electric field. And that means our Hamiltonian is now changed. So I will consider the case in which we have an electric field in the z direction-- a positive electric field in the z direction-- so that E is equal to E times z. And mu would be equal to minus mu times z, because it points down. We've assumed that the dipole is down. And the dipole is down for the case of spin in the z direction. So look what we get here-- this energy contribution is essentially mu E. And it's the energy that is acquired by the state in which the nitrogen is up. This is for nitrogen up. So what we've discovered-- if we want to model this in the Hamiltonian is that we can take the Hamiltonian that we have-- E0 minus delta, e0 minus delta, and the energy of the state up with nitrogen up, is this one-- mu E. So we add mu E. And the one with the spin down will be the opposite, so it will be minus mu E. And this is our reasonable expectation for the Hamiltonian of this molecule inside an electric field. So this is the NH3 in E field. So again, we can wonder what kind of thing happens here. And the best thing is to first say this is E0 1 minus delta sigma 1. And then you see, oh, it's mu E sigma 3. So getting to diagonalize this is a little more painful than the other one. And we don't have to do it, because we've solved the general problem. And the energies, this time, are going to be E of the excited one, and E of the lower one-- ground state. It's going to be E0 plus g. And g was the magnitude of the vector g. So it's the magnitude of the vector g that now has components minus delta 0 and mu E. So here we get plus square root of delta squared plus mu E squared. And here is 0 minus square root of delta squared plus mu E squared. So there we go. If we know how the energies behave, even if we have some electric field-- and typically delta is such, and mu is such that, for most electric fields that you ever have in the lab-- this is very small compared to that. The dipole moment is sufficiently small that the energies that you get from here are pale compared to the difference of energies over there. So you can approximate. This is E0 plus delta plus 1/2 mu E squared over delta. This is for mu E small. Here E0 minus delta minus 1/2 mu E over delta squared. And this is when mu E is much smaller than delta. Now, the only reason I point this out is because it does provide a technological opportunity to separate a beam of particles into excited- and the ground-state level. Sort of like Stern-Gerlach experiment, you put, now, this beam that has this ammonia molecules. And you put them inside an electric field that has a gradient. In a gradient, this state is going to try to go to minimize its energy. So it's going to go to the regions of the electric field where the electric field is small. This particle minimizes its energy when it goes to the regions of the electric field when the electric field is big. So it's like putting it in a Stern-Gerlach experiment. You have your beam, and you separate them. You have your beam and you manage to separate the things that can be in an excited state, and the things that are in the ground state. And now what you do is insert these excited states into a resonant cavity. Have a little hole here, and a little hole here, and E comes in, and something comes out. So we're getting now to the design of the maser. The idea that we're trying to do is that we try to make a cavity tuned to 23.7 gigahertz-- the frequency associated with a gap. And we just insert those Es over there, these excited states over there, and hope that by the time they go out, they become a g. Because if they go from E-- say there was an electric field here to separate them, and then E's over here. This is the excited state. There's no more electric fields over here. It just comes into the cavity as an excited state. The excited state has energy E0 plus delta. And then, if it manages to go out of the cavity as the ground state, then it would have energy E0 minus delta. It must've lost energy to delta. That can go nicely into the electromagnetic field and become one photon-- a one photon state in the cavity of the right frequency, because the cavity is tuned to that. The only difficulty with this assumption is that E is an energy eigenstate. So energy eigenstates are lazy. They're stationary states. They don't like to change. So there's no reason why it should go out as g. It's excited state. It's perfectly happy to remain excited forever. So what must happen somehow is that there's an electric field here in the cavity, and that stimulates this thing to make the transition, because once there's an electric field, E is not anymore an energy eigenstate. The E of the original system is not anymore an energy eigenstate, and nor is this. So then it's going to change in time. So the problem is a delicate one in which we want to somehow have an electric field here that is self-consistent with the idea that this excited state goes out as the ground state. And that's why it's microwave amplification by stimulated emission of radiation, because you're going to amplify a signal here. It's a microwave, 1 centimeter wavelength-- that's a microwave. And the stimulation is the fact that this wouldn't do it unless there's some electric field already. So you could say, well, so how does it get started? There's no electric field to begin with. Well, you know quantum mechanics, and you know that in general, there are little fluctuations, and there's energies-- small photons, one or two photons that suddenly appear because of anything. Any motion of charges in here produces an electromagnetic wave. So at the beginning, yes-- there's no many photons here. But somehow, by having it resonate at that frequency, it's very easy to get those photons. And a few appear, and a few molecules start to turn in, and then very soon this is full with energy, in which there's a consistent configuration of some electric field oscillating and producing precisely the right transitions here. So I want to use the next 50 minutes to describe that math. How do we do this? Because it just shows really the sort of hard part of the problem. How do you get consistently a field, and the radiation going on? So maybe I should call this E prime and g prime. They shouldn't be confused. E and g are these states that we had before. And E prime and g prime, we never wrote those states, but they are deformed states due to an electric field. OK, so what do we have to do? Well here is E and g. And we had a Hamiltonian. There's going to be an electric field here, so this Hamiltonian is the relevant one. The only problem with this Hamiltonian is that this is going to be a time-dependent field, something that we're a little scared of-- Hamiltonians with time dependence-- for good reason, because they're hard. But anyway, let's try to see. Today's all about Hamiltonians with time dependence. So there's going to be a time-dependent is going to be the wave here. So that's the relevant Hamiltonian. But it's the Hamiltonian in the 1 2 basis, in that up nitrogen, down nitrogen basis. I want that Hamiltonian in the Eg basis. It's better. It's more useful. So let's try to see how that looks-- Hamiltonian in the Eg basis. H prime in E equal 1 g equal 2 primes, maybe put basis. So here is the Hamiltonian in this basis. In the 1 2 basis, I have to pass to the other basis, the Eg basis. So it's not complicated. It takes a little work, but it's nothing all that difficult. For example, in the 1 prime h 1 prime, which would be the 1 1 element of this matrix, I'm supposed to put here 1 prime is EhE. And now I'm supposed to say, OK, what is this? Well, E is 1 over square root of 2. E was 1 over square root of 2, 1 minus 1. H is the original H, so it's E0 plus mu E minus delta minus delta E0 minus mu E. And E is 1 1, 1 minus 1, again. And there's also the square root of 2. So at the end, this is a 1/2. So this is the kind of thing you have to do to pass to this basis. So I think I'll do that in the notes. And this calculation is simple. In this case, it gives E0 plus delta. And in retrospect, that sort of is pretty reasonable. This is E0 plus delta, and this is E0 minus delta. And if you didn't have an electric field-- indeed in this basis, the first state is the excited state who has this energy. The second state is the ground state, and has this energy. And that makes sense if there's no mu E. Well, the mu E still shows up, and it shows up here. So that is the Hamiltonian in this basis, and the general state in this basis is the amplitude to be excited, and the amplitude to be in the ground state. This is the general psi of t. So your Schrodinger equation in this mu basis-- not mu basis, in the Eg basis, you see it's-- Eg basis is the basis of energy eigenstates if you don't have electric field. But once you have an electric field, it's not anymore energy eigenstates, and much worse if you have a time-dependent electric field. So the Schrodinger equation is i h bar d dt of CE Cg is equal to this mu matrix. And now, E0 is totally irrelevant for everything. It's a constant of the unit matrix. Let's put E0 to 0. There's no need to keep it. E0 equal zero. So we have delta mu E minus delta times CECg. I d dt of psi, h psi-- the Schrodinger equation. Now, the real difficulty that we have is that E is a function of time. So this is not all that trivial. So what you do to solve this is simplify the equation by saying what would the solution be if you didn't have a function of time? Then you would have-- if you didn't have E, CE Cg of time would be E to the minus i ht. So this would be i delta t-- the energy of this, if there's no electric field, the Hamiltonian will be delta minus delta. So here, I have my psi ht over h bar. And for the lower state, you would have E to the plus i delta t over h bar as solutions, if you didn't have this. This would be the solutions. But we want now better. So what we're going to say is well, that's a solution if I neglect this. So this cannot be the real solution. So I'll put here beta sub E of t, and a beta sub g of t. And sure-- if no electric field, betas are 1. They're not necessary. But if there is an electric field, the betas are going to be complicated, so we need them there. So this is like an [INAUDIBLE]. Now you could plug this back, and calculate what you get. And it should not be too surprising that you're going to get in here something in which these deltas are going to disappear, because this thing takes care of that. So there's a little bit of algebra here, maybe two, three lines of algebra. And let me give you what you get-- I h bar d dt of beta E beta g. Now the equation is really for this quantities-- and it's 0 E to the i omega 0t times mu E, E to the minus i omega 0t times mu E0, beta E, beta g, where omega 0 is the Larmor frequency, or 2 delta over h bar. So some calculation involved, but I hope you don't lose track of the physics here. The physics is that the amplitude to be in E and the amplitudes to be in g have now been replaced by beta and beta g, which are expected to be simpler things to calculate. And in fact, since the probability to be in E is the norm of to this thing squared, beta is as good as C to know how likely is the particle to be in E, or how likely is the particle to be in the ground state. You could use beta, because they differ by a phase. So betas have still the physical significance of the amplitude to be in E or g. And we're still having an electric field that is time dependent here. So it's time to put the time dependence, and what are we going to do? Well we're going to find a sort of self-consistent operation of this device, in which we will have an electromagnetic field, E. E of t will be 2 E-- letter E, that now is going to be a constant with an E0, cosine omega 0 t-- again the Larmor frequency or the photon frequency that is emitted by the possible transition. So we will consider the case when the cavity has already that electric field that is going precisely at the right speed to do things well. So this Et, with the 2 conveniently put here, is equal to E0 E to the i omega naught t plus e to the minus i omega naught t. So when you multiply these things, what do you get? Let me do one of them for you to see. You get this i-- the top one-- h I'm going to put to the other side. Beta E is going to be beta E dot, is going to couple with this to beta g. So that thing is going to be this electric field times mu, so mu e0, the h bar from the left-hand side, and you have E to the i omega naught t multiplying this, so it's 1 plus E to the 2i omega naught t times beta g. So that's the first equation. Not all that easy, for sure. Second equation-- i beta g dot is equal to mu E0 over h bar, 1 plus e to the minus 2i omega naught t beta E of t. And now you have to think maybe a little like engineers, or physicists, or mathematicians-- whatever you prefer. But you want to convince yourself that what you want to do is true. And what do you want to do? I like to forget about these curves, basically. That's what we want to do. Why would that be true? Well, here's a reason why it's true. This is a number that sort of goes between 1 and minus 1, with some phase. Mu E0, however, over h bar, is a very small number. Mu E0 we're thinking of-- we're saying compared to the natural scales of the problem, this energy, mu E0, is much smaller than delta. And delta is the thing that is related to omega naught, which h bar omega naught is equal to 2 delta. So essentially, mu E0 is an energy which is very, very slow compared to h omega naught. Now, being very small, whatever this is, this time derivative is going to be very small. So beta E and beta g are going to be very small-time derivatives-- going to move slowly. If they move slowly over the time that this oscillates, this hasn't changed a lot. And therefore, the average of this function over that time is 0, and it doesn't contribute to the differential equation. It's actually a good argument. You can try to convince yourselves, or maybe I'll try better in some way, or something. But it is right. I've tried it, actually, out with computers and with Mathematica, and things like that. And it's really absolutely true-- that if you think of differential equations as integrals, that you integrate with this in the right-hand side, you can see that if this-- really, the time derivative is controlled by this, that corresponds to a frequency much smaller than omega naught. These ones don't matter. So it's really interesting, and still not trivial. But this is a case where we end up ignoring part of the thing. So what do we get, then? i beta E dot is equal mu E0 over h bar beta g. And the second equation-- i beta g dot equal mu E0 over h bar beta E. Therefore, if you multiply by another i here, and differentiate i-- i beta double dot of E is equal mu E0 over h bar. The i extra that we borrowed, the dot here. You can use the second equation. So you get mu E0 over h-bar squared beta E. Therefore, beta E double dot is equal to minus mu E0 over h-bar beta E. And you, see your you're rotating with the a of-- put the h-bar-- with a frequency that i much smaller than omega 0. So this as a frequency, this is like omega 0. So indeed, the rate of change of this thing goes with the frequency that's much smaller. And it's all right, actually. So, we've had that, and then we can write the solution, finally. So, what is it? Beta E of T is cosine mu E0 t over h-bar. And the probability to be in the E state is the square of that amplitude, so the probability to be in the E state at time t is the square of that. So it's cos squared mu E 0 t over h-bar. Again, I'm sorry, this beta is here, and the probability to be in the excited state is this square of ce, but the square of ce is the square of beta E, so I just square this. So there you go. We have this thing, and we now have an understanding of how this goes as time goes by. In fact, this mu e 0 T or h-bar under bar goes by. This starts to as a cosine squared, and then it goes like this. And this is the place when this is equal to pi over 2. So, what do we need? We need the place where this happens-- we can call that time T-- be such that mu E 0 T over h-bar be equal to pi over 2, or 3 pi over 2. For those values of time, the probability to be excited is zero, and therefore you must be in the ground state. These two probabilities squared added go to 1. So you're either E or G. So if you have zero probability to be in E, you will be in the state G. So, the whole issue is basically at this moment that you must give the right speed, or for a given the speed of the molecules, there will be a time that it takes to traverse. That time is related to the steady state value of this electric field by this relation. So you need the right velocity-- the molecules have to be at the right temperature, that's when they'll have a velocity-- so that they travel in such a way that is consistent with this. As they do that, each of these particles that goes from E to G gives out one photon, one quantum of the electromagnetic field, and helps build the time dependent electric field that we started at the beginning. Now if you, at some point for some speed of the molecules, you saturate this and you build some electric field, and then you have your cavity operating at the nominal way. And then, of course, you want to use this for something, so you shine, let it go out, and shine those microwaves or do something. And if you want to recharge it, you keep adding nitrogen molecules. Now, this was a great discovery, actually. Charles Townes, Gordon, and Zeiger, built this ammonia maser in 1953. They got the Nobel Prize. Nobel Prize, Charles Townes got it in 1964. And he emphasized that this masers do the most perfect amplification consistent with the uncertainty principle. This is a coherent state of light that is built here, and it's much better than any vacuum amplifier or anything like that, because the thing that this giving out those photons is a molecule that this uncharged. So it doesn't disturb the electromagnetic field in the cavity as it goes through. Many times you use an electron, for example, to give out some energy. But the electron itself is charged, so it produces additional electromagnetic fields, shot noise, all kinds of noise. This is absolutely quiet device in which is this molecule turns from one state to another smoothly, stimulated by the electric field, because if there was no electric field would be an energy eigenstate and then it gives out photon after photon. So the uncertainties actually, in this noble lecture, which is fun to read-- in fact, you remember delta N delta phi supposed to be greater than 1/2. Well, for coherent states as we more or less discussed, this thing is saturated. And what do you have now? You have a coherent state of light. You may remember that the expectation value of N in a coherent state alpha was alpha squared. And you call this the number n of photons. And the uncertainty in N was, in fact, alpha. So, it's square root of n. So for this thing, we have the situation in which we are working with a coherent state that saturates this. So delta n times delta phi is about one half. And delta n is square root of the number of photons. So delta phi is about to 1 over 2 square root of the number of photons. And you can imagine the cavity easily can have 10 to the 12 photons, and to 15 photons. Something fairly big. And you get an uncertainty phase. The thing is coherent. All the pieces of that electromagnetic wave, the phases are very coherent up to an incredibly great accuracy. So it's a great discovery and the beginning of many things that were done here, in fact, by Professor Kleppner and others in the '60s with other type of lasers and masers and this whole thing. So, that's pretty much what's I wanted to say about the ammonia and these things, and we're going to use the last 15 minutes to begin something else, NMR. And, so are there any questions so this Point? Yes? AUDIENCE: Imagine this piece of ammonia traveling through the cavity. If it started changing between states, I sort of imagine a photon being omitted and absorbed, what exactly is happening then? Is it somehow being omitted and absorbed, and omitted and absorbed, and just happened to catch it in the right moment. PROFESSOR: Well, in this case, it's basically omitted all the time, because we've tuned the cavity in such a way that if it comes as E, goes out as G. So the whole process by the time that it entered and it went out, it has to have omitted one photon. AUDIENCE: In the midst of the process, how does it get back somehow, because it's sort of oscillating between states. PROFESSOR: It doesn't get it back. If the cavity would be badly designed in such a way that it may be twice as long, it would out still come as E. So it would make the one transition and then absorb another thing, it would just not generate anything. AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah. AUDIENCE: Interesting. PROFESSOR: And you know, a more complete discussion of this, of course, if you really want to do everything, you would have to treat the photon states. Here, we treated this as a wave coupled to the quantum system of the molecule. You could treat the photons themselves as quanta and do quantum mechanics of the photon field on that. And that would be a wave that you could calculate things more completely . But this is not what we do now. Any other questions? Yes? AUDIENCE: So physically, are the nitrogen molecules all fixed in the same orientation going in to this device? How does that happen? PROFESSOR: Are the what? AUDIENCE: The molecules all fixed in the same orientation? PROFESSOR: Yes. Essentially it's not quite like an orientation. It's an energy eigenstate. So basically what you have to do is have this beam of ammonia molecules and do this beam splitter that we talked about with the electric field. And you split the beam into some that are all excited, and some that are ground. And that's it. You need that everything that enters here is excited state. OK. So another time dependent problem that we're going to discuss today and continue next time is the NMR problem, nuclear magnetic resonance. So this is a pretty interesting problem. Our nuclear magnetic resonance. And it all begins by having a magnetic field that has a big component here, B0, I think. It's a good name for it. B0 Z, indeed. And then you're going to have some magnetic field that is going to be rotating here in time, in the xy plane. We'll assume that time equals 0, is here, and then it's rotating with some angular frequency omega. So the total magnetic field is B0 Z hat plus some number, B 1-- I'll write it here-- plus B 1 cosine omega t times x hat minus sine omega t times y hat. So, indeed, in the xy plane, seen like this, you have it here and it's rotating with angular frequency omega this way, clockwise. All right. So we have this magnetic field. And of course we're going to try to figure out what spins do in it. And the magnetic field is time dependent. So we're in risk of getting a time dependent Hamiltonian. So what is the Hamiltonian? This possibly time dependent, it's supposed to be minus gamma B times the spin. So what is that? It's minus gamma B0 Sz-- The z component matches with the z component of the spin-- plus B1, Sx cosine omega t minus Sy sine omega t. And, well, it's as bad as you could imagine, pretty much. This Hamiltonian is time dependent. And there's some good news if even if it's time dependent, they commute at different times. The time evolution is easy. But no, they don't commute at different times. Time equal 0, for example, you have Sz and Sx, but at a later time, you will have Sz and Sy, and they just don't commute. So we have no cookbook recipe to solve this problem. We have to figure it out. And there are several ways to figure it out. I had a way to figure it out that I explained in previous years, but today I suddenly thought I could explain it in a different way that is maybe a little harder conceptually, but explains more of what's going on. So this is what I want to try to do now. And basically what we're going to try to do is get the main intuition going for this problem. I have the Schrodinger equation for this problem that is very complicated with a time dependent Hamiltonian. So if you have a wave function-- and now I'm going to be a little rough for a little while. A h-bar D T H psi. And it-- oops. What is it? Let's see which lights-- Lights, general. OK. So if you have a problem like this, you might say, well, I would like to do something non-trivial to this, so I want to maybe change the Hamiltonian and do the same physical problem. Now that's not easy, because if it's the same physical problem, how can you change the Hamiltonian all that much? So one thing you could do is you can try to change the states by doing a unitary transformation, and then hope that the unitary transformation acts on this Hamiltonian. So this will be the new states. And you would hope that this unitary transformation would somehow, working with these new states, would simplify the Hamiltonian. But unitary transformations in general is just like a change of bases. It's not going to do all that much, unless the unitary transformation has time dependence, in which case it doesn't just rotate the Hamiltonian but messes up this term. But this is really your only possibility of fixing things, is to try to do a time dependent unitary transformation that somehow you started with a spin Hamiltonian, and you're going to do a unitary transformation, and perhaps it's going to be a time independent one. But this U is going to depend on time. And this is not the whole Hamiltonian, because there's a problem with this part. So the idea is sort of vague, but you're getting a little of the gist of what we try to do. So the first thing I tried to do is something maybe intuitive about though this system. I could say that suppose I have a system here, and the Hamiltonian is 0. Nothing. Nothing happens in that system. Now the thing that this curious about this, that this magnetic field is rotating. So let's try to imagine how would physics look if you have the xy axis, and you have the xy axis rotating with angular velocity omega. So the xy in the plane is rotating with angular velocity omega. What would happen? So there's an Hs that is originally 0. You say it's 0, and any spin state stays there. There's no time evolution. H is equal to 0. Nevertheless, if you jump into this rotating frame, all the spin states that were static, for you they're rotating. And in fact, for you, they are processing around the z-axis if you're in the rotating frame. Therefore, in the rotating frame, you must have some Hamiltonian, even though there's no Hamiltonian in the static frame. Because in the rotating frame, you feel things spinning. So in the rotating frame, if there's a static spin along the x direction, you now see it spinning around the z-axis with frequency omega, in fact, going the plus z direction. So in the rotating frame you can ask, what is the new Hamiltonian in the rotating frame? I mean, the rotating frame, the Hamiltonian should be such that it rotates the spins with angular velocity omega around the z-axis. You may recall that this is done by all the unitary operator E to the minus i omega t Sz hat over h-bar. This is the operator that rotates, spins, with omega t around the z direction, which is precisely what you want this Hamiltonian to do. So the Hamiltonian must be e to the minus i ht. So this rotating Hamiltonian must be omega Sz hat. So the Hamiltonian in the new frame is that because that Hamiltonian produces the rotation of states that you see because your friend that is not rotating is telling you that no state is moving. H is 0. So the intuition is that you've passed from the original Hamiltonian to this one, and you have to add this. But what we'll see now is that you have to add the little-- you have to do other things a little if this is not equal to 0. Here is what we want to do. Now there's also a couple of ways of doing this, but let me try a little calculation to do this second part. So, we're going to take, therefore, to have, for example of psi R a rotating wave function that is going to be given by a unitary operator times the physical wave function you want to solve. This is what you want, and this is what you hope has a simpler dynamics, psi R. So, let's try to figure out what is the Schrodinger equation for psi R if you know the Schrodinger equation for psi. So, I dt I h-bar dt psi is equal to Hs psi, let's see what is the Schrodinger equation satisfied by this one. So I h-bar dt of psi R is I h-bar dt of U psi. So let's differentiate this. This is I h-bar dt of U, and I will have the psi. But then I will put them a U dagger, there and another U acting on psi so that it gives me psi R. So the first derivative is acting just on the U. I acted, and then I put U dagger U and recreated the psi R. The other terms is in which it acts on psi, so I get plus I h-bar. The U goes away, and now we get dt of psi, which is I h-bar dt of psi is Hs I psi, so I must delete this. So the second term, when I acted here, I act with the whole thing. I ddt, I h-bar ddt on psi. The U is out, and I put Hs psi. And here, I can, of course, put a U dagger U and put back the psi R, so I will do that as well. U dagger psi R. So actually you have now a the Schrodinger equation which is of the form I h-bar dt of psi R is equal to U Hs U dagger plus I h-bar dt of U U dagger of psi R. Now that's it. This is the new Hamiltonian. This is the rotating Hamiltonian, essentially, that we're trying to figure out that is going to be simpler, we hope. And there we go here. We have this similarity transformation of the Hamiltonian and we have this extra term. Now suppose the original Hamiltonian had been 0. We want the new Hamiltonian, given our argument, to be a this for the rotating systems. So I will say that this U is such that you get I h-bar dt U U dagger being precisely omega Sz. And for that the U is nothing else but e to the minus I omega tSz over h-bar, which is in fact that thing that we had there. So that's what U is. And in that way, this whole term becomes just omega Sz hat. So we're almost, in a sense, done with this thing, because we have made some good progress. Except that we still don't know if everything has worked out or not. I'll continue here. And just one minute to close up the discussion by saying what has happened. So, what has happened is that we have found that there's a new Hamiltonian, HR equal U Hs U plus I h-bar dtU U dagger. And then U is given by e to the minus I omega tSz hat over h-bar. And psi-- as we said, psi of t is equal to e to the I omega tSz hat over h-bar psi R of t. So this came because we said that psi R was going to be U psi, and therefore I took U and the inverse and you get this. So look what the problem has turned into. It has turned into a problem for psi R with a Schrodinger equation that has a Hamilton HR in which in this piece is very simple. It's omega Sz hat. And now the crucial point is whether-- here I have U dagger-- whether this thing is time independent. And this should be time independent if we got our physics right, and that's exactly where we'll take on the next time, and prove that it's time independent and then we can solve the complete problem. All right, there will be notes posted on this soon. Good luck finishing your homework, and I think Monday is a holiday. Is that right? Well, no class Monday. We'll see you on Wednesday.

MIT_805_Quantum_Physics_II_Fall_2013

18_Two_State_Systems_continued_Multiparticle_States_and_Tensor_Products.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: It's time to get started. Today, we'll complete our discussion of nuclear magnetic resonance. And we started that last time. And after that, we will talk about multi-particle states, the kind of quantum mechanical states that you need if your system has more than one particle. That will bring us naturally to the idea of tensor products of vector spaces, which is the key mathematical idea behind operations such as addition of angular momentum. By introducing a little before, we really talk about addition of angular momentum, the idea of tensor products of vector spaces. I really hope that this subject will look a lot less mysterious in a couple of weeks when we will really be adding angular momentum. So you will have seen the main mathematical structures. And you will know what you're supposed to do when you have a formula. So that should be of some help. So last time, we were talking about nuclear magnetic resonance. And I described the problem. Here is a summary of where more or less we got last time. We said in the set up for nuclear magnetic resonance, we have a large magnetic field, the longitudinal magnetic field it's sometimes called, along the z direction. In addition to that longitudinal magnetic field, there's a rotating magnetic field in the xy plane. And it's rotating in the direction that you see here. As seen on the xy plane, it's clockwise on the plane. In terms of a rotation around the z direction, it's negative in the sense that the positive rotation around the z plane would go like this. And this magnetic field is rotating in the other direction. So I wrote it here. The magnitude of this radio frequency component is B1. And typically, it's much smaller than B0 in the problems of interest. Then, as you recall, the spin Hamiltonian is always given by this constant gamma that relates to the dipole moment of a particle to its spin angular momentum B dot S, the S operator. And the S operator has components Sx, Sy, Sz multiplied by B. Well, the z component picks up a z, the x component Sx, the y component Sy. And this is your Hamiltonian. And as we emphasized last time, this Hamiltonian is time dependent. Moreover, at different times, the Hamiltonians don't commute. So the HS at time T1, HS at time T2, they don't commute. Therefore, any of the formulas we have for solving problems like this don't apply. The only formula that applies is this time ordered exponential that we spoke about that in general is very difficult to use. The time ordered exponential just never quits. And we're writing terms, term after term. And then, you have to add it all up. And adding it all up is difficult. So most of the time, it's not a great help. So what did we say we would do? We wondered if we could somehow do quantum mechanics, or think of quantum mechanics, in the rotating frame, in a frame that is rotating with a magnetic field in such a way that perhaps in that frame the physics would simplify. And physics simplified would be that the Hamiltonian becomes time independent. So we're trying to figure out, how would the Hamiltonian look in another frame? Now, we've not discussed in general very explicitly transformations of Hamiltonians within different frames. But we can sort of roughly try to guess what it would have to do. So what we said last time is that we would imagine that you have an original system and no magnetic field, nothing whatsoever. So forget about all that-- no magnetic field. You put the spin state. It stays there forever, doesn't precess. It doesn't precess because there's no magnetic field. Then, you jump into a rotating frame. So this was a static frame, no magnetic field. And then, you start rotating. And then, you look at the spins. And the spins are not moving. But you say, oh, I'm rotating. So for me, they seem to be precessing, in fact precessing with the angular frequency that I am rotating. And therefore, in my rotating frame, I would have a Hamiltonian that I would call the rotating Hamiltonian, HR. And how would it look? We claim that in fact spin states in my rotating frame would have to be rotating really this way. Because I'm rotating with the magnetic field. So for me, they're rotating in the positive z direction with angular frequency omega. And this would be the operator that makes them rotate exactly that way. Because if you remember, an operator here minus i angle Sz over h bar, this is the angle as a function of time. Omega t, that's how these operators rotate states. So this U of t is the U of t that rotates states the way I want them in my rotating frame. Now, you remember for time independent systems, the unitary operator is e to the minus iht over h bar. So you identify the Hamiltonian as omega Sz. In general, given a unitary operator, the Hamiltonian is obtained this way. You take the time derivative of it multiplied by U dagger. And that's the formula for the Hamiltonian given the time evolution operator, something we did about a couple of weeks ago. So that's where we stood-- some idea about the rotating frame. So the way we now make this complete is to say, well, I'm going to attempt to do the following. I'm going to act with U, this U of t, on this complicated state that I don't know how it behaves. And this I'm going to call the rotated wave function. It's a definition. We're going to use this operator that seems to induce a rotation of the frames and act it on the state to find this. And we're going to hope that the Schrodinger equation for this one is simpler. So actually, last time, I actually did a computation and found what is the Schrodinger Hamiltonian for this, psi R. I want to do it in a slightly different way. I think it's a little easier perhaps, or maybe less familiar but more interesting. I write this like that-- U of t. And I say, look, this state evolves due to its own Hamiltonian. U of t is an extra thing I've put here. So here is U of t. But how do I distinguish it? I put an S for the spin system, for HS, psi 0. So psi of t is being evolved by the Schrodinger Hamiltonian, the unitary operator associated to the Hamiltonian that we want times time of 0. So this is the total unitary operator that evolves this state. Therefore, I'm going to say, I'm going to recover the Hamiltonian using this formula. So I will simply say that the rotating Hamiltonian is going to be HR is ih bar dt of this whole thing, U of t US of t, times its dagger. Because this is the general formula to obtain the Hamiltonian when you know the time evolution operator. So the dagger of this would be US dagger U of t dagger. Now we have to evaluate this derivative. It's is not all that bad. If the time derivative hits the U, this one, then U and U dagger annihilate each other. So you're left with U dagger of t here. So you get ih bar dtU U dagger. Now, when it acts on this second one, what do we get? Well, I'll write it this way. Plus U of t, because this will be first. Then, ih bar dtUS. And then, you have US dagger, and then U dagger of t. Now, this thing is how much? Well, in this thing is in fact this Hamiltonian associated to what we called U, which is just omega Sz. So actually, I don't know if I have-- yeah, I have this notation here. I think it's useful-- H of U, the Hamiltonian associated to whatever U it is. So you get a nice formula-- plus U of t. This is the Hamiltonian HS U dagger of t. So this formula we derived also last time by looking at the differential equations satisfied by psi R. So it's a nice formula. If somebody gives you something like this and says, here's a unitary operator, what is the new Hamiltonian? The new Hamiltonian is equal to the transformed old Hamiltonian plus the Hamiltonian associated to this time evolution operator. It's a nice formula. Now, more concretely now, we can see what this is supposed to be. So here is the important crucial step. Let's see, HR is supposed to be what? We now know we're going to choose U to be this thing. Therefore, HU is omega Sz hat. But now we have U of t, U dagger of t. So I have plus e to the minus i omega tSz over h bar times the Hamiltonian that we have over there, this full Hamiltonian HS. So what is it? Minus gamma big parentheses B0 Sz hat plus B1 cosine omega tSx minus sine omega tSy-- like that-- e to the i omega tSz over h bar. OK, that's supposed to be our new Hamiltonian. And unless it simplifies, we have not gained all that much. So before simplifying, are there questions, anything we've gotten so far? Yes. AUDIENCE: So U of t is the unitary operator that enacts the rotation, and it's not corresponding to any Hamiltonian of the spin? PROFESSOR: Right, U of t is the thing that essentially moves us into the rotating frame. This is the hope that we have. Any unitary operator, if it's what evolves your state, it corresponds to Hamiltonian. But that's not anything a priori to do with our original Hamiltonian. It's just another Hamiltonian. So you choose your U of t behind some Hamiltonian. But what you learn is that the Hamiltonian for your new system, or for the new wave function, will involve that other Hamiltonian. Yes? AUDIENCE: HR equals [INAUDIBLE]? PROFESSOR: This is-- yeah. It's HU. I think you are probably right. I should call it just HU for-- yeah, it's a bad idea to call it HR. U, HU, much better-- thank you. Here, HR is the Hamiltonian associated to the Schrodinger equation here. So the claim behind this calculation is that if you calculated ih bar d dt of psi R of t, you would find HR psi R of t. So that's the calculation I think we did last time in which we just calculated these derivatives. And you find this answer. OK, so let's continue here and see what happens. Well, I have an Sz and a Sz similarity transformation here. But here is Sz. So that Sz doesn't care. And it can just go out. So what do we have? We have omega minus gamma B0 Sz hat like that. Then, the rest would be what? I can take out the numbers gamma, the B1. And then, I have this exponential-- e to the minus i omega t Sz hat over h bar. I'm just having here cosine omega tSx hat minus sine omega tSy hat, e to the i omega tSz hat over h bar. OK, now there's two ways to do this. You have an exponential here and another exponential. This is the kind of thing you've done quite a few times in different ways. You could expand exponentials and just multiply. Or you can sort of do this usual formula of e to the AB e to the minus A for this term and for this term, and simplify, and do many things. So let's do something different. When one looks at this, one could say, well, here's my problem. HR-- I know U of t. I fixed it. So knowing psi is knowing psi R. But to know psi R, I need to know HR. And HR still looks very complicated unless somehow this whole idea has worked out very well and this simplifies. And the hope is that it simplifies. Because in the rotating frame, the field is not rotating anymore. So I'll call this M of t. And I think one way of doing this, doing something, if you're in a rush and you don't have a formula sheet or any of those things, is to just take its derivative, its time derivative. Now, if you have a function of time, and you don't know what it is, take its time derivative. And maybe you'll see a differential equation or see something. So let's take the d dt of M of t. OK, so what do we get? Well, we get the following-- e to the minus i omega tSz over h bar. And I'll put the big parentheses here. OK, here I have an object. This is something I think you are familiar. You've been computing these things for coherence state, for squeeze state, and for all that. So this will certainly sound familiar. If you take a time derivative of this exponential, it will bring an operator down. When you take the time derivative of this operator, it will bring the operator down but with a different sign. And therefore, it will form the commutator with the thing like this in the middle. So the first thing in this thing is the commutator of-- you take the time derivative of this-- minus i omega Sz hat over h bar with the rest of this thing, with cosine omega tSx minus sine omega tSy. Because the first derivative brings down the minus i omega this over that. So I'll actually take the minus i omega over h bar out. OK, that's the first term that you get from differentiating this and that. And then, you have to differentiate the term in the middle. So then, you get minus omega sine omega tSx hat minus omega cosine omega tSy hat. And now, we can close the big parentheses and put the other exponential like that. OK, so let's see this. Sz with Sx is ih bar Sy. So you get here minus i omega over h bar, ih bar Sy cosine omega t. Because Sz with Sx-- you should remember that I think. Otherwise, you'll waste time. x, y, and z, when you have Sz with Sx, you get Sy. Sx with Sy, you get a z. And that order helps. So ih bar this, and then the other term would be minus sine omega t. But now, Sz with Sy is minus ih bar, so plus ih bar Sx. OK, I won't copy the other term. And let's see if it worked out. Minus i and i is a plus. The h bar cancels, so plus omega Sy cosine omega t minus omega cosine omega tSy. So this term cancels with this. And this-- there's an ih bar. The sign is the same, so it's a plus omega sine Sx minus omega sine. So these two cancel. And happily, this whole thing is 0. So that is the signal your strategy worked. Why? Because M of t looks like it has lots of time dependence. But in fact, it has none. Its derivative is 0. So if its derivative is 0, you can evaluate it at any time you want. Well, the best time to evaluate it is time equals 0 now. And you put here 0-- disappears, disappears. This term disappears. The whole thing is just Sx hat, nothing else. So you may want to do it the other way by doing this similarity on this and on that, or do it whichever way you want. But this is good enough. So we have that this whole Hamiltonian has become very simple. So HR has become-- and I'll copy it here, I want to make sure-- minus gamma B0 plus omega Sz hat minus gamma B1Sx hat. Very nice Hamiltonian-- just two pieces. The Sz part of the Hamiltonian that used to be this got this extra piece from the rotation. That's our omega Sz. And this piece over here is all that is left of the rotating one. Because in the rotating frame, the magnetic field is always along the x-axis as it was at time equals 0. So it's a perfectly nice Hamiltonian. Let's factor a few things. Let's remember the notation that omega 0 is the Larmor frequency associated to B0. So I'll write here, this is equal to minus gamma B0 minus omega over gamma Sz hat plus B1Sx hat, which is minus gamma B0 1 minus omega over omega 0 Sz hat plus B1Sx hat. So let me look at what we've done. First step, I just factor a gamma here, so nothing else. And the same gamma went out. Next step was to use this equation, omega 0 equals gamma over B0 to eliminate gamma. So you eliminated gamma by that. So it's equal to omega 0 over B0. So the B0 went out. And you have 1 minus this thing. And then, you can think of this HR as minus gamma BR times S. In the usual notation for spin Hamiltonians, minus gamma BS, this is the rotating B, the so-called rotating B. But of course, it's not rotating anymore, happily. This is just B0 times 1 minus omega over omega 0 Sz plus B1Sx. And then, how about our answer? What is the answer to this problem? Well, the answer is the psi and t. So psi t is U dagger times psi R of t. And U dagger was e to the i omega tSz over h bar-- omega t, yeah. And what is the other U? Well, the other U associated to a time evolution is just e to the minus iHRt over h bar. Because HR is time independent. So this equation that tells us how psi R evolves is with HR. HR is time independent. And therefore, you'll have here e to the minus iHR, which is this. So it becomes plus i gamma BR dot S over h bar t acting on the state psi at time equals 0. You see, the state psi R at time equals 0 is the same as the state psi at time equals 0 because the unitary operator is the same. So let me box this as well. This is the complete solution for the problem of the rotating spins. So the quantum mechanical problem is you've gotten this time dependent magnetic field. You want to know the spin evolution. Well, the spin evolution is a little complicated. It's this whole thing. And let me just make sure everything is good here. Perfect. Fine. All right. Questions. Yes? AUDIENCE: So you have x hat, so xc hat and xs hat-- AUDIENCE: BR. BARTON ZWIEBACH: Oh. I'm sorry. It should be BR. I'm sorry. What is the mistake here? Sorry. This is terrible. z hat, x hat. Thank you. That's a magnetic field. Another question. Yes? AUDIENCE: This is a question [INAUDIBLE].. But when we take d dt of f of t, how can we take the computation relation with hu in that computation? Why is that the Hamiltonian that we take in the computation? BARTON ZWIEBACH: I don't understand. When we take the time derivative of m of t, here was m of t from here to here. AUDIENCE: [INAUDIBLE]. BARTON ZWIEBACH: Why is this thing here? AUDIENCE: Yeah, omega sc. BARTON ZWIEBACH: It's because when you take the time derivative of this-- let me give you a formula that you should check. tA minus tA. d dt of this is equal to e to the tA, A, B, e to the minus tA. That's the formula we used. It should be all right. So we have our time dependent states, and let's analyze it and see what it's doing. Yes? One more question. AUDIENCE: So [INAUDIBLE] that we have these magnetic fields, but somehow, in our Hamiltonian, it all boils down to just the spin operating in the x direction. BARTON ZWIEBACH: Well, they're still in the z direction. AUDIENCE: There's still the z component, but the time variant components in the y and x direction-- BARTON ZWIEBACH: Has become something like in the x direction. Right. AUDIENCE: It's hard to say physically, but what does the system actually look like? BARTON ZWIEBACH: That's what we're going to do right now. AUDIENCE: All right. BARTON ZWIEBACH: This formula is very pleasant to look at. You say, oh, I'm very accomplished. I solved it. But what does that do? What does that describe? That's what we have to do now. And that's the interesting part, of course. So applications always have B1 much smaller than B0. This is the longitudinal component, and that's the case. Let me describe for you what this is doing in case one, omega much smaller than omega 0. You see, given B0, there is an omega 0, the Larmor one over there. And B0 is very large, so omega 0 is very large as well. So omega very slow is reasonable. The magnetic field is rotating much smaller than whatever rotation this other magnetic field would create on the things. Moreover, if omega is much smaller than omega 0, BR is sort of like B0 z plus B1 x, roughly that. So what does this do? Well, this is a magnetic field mostly along the z-axis. So here is x, y, z. This is a magnetic field that's like this, BR. Now, this BR, let's assume also for an assumption that at t equals 0, the spin of whatever spin state you have is up plus z. So what does this magnetic field do to the spin? Well, you have it here. So this is going to rotate the spin states around the axis of BR. The rotation operator here, this is minus the angular velocity, so there's a minus sign here that you always must think about. But forget about it for a second. This is rotating around BR, so this, supposed to have a spin state here in the z direction, is going to start to rotating in a little cone like that. It's very close to the BR, so it's just going to rotate around it like that. And that's what this part is going to do. But you say, but that's not the whole thing. The whole thing is this as well. Well, intuitively, this rotation that this produces a rotation around the z-axis, but it's much slower than this rotation because this BR field, its magnitude is like B0 roughly. Therefore, it produces a very fast rotation. So what you must imagine is this little spin state generating this cone here and rotating a million times a second, but this whole thing is still being rotated by this exponential around the z-axis. So this whole cone, in some sense, is precessing around the z-axis. Now, if you want to know where the spin is at any instant of time, you do the following thing. You say, time equals one second. So you come here and say, one second, a billion turns and a quarter. It ends up here. But in one second, this rotates it by five degrees, so then it rotates a little more. You could do it that way. That's a good way to think about it. But in the approximation, which is so fast, basically, this cone has now, by the second exponential, been rotated like that. Now let's do the case that is really the important one, the case, as you could imagine in which we know some resonance and omega is set equal to omega 0. So the physicists know what omega 0 is for the spins, and then they make the radio frequency coincide with that omega 0. In that case, you lose completely the z component here of this BR. So BR is going to be B1 x. And then, here is what's going to happen. You have a B1 x here, a B1 over here in the x direction. The spin state is sitting at time equals 0 here. And since normal exponentials that do time evolutions have a minus sign here, I claim that instead of rotating the spin around B1, it's really rotating it around minus B1. So the spin will just go down here. Rotate, rotate, rotate, rotate. Now, it's rotating with B1, which is much smaller than B0. So it's rotating with an angular frequency that is much smaller than omega 0, but it's rotating down here. And now, what's really happening is the following. Let me draw this if I can. Here it is. The spin is up, and it begins to go into the y-axis. Here is z. B1 is turning into the y-axis, so it rotates a little. So let me ask you, what is the second exponential going to do? It's going to rotate it further, but this time around the z-axis. So this one goes down a little bit in some time, but then this z exponential, the other exponential, is going to rotate it around the z-axis. So this is going to go down a little but rotate. So actually, what's going to happen is that it's going to do a spiral. As it begins to go down, the other one rotates. It goes down a little, the other one rotates it. In fact, this, since B1 is much smaller than B0 or omega 0, here now, omega is equal to omega 0, so this is rotating around the z direction with omega 0. But B1 is much smaller than B0, so this rotation is very slow. As it goes down a little, it's rotating very fast and fills out the spiral until it gets down here. So that's what the poor spin is doing in this thing. It's actually of interest to time the radio frequency signals, to time the systems of B1 so that you get it to go into the plane so that the spin is maximally perpendicular to the original direction. For that, you choose omega 1 times time, which is the Larmor frequency associated to B1 times time, to be equal to pi over 2. And omega 1, just like any omega, is gamma B1 pi over 2. So t is pi over 2 gamma B1. It's called the half pulse, something like that. No, a 90 degree pulse. That's a normal name for it. So if you keep the radio frequency on for this much time, then the spin, at the end of the day, it got to the equator of the sphere, and B1 has disappeared. So B1 is gone, BR is gone, and then it keeps rotating because there is the other magnetic field still, the longitudinal magnetic field. One exercise that you still have in the homework is to figure out the equation of the spiral that comes out of here. You will find it sounds complicated, but it's a couple of lines. It's very simple, actually, to do it, after you think about it for a second. That's one thing you'll have to do. So this, in fact, is basically the solution of the problem, and let's just talk for a few minutes about what it's used for. So this is basically the technique that is exactly used for Magnetic Resonance Imaging, MRIs. If you have to say one of the interesting applications of quantum mechanics to technology, MRIs, Magnetic Resonance Imaging, is one of the great ones. So how does it work? Well, it's a very useful device. In fact, it revolutionized medicine because it goes much beyond what you can do with x-rays. It's very popular. I imagine a good fraction of you have had an MRI. Let's see. How many people have had an MRI in their lives? We're pretty much, I think, 60% of people here. So you remember getting into this cavity? Well, you've experienced there a large-- if you worry about cell phone radiation, well, there, you were with two Tesla, 20,000 Gauss, a big, big magnet. It's big enough that it has to be cooled with liquid helium at 3 degrees Kelvin so you can get the currents big enough and the magnetic field big enough. So they put you into this cylinder, a solenoid, and there you go. And it's not dangerous unless you forget some metal or some device like that. In fact, it says in WebMD that if you have some sort of tattoos, they used iron ink, it can burn your skin or you could have some problems. Anyway, if you're claustrophobic, may have MRIs that are open air, but they're less powerful, less strong magnetic field. So this is two Tesla, roughly. And so what happens is that basically, this thing just is trying to figure out the local concentration of water. It begins like that. The magnetic fields react with the hydrogen atoms, in fact, with the protons inside there. Each proton has a magnetic dipole moment. It gets roughly aligned to the magnetic field, to this B0. So the protons get aligned to the B0 that is going in here, B0. So the proton spin is up there. In fact, because of temperature, not all of your protons in your body get aligned. Maybe one in a million does, but that's high enough. And then they send this 90 degree pulse. So this thing starts spiraling, and then it finally goes into this direction and it rotates, rotates like crazy. Then, as it rotates like crazy, a rotating dipole moment is like a generator of electromagnetic waves. So it generates an electromagnetic wave, and there are detectors all over that pick up this signal. The strength of that signal is proportional to the concentration of water. So it gives you an idea, not to distinguish solid matter versus soft matter, but all kinds of liquids. So it's very, very useful. You pick a signal over here and you get it from a receiver, and it tells you about the local concentration of water. And therefore, it allows you to distinguish different tissues. Some tissues have lots of water, some tissues have less water, so it begins to distinguish different tissues. Now, there's two more things that people do. This pin is rotating very fast, but there's a relaxation time. After it rotates for a while, there's a time, T2, is relaxation of this rotation. Relaxation time. And then there's another time, T1, which is the time after this is turned off that it takes the spin to go back again to the original position. So two times, the relaxation time in which you lose your rotation here because it's interacting with other spins. It's called spin, spin, relaxation. And then an interaction with a whole set of neighboring atoms that brings it eventually back up and aligns it to the magnetic field. So you measure two things, T1 and T2, and those are very good clues because you can put any liquid in the machine and measure its T1 and T2. And then you place a table of T1's and T2's, and then if you want to figure out what kind of thing you have in your body, they look it up, and immediately, they may know what are the possible candidates. In fact, T2 is actually enough to distinguish white matter, gray matter, and fluids in your brain. Totally different T2's. T1 is helpful to discuss all kinds of things as well. So basically, people have figured out how to do that. Finally, one last thing. When you go into the machine, it sometimes makes noises. Big noises. And those are gradient magnets that are being moved. You see, basically, if this thing would be like that, you would pick up a signal and you wouldn't know where it comes from. So what these people do now-- this has gotten extremely sophisticated. They put a gradient magnet which changes the value of B0 locally by increasing it. For example, the B0 is not really constant in z, but it changes. So the omega 0 of rotation of the spins changes as well. So with a sufficiently high gradient of magnetic field, they can have spatial resolution. If they pick a signal a little bit higher frequency, they know it's a little bit higher up in your body. So they get resolutions with these magnets of about how much? Very high resolutions. Half a millimeter. So at the end of the day, it's a great machine, and lots of mathematics in reconstructing images, lots of computation, lots of experimental analysis of constants, much of it phenomenological. It would be hard to predict those constants, but you can measure them. So it's very, very practical and very nice. So it's a nice application, and we'll leave it at that. Are there any questions? Not that I know much more about that. But it's a junior lab experiment as well. So I believe the calculation is fairly straightforward and the technology is just amazing. It's great. Yes? AUDIENCE: So about getting your spin back up along the z direction, is that mechanism the same as when you relax [INAUDIBLE]? BARTON ZWIEBACH: When you relax what? AUDIENCE: So is the mechanism for getting your spin back along the z direction to find T1, is that the same mechanism as when you do all of this [INAUDIBLE]? BARTON ZWIEBACH: No. It's sort of a different mechanism. We don't analyze it. It's not being driven by Schrodinger time evolution. All these relaxations are more complicated phenomena. In fact, if you maintain the magnetic field, this is supposed to go on forever and never stop rotating. But due to interactions with other things, it's sort of stops rotating at some stage, and then eventually, many of them also go back up. So these are not easily calculable things within our analysis. These are phenomenological constants that need to be measured or done by experiment. So last part of today's lecture is multi-particle states and tensor products. Let's get on with that. Let's see we have there. So multi-particle states and tensor products. The idea is the following. You have a system with more than one particle. Let's talk two particles. It won't matter at this moment whether they're distinguishable, not distinguishable. Those are things that come later, and we will probably not discuss much of that in this semester. This will be more in 806. But let's consider if we have particle one, and we'll keep the possibility that this is completely distinguishable, in fact. Particle one. Its quantum mechanics is described by a complex vector space, v. And you have some operators, T1, T2. Particle two, complex vector space, w, and the operators, S1, S2, all that. You see, the list of operators is something that you are already familiar with. Quantum mechanics operators can include momentum, position, if it's three dimensional, three positions, three momenta, angular momentum, spin, Hamiltonians, all those things, and those exist for both particles. So the question is, how do we describe the composite system, the system of the two particles that exist at the same time, that possibly could interact even with each other? So we need a description of these two things. Well, a description of particle one is described by some vector, v, in the vector space. Description of particle two is some state, some vector omega, in the vector space w. So we imagine that it's a reasonable thing to give you those two vectors and say, look, that's what particle one is doing, that's what particle two is doing, and that is correct. It's a bit more subtle than that, as we'll see now, but we could list v and list w, and this is the information about each particle. And this is, in some ways, too naive, as we will see. It has the right idea, but the amazing possibilities that can take place when you have these two particles are not quite reflected on that thing yet. So to make this idea clearer, we'll use a notation. So I will say that I will encode these things and I will write them as v tensor product w. It's reflecting that we're going to do something more than just view this as the possibility. You have a system. You know what the particle one is doing, you know what the second particle is doing. List those two, that's all that can happen. Not quite true. So let's put it like that, and this will be the information. BARTON ZWIEBACH: I'm not multiplying this in any obvious way. It's not that I'm supposed to multiply and do a calculation here. I'm just putting the two pieces of data but putting this curly ball here as to say that it's the two together, and we'll have some ways of dealing with this object. And this will be said to be an element of a new, complex vector space, v tensor w. So what we are saying here, with v belonging to v and w belonging to w, this thing belongs to v plus w. Tensor product of the vector spaces. At this moment, this is a little funny, but let me ask you some things about it. We have v w. So this is an element of that space. I put a vector of the first vector space here and I put a vector of the second vector space, and this will be a vector here. I'm not saying it's the most general vector there, or how we operate with them, but it's a vector there. Now, you say, look, when I had states, you had states plus minus, you put constants in front of them. So I'll put a constant in front of this v, alpha. Well, is this something related? We had this vector that was a vector in the tensor product. How is this vector related to this vector? Well, unless you declare that these things are somewhat related, this object is going to be very, very large because if this vector has nothing to do with this, this is going to be yet another vector linearly independent with this. So it's, again, your choice what you're going to declare. If you will be constructing what is called a direct product, not a tensor product, and that's apologies to Shankar. He uses the word "direct products" wrong mathematically. Mathematicians don't call this a direct product. Direct product is something in which this has nothing to do with this. But in physics, you're thinking of the amplitude to have a first particle doing that, and suddenly, that amplitude becomes twice as big and this other one doesn't change. Well, the amplitude defining this one here and that one there has become twice as big just because this one is twice more likely to be here and that's the same. So we're going to say that this is the same as that. The alpha can go out and it's the same thing, and it's also the same thing as v tensor alpha w. So numbers go out, and they don't go out with a complex conjugate. They go out just like that. So that's one thing we declare that will happen with these things, and that's a property that we impose. It's what is usually understood to take place in tensor products. Now, if this is a vector in this space, then if, for example, v1 is a vector and v2 cross w2 is a vector in v cross w, then this is supposed to be a linear vector space. So it should be true that alpha v1 omega 1 plus beta v2 omega 2 also belongs to this vector space. And suddenly, you see why this was a little too naive. If you tell me, this is what particle one is doing and this is what particle is doing, end of story, this list, well, quantum mechanics seems to say that there's more interesting possibilities in which the state is a superposition now in which first particle is doing this, which may be a superposition itself, and this one is doing this, plus first particle is doing that and second particle is doing that. So it's not enough to say to the state of two particles, say, you just need to know the state of one, state of the other, list the two. That is a possible state of the system, but it's not the most general. The most general is a superposition in which particle one does something, particle two does something, plus another possibility, particle one is doing something else, particle two is doing another thing. This is the origin of this famous idea of entanglement, as we will see next time. But let's just continue here. These particles, roughly speaking, seem to be entangled, in which you can't say what particle one is doing without knowing what particle two is doing and things like that. So something new has happened by taking this idea slowly and figuring out what's going on, but we need yet one more thing to cut down this tensor product. This tensor product is still a little bigger than what you may want it to be in the following sense. Suppose you have v1 plus v2, which is another vector, tensor w, another vector. Unless you tell me what this is, I don't know that this is actually v1 plus v2 unless you declare it. This is just another vector here and that. Why should that be equal to this? Well, it's the way you feel about quantum mechanics. That's how you should think of the composite system. If the first state can be either of two possibilities while the other one is not, well it's a superposition in which, yes, the first state is doing this and the first state is doing something different with the second doing the same. So this is part of also an axiom of the tensor product. Whenever you have tensor products, you will simplify in that way. It looks almost trivial, but it's certainly something that has to be stated. If you had a direct product, you form a vector space by just putting first vector, second vector, and you don't explain anything. This would not be true. So similarly, we'll also have v1 w1 plus w2 equals v1 w1 plus v1 w2. So these are our last operations. So with these two operations, we've defined completely how to calculate with this two particle Hilbert space, and how to define the states. Let me add some intuition to this by stating the following. So what is this space, v tensor w, is the space spanned by things like this. So with these rules, here comes the thing that puts it all, in a sense, together. If e1 up to en is a basis for v, and f1 up to fm is a basis for w, this set of things, ei tensor fj, and how many of them are there? There are n of this and m of those. This for i from 1 up to n, j from 1 up to m, form a basis for v tensor w. So basically, if you have a four dimensional vector space v and a 10 dimensional vector space w, v cross w is 40 dimensional. You multiply the dimensions. You don't sum them. In the tensor product, the dimensions multiply. You see, the elements of this tensor product was one vector here, one vector there. So it behooves you that you would expect that, well, you can get all vectors that can sit here with all the ei's, all the vectors here, but only if you have these rules of superposition and linearity. So this is the whole basis of the vector space. And dimension of v cross w would be equal to the dimension of v times the dimension of w. Questions? Yes? AUDIENCE: So for the first box thing up there, so v and w are two different vector spaces. Let's say that they're vector spaces over different fields. How do you know that-- BARTON ZWIEBACH: Oh no. Different fields, no. Both are over complex numbers. AUDIENCE: Oh, OK. They're both complex. BARTON ZWIEBACH: Both spaces are complex vector spaces. Did I say it? Complex vector space, complex vector space. Then v cross w is an element of a new complex vector space. So yes, everybody's complex here. Let me continue a little more because we need to get a little further. I would like to get a little further here today. We still have about 10 minutes. These are really, in some ways simple but in some ways very subtle ideas. I hope you appreciate it. At some moments, you think, this is obvious. It's ridiculous. It's taking too long to explain it. At some point, you step back and say, I'm now confused. What does that mean? It is subtle. So let's see now what happens. We need to do operators on v cross w. So suppose T is an operator in v, and s is an operator in w. So we call them like that, T operators and the S operator on w. So here it is. We define an s tensor T that will be an operator on the tensor product, v cross w. Let's see how it is. So if this is an operator on the tensor product, it should know how to act on v any element. In any basic element or element of this form, since all the elements can be written as superposition of these things, if this is a linear operator and I define it acting on this, we've got it. Now, what I'm going to write in the right hand side is a matter of common sense. It's a definition, but it's hardly possible to have any other definition than what I'm going to do. Lights. Main screen, window shades. Oh, lights here. There we go. Well, you know, I got the order wrong here, T cross S. And the only thing you could do is if T, you know how it acts on v. Well, let it act on v. And it will be a vector in v, and you know how S acts on omega, so let it act on omega, and that's the definition. Not that complicated, really, in a sense. Each operator acts on its thing, so you just let it act. Whoever has to act, acts wherever it can do it. And it's linear operator, and if you act on more things, it will be linear. It will all be fine here. Now, this is a fine operator. Now, the operators that are a little more surprising, perhaps, at first sight are the following. Suppose you have an operator, T1, that is an operator on v, and you want it to act on the tensor product. You say, well, I need another operator. This acts on v. I need another operator attached on the tensor product, but you don't give me one, so what am I to do? So this is what is called upgrading an operator that acts on one vector space to an operator that acts on the whole thing because you need everything to act now on the tensor product. So what you do is the following. You let the operator become T1 that acts on v, and you put tensor product with the identity operator. This is more or less what you would imagine. You have now an operator that belongs to the linear operators on the tensor product, but it's the operator you got times that. If you had an operator, S1, belonging to L of w, it would go into 1 times S1. Now I ask you the question, do these operators commute? Strange question, but it's a fundamental question. It's really the basis of most of your intuition about two particle systems. Well, let's see if they commute. You have T1 tensor 1 multiplied by 1 tensor S1. It's going to act on v tensor w. Or you have it in the other order, 1 S1 T1 1. Well, when it acts, the first part, it gives me T1 tensor 1, but now acting on this, this is v tensor Sw. And then when I act on this one, I get now T1 on v tensor S1 of w. And when I act here, the first step, it gives me a T1 on v, and the second gives me the S1 on w, so it gives me the same thing. T1 on v, S1 on w. So these two operators, because they originated, they've operators of the first particle, operator of the second particle, they can act on the whole system, but they still commute. They don't know about each other, and the communication now is the calculation which you just have seen. What this is a good example of this thing is that when you try to write the Hamiltonian of the whole system, H total, you would say, oh, it's the Hamiltonian of the first system, tensor 1, plus 1 tensor the Hamiltonian of the second system. Time for an example. So the example is a famous one, and this is a great example because it's at the basis of combining angular momentum. So it's an example you're going to see now and see many times. Two spin one 1/2 particles. The first particle has a state plus and a state minus for the first particle. The second particle has a state plus and a state minus. So how do we form the tensor product? Well, we say, these are our basis states for the first Hilbert space, the basis states for the second Hilbert space. We're supposed to take the product state. So our tensor product is going to be spanned, so two spins. The tensor vector space is spanned by a vector in the first times a vector in the second, the basis vectors. You could have plus in 1, minus in 2, minus in 1, plus in 2, and minus in 1, minus in 2. So two spin states form a four-dimensional complex vector space. That's how you describe them, with those little products. And the most general state is the following. Most general state is a psi, which is alpha plus plus-- you could put 1 and 2-- plus here's alpha 1, alpha 2, plus, minus, plus alpha 3 minus, plus, plus alpha 4 minus, minus. Let's do just one simple computation, be done for today. I want you to try to figure out what is the result of acting with a total z component of angular momentum on this state, whatever that means. Total z component of angular momentum. So naively speaking, the total z component of angular momentum would be the z component of angular momentum of the first particle plus the z component of the angular momentum of the second particle, but you know that's not the way we should write it. So how is it really? It should be Sz of the first particle tensor product with 1 plus 1 tensor product with Sz of the second particle. You can say 2 or 1 here. This is really what it is. You see, how do we upgrade an operator in the first Hilbert space to an operator on the tensor product? You just let it act on the first state but do nothing on the second. So summing the two angular momenta really means constructing this new operator in the new, larger, more extended space in which it acts in this way. So let's just calculate what that is and we'll stop there. Let's see. For example, we'll have the first term, Sz 1 tensor 1 acting on psi on the whole state. Well, it acts on the first term, alpha 1. Now, this operator is supposed to act on this thing, so this acts on that and that acts on that. So you get-- I'll write this whole thing-- Sz hat on plus tensor plus for the first term. This term just acted on that. We can act with the second one now. Well, I was putting the first, so let me leave the first. Let's do the second term, plus alpha 2. It still acts on the first only, Sz hat plus tensor minus plus alpha 2 Sz hat on minus tensor plus, plus alpha 4 Sz hat on minus tensor minus. Now, what is this? Well, this thing is h bar over 2 times plus, and the number, of course, goes out of the tensor product. You don't worry about that. There's h bar over 2 everywhere. Alpha 1 plus, plus. Here the same, another plus, so plus alpha 2 plus, minus. And here is minus, so minus alpha 3 minus, plus, minus alpha 4, also minus, minus, minus. So that's what it is. If I do the other one, you could do it with me now quickly. 1 tensor Sz 2 on psi. Once you get accustomed, these are pretty direct. So I have to act with Sz hat on these ones, so I just act on the second one, on the second state. So I get h bar over 2. For the first one, you get a plus, because it's acting on this thing, so you get alpha 1 plus, plus. For the second one, however, you get a minus because it's the second operator, so minus alpha 2 plus, minus. For the third one, it's a plus, so you get a plus alpha 3 minus, plus. And for the last one, it's a minus, so you get minus alpha 4 minus, minus. So when you add them together, these two pieces, it's the total, Sz total, acting on the state, and what did you get? You get h bar, this one's at, alpha plus, plus alpha 1. These two cancel, these two cancel. Minus alpha 4 minus, minus. That's the whole action of Sz on this thing. And if you wanted to have a state with total Sz equals 0, then you would have to put alpha 1 and alpha 4 to 0. You will, if you want, at some stage, calculate, maybe in recitation, how much is Sy at this state and how much is Sx at this state, and try to figure out if there is a state whose total spin angular momentum is 0. It's just a calculation like this. So next week, we'll continue with this and with teleportation and Bell inequalities.

MIT_805_Quantum_Physics_II_Fall_2013

5_Linear_Algebra_Vector_Spaces_and_Operators.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Last time we talked about the spin operator pointing in some particular direction. There were questions. In fact, there was a useful question that I think I want to begin the lecture by going back to it. And this, you received an email from me. The notes have an extra section added to it that is stuff that I didn't do in class last time, but I was told in fact some of the recitation instructors did discuss this matter And I'm going to say a few words about it. Now, I do expect you to read the notes. So things that you will need for the homework, all the material that is in the notes is material that I kind of assume you're familiar with. And you've read it and understood it. And I probably don't cover all what is in the notes, especially examples or some things don't go into so much detail. But the notes should really be helping you understand things well. So the remark I want to make is that-- there was a question last time that better that we think about it more deliberately in which we saw there that Pauli matrices, sigma 1 squared was equal to sigma 2 squared equal to 2 sigma 3 squared was equal to 1. Well, that, indeed, tells you something important about the eigenvalues of this matrices. And it's a general fact. If you have some matrix M that satisfies an equation. Now, let me write an equation. The matrix M squared plus alpha M plus beta times the identity is equal to 0. This is a matrix equation. It takes the whole matrix, square it, add alpha times the matrix, and then beta times the identity matrix is equal to 0. Suppose you discover that such an equation holds for that matrix M. Then, suppose you are also asked to find eigenvalues of this matrix M. So suppose there is a vector-- that is, an eigenvector with eigenvalue lambda. That's what having an eigenvector with eigenvalue lambda means. And you're supposed to calculate these values of lambda. So what you do here is let this equation, this matrix on the left, act on the vector v. So you have M squared plus alpha M plus beta 1 act on v. Since the matrix is 0, it should be 0. And now you come and say, well, let's see. Beta times 1 on v. Well, that's just beta times v, the vector v. Alpha M on v, but M on v is lambda v. So this is alpha lambda v. And M squared on v, as you can imagine, you act with another M here. Then you go to this side. You get lambda Mv, which is, again, another lambda times v. So M squared on v is lambda squared v. If acts two times on v. Therefore, this is 0. And here you have, for example, that lambda squared plus alpha lambda plus beta on v is equal to 0. Well, v cannot be 0. Any eigenvector-- by definition, eigenvectors are not 0 vectors. You can have 0 eigenvalues but not 0 eigenvectors. That doesn't exist. An eigenvector that is 0 is a crazy thing because this would be 0, and then it would be-- the eigenvalue would not be determined. It just makes no sense. So v is different from 0. So you see that lambda squared plus alpha lambda plus beta is equal to 0. And the eigenvalues, any eigenvalue of this matrix, must satisfy this equation. So the eigenvalues of sigma 1, you have sigma 1 squared, for example, is equal to 1. So the eigenvalues, any lambda squared must be equal to 1, the number 1. And therefore, the eigenvalues of sigma 1 are possibly plus or minus 1. We don't know yet. Could be two 1's, 2 minus 1's, one 1 and one minus 1. But there's another nice thing, the trace of sigma 1. We'll study more the trace, don't worry. If you are not that familiar with it, it will become more familiar soon. The trace of sigma 1 or any matrix is the sum of elements in the diagonal. Sigma 1, if you remember, was of this form. Therefore, the trace is 0. And in fact, the traces of any of the Pauli matrices are 0. Another little theorem of linear algebra shows that the trace of a matrix is equal to the sum of eigenvalues. So whatever two eigenvlaues sigma 1 has, they must add up to 0. Because the trace is 0 and it's equal to the sum of eigenvalues. And therefore, if the eigenvalues can only be plus or minus 1, you have the result that one eigenvalue must be plus 1. The other eigenvalue must be minus 1, is the only way you can get that to work. So two sigma 1 eigenvalues of sigma 1 are plus 1 and minus 1. Those are the two eigenvalues. So in that section as well, there's some discussion about properties of the Pauli matrices. And two basic properties of Pauli matrices are the following. Remember that the spin matrices, the spin operators, are h bar over 2 times the Pauli matrices. And the spin operators had the algebra for angular momentum. So from the algebra of angular momentum that says that Si Sj is equal to i h bar epsilon i j k Sk, you deduce after plugging this that sigma i sigma j is 2i epsilon i j k sigma k. Moreover, there's another nice property of the Pauli matrices having to deal with anticommutators. If you do experimentally try multiplying Pauli matrices, sigma 1 and sigma 2, you will find out that if you compare it with sigma 2 sigma 1, it's different. Of course, it's not the same. These matrices don't commute. But they actually-- while they fail to commute, they still fail to commute in a nice way. Actually, these are minus each other. So in fact, sigma 1 sigma 2 plus sigma 2 sigma 1 is equal to 0. And by this, we mean that they anticommute. And we have a brief way of calling this. When this sign was a minus, it was called the commutator. When this is a plus, it's called an anticommutator. So the anticommutator of sigma 1 with sigma 2 is equal to 0. Anticommutator defined in general by A, B. Two operators is AB plus BA. And as you will read in the notes, a little more analysis shows that, in fact, the anticommutator of sigma i and sigma j has a nice formula, which is 2 delta ij times the unit matrix, the 2 by 2 unit matrix. With this result, you get a general formula. Any product of two operators, AB, you can write as 1/2 of the anticommutator plus 1-- no, 1/2 of the commutator plus 1/2 of the anticommutator. Expand it out, that right-hand side, and you will see quite quickly this is true for any two operators. This has AB minus BA and this has AB plus BA. The BA term cancels and the AB terms are [INAUDIBLE]. So sigma i sigma j would be equal to 1/2. And then they put down the anticommutator first. So you get delta ij times the identity, which is 1/2 of the anticommutator plus 1/2 of the commutator, which is i epsilon i j k sigma k. It's a very useful formula. In order to make those formulas look neater, we invent a notation in which we think of sigma as a triplet-- sigma 1, sigma 2, and sigma 3. And then we have vectors, like a-- normal vectors, components a1, a2, a3. And then we have a dot sigma must be defined. Well, there's an obvious definition of what this should mean, but it's not something you're accustomed to. And one should pause before saying this. You're having a normal vector, a triplet of numbers, multiplied by a triplet of matrices, or a triplet of operators. Since numbers commute with matrices, the order in which you write this doesn't matter. But this is defined to be a1 sigma 1 plus a2 sigma 2 plus a3 sigma 3. This can be written as ai sigma i with our repeated index convention that you sum over the possibilities. So here is what you're supposed to do here to maybe interpret this equation nicely. You multiply this equation n by ai bj. Now, these are numbers. These are matrices. I better not change this order, but I can certainly, by multiplying that way, I have ai sigma i bj sigma j equals 2 ai bj delta ij times the matrix 1 plus i epsilon i j k ai bj sigma k. Now, what? Well, write it in terms of things that look neat. a dot sigma, that's a matrix. This whole thing is a matrix multiplied by the matrix b dot sigma gives you-- Well, ai bj delta ij, this delta ij forces j to become i. In other words, you can replace these two terms by just bi. And then you have ai bi. So this is twice. I don't know why I have a 2. No 2. There was no 2 there, sorry. So what do we get here? We get a dot b, the dot product. This is a normal dot product. This is just a number times 1 plus i. Now, what is this thing? You should try to remember how the epsilon tensor can be used to do cross products. This, there's just one free index, the index k. So this must be some sort of vector. And in fact, if you try the definition of epsilon and look in detail what this is, you will find that this is nothing but the k component of a dot b. The k-- so I'll write it here. This is a cross b sub k. But now you have a cross b sub k times sigma k. So this is the same as a cross b dot sigma. And here you got a pretty nice equation for Pauli matrices. It expresses the general product of Pauli matrices in somewhat geometric terms. So if you take, for example here, an operator. No. If you take, for example, a equals b equal to a unit vector, then what do we get? You get n dot sigma squared. And here you have the dot product of n with n, which is 1. So this is 1. And the cross product of two equal vectors, of course, is 0 so you get this, which is nice. Why is this useful? It's because with this identity, you can understand better the operator S hat n that we introduced last time, which was n dot the spin triplet. So nx, sx, ny, sy, nz, sc. So what is this? This is h bar over 2 and dot sigma. And let's square this. So Sn vector squared. This matrix squared would be h bar over 2 squared times n dot sigma squared, which is 1. And sigma squared is 1. Therefore, this spin operator along the n direction squares to h bar r squared over 2 times 1. Now, the trace of this Sn operator is also 0. Why? Because the trace means that you're going to sum the elements in the diagonal. Well, you have a sum of matrices here. And therefore, you will have to sum the diagonals of each. But each of the sigmas has 0 trace. We wrote it there. Trace of sigma 1 is 0. All the Pauli matrices have 0 trace, so this has 0 trace. So you have these two relations. And again, this tells you that the eigenvalues of this matrix can be plus minus h bar over 2. Because the eigenvalues satisfy the same equation as the matrix. Therefor,e plus minus h bar over 2. And this one says that the eigenvalues add up to 0. So the eigenvalues of S hat n vector are plus h bar over 2 and minus h bar over 2. We did that last time, but we do that by just taking that matrix and finding the eigenvalues. But this shows that its property is almost manifest. And this is fundamental for the interpretation of this operator. Why? Well, we saw that if n points along the z-direction, it becomes the operator sz. If it points about the x-direction, it becomes the operator sx. If it points along y, it becomes sy. But in an arbitrary direction, it's a funny thing. But it still has the key property. If you measured the spin along an arbitrary direction, you should find only plus h bar over 2 or minus h bar over 2. Because after all, the universe is isotopic. It doesn't depend on direction. So a spin one-half particle. If you find out that whenever you measure the z component, it's either plus minus h bar over 2. Well, when you measure any direction, it should be plus minus h bar over 2. And this shows that this operator has those eigenvalues. And therefore, it makes sense that this is the operator that measures spins in an arbitrary direction. There's a little more of an aside in there, in the notes about something that will be useful and fun to do. And it corresponds to the case in which you have two triplets of operators-- x1, x2, x3. These are operators now. And y equal y1, y2, y3. Two triplets of operators. So you define the dot product of these two triplets as xi yi summed. That's the definition. Now, the dot product of two triplets of operators defined that way may not commute. Because the operators x and y may not commute. So this new dot product of both phase operators is not commutative-- probably. It may happen that these operators commute, in which case x dot y is equal to y dot x. Similarly, you can define the cross product of these two things. And the k-th component is epsilon i j k xi yj like this. Just like you would define it for two number vectors. Now, what do you know about the cross product in general? It's anti-symmetric. A cross B is equal to minus B cross A. But this one won't be because the operators x and y may not commute. Even x cross x may be nonzero. So one thing I will ask you to compute in the homework is not a long calculation. It's three lines. But what is S cross S equal to? Question there? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yes, it's the sum [INAUDIBLE]. Just in the same way that here you're summing over i's and j's to produce the cross product. So whenever an index is repeated, we'll assume it's summed. And when it is not summed, I will put to the right, not summed explicitly-- the words-. Because in some occasions, it matters. So how much is this? It will involve i, h bar, and something. And you will try to find out what this is. It's a cute thing. All right, any other questions? More questions? Nope. OK. So now, finally, we get to that part of the course that has to do with linear algebra. And I'm going to do an experiment. I'm going to do it differently than I did it in the previous years. There is this nice book. It's here. I don't know if you can read from that far, but it has a pretty-- you might almost say an arrogant title. It says, Linear Algebra Done Right by Sheldon Axler. This is the book, actually, MIT's course 18.700 of linear algebra uses. And when you first get the book that looks like that, you read it and open-- I'm going to show you that this is not that well done. But actually, I think it's actually true. The title is not a lie. It's really done right. I actually wish I had learned linear algebra this way. It may be a little difficult if you've never done any linear algebra. You don't know what the matrix is-- I don't think that's the case anybody here. A determinant, or eigenvalue. If you never heard any of those words, this might be a little hard. But if you've heard those words and you've had a little linear algebra, this is quite nice. Now, this book has also a small problem. Unless you study it seriously, it's not all that easy to grab results that you need from it. You have to study it. So I don't know if it might help you or not during this semester. It may. It's not necessary to get it. Absolutely not. But it is quite lovely. And the emphasis is quite interesting. It really begins from very basic things and logically develops everything and asks at every point the right questions. It's quite nice. So what I'm going to do is-- inspired by that, I want to introduce some of the linear algebra little by little. And I don't know very well how this will go. Maybe there's too much detail. Maybe it's a lot of detail, but not enough so it's not all that great. I don't know, you will have to tell me. But we'll try to get some ideas clear. And the reason I want to get some ideas clear is that good books on this subject allow you to understand how much structure you have to put in a vector space to define certain things. And unless you do this carefully, you probably miss some of the basic things. Like many physicists don't quite realize that talking about the matrix representation, you don't need brass and [INAUDIBLE] to talk about the matrix representation of an operator. At first sight, it seems like you'd need it, but you actually don't. Then, the differences between a complex and a vector space-- complex and a real vector space become much clearer if you take your time to understand it. They are very different. And in a sense, complex vector spaces are more powerful, more elegant, have stronger results. So anyway, it's enough of an introduction. Let's see how we do. And let's just begin there for our story. So we begin with vector spaces and dimensionality. Yes. AUDIENCE: Quick question. The length between the trace of matrix equals 0 and [INAUDIBLE] is proportional to the identity. One is the product of the eigenvalues is 1 and the other one was the sum is equal to 0. Are those two statements related causally, or are they just separate statements [INAUDIBLE]? PROFESSOR: OK, the question is, what is the relation between these two statements? Those are separate observations. One does not imply the other. You can have matrices that square to the identity, like the identity itself, and don't have 0 trace. So these are separate properties. This tells us that the eigenvalue squared are h bar over 2. And this one tells me that lambda 1 plus lambda 2-- there are two eigenvalues-- are 0. So from here, you deduce that the eigenvalues could be plus minus h bar over 2. And in fact, have to be plus minus h bar over 2. All right, so let's talk about vector spaces and dimensionality. Spaces and dimensionality. So why do we care about this? Because the end result of our discussion is that the states of a physical system are vectors in a complex vector space. That's, in a sense, the result we're going to get. Observables, moreover, are linear operators on those vector spaces. So we need to understand what are complex vector spaces, what linear operators on them mean. So as I said, complex vector spaces have subtle properties that make them different from real vector spaces and we want to appreciate that. In a vector space, what do you have? You have vectors and you have numbers. So the two things must exist. The numbers could be the real numbers, in which case we're talking about the real vector space. And the numbers could be complex numbers, in which case we're talking about the complex vector space. We don't say the vectors are real, or complex, or imaginary. We just say there are vectors and there are numbers. Now, the vectors can be added and the numbers can be multiplied by vectors to give vectors. That's basically what is happening. Now, these numbers can be real or complex. And the numbers-- so there are vectors and numbers. And we will focus on just either real numbers or complex numbers, but either one. So these sets of numbers form what is called in mathematics a field. So I will not define the field. But a field-- use the letter F for field. And our results. I will state results whenever-- it doesn't matter whether it's real or complex, I may use the letter F to say the numbers are in F. And you say real or complex. What is a vector space? So the vector space, V. Vector space, V, is a set of vectors with an operation called addition-- and we represent it as plus-- that assigns a vector u plus v in the vector space when u and v belong to the vector space. So for any u and v in the vector space, there's a rule called addition that assigns another vector. This also means that this space is closed under addition. That is, you cannot get out of the vector space by adding vectors. The vector space must contain a set that is consistent in that you can add vectors and you're always there. And there's a multiplication. And a scalar multiplication by elements of the numbers of F such that a, which is a number, times v belongs to the vector space when a belongs to the numbers and v belongs to the vectors. So every time you have a vector, you can multiply by those numbers and the result of that multiplication is another vector. So we say the space is also closed under multiplication. Now, these properties exist, but they must-- these operations exist, but they must satisfy the following properties. So the definition is not really over. These operations satisfy-- 1. u plus v is equal to v plus u. The order doesn't matter how you sum vectors. And here, u and v in V. 2. Associative. So u plus v plus w is equal to u plus v plus w. Moreover, two numbers a times b times v is the same as a times bv. You can add with the first number on the vector and you add with the second. 3. There is an additive identity. And that is what? It's a vector 0 belonging to the vector space. I could write an arrow. But actually, for some reason they just don't like to write it because they say it's always ambiguous whether you're talking about the 0 number or the 0 vector. We do have that problem also in the notation in quantum mechanics. But here it is, here is a 0 vector such that 0 plus any vector v is equal to v. 4. Well, in the field, in the set of numbers, there's the number 1, which multiplied by any other number keeps that number. So the number 1 that belongs to the field satisfies that 1 times any vector is equal to the vector. So we declare that that number multiplied by other numbers is an identity. [INAUDIBLE] identity also multiplying vectors. Yes, there was a question. AUDIENCE: [INAUDIBLE]. PROFESSOR: There is an additive identity. Additive identity, the 0 vector. Finally, distributive laws. No. One second. One, two, three-- the zero vector. Oh, actually in my list I put them in different orders in the notes, but never mind. 5. There's an additive inverse in the vector space. So for each v belonging to the vector space, there is a u belonging to the vector space such that v plus u is equal to 0. So additive identity you can find for every element its opposite vector. It always can be found. And last is this [INAUDIBLE] which says that a times u plus v is equal to au plus av, and a plus b on v is equal to av plus bv. And a's and b's belong to the numbers. a and b's belong to the field. And u and v belong to the vector space. OK. It's a little disconcerting. There's a lot of things. But actually, they are quite minimal. It's well done, this definition. They're all kind of things that you know that follow quite immediately by little proofs. You will see more in the notes, but let me just say briefly a few of them. So here is the additive identity, the vector 0. It's easy to prove that this vector 0 is unique. If you find another 0 prime that also satisfies this property, 0 is equal to 0 prime. So it's unique. You can also show that 0 times any vector is equal to 0. And here, this 0 belongs to the field and this 0 belongs to the vector space. So the 0-- you had to postulate that the 1 in the field does the right thing, but you don't need to postulate that 0, the number 0, multiplied by a vector is 0. You can prove that. And these are not difficult to prove. All of them are one-line exercises. They're done in that book. You can look at them. Moreover, another one. a any number times the 0 vector is equal to the 0 vector. So in this case, those both are vectors. That's also another property. So the 0 vector and the 0 number really do the right thing. Then, another property, the additive inverse. This is sort of interesting. So the additive inverse, you can prove it's unique. So the additive inverse is unique. And it's called-- for v, it's called minus v, just a name. And actually, you can prove it's equal to the number minus 1 times the vector. Might sound totally trivial but try to prove them. They're all simple, but they're not trivial, all these things. So you call it minus v, but it's actually-- this is a proof. OK. So examples. Let's do a few examples. I'll have five examples that we're going to use. So I think the main thing for a physicist that I remember being confused about is the statement that there's no characterization that the vectors are real or complex. The vectors are the vectors and you multiply by a real or complex numbers. So I will have one example that makes that very dramatic. As dramatic as it can be. So one example of vector spaces, the set of N component vectors. So here it is, a1, a2, up to a n. For example, with capital N. With a i belongs to the real and i going from 1 up to N is a vector space over r, the real numbers. So people use that terminology, a vector space over the kind of numbers. You could call it also a real vector space, that would be the same. You see, these components are real. And you have to think for a second if you believe all of them are true or how would you do it. Well, if I would be really precise, I would have to tell you a lot of things that you would find boring. That, for example, you have this vector and you add a set of b's. Well, you add the components. That's the definition of plus. And what's the definition of multiplying by a number? Well, if a number is multiplied by this vector, it goes in and multiplies everybody. Those are implicit, or you can fill-in the details. But if you define them that way, it will satisfy all the properties. What is the 0 vector? It must be the one with all entries 0. What is the additive inverse? Well, change the sign of all these things. So it's kind of obvious that this satisfies everything, if you understand how the sum and the multiplication goes. Another one, it's kind of similar. 2. The set of M cross N matrices with complex entries. Complex entries. So here you have it, a1 1, a1 2, a1 N. And here it goes up to aM1, aM2, aMN. With all the a i j's belonging to the complex numbers, then-- I'll erase here. Then you have that this is a complex vector space. Is a complex vector space. How do you multiply by a number? You multiply a number times every entry of the matrices. How do sum two matrices? They have the same size, so you sum each element the way it should be. And that should be a vector space. Here is an example that is, perhaps, a little more surprising. So the space of 2 by 2 Hermitian matrices is a real vector space. You see, this can be easily thought [INAUDIBLE] naturally thought as a real vector space. This is a little surprising because Hermitian matrices have i's. You remember the most general Hermitian matrix was of the form-- well, a plus-- no, c plus d, c minus d, a plus ib, a minus ib, with all these numbers c, d, b in real. But they're complex numbers. Why is this naturally a real vector space? The problem is that if you multiply by a number, it should still be a Hermitian matrix in order for it to be a vector space. It should be in the vector. But if you multiply by a real number, there's no problem. The matrix remains Hermitian. You multiplied by a complex number, you use the Hermiticity. But an i somewhere here for all the factors and it will not be Hermitian. So this is why it's a real vector space. Multiplication by real numbers preserves Hermiticity. So that's surprising. So again, illustrates that nobody would say this is a real vector. But it really should be thought as a vector over real numbers. Vector space over real numbers. Two more examples. And they are kind of interesting. So the next example is the set of polynomials as vector space. So that, again, is sort of a very imaginative thing. The set of polynomials p of z. Here, z belongs to some field and p of z, which is a function of z, also belongs to the same field. And each polynomial has coefficient. So any p of z is a0 plus a1 z plus a2 z squared plus-- up to some an zn. A polynomial is supposed to end That's pretty important about polynomials. So the dots don't go up forever. So here it is, the a i's also belong to the field. So looked at this polynomials. We have the letter z and they have these coefficients which are numbers. So a real polynomial-- you know 2 plus x plus x squared. So you have your real numbers times this general variable that it's also supposed to be real. So you could have it real. You could have it complex. So that's a polynomial. How is that a vector space? Well, it's a vector space-- the space p of F of those polynomials-- of all polynomials is a vector space over F. And why is that? Well, you can take-- again, there's some implicit definitions. How do you sum polynomials? Well, you sum the independent coefficients. You just sum them and factor out. So there's an obvious definition of sum. How do you multiply a polynomial by a number? Obvious definition, you multiply everything by a number. If you sum polynomials, you get polynomials. Given a polynomial, there is a negative polynomial that adds up to 0. There's a 0 when all the coefficients is 0. And it has all the nice properties. Now, this example is more nontrivial because you would think, as opposed to the previous examples, that this is probably infinite dimensional because it has the linear polynomial, the quadratic, the cubic, the quartic, the quintic, all of them together. And yes, we'll see that in a second. So set of polynomials. 5. Another example, 5. The set F infinity of infinite sequences. Sequences x1, x2, infinite sequences where the x i's are in the field. So you've got an infinite sequence and you want to add another infinite sequence. Well, you add the first element, the second elements. It's like an infinite column vector. Sometimes mathematicians like to write column vectors like that because it's practical. It saves space on a page. The vertical one, you start writing and the pages grow very fast. So here's an infinite sequence. And think of it as a vertical one if you wish. And all elements are here, but there are infinitely many in every sequence. And of course, the set of all infinite sequences is infinite. So this is a vector space over F. Again, because all the numbers are here, so it's a vector space over F. And last example. Our last example is a familiar one in physics, is the set of complex functions in an interval. Set of complex functions on an interval x from 0 to L. So a set of complex functions f of x I could put here on an interval [INAUDIBLE]. So this is a complex vector space. Vector space. The last three examples, probably you would agree that there are infinite dimensional, even though I've not defined what that means very precisely. But that's what we're going to try to understand now. We're supposed to understand the concept of dimensionality. So let's get to that concept now. So in terms of dimensionality, to build this idea you need a definition. You need to know the term subspace of a vector space. What is a subspace of a vector space? A subspace of a vector space is a subset of the vector space that is still a vector space. So that's why it's called subspace. It's different from subset. So a subspace of V is a subset of V that is a vector space. So in particular, it must contain the vector 0 because any vector space contains the vector 0. One of the ways you sometimes want to understand the vector space is by representing it as a sum of smaller vector spaces. And we will do that when we consider, for example, angular momentum in detail. So you want to write a vector space as a sum of subspaces. So what is that called? It's called a direct sum. So if you can write-- here is the equation. You say V is equal to u1 direct sum with u2 direct sum with u3 direct sum with u m. When we say this, we mean the following. That the ui's are subspaces of V. And any V in the vector space can be written uniquely as a1 u1 plus a2 u2 plus a n u n with ui [INAUDIBLE] capital Ui. So let me review what we just said. So you have a vector space and you want to decompose it in sort of basic ingredients. This is called a direct sum. V is a direct sum of subspaces. Direct sum. And the Ui's are subspaces of V. But what must happen for this to be true is that once you take any vector here, you can write it as a sum of a vector here, a vector here, a vector here, a vector everywhere. And it must be done uniquely. If you can do this in more than one way, this is not a direct sum. These subspaces kind of overlap. They're not doing the decomposition in a minimal way. Yes. AUDIENCE: Does the expression of V have to be a linear combination of the vectors of the U, or just sums of the U sub i's? PROFESSOR: It's some linear combination. Look, the interpretation, for example, R2. The normal vector space R2. You have an intuition quite clearly that any vector here is a unique sum of this component along this subspace and this component along this subspace. So it's a trivial example, but the vector space R2 has a vector subspace R1 here and a vector subspace R1. Any vector in R2 is uniquely written as a sum of these two vectors. That means that R2 is really R1 plus R1. Yes. AUDIENCE: [INAUDIBLE]. Is it redundant to say that that-- because a1 u1 is also in big U sub 1. PROFESSOR: Oh. Oh, yes. You're right. No, I'm sorry. I shouldn't write those. I'm sorry. That's absolutely right. If I had that in my notes, it was a mistake. Thank you. That was very good. Did I have that in my notes? No, I had it as you said it. True. So can be written uniquely as a vector in first, a vector in the second. And the a's are absolutely not necessary. OK. So let's go ahead then and say the following things. So here we're going to try to get to the concept of dimensionality in a precise way. Yes. AUDIENCE: [INAUDIBLE]. PROFESSOR: Right, the last one is m. Thank you. All right. The concept of dimensionality of a vector space is something that you intuitively understand. It's sort of how many linearly independent vectors you need to describe the whole set of vectors. So that is the number you're trying to get to. I'll follow it up in a slightly rigorous way to be able to do infinite dimensional space as well. So we will consider something called a list of vectors. List of vectors. And that will be something like v1, v2 vectors in a vector space up to vn. Any list of vectors has finite length. So we don't accept infinite lists by definition. You can ask, once you have a list of vectors, what is the vector subspace spanned by this list? How much do you reach with that list? So we call it the span of the list. The span of the list, vn. And it's the set of all linear combinations a1 v1 plus a2 v2 plus a n vn for ai in the field. So the span of the list is all possible products of your vectors on the list are-- and put like that. So if we say that the list spans a vector space, if the span of the list is the vector space. So that's natural language. We say, OK, this list spans the vector space. Why? Because if you produce the span of the list, it fills a vector space. OK, so I could say it that way. So here is the definition, V is finite dimensional if it's spanned by some list. If V is spanned by some list. So why is that? Because if the list is-- a definition, finite dimensional. If it's spanned by some list. If you got your list, by definition it's finite length. And with some set of vectors, you span everything. And moreover, it's infinite dimensional if it's not finite dimensional. It's kind of silly, but infinite-- a space V is infinite dimensional if it is not finite dimensional. Which is to say that there is no list that spans the space. So for example, this definition is tailored in a nice way. Like let's think of the polynomials. And we want to see if it's finite dimensional or infinite dimensional. So you claim it's finite dimensional. Let's see if it's finite dimensional. So we make a list of polynomials. The list must have some length, at least, that spans it. You put all these 730 polynomials that you think span the list, span the space, in this list. Now, if you look at the list, it's 720. You can check one by one until you find what is the one of highest order, the polynomial of highest degree. But if the highest degree is say, z to the 1 million, then any polynomial that has a z to the 2 million cannot be spanned by this one. So there's no finite list that can span this, so this set-- the example in 4 is infinite dimensional for sure. Example 4 is infinite dimensional. Well, example one is finite dimensional. You can see that because we can produce a list that spans the space. So look at the example 1. It's there. Well, what would be the list? The list would be-- list. You would put a vector e1, e2, up to en. And the vector e1 would be 1, 0, 0, 0, 0. The vector e2 would be 0, 1, 0, 0, 0. And go on like that. So you put 1's and 0's. And you have n of them. And certainly, the most general one is a1 times e1 a2 times e2. And you got the list. So example 1 is finite dimensional. A list of vectors is linearly independent. A list is linearly independent if a list v1 up to vn is linearly independent, If a1 v1 plus a2 v2 plus a n vn is equal to 0 has the unique solution a1 equal a2 equal all of them equal 0. So that is to mean that whenever this list satisfies this property-- if you want to represent the vector 0 with this list, you must set all of them equal to 0, all the coefficients. That's clear as well in this example. If you want to represent the 0 vector, you must have 0 component against the basis vector x and basis vector y. So the list of this vector and this vector is linearly independent because the 0 vector must have 0 numbers multiplying each of them. So finally, we define what is a basis. A basis of V is a list of vectors in V that spans V and is linearly independent. So what is a basis? Well, you should have enough vectors to represent every vector. So it must span V. And what else should it have? It shouldn't have extra vectors that you don't need. It should be minimal. It should be all linearly independent. You shouldn't have added more stuff to it. So any finite dimensional vector space has a basis. It's easy to do it. There's another thing that one can prove. It may look kind of obvious, but it requires a small proof that if you have-- the bases are not unique. It's something we're going to exploit all the time. One basis, another basis, a third basis. We're going to change basis all the time. Well, the bases are not unique, but the length of the bases of a vector space is always the same. So the length of the list is-- a number is the same whatever base you choose. And that length is what is called the dimension of the vector space. So the dimension of a vector space is the length of any bases of V. And therefore, it's a well-defined concept. Any base of a finite vector space has the same length, and the dimension is that number. So there was a question. Yes? AUDIENCE: Is there any difference between bases [INAUDIBLE]? PROFESSOR: No, absolutely not. You could have a basis, for example, of R2, which is this vector. The first and the second is this vector. And any vector is a linear superposition of these two vectors with some coefficients and it's unique. You can find the coefficients. AUDIENCE: [INAUDIBLE]. PROFESSOR: Yes. But you see, here is exactly what I wanted to make clear. We're putting the vector space and we're putting the least possible structure. I didn't say how to take the inner product of two vectors. It's not a definition of a vector space. It's something we'll put later. And then, we will be able to ask whether the basis is orthonormal or not. But the basis exists. Even though you have no definition of an inner product, you can talk about basis without any confusion. You can also talk about the matrix representation of an operator. And you don't need an inner product, which is sometimes very unclear. You can talk about the trace of an operator and you don't need an inner product. You can talk about eigenvectors and eigenvalues and you don't need an inner product. The only thing you need the inner product is to get numbers. And we'll use them to use [INAUDIBLE] to get numbers. But it can wait. It's better than you see all that you can do without introducing more things, and then introduce them. So let me explain a little more this concept. We were talking about this base, this vector space 1, for example. And we produced a list that spans e1, e2, up to en. And those were these vectors. Now, this list not only spans, but they are linearly independent. If you put a1 times this plus a2 times this and you set it all equal to 0. Well, each entry will be 0, and all the a's are 0. So these e's that you put here on that list is actually a basis. Therefore, the length of that basis is the dimensionality. And this space has dimension N. You should be able to prove that this space has been dimension m times N. Now, let me do the Hermitian-- these matrices. And try to figure out the dimensionality of the space of Hermitian matrices. So here they are. This is the most general Hermitian matrix. And I'm going to produce for you a list of four vectors. Vectors-- yes, they're matrices, but we call them vectors. So here is the list. The unit matrix, the first Pauli matrix, the second Pauli matrix, and the third Pauli matrix. All right, let's see how far do we get from there. OK, this is a list of vectors in the vector space because all of them are Hermitian. Good. Do they span? Well, you calculate the most general Hermitian matrix of this form. You just put arbitrary complex numbers and require that the matrix be equal to its matrix complex conjugate and transpose. So this is the most general one. Do I obtain this matrix from this one's? Yes I just have to put 1 times c plus a times sigma 1 plus b times sigma 2 plus d times sigma 3. So any Hermitian matrix can be obtained as the span of this list. Is this list linearly independent? So I have to go here and set this equal to 0 and see if it sets to 0 all these coefficients. Well, it's the same thing as setting to 0 all this matrix. Well, if c plus d and c minus d are 0, then c and d are 0. If this is 0, it must be a 0 and b 0, so all of them are 0. So yes, it's linearly independent. It spans. Therefore, you've proven completely rigorously that this vector space is dimension 4. This vector space-- I will actually leave it as an exercise for you to show that this vector space is infinite dimensional. You say, of course, it's infinite dimensional. It has infinite sequences. Well, you have to show that if you have a finite list of those infinite sequences, like 300 sequences, they span that. They cannot span that. So it takes a little work. It's interesting to think about it. I think you will enjoy trying to think about this stuff. So that's our discussion of dimensionality. So this one is a little harder to make sure it's infinite dimensional. And this one is, yet, a bit harder than that one but it can also be done. This is infinite dimensional. And this is infinite dimensional. In the last two minute, I want to tell you a little bit-- one definition and let you go with that, is the definition of a linear operator. So here is one thing. So you can be more general, and we won't be that general. But when you talk about linear maps, you have one vector space and another vector space, v and w. This is a vector space and this is a vector space. And in general, a map from here is sometimes called, if it satisfies the property, a linear map. And the key thing is that in all generality, these two vector spaces may not have the same dimension. It might be one vector space and another very different vector space. You go from one to the other. Now, when you have a vector space v and you map to the same vector space, this is also a linear map, but this is called an operator or a linear operator. And what is a linear operator therefore? A linear operator is a function T. Let's call the linear operator T. It takes v to v. In which way? Well, T acting u plus v, on the sum of vectors, is Tu plus T v. And T acting on a times a vector is a times T of the vector. These two things make it into something we call a linear operator. It acts on the sum of vectors linearly and on a number times a vector. The number goes out and you act on the vector. So all you need to know for what a linear operator is, is how it acts on basis vectors. Because any vector on the vector space is a superposition of basis vectors. So if you tell me how it acts on the basis vectors, you know everything. So we will figure out how the matrix representation of the operators arises from how it acts on the basis vectors. And you don't need an inner product. The reason people think of this is they say, oh, the T i j matrix element of T is the inner product of the operator between i and j. And this is true. But for that you need [? brass ?] and inner product, all these things. And they're not necessary. We'll define this without that. We don't need it. So see you next time, and we'll continue that. [APPLAUSE] Thank you.

MIT_805_Quantum_Physics_II_Fall_2013

22_Angular_Momentum_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right, it is time to get started. Thanks for coming for this cold and rainy Wednesday before Thanksgiving. Today we're supposed to talk about the radial equation. That's our main subject today. We discussed last time the states of angular momentum from the abstract viewpoint, and now we make contact with some important problems, and differential equations, and things like that. And there's a few concepts I want to emphasize today. And basically, the main concept is that I want you to just become familiar with what we would call the diagram, the key diagram for the states of a theory, of a particle in a three dimensional potential. I think you have to have a good understanding of what it looks, and what is special about it, and when it shows particular properties. So to begin with, I'll have to do a little aside on a object that is covered in many courses. I don't know to what that it's covered, but it's the subject of spherical harmonics. So we'll talk about spherical harmonics for about 15 minutes. And then we'll do the radial equation. And for the radial equation, after we discuss it, we'll do three examples. And that will be the end of today's lecture. Next time, as you come back from the holiday next week, we are doing the addition of angular momentum basically. And then the last week, more examples and a few more things for emphasis to understand it all well. All right, so in terms of spherical harmonics, I wanted to emphasize that our algebraic analysis led to states that we called jm, but today I will call lm, because they will refer to orbital angular momentum. And as you've seen in one of your problems, orbital angular momentum has to do with values of j, which are integers. So half integers values of j cannot be realized for orbital angular momentum. It's a very interesting thing. So spin states don't have wave functions in the usual way. It's only states of integer angular momentum that have wave functions. And those are the spherical harmonics. So I will talk about lm, and l, as usual, will go from 0 to infinity. And m goes from l to minus l. And you had these states, and we said that algebraically you would have L squared equals h squared l times l plus 1 lm. And Lz lm equal hm lm. Now basically, the spherical harmonics are going to be wave functions for these states. And the way we can approach it is that we did a little bit of work already with constructing the l squared operator. And in last lecture we derived, starting from the fact that L is r cross p and using x,y, and z, px, py, pz, and passing through spherical coordinates that L squared is the operator minus h squared 1 over sine theta d d theta sine theta d d theta again plus 1 over sine squared theta d second d phi squared. And we didn't do it, but Lz, which you know is h bar over i x d dy minus y d dx can also be translated into angular variables. And it has a very simple form. Also purely angular. And you can interpret it Lz is rotations around the z-axis, so they change phi. So it will not surprise you, if you do this exercise, that this is h over i d d phi. And you should really check it. There's another one that is a bit more laborious. L plus minus, remember, is Lx plus minus i Lz. We have a big attendance today. Is equal to-- more people. h bar e to the plus minus i phi i cosine theta over sine theta d d phi plus minus d d theta. And that takes a bit of algebra. You could do it. It's done in many books. It's probably there in Griffith's. And these are the representations of these operators as differential operators that act and function on theta and phi and don't care about radius. So in mathematical physics, people study these things and invent these things called spherical harmonics Ylm's of theta and phi. And the way you could see their done is in fact, such that this L squared viewed as this operator, differential operator, acting on Ylm is indeed equal to h squared l times l plus 1 Ylm. And Lz thought also as a differential operator, the one that we've written there. On the Ylm is h bar m Ylm. So they are constructed in this way, satisfying these equations. These are important equations in mathematical physics, and these functions were invented to satisfy those equations. Well, these are the properties of those states over there. So we can think of these functions as the wave functions associated with those states. So that's interpretation that is natural in quantum mechanics. And we want to think of them like that. We want to think of the Ylm's as the wave functions associated to the states lm. So lm. And here you would put a position state theta phi. This is analogous to the thing that we usually call the wave function being a position state times the state side. So we want to think of the Ylm's in this way as pretty much the wave functions associated to those states. Now there is a little bit of identities that come once you accept that this is what you think of the Ylm's. And then the compatibility of these equations. Top here with these ones makes in this identification natural. Now in order to manipulate and learn things about those spherical harmonics the way we do things in quantum mechanics, we think of the completeness relation. If we have d cube x x x, this is a completeness relation for position states. And I want to derive or suggest a completeness relation for these theta phi states. For that, I would pass this integral to do it in spherical coordinates. So I would do dr rd theta r sine theta d phi. And I would put r theta phi position states for these things. And position states r theta phi. Still being equal to 1. And we can try to split this thing. It's natural for us to think of just theta phi, because these wave functions have nothing to do with r, so I will simply do the integrals this way. d theta sine theta d phi. And think just like a position state in x, y, z. It's a position state in x, in y, and in z multiplied. We'll just split these things without trying to be too rigorous about it. Theta and phi like this. And you would have the integral dr r squared r r equal 1. And at this point, I want to think of this as the natural way of setting a completeness relation for theta and phi. And this doesn't talk to this one, so I will think of this that in the space of theta and phi, objects that just depend on theta and phi, this acts as a complete thing. And if objects depend also in r, this will act as a complete thing. So I will-- I don't know. Maybe the right way to say is postulate that we'll have a completeness relation of this form. d theta sine theta d phi theta phi theta phi equals 1. And then with this we can do all kinds of things. First, this integral is better written. This integral really represents 0 to pi d theta sine theta 0 to 2 pi d phi. Now this is minus d cosine theta. And when theta is equal to 0, cosine theta is 1 to minus 1 integral d phi 0 to 2 pi. So this integral, really d theta sine theta d phi this is really the integral from minus 1 to 1. Change that order of d cos theta integral d phi from 0 to 2 pi. And this is called the integral over solid angle. That's a definition. So we could write the completeness relation in the space theta phi as integral over solid angle theta phi theta phi equals 1. Then the key property of the spherical harmonics, or the lm states, is that they are orthogonal. So delta l, l prime, delta m, m prime. So the orthogonality are of this state is guaranteed because Hermitian operators, different eigenvalues, they have to be orthogonal. Eigenstates of Hermitian. Operators with different eigenvalues. Here, you introduce a complete set of states of theta phi. So you put l prime m prime theta phi theta phi lm. And this is the integral over solid angle of Yl prime m prime of theta phi star. This is in the wrong position. And here Ylm of theta phi being equal delta l l prime delta m m prime. So this is orthogonality of the spherical harmonics. And this is pretty much all we need. Now there's the standard ways of constructing these things from the quantum mechanical sort of intuition. Basically, you can try to first build Yll, which corresponds to the state ll. Now the kind of differential equations this Yll satisfies are kind of simple. But in particular, the most important one is that L plus kills this state. So basically you use the condition that L plus kills this state to find a differential equation for this, which can be solved easily. Not a hard differential equation. Then you find Yll. And then you can find Yll minus 1 and all the other ones by applying the operator L minus. The lowering operator of m. So in principle, if you have enough patience, you can calculate all the spherical harmonics that way. There's no obstruction. But the form is a little messy, and if you want to find the normalizations so that these things work out correctly, well, it takes some work at the end of the day. So we're not going to do that here. We'll just leave it at that, and if we ever need some special harmonics, we'll just hold the answers. And they are in most textbooks. So if you do need them, well, you'll have to do with complicated normalizations. So that's really all I wanted to say about spherical harmonics, and we can turn then to the real subject, which is the radial equation. So the radial equation. So we have a Hamiltonian H equals p squared vector over 2m plus v of r. And we've seen that this is equal to h over 2m 1 over r d second dr squared r plus 1 over 2mr squared L squared plus v of r. So this is what we're trying to solve. And the way we attempt to solve this is by separation of variables. So we'll try to write the wave function, psi, characterized by three things. Its energy, the value of l, and the value of m. And it's a function of position, because we're trying to solve H psi equal E psi. And that's the energy that we want to consider. So I will write here to begin with something that will not turn out to be exactly right, but it's important to do it first this way. A function of art r that has labels E, l, and m. Because it certainly could depend on E, could depend on l, and could depend on m, that radial function. And then the angular function will be the Ylm's of theta and phi. So this is the [INAUDIBLE] sets for the equation. If we have that, we can plug into the Schrodinger equation, and see what we get. Well, this operator will act on this f. This will have the operator L squared, but L squared over Ylm, you know what it is. And v of r is multiplicative, so it's no big problem. So what do we have? We have minus h squared over 2m 1 over r. Now I can talk normal derivatives. d r squared r times fElm plus 1 over 2mr squared. And now have L squared acting on this, but L squared acting on the Ylm is just this factor. So we have h squared l times l plus 1 times the fElm. Now I didn't put the Ylm in the first term because I'm going to cancel it throughout. So we have this term here plus v of r fElm equals E fElm. That is substituting into the equation h psi equal E psi. So first term here. Second term, it acted on the spherical harmonic. v of r is multiplicative. E on that. But then what you see immediately is that this differential equation doesn't depend on m. It was L squared, but no Lz in the Hamiltonian. So no m dependent. So actually we were overly proven in thinking that f was a function of m. What we really have is that psi Elm is equal to a function of E and l or r Ylm of theta phi. And then the differential equation is minus h squared over 2m. Let's multiply all by r. d second dr squared of r fEl. Plus look here. The r that I'm multiplying is going to go into the f. Here it's going to go into the f. Here it's going to go into the f. It's an overall thing. But here we keep h squared l times l plus 1 over 2mr squared rfEl plus v of r fEl rfEl equal e times rfEl. So what you see here is that this function is quite natural. So it suggests the definition of uEl to be rfEl. So that the differential equation now finally becomes minus h squared over 2m d second dr squared of uEl plus there's the u here, the u here, and this potential that has two terms. So this will be v of r plus h squared l times l plus 1 over 2mr squared uEl equals E times eEl. And this is the famous radial equation. It's an equation for you. And here, this whole thing is sometimes called the effective potential. So look what we've got. This f, if you wish here, is now of the form uEl of r Ylm over r theta phi. f is u over r. So this is the way we've written the solution, and u satisfies this equation, which is a one dimensional Schrodinger equation for the radius r. One dimensional equation with an effective potential that depends on L. So actually the first thing you have to notice is that the central potential problem has turned into an infinite collection of one dimensional problems. One for each value of l. For different values of l, you have a different potential. Now they're not all that different. They have different intensity of this term. For l equals 0, well you have some solutions. And for l equal 1, the answer could be quite different. For l equal 2, still different. And you have to solve an infinite number of one dimensional problems. That's what the Schrodinger equation has turned into. So we filled all these blackboards. Let's see, are there questions? Anything so far? Yes? AUDIENCE: You might get to this later, but what does it mean in our wave equations, in our wave function there, psi of Elm is equal to fEl, and the spherical harmonic of that one mean that one has an independence and the other doesn't. Can they be separated on the basis of m? PROFESSOR: So it is just a fact that the radial solution is independent of n, so it's an important property. n is fairly simple. The various state, the states with angular momentum l, but different m's just differ in their angular dependence, not in the radial dependence. And practically, it means that you have an infinite set of one dimensional problems labeled by l, and not labeled by m, which conceivably could have happened, but it doesn't happen. So just a major simplicity. Yes? AUDIENCE: Does the radial equation have all the same properties as a one dimensional Schrodinger equation? Or does the divergence in the effect [INAUDIBLE] 0 change that? PROFESSOR: Well, it changes things, but the most serious change is the fact that, in one dimensional problems, x goes from minus infinity to infinity. And here it goes from 0 to infinity, so we need to worry about what happens at 0. Basically that's the main complication. One dimensional potential, but it really just can't go below 0. r is a radial variable, and we can't forget that. Yes? AUDIENCE: The potential v of r will depend on whatever problem you're solving, right? PROFESSOR: That's right. AUDIENCE: Could you find the v of r [INAUDIBLE]? PROFESSOR: Well that doesn't quite make sense as a Hamiltonian. You see, if you have a v of r, it's something that is supposed to be v of r for any wave function. That's the definition. So it can depend on some parameter, but that parameter cannot be the l of the particular wave function. AUDIENCE: [INAUDIBLE] or something that would interact with the-- PROFESSOR: If you have magnetic fields, things change, because then you can split levels with respect to m. Break degeneracies and things change indeed. We'll take care of those by using perturbation theory mostly. Use this solution and then perturbation theory. OK, so let's proceed a little more on this. So the first thing that we want to talk a little about is the normalization and some boundary conditions, because otherwise we can't really understand what's going on. And happily the discussion is not that complicated. So we want to normalize. So what do we want? Integral d cube x psi Elm of x squared equals 1. So clearly we want to go into angular variables. So again, this is r squared dr integral d solid angle, r squared Er. And this thing is now uEl squared absolute value over r squared. Look at the right most blackboard. uEl of r, I must square it because the wave function is squared. Over r squared. And then I have Ylm star of theta phi Ylm of theta phi. And if this is supposed to be normalized, this is supposed to be the number 1. Well happily, this part, this is why we needed to talk a little about spherical harmonics. This integral is 1, because it corresponds precisely to l equal l prime m equal m prime. And look how lucky or nice this is. r squared cancels with r squared, so the final condition is the integral from 0 to infinity dr uEl of r squared is equal to 1, which shows that kind of the u really plays a role for wave function and a line. And even though it was a little complicated, there was the r here, and angular dependence, and everything, a good wave function is one that is just think of psi as being u. A one dimensional wave function psi being u, and if you can integrate it square, you've got it. AUDIENCE: [INAUDIBLE]. PROFESSOR: Because I had to square this, so there was u over r. AUDIENCE: But that's [INAUDIBLE]. PROFESSOR: Oh, I'm sorry. That parenthesis is a remnant. I tried to erase it a little. It's not squared anymore. The square is on the absolute value is r squared. So this is good news for our interpretation. So now before I discuss the peculiarities of the boundary conditions, I want to introduce really the main point that we're going to illustrate in this lecture. This is the thing that should remain in your heads. It's a picture, but it's an important one. When you want to organize the spectrum, you'll draw the following diagram. Energy is here and l here. And it's a funny kind of diagram. It's not like a curve or a plot. It's like a histogram or kind of thing like that. So what will happen is that you have a one dimensional problem. If these potentials are normal, there will be bound states. And let's consider the case of bound states for the purposes of this graph, just bound states. Now you look at this, and you say OK, what am I supposed to do? I'm going to have states for all values of l, and m, and probably some energies. So m doesn't affect the radial equation. That's very important. But l does, so I have a different problem to solve for different l. So I will make my histogram here and put here l equals 0 at this region. l equals 1, l equals 2, l equals 3, and go on. Now suppose I fix an l. l is fixed. Now it's a Schrodinger equation for a one dimensional problem. You would expect that if the potential suitably grows, which is a typical case, E will be quantized. And there will not be degeneracies, because the bound state spectrum in one dimension is not degenerate. So I should expect that for each l there are going to be energy values that are going to appear. So for l equals 0, I expect that there will be some energy here for which I've got a state. And that line means I got a state. And there's some energy here that could be called E1, 0 is the first energy that is allowed with l equals 0. Then there will be another one here maybe. E-- I'll write it down-- 2,0. So basically I'm labeling the energies with En,l which means the first solution with l equals 0, the second solution with l equals 0, the third solution E 3,0. Then you come to l equals 1, and you must solve the equation again. And then for l equal 1, there will be the lowest energy, the ground state energy of the l equal 1 potential, and then higher and higher. Since the l equal 1 potential is higher than the l equals 0 potential, it's higher up. The energies should be higher up, at least the first one should be. And therefore the first one could be a little higher than this, or maybe by some accident it just fits here, or maybe it should fit here. Well, we don't know but know, but there's no obvious reason why it should, so I'll put it here. l equals 1. And this would be E1,1. The first state with l equals 1. Then here it could be E2,1. The second state with l equal 1 and higher up. And then for l equal-- my diagram is a little too big. E1,1. E2,1. And then you have states here, so maybe this one, l equals 2, I don't know where it goes. It just has to be higher than this one, so I'll put it here. And this will be E1,2. Maybe there's an E2,2. And here an E1,3. But this is the answer to your problem. That's the energy levels of a central potential. So it's a good, nice little diagram in which you put the states, you put the little line wherever you find the state. And for l equals 0, you have those states. Now because there's no degeneracies in the bound states of a one dimensional potential, I don't have two lines here that coincide, because there's no two states with the same energy here. It's just one state. And this one here. I cannot have two things there. That's pretty important to. So you have a list of states here. And just one state here, one state, but as you can see, you're probably are catching me in a little wrong play of words, because I say there's one state here. Yes, it's one state, because it's l equals 0. One state, one state. But this state, which is one single-- this should be called one single l equal 1 multiplet. So this is not really one state at the end of the day. It's one state of the one dimensional radial equation, but you know that l equals 1 comes accompanied with three values of m. So there's three states that are degenerate, because they have the same energy. The energy doesn't depend on l. So this thing is an l equal 1 multiplet, which means really three states. And this is three states. And this is three states. And this is 1 l equal 2 multiplet, which has possibility of m equals 2, 1, 0 minus 1 and minus 2. So in this state is just one l equal 2 multiplet, but it really means five states of the central potential. Five degenerate states, because the m doesn't change the energy. And this is five states. And this is seven states. One l equal 3 multiplet, which contains seven states. OK, so questions? This is the most important graph. If you have that picture in your head, then you can understand really where you're going with any potential. Any confusion here above the notation? Yes? AUDIENCE: So normally when we think about a one dimensional problem, we say that there's no degeneracy. Not really. No multiple degeneracy, so should we think of the radial equation as having copies for each m value and each having the same eigenvalue? PROFESSOR: I don't think it's necessary. You see, you've got your uEl. And you have here you solutions. Once the uEl is good, you're supposed to be able to put any Ylm. So put l, and now the m's that are allowed are solutions. You're solving the problem. So think of a master radial function as good for a fixed l, and therefore it works for all values of m. But don't try to think of many copies of this equation. I don't think it would help you. Any other questions? Yes? AUDIENCE: Sorry to ask, but if you could just review how is degeneracy built one more time? PROFESSOR: Yeah. Remember last time we were talking about, for example, what is a j equal to multiplet. Well, these were a collection of states jm with j equals 2 an m sum value. And they are all obtained by acting with angular momentum operators in each other. And there are five states. The 2,2, the 2,1, the 2,0, the 2, minus 1, and the 2, minus 2. And all these states are obtained by acting with, say, lowering operators l minus and this. Now all these angular momentum operators, all of the Li's commute with the Hamiltonian. Therefore all of these states are obtained by acting with Li must have the same energy. That's why we say that this comes in a multiplet. So when you get j-- in this case we'll call it l-- l equals 2. You get five states. They correspond to the various values of m. So when you did that radial equation that has a solution for l equals 2, you're getting the full multiplet. You're getting five states. 1 l equal 2 multiplet. That's why one line here. That is equivalent to five states. OK, so that diagram, of course, is really quite important. So now we want to understand the boundary conditions. So we have here this. So this probably shouldn't erase yet. Let's do the boundary conditions. So behavior here at r equals to 0. At r going to 0. The first claim is that surprisingly, you would think, well, normalization is king. If it's normalized, it's good. So just any number. Just don't let it diverge near 0, and that will be OK. But it turns out that that's not true. It's not right. And you need the limit as r goes to 0 of uEl of r be equal to 0. And we'll take this and explore the simplest case. That is corresponds to saying what if the limit of r goes to 0 or uEl of r was a constant? What goes wrong? Certainly normalization doesn't go wrong. It can be a constant. u could be like that, and it would be normalized, and that doesn't go wrong. So let's look at the wave function. What happens with this? I actually will take for simplicity, because we'll analyze it later, the example of l equals 0. So let's put even 0. l equals 0. Well, suppose you look at the wave function now, and how does it look? Psi of E0-- if l is equal to 0, m must be equal to 0-- would be this u over r times a constant. So a constant, because y 0, 0 is a constant. And then you uE0 of r over r. So when r approaches 0, psi goes like c prime over r, some other constant over r. So I'm doing something very simple. I'm saying if uE0 is approaching the constant at the origin, if it's uE0, well, this is a constant because it's 0,0. So this is going to constant. So at the end of the day, the wave function looks like 1 over r. But this is impossible, because the Schrodinger equation H psi has minus h squared over 2m Laplacian on psi plus dot dot dot. And the up Laplacian of 1 over r is minus 4 pi times a delta function at x equals 0. So this means that the Schrodinger equation, you think oh I put psi equals c over r. Well, if you calculate the Laplacian, it seems to be 0. But if you're more careful, as you know for [? emm ?] the Laplacian of 1 over r is minus 4 pi times the delta function. So in the Schrodinger equation, the kinetic term produces a delta function. There's no reason to believe there's a delta function in the potential. We'll not try such crazy potentials. A delta function in a one dimensional potential, you've got the solution. A delta function in a three dimensional potential is absolutely crazy. It has infinite number of bound states, and they just go all the way down to energies of minus infinity. It's a very horrendous thing, a delta function in three dimensions, for quantum mechanics. So this thing, there's no delta function in the potential. And you've got a delta function from the kinetic term. You're not going to be able to cancel it. This is not a solution. So you really cannot approach a constant there. It's quite bad. So the wave functions will have to vanish, and we can prove that, or at least under some circumstances prove it. And as all these things are, they all depend on how crazy potentials you want to accept. So we should say something. So I'll say something about these potentials, and we'll prove a result. So my statement will be the centrifugal barrier, which is a name for this part of the potential, dominates as r goes to 0. If this doesn't happen, all bets are off. So let's assume that v of r, maybe it's 1 over r, but it's not worse than 1 over r squared. It's 1 over r cubed, for example, or something like that. You would have to analyze it from scratch if it would be that bad. But I will assume that the centrifugal barrier dominates. And then look at the differential equation. Well, what differential equation do I have? Well, I have this and this. This thing is less important than that, and this is also less important, because this is u divided by r squared. And here is just u. So this is certainly less important than that, and this is less important than that, and if I want to have some variation of u, or understand how it varies, I must keep this. So at this order, I should keep just the kinetic term h squared over 2m d second dr squared u of El. And h squared l times l plus 1 over 2 mr squared. And I will try to cancel these two to explore how the wave function looks near or equal 0. These are the two most important terms of the differential equation, so I have the right to keep those, and try to balance them out to leading order, and see what I get. So all the h squared over 2m's go away. So this is equivalent to d second uEl dr squared is equal to l times l plus 1 uEl over r squared. And this is solved by a power uEl. You can try r to the s, some number s. And then this thing gives you s times s minus 1. Taking two derivatives is equal to l times l plus 1. As you take two derivatives, you lose two powers of r, so it will work out. And from here, you see that the possible solutions are s equals l plus 1. And s equals 2 minus l. So this corresponds to a uEl that goes like r to the l plus 1, or a uEl that goes like 1 over r to the l. This Is far too singular. For l equals 0, we argued that the wave function should go like a constant. I'm sorry, cannot go like a constant. Must vanish. This is not possible. It's not a solution. It must vanish. For l equals 0, uE0 goes like r and vanishes. So that's consistent, and this is good. For l equals 0, this would be like a constant as well and would be fine. But for l equals 1 already, this is 1 over r, and this is not normalizable. So this time this is not normalizable for l greater or equal than one. So this is the answer [INAUDIBLE] this assumption, which is a very reasonable assumption. But if you don't have that you have to beware. OK, this is our condition for u there. And so uEl goes like this as r goes to 0. It would be the whole answer. So f, if you care about f still, which is what appears here, goes like u divided by r. So fEl goes like cr to the l. And when l is equal to 0, f behaves like a constant. u vanishes for l equal to 0, but f goes like a constant, which means that if you take 0 orbital angular momentum, you may have some probability of finding the particle at the origin, because this f behaves like a constant for l equals 0. On the other hand, for any higher l, f will also vanish at the origin. And that is intuitively said that the centrifugal barrier prevents the particle from reaching the origin. There's a barrier, a potential barrier. This potential is 1 over r squared. Doesn't let you go to close to the origin. But that potential disappears for l equals 0, and therefore the particle can reach the origin. But only for l equals 0 it can reach the origin. OK, one more thing. Behavior near infinity is of interest as well. So what happens for r goes to infinity? Well, for r goes to infinity, you also have to be a little careful what you assume. I wish I could tell you it's always like this, but it's not. It's rich in all kinds of problems. So there's two cases where there's an analysis that is simple. Suppose v of r is equal to 0 for r greater than some r0. Or r times v of f goes to 0 as r goes to infinity. Two possibilities. The potential is plane 0 after some distance. Or the potential multiplied by r goes to 0 as r goes to infinity. And you would say, look, you've missed the most important case. The hydrogen atom, the potential is 1 over r. r times v of r doesn't go to 0. And indeed, what I'm going to write here doesn't quite apply to the wave functions of the hydrogen atom. They're a little unusual. The potential of the hydrogen atom is felt quite far away. So never the less, if you have those conditions, we can ignore the potential as we go far away. And we'll consider the following situation. Look that the centrifugal barrier satisfies this as well. So the full effective potential satisfies. If v of r satisfies that, r times 1 over r squared of effective potential also satisfies that. So we can ignore all the potential, and we're left ignore the effective. And therefore we're left with minus h squared over 2m d second uEl dr squared is equal to EuEl. And that's a very trivial equation. Yes, Matt? AUDIENCE: When you say v of r goes to 0 for r greater than [INAUDIBLE] 0. Are you effectively [INAUDIBLE] the potential? PROFESSOR: Right, there may be some potentials like this. A potential that is like that. An attractive potential, and it vanishes after some distance. Or a repulsive potential that vanishes after some distance. AUDIENCE: But say the potential was a [INAUDIBLE] potential. Are you just approximating it to 0 after it's [INAUDIBLE]? PROFESSOR: Well, if I'm in the [INAUDIBLE] potential, unfortunately I'm neither here nor here, so this doesn't apply. So the [INAUDIBLE] potential is an exception. The solutions are a little more-- AUDIENCE: The conditions you're saying. [INAUDIBLE]. PROFESSOR: So these are conditions that allow me to say something. If they're not satisfied, I sort of have to analyze them case by case. That's the price we have to pay. It's a little more complicated than you would think naively. Now here, it's interesting to consider two possibilities. The case when E is less than 0, or the case when E is greater than 0. So scattering solutions or bound state solutions. For these ones, if the energy is less than 0 and there's no potential, you're in the forbidden zone far away, so you must have a decaying exponential. El goes like exponential of minus square root of 2m E over h squared r. That solves that equation. You see, the solution of these things are either exponential decays or exponential growths and oscillatory solutions, sines and cosines, or E to the i things. So here we have a decay, because with energy less than 0, the potential is 0. So you're in a forbidden region, so you must decay like that. In this hydrogen atom what happens is that there's a power of r multiplying here. Like r to the n, or r to the k or something like that. If E is less than 0, you have uE equal exponential of plus minus ikr, where k is square root of 2m E over h squared. And those, again, solve that equation. And they are sort of wave solutions far away. Now with this information, the behavior of the u's near the origin, the behavior of the u's far away, you can then make qualitative plots of how solutions would look at the origin. They grow up like r to the l. Then it's a one dimensional potential, so they oscillate maybe, but then decay exponentially. And the kind of thing you used to do in 804 of plotting how things look, it's feasible at this stage. So it's about time to do examples. I have three examples. Given time, maybe I'll get to two. That's OK. The last example is kind of the cutest, but maybe it's OK to leave it for Monday. So are there questions about this before we begin our examples? Andrew? AUDIENCE: What is consumption of [INAUDIBLE] [? barrier ?] dominates. But why is that a reasonable assumptions? PROFESSOR: Well, potentials that are just too singular at the origin are not common. Just doesn't happen. So mathematically you could try them, but I actually don't know of useful examples if a potential is very singular at the origin. AUDIENCE: [INAUDIBLE] in the potential [INAUDIBLE] the centrifugal barrier. That [INAUDIBLE]. PROFESSOR: Right. An effective potential, the potential doesn't blow up-- your potential doesn't blow up more than 1 over r squared or something like that. So we'll just take it like that. OK, our first example is the free particle. You would say come on. That's ridiculous. Too simple. But it's fairly non-trivial in spherical coordinates. And you say, well, so what. Free particles, you say what the momentum is. You know the energy. How do you label the states? You label them by three momenta. Or energy and direction. So momentum eigenstates, for example But in spherical coordinates, these will not be momentum eigenstates, and these are interesting because they allow us to solve for more complicated problems, in fact. And they allow you to understand scattering out of central potential. So these are actually pretty important. You can label these things by three numbers. p1, p2, p3. Or energy and theta and phi, the directions of the momenta. What we're going to label them are by energy l and m. So you might say how do we compare all these infinities, but it somehow works out. There's the same number of states really in either way. So what do we have? It's a potential that v is equal to 0. So let's write the differential equation. v is equal to 0. But not v effective. So you have minus h squared over 2m d second uEl dr squared plus h squared over 2m l times l plus 1 over r squared uEl equal EuEl. This is actually quite interesting. As you will see, it's a bit puzzling the first time. Well, let's cancel this h squared over 2m, because they're kind of annoying. So we'll put d second uEl over dr squared with a minus-- I'll keep that minus-- plus l times l plus 1 over r squared uEl. And here I'll put k squared times uEl. And k squared is the same k as before. And E is positive because you have a free particle. E is positive. And k squared is given by this, 2m E over h squared. So this is the equation we have to solve. And it's kind of interesting, because on the one hand, there is an energy on the right hand side. And then you would say, look, it looks like this just typical one dimensional Schrodinger equation. Therefore that energy probably is quantized because it shows in the right hand side. Why wouldn't it be quantized if it just shows this way? On the other hand, it shouldn't be quantized. So what is it about this differential equation that shows that the energy never gets quantized? Well, the fact is that the energy in some sense doesn't show up in this differential equation. You think it's here, but it's not really there. What does that mean? It actually means that you can define a new variable rho equal kr, scale r. And basically chain rule or your intuition, this k goes down here. k squared r squared k squared r squared, it's all rho. So chain rule or changing variables will turn this equation into a minus d second uEl d rho squared plus l times l plus 1 rho squared is equal to-- times uEl-- is equal to uEl here. And the energy has disappeared from the equation by rescaling, a trivial rescaling of coordinates. That doesn't mean that the energy is not there. It is there, because you will find solutions that depend on rho, and then you will put rho equal kr and the energies there. But there's no quantization of energy, because the energy doesn't show in this equation anymore. It's kind of a neat thing, or rather conceptually interesting thing that energy is not there anymore. And then you look at this differential equation, and you realize that it's a nasty one. So this equation is quite easy without this. It's a power solution. It's quite easy without this, it's exponentials are this. But whenever you have a differential equation that has two derivatives, a term with 1 over x squared times the function, and a term with 1 times the function, you're in Bessel territory. All these functions have Bessel things. And then you have another term like 1 over x d dx. That is not a problem, but the presence of these two things, one with 1 over x squared and one with this, complicates this equation. So Bessel, without this, would be exponential solution without this would be powers. In the end, the fact is that this is spherical Bessel, and it's a little complicated. Not terribly complicated. The solutions are spherical Bessel functions, which are not all that bad. And let me say what they are. So what are the solutions to this thing? In fact, the solutions that are easier to find is that the uEl's are r times the Bessel function jl is called spherical Bessel functions. So it's not capital j that people use for the normal Bessel, but lower case l. Of kr. As you know, you solve this, and the solutions for this would be of the form rho jl for rho. But rho is kr, so we don't care about the constant, because this is a homogeneous linear equation. So some number here. You could put a constant if you wish. But that's the solution. Therefore your complete solutions is like the psi's of Elm would be u divided by r, which is jl of kr times Ylm's of theta phi. These are the complete solutions. This is a second order differential equation. Therefore it has to have two solutions. And this is what is called a regular solution at the origin. The Bessel functions come in j and n type. And the n type is singular at the origins, so we won't care about it. So what do we get from here? Well, some behavior that is well known. Rho jl of rho behaves like rho to the l plus 2 over 2l plus 1 double factorial as rho goes to 0. So that's a fact about these Bessel functions. They behave that way, which is good, because rho jl behaves like that, so u behaves like r to the l plus 1, which is what we derived a little time ago. So this behavior of the Bessel function is indeed consistent with our solution. Moreover, there's another behavior that is interesting. This Bessel function, by the time it's written like that, when you go far off to infinity jl of rho, it behaves like sine of rho minus l pi over 2. This is as rho goes to infinity. So as rho goes to infinity, this is behaving like a trigonometric function. It's consistent with this, because rho-- this is rho jl is what we call u essentially. So u behaves like this with rho equal kr. And that's consistent. This superposition of a sine and a cosine. But it's kind of interesting though that this l pi over 2 shows up here. You see the fact that this function has to vanish at the origin. It vanishes at the origin and begins to vary. And by the time you go far away, you contract. And the way it behaves is this way. The face is determined. So that actually gives a lot of opportunity to physicists because the free particle-- so for the free particle, uEl behaves like sine of kr minus l pi over 2 as r goes to infinity. So from that people have asked the following question. What if you have a potential that, for example for simplicity, a potential that is localized. Well, if this potential is localized, the solution far away is supposed to be a superposition of sines and cosines. So if there is no potential, the solution is supposed to be this. Now another superposition of sines and cosines, at the end of the day, can always be written as some sine of this thing plus a change in this phase So in general, uEl will go like sine of kr minus l pi over 2 plus a shift, a phase shift, delta l that can depend on the energy. So if you haven't tried to find the radial solutions of a problem with some potential, if the potential is 0, there's no such term. But if the potential is here, it will have an effect and will give you a phase shift. So if you're doing particle scattering experiments, you're sending waves from far away and you just see how the wave behaves far away, you do have measurement information on this phase shift. And from this phase shift, you can learn something about the potential. So this is how this problem of free particle suddenly becomes very important and very interesting. For example, as a way through the behavior at infinity learning something about the potential. For example, if the potential is attractive, it pulls the wave function in and produces some sign of delta that the corresponds to a positive delta. If the potential is repulsive, it pushes the wave function out, repels it and produces a delta that is negative. You can track those signs thinking carefully. But the potentials will teach you something about delta. The other case that this is interesting-- I will just introduce it and stop, because we might as well stop-- is a very important case. The square well. Well, we've studied in one dimension the infinite square well. That's one potential that you now how to solve, and sines and cosines is very easy. Now imagine a spherical square well, which is some sort of cavity in which a particle is free to move here, but the potential becomes infinite at the boundary. It's a hollow sphere, so the potential v of r is equal to 0 for r less than a. And it's infinity for r greater than a. So it's like a bag, a balloon with solid walls impossible to penetrate. So this is the most symmetric simple potential you could imagine in the world. And we're going to solve it. How can we solve this? Well, we did 2/3 of the work already in solving it. Why? Because inside here the potential is 0, so the particle is free. So inside here the solutions are of the form uEl go like rjl of kr. And the only thing you will need is that they vanish at the end. So you will fix this by demanding that ka is a number z such-- well, the jl of ka will be 0. So that the wave function vanishes at this point where the potential becomes infinite. So you've solved most of the problem. And we'll discuss it in detail, because it's an important one. But this is the most symmetric potential, you may think. This potential is very symmetric, very pretty, but nothing to write home about. If you tried to look-- and we're going to calculate this diagram. You would say well it's so symmetric that something pretty is going to happen here. Nothing happens. These states will show up. And these ones will show up, and no state ever will match another one. There's no pattern, or rhyme, or reason for it. On the other hand, if you would have taken a potential v of r of the form beta r squared, that potential will exhibit enormous amounts of degeneracies all over. And we will have to understand why that happens. So we'll see you next Monday. Enjoy your break. Homework will only happen late after Thanksgiving. And just have a great time. Thank you for coming today, and will see you soon. [APPLAUSE]

MIT_805_Quantum_Physics_II_Fall_2013

25_Addition_of_Angular_Momentum_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So let's get started. So I'm going to lecture today, Professor Zweback's away. And I just wanted to say a couple of things, just in case you haven't noticed. We posted the solutions for P-set 11. And then also later in the week, we'll post the solutions for the extra problems that came along with P-set 11, so you can look at those. And also, there's two past exams with solutions also on the website now. So you can start going through those. And also, there's a formula sheet there. And if you've got suggestions for things that you think should be on there that aren't, let us know and they probably can be put on there. So I want to turn back to what we were doing at the end of last lecture, which was talking about the spin-orbit coupling. And so this is a contribution to our Hamiltonian that looks like spin of the electron dotted into the angular momentum that the electron has around the proton in the hydrogen atom. And so because of this term we had to change the complete set of commuting observables that we wanted to talk about. So we have now this full Hamiltonian that includes this piece that has the Se dot L term in it. We have L squared, we have the spin squared. But because of this piece, Lz, which was previously one of the quantum numbers we used to classify things by, that doesn't commute with this term, right? So here we have to throw that one away. Similarly, we have this throw away the z component of the electron spin. That doesn't commute with this either. And what we replace those by is actually the J squared and the Z component of J. So J is the vector sum of the angular momentum and the spin of the electron. And this is very interesting. This term does something interesting. So if we look at-- let me go up here. If we remember the hydrogen states when we don't have this term, there's a state that has n equals 2 and l equals 1. And you can think of that as three states. And then we've got to tensor that with the spin, so the spin of the electron could be spin up or spin down. So there's a spin a half, so this is two states. And so you've got a total of six states you're going to talk about. And now what we have to do is classify these according to the quantum numbers that are actually preserved by the system. So we can't use Lz or Sz. We have to use J squared and Jz. So we've got a J equals 3/2 multiplet-- and that's four states-- plus a J equals 1/2. And you can see the number of states works out. We've got 3 times 2 is equal to 4 plus 2. And so this L dot S term takes these original six states, which without this interaction degenerate, and it splits them into the four states up here, and then two states down here. The J equals 1/2, J equals 3/2. And we also worked out the splittings. If I do this, this is plus h bar squared over 2. And this is minus h bar squared. So this gives you a splitting. Now this is not the only thing that happens in hydrogen, because you probably all know that the proton itself has spin. The proton has a spin 1/2 particle, just like the electron. It's even more complicated because it's a composite object. But that leads to additional splittings in hydrogen. And so these ones, this one here is called the defined structure. Or we can also talk about the type hyperfine structure. So this is going to be a small effect on top of this one. So we have the proton that's spin 1/2, we have the electron spin a half, and then we have the relative orbital angular momentum. And so the total angular momentum, which is J, which is going to be the sum of L plus the spin of the electron plus the spin of the proton, this is conserved. And the thing we were talking about here is actually not conserved. So once you worry about the spin of the proton you've got to look at the total angular momentum. And that's what will be conserved. And so our complete set of commuting observables is going to be a four Hamiltonian, which we'll get to in a moment, L squared the spin squareds of the proton and the electron, and then J squared, and finally Jz is the things that we're going to end up classifying states by. So we originally thought about these two here, and did a coupling between those. It's pretty natural to assume that there maybe couplings between the angular momentum and the spin of the proton, which there are. But also there's going to be a coupling between the spins of the electron and the proton. And that's the one we're going to talk about at the moment. The other one is there but we won't go over it in any detail. So the proton and the electron both spin 1/2 particles, and they both have magnetic dipole moments, which are proportional to their spin. And so it's really a coupling between these moments that tells us what the effect of this interaction is going to be. So we have the mu of the electron is equal to e over me-- minus me-- times the spin of the electron. And mu of the proton is, let me just write it as gp. And gp happens to have the value of about 5.6. And this is actually kind of interesting. So if you look at the formula up here, really I could have written this as a g over 2, with g being 2. So for the electron, the g factor is very close to 2. This is because the electron is essentially a fundamental particle, with no substructure. But the proton, which is made up of quarks and gluons flying around inside some region, has a lot of structure. And so this is really indicative of it being a composite particle. Because a fundamental spin 1/2 particle should have this g being 2. So we've got these two dipole moments. And one way to think about this is you've got this dipole of the proton. We're going to think about the proton having a little dipole charge-- sorry, dipole magnetic moment-- and this produces a magnetic field. And the electron is sitting in that magnetic field. And its spin can couple to the field. So we're going to have a Hamiltonian, a hyperfine Hamiltonian, that looks like minus mu of the electron dotted into a magnetic field produced by the proton, which is going to depend on r, on where the electron is. And you can simplify this as just e over m spin of the electron dotted into this B of the proton. So we need to know what this dipole field is. And for that you really have to go back to electromagnetism. And you've probably seen this before. But let me just write it down, and we won't derive it here. But let's go down here. This has a kind of complicated form. So there's this piece, and then there's another piece that looks like 8 pi over 3c squared mu p times the delta function at the origin. And so you think about the dipole field arising from a spinning charge distribution here. So we've got a magnetic dipole moment pointing up. This produces a field like this, a dipole type field going this way. So this is our B. And then inside here, you should really think of taking the limit as this thing goes to 0 size. And so in order to get the right field in the middle, you need to have this term here. And so if you want to see this being derived you can look in Griffiths. That does the derivation of this. But we will skip that. So we've got the field. And now we can put it into our Hamiltonian. So it's mu e. So I could replace my mu's with the spins. So I get some factor out the front that looks like ge squared over 2 Me Mp c squared. And then I get 1 over r cubed plus-- So just plugging those in we get this Hamiltonian here. And let me just simplify a little bit. Let's just call this thing q. And so this Hamiltonian is going to be given by q. And I can write it as the i-th component of the electron spin, the j-th component of the proton spin, dotted into r hat i hat j minus-- So just taking the common factors of the spins components out the front. So if we've got this, we want to ask what it's going to do to the energy of the ground state of hydrogen. So we're going to take matrix elements of this between the hydrogen wave functions. So does anyone have questions so far? Yes. AUDIENCE: Can you use r as a [INAUDIBLE]? PROFESSOR: Right right. So these are unit vectors in the r direction. And this r is the length of the vector, r vector. The usual thing. So what we're going to try and evaluate is the expectation value. So we're going to do this. Because going back to the start of last lecture, this is going to be a small correction. And so we can work out its contribution to the energy by using the original wave functions, but just calculating its matrix elements. So we're going to calculate-- and let me just give this a name. This can be-- So this is q, and the ground state has no angular dependents. So in fact, for the ground state, I can just write this is a function of r squared. For overtly excited states I can't do that. But for the ground state that works. And then we have, so we've got the wave function. And then in between them we have to put this stuff over here. So let's put the there. So one of these terms is very easy to evaluate. With this [INAUDIBLE] function I just get the wave function at the origin. And the second term is actually also relatively easy to evaluate. Who can tell me what this integral over all three directions of just one direction? What's that? AUDIENCE: 0. PROFESSOR: 0. And you can argue that by just asking, well what can it be? It's got to carry an index, because there's an index on this side of the equation. And there's no other vectors around in this problem. So the only thing it can be is 0. So if I do integral d3r of ri rj, what can that be? Sorry? AUDIENCE: 1. PROFESSOR: 1? No. So it's got two indices. So the thing on this side of the equation also has to have two indices. AUDIENCE: Delta ij? PROFESSOR: Delta ij, very good. So the only thing that can carry two indices is delta ij. And then there might be some number here. And it actually turns out that you can do an even more complicated integral. We can look at integral d3r of ri rj sum f of r squared. And that is also just some number, which depends on what f is, times delta ij. And if you go along these lines and actually look at this, the difference between the integral of this piece and the integral of this piece is actually a factor of 1/3. And so this actually integrates to 0. So when I integrate over this one, I get something times delta ij. And that something is actually 1/3. And so this term and this term cancel in the integral. And so you just get the delta function contributions. So you get some number times Sei S delta ij. So it becomes Se dotted into Sp. 8 pi over 3. And then it's psi 100 at the origin. So this we know, we've already computed these radial wave functions, and saw at the origin this one is actually 1 over pi times the Bohr constant. And if you plug-in what Q is, and what the Bohr constant is, you can just find out that this whole thing ends up looking like 4/3 this gp and this we can call delta e hyperfine. So you end up with a very simple thing. And it's just proportional to the dot product of the two spins. So you've seen, essentially, you saw this term in your homework. So we just assume that this thing here came out of nowhere and was just some number times Se dot Sp, and this was a contribution to your Hamiltonian. But now we actually know where that comes from. And interestingly, this thing here, this whole thing, it's still an operator because it's got these spins in it. And that's-- put a star next to that because it's important. So now we need to ask, well what are the real states of hydrogen so they're where we've got two spins? The spin of the proton, they could be aligned, or they could be anti-aligned. Oh, sorry. We have a question up there. AUDIENCE: Is that np over np, or mu u? PROFESSOR: No, me, mass. Mass of the electron over mass of the proton. So you have to remember that the spins of the proton and the electron to can parallel or they can be anti-parallel. Or they can be both down. And so we have to go back and work out-- we have to realize that because of these terms the z components of those spins are not good quantum numbers. The only z component that appears in our list is Jz, so the total z component of angular momentum. So we need to go back and do what you-- you probably have done this to the P-set. But let's just do it very quickly. We'll take those two spin 1/2 things and so let's make this J1 and this is J2. And we're going to have J. So if I've got these two spins I can make various things. I can write down-- And if I've done this than I should also write that the m, the m quantum number that goes with the J quantum number is going to be equal to m1 plus m2. So this state here, because both of the spins are pointing up, this is an m equals 1 state. And then we can also have something like 1/2, 1/2. You could have these two states. So they both have m equals 0. And then there's an m equals minus 1, which is 1/2 this guy. So since this has m equals 1, and 1/2 cross 1/2 is going to give us a spin 0 multiplet and a spin 1 multiplet, because it's got m equals 1, this has to be J equals 1 as well. And this one has to be J equals 1. But the two states in the middle, we don't know what those are. There's going to be a J equals 1, m equals 0 state, which is going to be some linear combination of these two. So let's just go over here. We don't need any of this. And we need to work out what the linear combination is. So something to remember is this. The J plus or minus acting on Jm is this funny square root thing. So these are the raising and lowering operators. They take us from one state to the one with a different m value. And so we can use that to start with. We could basically take J minus on our state. And according to this formula, this will give us the square root of 1 times 2 minus m is 1. And this should be 0, right? I think I've got this sign up the wrong way. I think this is minus plus. No, sorry, that's right. It should be-- I'm doing the J minus so I have 1 times 1 minus-- yeah, right, so it's this. So this is square root 2 times Jm minus 1. But we also know that J minus a is equal to J1 minus plus J2 minus because J is just the vector sum of the two J's. So we can ask what J minus on the state is. But this state we can write in terms of the tensor product. So this is equal to J1 minus 1. If we use this formula for lowering something with spin 1/2 we get 1/2 times 3/2 minus 1/2 times minus 1/2, which is actually 1 under that square root. And so this actually equals 1/2 minus 1/2 tensor 1/2, 1/2. So these two things are equal. And so that tells us, in fact, that the 1, 0 state, which is what's over here-- oh, sorry. Oh, why did I do that? This should be J equals 1 and M equals what it was, minus 1, this. So the 1, 0 state, if we bring that 1 over root 2 on the other side is this combination. So it's one linear combination of those two pieces. We also want the other one. So we've got three of our states. The fourth state is then, of course, the other linear combination of the two states over there. And so that's going to be our J equals 0, M equals 0 state. So this state is going to be orthogonal to the one we've just written here. And so this is pretty easy to work out. Since there's only two terms, all we do is change the sign of one of them and it becomes orthogonal, because these states here are normalized. So this becomes 1/2, 1/2 tensor-- let me just write it in the same way that-- 1/2 minus 1/2 minus-- And so our four states, so we can condense our notation so we can say that this state we can just label as this. And we can just label as a down arrow. And then something like we can label as just up down, just to make everything compact. You just have to remember that this is referring to the first spin, this is always referring to the second spin. And so then we can write our multiplets. J equals 1 has three states. It has up, up. It has up, down plus down, up. And it has down, down. So those are our three states that have J equals 1. And then we have J equals 0, which just has 1 over square root 2 up, down minus down, up. And so the two spins in our hydrogen atom, the spin of the proton and the electron, can combine to be a J equals 1 or a J equals 0 system. And since we're talking about the ground state of hydrogen, it has 0 angular momentum. And so I'm really just talking about J total, here. So if we now have this Hamiltonian, which is still an operator in spin-- we've dealt with the spacial dependence of the wave functions, but it's still an operator in spin-- we can now evaluate this. So we can take its expectation value in either the J equals 1 multiplet or the J equals 0 multiplet. So let's just write it out again. So we have h hyperfine 1, 0, 0. This is equal to some delta E HF spin of the electron dotted into the spin of the proton. We can rewrite this using something we did last time. We can write this as J squared minus Se squared minus Sp squared, with the 1/2 out the front. So here, because l equals 0 because we're in the ground state, then J equals Se plus Sp. And so J squared is going to give us Se squared, Sp squared, and then the dot product. So great. So what is this, the spin squared of the electron? What's the eigenvalue of J squared, always? J, J plus 1 times h bar squared. And what is J for the electron? 1/2. And what about the proton? AUDIENCE: 1/2 as well. PROFESSOR: 1/2. So we've got 1/2 times 1/2 plus 1, so 3/2. So this gives us minus 3/4. This gives us minus 3/4. So this just looks like delta e HF over 2 J squared minus 3/2 h bar squared. OK? Anyone lost doing that? Or is that OK? AUDIENCE: [INAUDIBLE] PROFESSOR: Yep. AUDIENCE: So, when you define delta e [INAUDIBLE] over there, that exudes energy? PROFESSOR: Oh, you're right. You're very right. I've messed up. I've-- AUDIENCE: [INAUDIBLE] PROFESSOR: Let me see. Yeah, really I have an h bar squared here. I think I should have had an h bar squared over there as well. Yeah. That should be over h bar squared here. Thank you. OK so-- AUDIENCE: [INAUDIBLE] PROFESSOR: Sorry? AUDIENCE: When does it get [INAUDIBLE] PROFESSOR: That was just in the algebra going from this expression, writing it in terms of alpha, things like that. So it's just some algebra. OK, anything else? No? Good. OK so now we can easily evaluate these things. We can now take J equals 1 and some M-- and this is for M equals all three states here-- and just evaluate this. And all that means is we have this J squared operator acting on this state here. And this gives us h bar squared 1 times 1 plus 1, or 2h bar squared. So this will give us delta e hyperfine over 2. And then we've got, let's pull the-- sorry there's still and h bar squared here, and an h bar squared there. But now we can evaluate. The h bar squared here cancels that one, and we get a 1 times a 1 plus 1 minus 3/2. And that's just one quarter, which is-- And similarly we can take the J equals 0 state, and this one gives us delta e hyperfine over 2. And then it's 0 time 1 minus 3/2. And so that equals minus 3/4 EHF. So what we're doing is evaluating these in these particular J states. And now we end up with something that's just a number. It's no longer an operator. It's an energy that we can measure. Yeah? AUDIENCE: So, this expectation value h hyperfine 1, 0, 0, is still an operator. Is that because we only took the expectation value over the angular [INAUDIBLE] PROFESSOR: We took over the spacial wave function. We did the r integral, right? But we didn't-- AUDIENCE: [INAUDIBLE] PROFESSOR: Right, right. Yeah. So this is actually a really important system. So let's just draw the energy level diagram here. And here we have four states. We have the spin 1/2 times spin 1/2. So 2 times 2 states. So we get a triplet and a singlet. And what this hyperfine splitting does is take those four states and split the triplet up here, and split the singlet down here. And this gap we can see is-- oops, so this should be a delta HF. So this gap is delta E HF. So it's 1/4 and minus 3/4. And if you plug numbers in, delta E HF, this actually ends up being 5.9 times 10 to the minus 6 electron volts, which is a pretty small scale. So you should be comparing that to the binding energy of the ground state of hydrogen of 13.6 electron volts. So this is a very small effect. And you can really think about the relative size. So the Bohr energy, so that 13.6, formally this goes like, alpha squared times Me c squared. Then last time we talked about the S coupling, so the spin orbit, or fine structure. And so this one we found went like alpha to the fourth Me C squared. So smaller than the binding energy by a factor of 1 over 137 squared, or about 20,000. And then this one that we're talking about here, the hyperfine, this, if you look over here, this is going like alpha to the fourth Me C squared times an additional factor of Me over Mp. And the mass of the proton is about 2,000 times the mass of the electron. And so this again is-- oh, sorry. This is alpha to the fourth. So this is suppressed by about another factor of 2,000. You can go further. There are further corrections to this in something called the Lamb shift, which we won't say anything else about. This goes like alpha to the fifth Me C squared. And there's a whole host of higher order corrections. People actually calculate these energy levels to very high precision. But we won't do any more. So this transition here is actually astrophysically extremely important. So if we think about something sitting in the state here, it can decay down to the ground state by emitting a photon. So we can decay from J equals 1 to J equals 0 by a photon. And that photon will have a wavelength that corresponds exactly to this energy difference. And so that wavelength is going to be, we can write it as c over the frequency, or hc-- oh, hc not h bar c-- hc over this delta e hyperfine. If you plug numbers into this you find out that this is approximately 21.1 centimeters. And the frequency is 1,420 megahertz. And so right in the middle-- well, at the end-- of the FM band in radio. So theses are radio waves. So the size of this wavelength is firstly important because it's large compared to the size of dust in the universe. So dust is little stuff. So this is essentially goes straight through dust. So these photons will go straight through dust. The other important thing is that you probably know that there's a cosmic microwave background radiation in the universe, that's essentially very close to constant everywhere. And so we have, essentially, we have a temperature of 2.7 Kelvin. That corresponds to photons with an energy kT, which is about 0.2 times 10 to the minus 3 electron volts. So milli electron volts. But if you compare this number to what we have here, this cosmic background microwave radiation can excite hydrogen from here up to here. There's enough energy for one of those photons to come along, knock the hydrogen atom, and excite it up to here. And then it will decay and will emit this beautiful 21 centimeter line, which will go through all the dust. And so we can actually see the universe in this 21 centimeter line. Even more remarkable is, we can't calculate this at the moment, but you can show that the lifetime for this transition to happen is about 10 to the 7 years. So we can never measure that in a lab. But because these hydrogen atoms can be wandering around the universe, not interacting for that long, then they can emit. And so we can see that. This was first observed in about 1951, and is the first way that we actually saw that the galaxy had spiral shaped arms. So it's pretty important. And another nice thing about this is if you think about another galaxy-- so let me just draw another galaxy, a spiral galaxy somewhere else, like this. Let's have us over here looking at this galaxy from side on. This galaxy is rotating. So this one's moving this way, this one to moving this way. There's hydrogen over here and over here. And so we get these photons coming over here, and photons coming to us from there. But what's going to be different about these? AUDIENCE: [INAUDIBLE] PROFESSOR: Their rate shifted, right? The Doppler shifted. So this is my 21 centimeter photon. But they get Doppler shifted. And so we can measure the difference in the frequencies of those. What does that tell us? AUDIENCE: [INAUDIBLE] PROFESSOR: How fast this galaxy is spinning, right? And so one very interesting thing you find from that is if you look at the galaxy and count how many stars are in it, and essentially work out how massive that galaxy is, the speed of rotation here is actually-- that you measure from these hydrogen lines-- is that it's actually faster than the escape velocity of the matter. And so if all that was there was the visible matter, then the thing would just fall apart. And so this actually tells you that there's dark matter that doesn't interact with visible light, that's kind of all over here. So that's kind of a pretty interesting thing. So that, I think, yeah. So any questions about that? We're going to move on to another topic. Yep? AUDIENCE: You said earlier [INAUDIBLE] it's 10 to the 7 years. Does that mean it takes an average of 10 to the 7 years for the cosmic microwave background energy to shift back [INAUDIBLE] PROFESSOR: No, it's really, if I just took hydrogen in this state, and took a sample of it, that's how long it would take for half of it to have gone and made the decay. So it can happen much faster. And there's a lot of it in the universe. So there's many more than 10 to the 7 atoms of hydrogen in the universe. So we see more than one of these things every year. So if you were just looking at one of them you would have to wait a long time. AUDIENCE: But the thing about the cosmic microwave background is to go from [INAUDIBLE] from 0 to 0 up to get [INAUDIBLE] PROFESSOR: Right so I mean the energy is large compared to that. So it will typically knock you up into an even higher state. And then you will kind of decay down. But then this last decay is-- because this lifetime is very long, the width of this line is also very, very narrow. So now let's talk more about adding angular momenta. Oh, maybe I should have left that up. Too late. So we're going to do this in a more general sense. So we're going to take J1, some spin J1 that has states J1, M1, with M1 equals minus J up to J. And so we're sort of talking about something like the electron in the hydrogen atom. And so that's not in any particular orbital angular momentum. So we can talk about that the Hilbert space that this thing lives in. So we can think about particle 1 with angular momentum J1. And this is basically spanned by the states J1, M1 of these [INAUDIBLE]. We can take another system with another J2, and this is going to have states J2, M2, with M2-- that should be J1's there. And that would similarly talk about some Hilbert space of some fixed angular momentum. If we want to talk about the electron in a hydrogen atom, where it doesn't have a fixed angular momentum, what we really want to talk about is the Hilbert space H1, which is the sum over J1 of these Hilbert spaces. And so this is talking about-- this Hilbert space contains every state the electron can have in a hydrogen atom. It can have all the different angular momenta. And similarly we could do that for J2. We can define J, which is J1 plus J2, as you might guess. And really this you should think of as J1 tensor the identity plus the identity tensor J2, where this one is acting on things in this Hilbert space, and the 1 here is acting on things in this Hilbert space. And similarly there's an H2, J2 that goes along with these guys. And so this operator, this big J, is something that acts on vectors in things in this tensor product space. Actually I should label this with a J1. It also acts on things in the full space, but we can talk just about that one. So now we might want to construct a basis for this space. And we conversely construct an uncoupled basis which is just take the basis elements of each of the spaces and multiply them. So we would have J1, J2, M1, M2. We'd have the states here. And if we just ask what our various-- J1 just gives us h bar squared J, J plus 1. And this one gives us h bar M1 h bar squared times our state. And so we can think about all of these. And this is what we label our state with. And that's because these form a complete set of commuting observables. And we'll just call this the A set. We can also talk about our operator J and use that to define our basis. And let's just be a little explicit about what J squared is going to be. So this is J1 tensor identity plus 1 tensor J2. And the same thing here. If you expand this out you get J1 squared tensor identity plus 1 tensor J2 squared plus the dot product, which we can write as the sum of J1k tensor J2k. And because of this piece here, J squared doesn't commute with J1z, for example. So we can't add this operator to our list of operators over there. And similarly J2z J squared is not equal to 0. So if we want to talk about this operator we have to throw both of those away. But there is an operator total Jz that commutes with J squared. And it also commutes with J1 squared and J2 squared. And so we can have another complete set of commuting observables B that's equal to J1 squared, J2 squared, J squared, and Jz. And so if there are observables then the natural basis is to label them by the eigenvalues here. So we're going to have a J1, a J2, then a J and an M. And so this is the coupled basis. Now both of these bases are equally good. They both span the full space. They're both orthogonal, orthonormal. And so we can actually write one basis into in terms of the other one. And that's the generic problem that we are trying to do when we're trying to write what we did over here before, when we did spin 1/2 cross spin 1/2. We're trying to write those products states in terms of the coupled basis. Well, they're both orthonormal basis. So I can expand J1, J2. Well actually, maybe I'll say one more thing first. So being orthonormal means that, for example, sum over J1, J2-- This is 1, right? You can resolve the identity in terms of these states. And this is the identity on this Hilbert space. I can also think about the identity just on this smaller Hilbert space, where the J1 and J2 are fixed. And so I can actually write it's the identity operator. So because every state in this space has J1 equal to some fixed value and J2 equal to some fixed value, then an identity in that thing is just somewhere over the M's, because they're the only things there. So using this, because I know that the state J1, J2, Jm has some fixed value of J1 and J2, I can write this as a sum. I can use this form of the identity. So I've written my coupled basis in terms of the uncoupled basis here. And these are just coefficients. These are called Clebsch-Gordan coefficients. They're just numbers like square root 2 and things like this. And I tell you how to do this decomposition. So they have various properties. Firstly, sometimes you also see them written as C of J1J2J colon M1M2M and various other notations. So basically things with six indices are probably going to be these guys. So they have various properties. The first property is they vanish if M is not equal to M1 plus M2. And this is actually very easy to prove. So remember that Jz is just going to be J1z plus J2z. So as an operator I can write Jz minus J1z minus J2z. And what is that operator? It's just 0, right? This is equal to that. So this is 0. So I can put this 0 anywhere I want and I'll still get 0. So let's put this between-- so this is 0-- put it between J1, J2, Jm on this side. So a coupled state here. And on this side I'll put it between the uncoupled state, J1, J2, M1, M2. So this state is an eigenstate of Jz. And this state is an eigenstate of J1z and J2z. So I can act to the right with Jz, and act to the left with these J1z and J2z, and they have mission operators. And I know because this is 0, this whole thing is 0. So then act this one on these guys and these two back this way. And so you see that gives me h bar. And then I get this one acting on here gives me M. And J1 acting on here gives me M1. And if M is not equal to M1 plus M2, then this term isn't 0. But the whole thing is so that has to be 0. So that's QED. The second property is that-- So they only allow values of J fall in this range here. And each J occurs once. Now one way to think about this is to think of these things as vectors. So you have vector J1, and then from the point of this you can have vector J2. But it can go into an arbitrary direction. So it can go up here, or it can go like this. These are meant to be the same length. And I can come all the way down here. But I can could only sit on integer points. And so this is kind of J2. And so the length of this thing here would be the length of J1 plus the length of J2. So it would be this. And then the length of up to here would be this one. And then all of the other ones are in between. But you can also just look at the multiplicities of the different states. So if we look at the uncoupled basis-- so the first state, which was J equals J1, there are two J1 plus 1 states, because it can have all of the M values from J1 down to minus J1. And the other one can have two J2 plus 1 states. So that's the total number of states that I expect to have. So now let's assume this is correct and ask what the N coupled is. So this would be the sum over J equals mod J1 minus J2 up to J1 plus J2 of 2J plus 1. And let's assume that J1 is greater than or equal to J2, just to stop writing absolute values all the time. So we can write this as the difference of two sums, J equals 0 to J1 of-- J1 plus J2-- of 2J plus 1 minus the sum of J equals 0 to J equals J1 minus J2 minus 1 of 2J plus 1. And if you go through-- so this is just N,N plus 1 over 2 for each of these things. You end up with, well, you end up with this. You end up with the same thing. And so this is at least consistent that the number of states that we have is consistent with choosing this. One other thing we can do is look at the top state, and just see if that works. See if that has the right properties. So because we know that the J1, J2, J equals J1 plus J2, M equals J1 plus J2. So the maximal state, the only way we can make this is to take J1, J2, M1 equals J1, M2 equals J2. Our spins are completely aligned in the total up direction. Yeah? AUDIENCE: Sir, would you be able to write a little larger? PROFESSOR: Yes, sorry. OK, yeah. That's why I like a big chalk. But we've run out of big chalk, so I'll try. So we know, also, that J squared is equal to J1 squared plus J2 squared plus the dot product. We can write that out as J1 squared plus J2 squared plus 2J1z J2z plus J1 plus J2 minus plus J1 minus J2 plus. And then we can ask what does J squared on this state give? And this is J1, J2, J1 plus J2, J1 plus J2. So J squared, so we know what that should be. That should return J1 plus J2 times J1 plus J2 plus 1 times h bar squared, because J is the good quantum number. But let's let it act on this piece. So this equals J1 squared plus J2 squared plus 2J1z J2z plus J1 plus J2 minus plus J1 minus J2 plus acting on J1, J2, J1, J2. So this state here. So we know how that acts. So this one gives us-- everything gives us an h bar squared. This gives us J1 J1 plus 1, for this term, plus J2 J2 plus 1 for the second term. Each of these gives us the M quantum numbers. But that's J1 J2. So this is plus 2J1, J2. And now what does this one do? J1 plus on this state. AUDIENCE: Kills it. PROFESSOR: Kills it, right. Because it's trying to raise the M component of 1, and it's already maximal. And this one, J2 plus, also kills it. So you get plus 0 plus 0 times the state. So if you rearrange all of this you actually find you can write this as J1 plus J2, J1 plus J2 plus 1 times the state, which is what you want. So the J squared operator acting in the coupled basis gives-- well, acting in the uncoupled basis gives you what you expect in the coupled basis. So now I need a big blackboard. So let's do an example of multiplying two things. So let's write out a multiplet. So we're going to take J1, and we're going to have J1 bigger than J2 here. So we've got J1 J1, J1 J1 minus 1. And then somewhere down here I've got J1 and 2J2 minus J1. And then all the way down to J1 minus J1. So this has two J1 plus 1 states. And I'm going to tensor that was another multiplet, with my J2 multiplet, which is going to be smaller. So I'm going to have J2 J2. Oh, maybe I'll put one more in. And down here we've got J2 comma minus J2. And so here we have two J2 plus 1 states. And importantly, this left hand side has two J1 minus J2 more states than the right hand side. Just counting those states that's pretty obvious. So now let's start multiplying these things, and forming states of particular values of M, the total M. So if we say we want M equals J1 plus J2 what can we do? How can we make that? So we have to take the top state in each case. Because if I take this one and I take this M value, I can't get up to this, right? So there's only one way to make this. So I'm going to draw a diagram of this. We're going to have a one state there. The next M value, J1 plus J2 minus 1, how can I make that? So I can start with this state, and I will multiply it by this one, right? Or, what else can I do? AUDIENCE: Start with the second down on the left and tensor with the top? PROFESSOR: That's right. So I take those two. So there are two states. And those two states are just two linear combinations. So let me draw two dots here, I can form two states. Keep going-- minus 2, I get three states. And let me try and draw lines here to guide this stuff. OK, I'm not going to keep going. But at some point we get-- what's the largest number of states of a given M I can make going to be? Can anyone see that? AUDIENCE: 2J2 plus 1? PROFESSOR: 2J2 plus 2, right, because I've got 2J2 plus 1 states here. And I'm taking one of these plus one of these will give me-- so down here I'll have an M equals-- what is it-- J1 minus J2. And here I have 2J2 plus 1 states. And so let me kind of draw some of these states in. And then dot, dot, dot. And then over here we end up with this guy. So if I go down to the next one, how many states? So to form those states I was taking this top state with the bottom state here. That gives me J1 minus J2, right? Or I was taking the second state here with the second to bottom state here, and so forth. And then all the way up to here. Now if I then start shifting things down in this side, but leave exactly the same things over there, then I'll lower J by 1. And I'll keep doing it until I hit the bottom. And because there's this number of states more in the right hand side than the left hand side-- hang on, let me just write this one. OK, we might need to go onto the next board. So this keeps on going until I get to M equals-- I don't remember the number-- M equals J2 minus J1. And there are 2J plus 2 plus 1 states of this. And then once I do that, then I start having fewer and fewer states. Because I've gone basically moving out the bottom of this multiplet. And so here we have, this is 2 J1 minus J2 plus 1 rows. And then I start contracting. So the next one, M equals J2 minus J1 minus 1 has two J2 states. So then we can keep going. And this is meant to continue this diagram up here. So then we keep going down, down, down. And then we'd have M equals minus J1 plus 1. And how many states can I make that have that? So I need to take this one, and I could take the-- oh, where is it? Sorry, not this. This is not what I mean. Minus J1 minus J2 plus 1. That's more obvious, right? So there's two states. And so this picture has kind of-- this line starts coming in. And now I've got my two states here. I have the next one, I've got three states. And then finally, M equals minus J1 minus J2, I have one state. And so I get this. And so you actually-- oh, that was not very well drawn. So if you look how many states there are in this first column, how many is going to be there? So it goes from plus J1 plus J2 to minus J1 minus J2. So there's two J1 plus J2 plus 1 states there. And here there is two J1 plus J2 minus 1 plus 1 states in this guy. And so this is a J equals J1 plus J2. This one is J equals J1 plus J2 minus 1. And if you are careful you'd find that this one here, the last one here, this has this many states in it, two J minus 1 plus 1 states. So this is a J equals J1 minus J2. And to be completely correct, we put an absolute value in case J2 is bigger than J1. So this is our full multiplet structure of this system. So all of the states in this column will transform into each other under rotations, and things like this. And same for each column, they all form separate multiplets. So just some last things before we finish. So another property of Clebsch-Gordan coefficients we can choose them to be real. They satisfy a recursion relation but don't have a nice, closed form. I think this is in Griffiths. It gives you what this recursion relation is. I think it does, at least many books do. And also, they're tabulated in lots of places. So if you need to know the values, you can just go and look them up, rather than trying to calculate them all necessarily. And I think that's all we've got time for. So are there any questions about that? Any questions about anything else? OK, great. So we will see you on Wednesday for the last lecture.

MIT_805_Quantum_Physics_II_Fall_2013

14_Quantum_Dynamics_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. BARTON ZWIEBACH: [INAUDIBLE] of today's lecture is coherent states of the harmonic oscillator. So let me begin by telling you about some things we've learned in the last lecture, and here they are. We learned how to calculate the so-called Heisenberg operators. Remember, if you have a Schrodinger operator, you subject it to this transformation with a unitary operator. That creates time evolution and that gives you the Heisenberg operator. We learned things about Heisenberg expectation values. If the Hamiltonian is time independent, h, time independent, the formula is quite simple and gives you the Heisenberg operator at the later time. So we did this. We found, in fact, Heisenberg operators satisfy equations of motion. And we calculated the Heisenberg operators for the harmonic oscillator. That was our main achievement last time, a formula for the time development of the x and p operators in the Heisenberg picture. And that really contains all the information of the dynamics, as you will see today, when we will be using this stuff. Now, I suggested that you read-- and you may do it later. There's no need that you've done it for today-- the information on the time development of the creation and annihilation operators. You see, the a and a dagger are different versions of x and p, are linear combinations. So the a and the a dagger operators also can be further Schrodinger operators that have no time dependence. And suddenly, if you go to the Heisenberg picture, the creation and annihilation operators become time dependent operators. So that's in the notes. You can read about it. So we define the time dependent operator, a hat, to be the Heisenberg version of a hat. And you're supposed to do a calculation and try it or read it, and the answer is very nice, simply a phase dependence. The a is a at time equals 0, the Schrodinger 1 times e to the minus i omega t. Then a dagger is just what you would expect, the dagger of this, which the phase has an opposite sign and a becomes a dagger. Finally, if you substitute this a and a daggers in this formula. For example, you could say x Heisenberg is a Heisenberg plus a dagger Heisenberg. And you substitute those Heisenberg values there, you will obtain this. Same for the momentum. If you put Heisenberg, Heisenberg, Heisenberg, remember, if you have an equality of Schrodinger operators, it also holds when you put Heisenberg in every operator. And therefore, if you put the Heisenberg a, a, and use those values, you will recover this equation. So in a sense, these equations are equivalent to these ones. And that's basically our situation. This is what we've learned so far, and our goal today is to apply this to understand coherent states of the harmonic oscillator. Now, why do we want to understand coherent states of the harmonic oscillator? You want to understand coherent states because the energy eigenstates are extraordinarily quantum. The energy eigenstates of the harmonic oscillator don't look at all-- and you've seen the expectation value of the position. It's time independent. It just doesn't change. Expectation value of any operator in a stationary state is a constant. It just doesn't change. So you have any eigenstate, any energy eigenstate of the harmonic oscillator, you ask, what is the position of this particle doing? Nothing. What is the momentum of this particle doing? Nothing. So nevertheless, of course, it's an interesting state, but we want to construct quantum mechanical states that behave a little like the classical states we're accustomed to. And that's what coherent states do. We'll have an application of coherent states to light, photons, coherent photons. What are they? We'll see it later this week. So that's the reason we want to understand coherent states, because we want some states that in some ways behave classically, or close to classically. So they have many applications, these states, and you will see some of them in this lecture. I'm going to try to keep this blackboard there, untouched, so that we can refer to these equations. So our first step is considering translation operators. So let's consider the unitary translation operator. So translation operators. So this translation operator that I will write as T sub x0 will be defined to be the exponential of e to the minus i p hat x0 over h bar. You have seen such operators before. We've seen a lot of them in the homework. So first of all, why is it unitary? well, it's unitary because x0 is supposed to be a real number. p is Hermitian. Therefore, this with the i is anti-Hermitian, and an exponential of anti-Hermitian operator is unitary. Now, it has, actually, a very simple property. The multiplication of two of those operators is what? Well, you have an exponential, e to the minus ipx0, and an exponential followed, e to the minus ipy0. Now, if you're well trained in 805, you should get a little nervous for a second because you don't know, can I treat it easily? And then you relax and say, yes, these two operators, whatsoever the numbers here, this with another one with a y0 would commute. Therefore, they can be put together in the exponential, and this is T of x0 plus y0. No combo Baker-Hausdorff needed here. It's just straightforward. So what is Tx0 dagger? T x0 dagger, if you take the dagger, you change this i for a minus i, so it's exactly the same as changing the sign of x0. So this is T of minus x0. And by this identity, T of minus x0 with a T of x0 would be T of 0, which is the unit operator. So T of minus x0 is the inverse of T of x0, confirming that the operator is unitary. The inverse is the dagger. So I used here that this is the inverse because T minus x0 times T x0 is T of 0 is equal to 1. So I could mention here, T of 0 is equal to the unit operator. So these are our translation operators, but you don't get the intuition of what they do unless you compute a little more. And a little more than you should compute is this. What is T x0 dagger x T x0? And what is T x0 dagger p T x0? Now, why do we ask for these particular things? Why don't I ask, what is x hat multiplied by T x0? Why do I ask this? It is because an operator acting on an operator always does this. If you say an operator is acting on another operator, the first operator that is acting, you put it here with its inverse. It happens to be unitary, so you put the dagger, and you put the operator here. And this is the right thing to do. It has a simple answer and a simple interpretation, as we'll see now. So what is T, this commutator, supposed to be? Well, you can probably imagine what this is. You've calculated it in homework, so I will not do it again. This is x plus x0. So you get the operator, x, plus x0 times the unit operator. That was done before. And here, you get just p. Why? Because p hat is the only operator that exists in this translation thing, so p commutes with p. So these two operators commute and the T tagger hits the T, and it's equal to 1, so that's a simple thing. So why is this reasonable? It's because of the following situation. If you have a state, psi, you can ask, for example, what is the expectation value of x in the state psi? And if this state represents a particle that is sitting somewhere here, roughly, the expectation value of x is basically that vector that tells you where the particle is. So you could ask, then, what is the expectation value of x in the state T x0 psi? So you want to know, what does T x0 really do? Here, it seems to say something, takes the operator and displaces it, but that seems abstract. If you ask this question, this seems more physical. You had a state, you act with an operator, it's another state. How does it look? Well, this expectation value would be the expectation value of x on T x0 psi, and the bra would be psi T x0 dagger. So actually, that expectation value builds precisely this combination, and that's why it's meaningful. And since you know what this is, this is psi x plus x0 psi. This is equal to the expectation value of x in the original state plus x0 times 1. So the expectation value of x in the new state, the x0 psi, is the expectation value of x in the old state plus x0. So indeed, if this is x, you could do this for vectors, and here is x0. Well, the expectation value of x in the new state, the T x0 operator, took the state and moved it by a displacement x0 so that the new expectation value is the old one plus x0. So that's physically why these things are relevant. A couple of other things you've shown in the homework, and you could retry doing them, is that T x0 on the x state, by this intuition, should be the x plus x0 state. It moves the state to the right. And if psi has a wave function, psi of x, T x0 of psi has a wave function, psi of x minus x0, since you know that psi of x minus x0 is the wave function translated by x0 to the right. The sign is always the opposite one. When you write psi of x minus x0, the function has been moved to the right x0. So this is our whole discussion and reminder of what the translation operators are. So we've got our translation operator. Let's see how we can use it. And we'll use it to define the coherent states. So here comes the definition of what the coherent state is. It's a beginning definition, or a working definition, until we understand it enough that we can generalize it. By the time we finish the lecture, this definition will be generalized in a very nice way, in a very elegant way. So coherent states. So here it goes. I'm going to take the vacuum state of the harmonic oscillator, the ground state of the harmonic oscillator, and simply displace it with a translation operator by x0. So this is going to be e to the minus i p hat x0 over h bar 0. And I want a name for this state, and that's the worst part of it. There's no great name for it. I don't know if any notation is very good. If it's very good, it's cumbersome, so I'll write it like this. A little misleading. I'll put a tilde over the state. You could say it's a tilde over the x, but it really, morally speaking, is a tilde over the whole state. It means that this thing, you should read there's an x0 here used for the translation operator that appears here. So that's the state, x tilde 0. Intuitively, you know what it is. You have the harmonic oscillator potential. Here is x. The ground state is some wave function like that. This state has been moved to position x0, and presumably some sort of wave function like that, because this translates the wave function. So the ground state moves it up there to the right. That's what it is. That's a coherent state. And there's no time dependence here so far, so this is the state at some instant of time. The coherent state, maybe call it at time equals zero. Let's leave time frozen for a little while until we understand what this state does. Then we'll put the time back. So a few remarks on this. x0 x0 is how much? Now, don't think these are position eigenstates. That's a possible mistake. That's not a position eigenstate. This is a coherent state. If these would be position eigenstate, you say delta of this minus that, but it's nothing to do with that. Can you tell without doing any computation what is this number? How much should be? Yes? AUDIENCE: 1. BARTON ZWIEBACH: It should be 1. Why? Because it's a unitary operator acting on this thing, so it preserve length. So this should be equal to 0 0, should be 1. Very good. No need to do the computation. It's just 1. Psi associated to this state is the ground state wave function at x minus x0. Where this refers to the wave function, x0 is psi 0 of x. So this is what I was saying here. The wave function has been translated to x0, the remark over there. So these are our coherent states and we want to understand the first few basic things about them so we can do the following simple computations. So if I have to do the following, if I have to compute the expectation value of any operator, A, on a coherent state, I use the fact that I want to go back to the vacuum, so I put T x0 dagger A T x0 0. Because that way, I trace back to what the vacuum is doing. It's much easier to do that than to try to calculate something from scratch. So for example, we have here that x0 x x0, well, you would replace it by T x T, T dagger x T, which you know is x hat plus x0. We calculated it a few seconds ago, top blackboard. And therefore, you got what is the expectation value of x on the ground state? x0, very good. And therefore, we just got x0, which is what you would expect. The expectation value of x on the coherent state is x0. You're there. You've been displaced. How about the momentum, x0 p hat x0? Well, p acted by the translation operator is unchanged. Therefore, we got 0 p 0, and again that 0, so this state still has no momentum. It represents a T equals 0, a state that is over here. And just by looking at it, it's just sitting there, has no momentum whatsoever. Another question that is interesting, what is the expectation value of the Hamiltonian on the coherent state? Well, this should be, now you imagine in your head, T dagger H T. Now, H is p squared over 2m, and that p squared over 2m gets unchanged. p squared over 2m is not changed because T dagger and T does nothing to it, T dagger from the left, T. Nevertheless. the Hamiltonian has a 1/2 m omega squared x hat, and x hat is changed by becoming x hat plus x0. Well, we don't want to compute too hard, do too much effort here. So first, we realize that here's the p squared over 2m and here's the m omega squared x hat squared, so that's the whole Hamiltonian. So we got 0 H 0 plus the extra terms that come here. But what terms come here? There's a product of an x0 and an x between 0 and 0. x0 is a number, so you have an x between 0 and 0, and that's 0. So the cross product here won't contribute to the expectation value, so the last term that is there is 1/2 m is a number omega squared, x0 squared. And what is the expectation value of the Hamiltonian on the vacuum? It's h omega over 2 plus 1/2 m omega squared, x0 squared. And you start seeing classical behavior. The expectation value of the energy at this point is a little quantum thing plus the whole cost of stretching something all the way to x0. 1/2 of k squared, k for the oscillator, x0 squared. So the energy of this thing is quite reasonably approximated, if x0 is large enough, by the second term, and this is the cost of energy of having a particle of the potential. So it's behaving in a reasonable way. You can do a couple more little exercises that I'll put here as things for you to check. Exercise. x0 tilde x squared x0 tilde. Just calculate. It's just useful to have. x0 squared plus h bar over 2m omega. And x0 tilde p squared x0 tilde is mh omega over 2. And finally, x0 tilde xp plus px x0 tilde is equal to 0. Any questions? These are exercises for you to practice a little these expectation values. Questions on what we've done so far? Yes? AUDIENCE: You said these coherent states is most significant only in the ground state, or is it also important to use them for other bound states. BARTON ZWIEBACH: Well, we've defined the coherent state by taking the ground state and moving it, and these are particularly interesting. You could try to figure out what would happen if you would take an excited state and you move it. Things are a little more complicated. PROFESSOR: And in a sense, they can all be understood in terms of what we do to the ground state. So we will not focus on them too much. In a sense, you will see when we generalize this how what we're doing is very special, in at least one simple way. So we'll always focus on translating the grounds. Other questions? Yes. AUDIENCE: Where does the term coherent arise, and why does it cohere when you translate it? PROFESSOR: OK, here is the thing of the coherent state. Is this an energy eigenstate at this moment? What do you think? Is this an energy eigenstate-- this state over here? No, it won't be an energy eigenstate. There's something funny about it. Energy eigenstates are always diffuse things. They never look like that. So this is not an energy eigenstate, and you've done things with non-energy eigenstates. They change shape. As they evolve, they change shape. What we will see very soon is that this state, if we let it go, it will start moving back and forth without changing shape. It's going to do an amazing thing. Energy eigenstates-- you super-pose two energy eigenstates. You get something that changes in time and the shape changes, and you've even done problems like that. But this state is so exceptional that even as we let it go in time, it's going to change, but the shape is not going to spread out. Do you remember when you considered a pulse in a free particle, how it disappears and stretches away? Well, in the harmonic oscillator, this has been so well prepared that this thing, as time goes by, will just move and oscillate like a particle. And it does so coherently. It doesn't change shape. When we talk about light, coherent light is what you get from lasers. And so if you want understand lasers, you have to understand coherent states. OK, so this brings us there to time evolution. So let's do time evolution. So what will happen? We'll have a state x0 goes to x0 comma t. So that's the notation. That's what we'll mean by the state at a later time. And how are we going to explore this? Well, we're all set with our Heisenberg operator, there. We'll take expectation values of things to figure out how things look. So what do we have here? We'll ask for X0 t, and we'll put the Schrodinger operator in between here-- X0 t, and this is what we'll call the expectation value of A as time goes by in the X0 0 state. This is what we call this. But then, we have the time evolution. So this is equal to the original state, Heisenberg operator of A-- original state. And if you wish, you could then put the t operator-- as we have in the top blackboard to the right-- and reduce it even more. But we've computed a lot of this coherent state expectation value, so let's leave it like that. So you could, if you wish, say this is equal to 0-- T X0 dagger A T X0 0. So you can ultimately reduce the expectation values of things on the vacuum. So OK, we're all set. Let's try to do one. And the reason this is a nice calculation is that the time evolution of this state is a little complicated. We'll figure it out later, but it's easier to work with the time evolved state. So here it goes-- what is the expectation value of X as a function of time on the X0 state? Well, it says here take the X0 state, and take the Heisenberg value of X. So we have it up there-- X hat cosine omega t plus b hat over M omega sine omega t X0. Forget about time evolution of the coherent states. We evolved the operator. On the other hand, we have that the expectation value of p is 0 in the coherent state, and the expectation value of X is X0. So end of story-- calculation over-- X0 cosine of omega t. That's expectation value in time. This thing is oscillating classically. That's nice as can be. So classical behavior again, of a quantum state. How about expectation value of p X0 of t? If it's oscillating, it better be moving, and it better have some momentum. So let's put the momentum operator here, the Heisenberg one. So we'll have p hat cosine omega t minus m omega x hat sine omega t X0 tilde. And this is 0, but X has X0 there, so minus m omega X0 sine of omega t, which is equal to m d dt of the expectation value of X. Here it is-- expectation value of X. m d dt of that is minus m omega X0 sine omega t. That's what it should be. And this thing is really oscillating classically-- not only the X position, but the momentum is doing that. Now, the other thing that we can compute-- and we want to compute-- is the key thing. You have this state. We said it's coherent evolution. So the ground state is this state that is a minimum uncertainty packet. It has a delta X uncertainty mix and a delta p. Their product saturates the uncertainty in equality. And when we move the state X0, well, the delta X will be the same. The delta p will be the same, and it's that. But now as it starts to move, we want to see if the shape is kept the same. Maybe it fattens up, and shrinks down, and does things in the middle. So the issue of coherency of this state is the issue whether the uncertainties remain the same. If the uncertainties remain the same, and they are saturated-- the product is saturating the inequality, you know that the shape has to be Gaussian, and it must be the same shape that is running around. So what we need to compute is the uncertainty in X, for example. So how do delta X of t and delta p of p behave? That's our question. And let's see how they do. Well, we have this computation-- actually, if you don't have the Heisenberg picture, it's kind of a nightmare. With the Heisenberg picture, it's a lot easier. Delta x squared of t would be the expectation value of X0 t of X squared, X0 t, minus the expectation value of X0 t X, X0 t squared. I wrote what the definition of the uncertainty squared is. It's the expectation value of the operator squared, minus the square of the expectation value of the operator. And of course, everything is going to turn Heisenberg immediately, so this thing-- maybe I can go one more line here-- would be X0 X Heisenberg squared of t, X0 minus-- this is simple-- this we've calculated. it's that expectation value at the top is the expectation value of X in time. It's that. So this is minus X0 squared cosine squared of omega t. So what do we have to do? We have to focus on this term. So this term is equal to X0. And you have X Heisenberg squared, so let's do it-- X squared cosine squared omega t plus p hat squared over m squared w squared sine squared omega t plus 1 over mw cosine omega t sine omega t X p plus pX X0 tilde. That shows that term, and I just squared that thing, but that I suggested a few exercises here. This is 0. In fact, it's 0 in the ground state as well, so this is 0. X squared gives you the top equation-- X0 squared plus h bar over 2 m omega cosine squared omega t-- plus p squared over m squared w squared, so p squared is m h omega over 2. And then you have m squared omega squared sine squared omega t. And that's this whole term. And the thing that we're supposed to do is subtract this here. You see that the X0 squared cosine squared of omega t cancels here. So what do we get? h bar over 2 mw cosine squared omega t. But this thing is also h bar over 2 mw sine squared omega t. So this whole thing, all the times have disappeared-- delta X squared-- the time dependence here has disappeared with that, and the cosine squared with sine squared have combined, and you get h bar over 2 m omega, which was-- this is, I'm sorry, of t. We work very hard to put the t there. We should leave it. The uncertainty as a function of time has not changed. It is the original uncertainty of the ground state. So this is moving in a nice way. You're supposed to compute now as well the uncertainty in p. I leave that as an exercise-- delta p squared of t equal m h bar omega over 2. So this is an exercise. Practice with coherent states. It's worth doing it, I think. Actually there's going to be a problem in the homework set, in which you're going to ask to do most of these things, including things I'm doing here on the blackboard. So you will practice this. So between these two, delta p, delta X-- delta X of t, delta p of t is, in fact, equal to h bar over 2. And this is a minimum uncertainty thing. And it behaves quite nicely. All right, so the name coherent now should make sense. You've produced a quantum state that has about the energy of a state that you're familiar with, and it moves classically, and it doesn't change shape as it moves, so it moves coherent. So our next task, therefore, will be to understand this in the energy basis. Because in the energy basis, it looks like a miracle. You've suddenly managed to produce a set of states of different energies, created the superposition, and suddenly, it moves in a nice way. Why does that happen? So we need to understand the energy basis. And as we do that, we'll understand how to generalize the coherent states completely. So let's go on with that, and let's explore this in the energy basis. So what do we have? We have the coherent state-- no need to put the time yet-- is e to the exponential of minus i p hat X0 over h bar. There is, as you've seen already, at length scale in the harmonic oscillator-- famous length scale, and we'll have an abbreviation for it. It's the length scale d0 squared h bar over m omega. You can use the parameters of the harmonic oscillator-- h bar and m and omega-- to produce a length scale. And that length scale is d0. It's essentially the uncertainty in the position in the ground state, up to the square root of 2. It's the way-- you want to construct a length-- there it is-- the only way you can construct a length. So I'm going to use that notation. So let me put what the p is into that formula, and simplify it. So this is on the vacuum-- I'm sorry, I stopped half the way. So this is the exponential of X0 over square root of 2d a dagger minus a on the vacuum. Plug in the p, get the h bars, and you will see that d enters in that way. It's the way it has to enter, because this exponential should have no units, and therefore X0 over d0 has no units, and the a's and the a daggers have no units. So it couldn't be any way different like that. The i also shouldn't be there, because this operator-- the i was there to make this anti-Hermitian. But this, with this real, is already anti-Hermitian. You see, you take the dagger. It becomes minus itself. So this is anti-Hermitian. No need for an i-- in fact, an i would be wrong, so there's no i. And that's this. Now, we want to figure out how this looks in the energy basis, so what are we going to do? We're going to have to do something with that exponential. We're going to have to reorder it. This is a job for Baker-Campbell-Hausdorff. Which one? Well, this one-- e to the X plus Y is equal to e to X, e to the Y, e to the minus 1/2-- I don't know this by heart-- XY, and it stops there. If and only if X commutator with Y commutes with X, and commutes with Y. There was a problem in the test that there was an operator with Xp plus pX acting on X, and after you commuted, you get X again, and you have to keep including terms. So this stops here if XY commutes with X and Y. And why do I want this? Because I actually want to split the creation and the manipulation operators. I want them in separate exponentials. We have that energy eigenstates are creation operators on the vacuum, but here I have creation minus destruction. So if I expand the exponential, I'm going to get lots of creation and destruction, and I'm going to spend hours trying to sort it out. If you did expand it, I bet you won't see it through so easily-- probably will take you forever, and it might not work out. So expanding an exponential is something that we should be reluctant to do. On the other hand, this is a nice option, because then you think of this as e to the X0 over square root of 2d0 a dagger minus X0 over square root of 2d0 a. And I chose this analogy X with this, and Y with this. It's like Y is this thing minus that, and X is that. You could have done it the other way around, but then you would run into trouble again. Why? Because I want a Y factor that has the manipulators to be to the right of the X factor that has the creation. Why? Because if I have annihilators closer to the vacuum, that's good. Annihilators closer to the vacuum is what you really want, because if you have creators close to the vacuum, they create state, but then you have the annihilators, and you have to start working them out. On the other hand, if the annihilators are close to the vacuum, they just kill the vacuum and you can forget them. So it's very important that you identify X with this, and Y with this whole thing. So that this is e to the X0 over square root of 2D0 a dagger, e to the minus X0 over square root of 2d0 a. And now you're supposed to do the commutator of these two things. And the commutator is the commutator of an a dagger with an a, and that is 1. So this commutator is a number-- crucial, because if it wasn't a number, if it would be an a or an a dagger, it would not commute with a, X, and Y, and you have to include more terms. So the fact that this commutator is a number allows you to use this formula. So now we'll put this factor here, minus 1/2. Then you have X with Y, and that's X0 minus X0 over square root of 2d squared, minus-- and this factor squared-- an a dagger with a, which is minus 1. So that's that whole operator. So let's write it. The coherent state, therefore, X0 tilde, is equal to e to the X0 over square root of 2d0 a hat dagger, e to the minus X0 over square root of 2d0 a, and this factor that seems to be e to the minus 1/4 X0 squared over d0 squared. And here is this nice vacuum. Yes, factor is right. So what is this? Well, this is a number, so I can pull it to the left. And here is the exponential of the annihilator operator. Now expand the exponential. It's 1. That survives, but the first term has an a-- kills it. The second term has an a squared-- kills it. Everything kills it. This thing acting on the vacuum is just 1. That's why this is simple. So what have we got? In the state X0 tilde is e to the minus 1/4 X0 squared over d0 squared times e to the X0 over square root of 2d0 a dagger on the vacuum. Well this is nice-- not quite energy eigenstates, but we're almost there. What is this? e to the minus 1/4, X0 squared over d0 squared. And now expand. This is the sum from n equals 1 to infinity, 1 over n factorial. X0 over square root of 2d0 to the n, a hat dagger to the n on the vacuum. And what was the nth energy eigenstate? You probably remember. The nth energy eigenstate is a dagger to the n on the vacuum over square root of n factorial. So we've got a little more than the square root of 2 n factorial. So maybe I'll do it here. We get e to the minus 1/4 X0 squared over d0 squared, sum from n equals 1 to infinity, 1 over square root of n factorial, X0 over the square root of 2 d0 to the n times the nth energy eigenstate. It's a little messy, but not so bad. I think actually I won't need that anymore. Well no, I may. I will. So let's write it maybe again. Well, it's OK. Let's write it as follows-- Cn n. OK, so I got some cn's and n's. So this is a very precise superposition of energy eigenstates, very delicate superposition of energy eigenstates. Let me write it in the following way-- cn squared. Why would I care about cn squared? cn squared is the probability to find the coherent state in the nth energy eigenstate. The amplitude to have it in the nth energy eigenstate is cn. So that probability to find it in the nth energy eigenstate is cn squared, is the probability 4x tilde 0 2b in the nth energy eigenstate. That is what? Exponential of minus 1/2 x0 squared over d0 squared. And I have to square that. So I have 1 over n factorial. I have to square that coefficient there. So it's x0 squared over 2d0 squared-- that's nice, it's the same one here-- to the n. So it's easier to think of this if you invent a new letter, lambda, to be x0 squared over 2d0 squared. Then, cn squared is equal to e to the minus lambda lambda to the n over n factorial. Yes. AUDIENCE: Is that something that we should expect to be true for any type of commutator, or is that something that we [INAUDIBLE]?? PROFESSOR: Well, let me say it this way. In a second, it will become clear that this was almost necessary. I actually don't know very deeply why this is true. And I'm always a little puzzled and uncomfortable at this point in 805. So what is really strange about this is that this is the so-called Poisson distribution. So there's something about this energy eigenstate that their Poisson distributed in a coherent state. So these are probabilities, as I claimed, to find an n. And indeed, let's check the sum of the cn squareds from n equals 1 to infinity. Let's see what it is. And you will see, you cannot tinker with this. This is e to the minus lambda the sum from n equals 1 to infinity lambda to the n over n factorial. And that sum-- it's not from n equals 1. It's 0 to infinity, I'm sorry. Did I write once anywhere? Yeah, it should be 0. OK, this is 0. There is the ground state, so from n equals 0 to infinity. And this is e to the minus lambda e to the lambda, which is 1. So yes, this is Poisson distributed. It's some sort of distribution like that. So if you have the n's, the cn's, Poisson distributions have to do with if you have a radioactive material. It has a lifetime. And you say, how many events should I-- the lifetime is five years. How many events should you expect to happen in a week? These are Poisson distributed. So it's a Poisson distribution. It's a very nice thing. So let me just make one more remark about it. And it's quite something. So one question that you could ask is, what is the most probable n? That's a good question. You have a coherent state. So it's going to have the superposition of the vacuum, the first. What is the most probable n, so the expectation value of n? Now, I'm thinking of it probabilistically. So I'm thinking this is a probability distribution. Then, I will show it for you that this is really computing what you want. But probabilistically, what is the expectation value of n? You should sum n times the probability that you get n. So this is sum over ne to the minus lambda lambda to the n over n factorial. So you got an n there. And the way you get an n there-- well, the e to the minus lambda goes out. And the n can be reproduced by doing lambda d d lambda on the sum. Because lambda d d lambda on this sum brings down this n, puts back the lambda so it gives you the thing you had, and that's what it is. So here, you get e to the minus lambda lambda d d lambda of e to the lambda. And that is lambda, OK? So the expectation value of the most sort of not the peak, but the expected value of n in this distribution, the level that you're going to be excited is basically lambda. So if x0 is 1,000 times bigger than d0, you've moved this thing 1,000 times the quantum uncertainty. Then, you're occupying most strongly the levels at 1 million. You get x0 over d0 controls which n is the most likely. Indeed, look, this n-- suppose you would compute x0 tilde n hat x0 tilde. This is what you would think is an occupation number. This sounds a little hand wavy. But this is the number operator, the expected value of the number operator, in the coherent state. But this is-- you have that the coherent state is this. So let's substitute that in there. So you get two sums over n and over m. And you would have cm star m N n cn. I've substituted x0 and x0 dagger. The c's in fact are real. And then, the number operator on here is little n. And then, you get the Kronecker delta. So this is sum over n and m cmcn-- it's real. And then, you get n delta m, n. So this is in fact the sum over ncn squared. So what we wrote here, this is really the expectation value of the number operator. And one can do more calculations here. A calculation that is particularly interesting to discover what these states look like is the uncertainty in the energy. So that's another sort of relevant measure. How big is the uncertainty in the energy? What are, basically, the delta E associated to the coherent state? How does it look like? Is it very sharp? So it's a good question. And it's in the notes. I leave it for you to try to calculate it. Delta E in the coherent state x0, how much is it? And it turns out to be the following-- h omega x0 over square root of 2d. So actually, maybe this is a little surprising. But delta E over h omega is equal to x0 over d. So actually, the energy uncertainty for a classical look in coherent state-- I'm sorry, I'm missing a square root of 2 there. So what is a classical looking coherent state? It's a state in which x0 is much bigger than the quantum d. So x0 is much bigger than d. So in that case, this is a large number for a classical state-- "classical" state. But in that case, the uncertainty in delta E is really big compared to the spacing of the harmonic oscillator. So you have-- here is the ground state. Here is h omega. Here is the coherent state, maybe. And you have a lot of energy levels that are excited. So if x0 over this is 1,000, well, at least 1,000 energy levels are excited. But you shouldn't fear that too much. Because at the same time, the expectation value of E over delta E-- the expectation of E is something we calculated at the beginning of the lecture. You have the oscillator displaced. So this is roughly 1/2 m omega squared x0 squared. Throw away the ground state energy. That's supposed to be very little. Delta E is h bar omega x0 over square root of 2. And this is, again, the same ratio. So yes, this state is very funny. It contains an uncertainty that measured in harmonic oscillator levels contains many levels. But still, the uncertainty is much smaller by the same amount than the average energy. So this state is a state of some almost definite energy, the uncertainty being much smaller. But even though it's much smaller, it still contains a lot of levels of the oscillator. So that I think gives you a reasonable picture of this. So you're ready for a generalization. This is a time to generalize the coherent states and produce the set of coherent states that are most useful eventually, and most flexible. And we do them as follows. We basically are inspired by this formula to write the following operator. And here, we change notation. This x0 was here. But now we'll introduce what is called the alpha coherent state. Most general coherent state is going to be obtained by acted with a unitary operator on the vacuum. So far so good-- D of alpha unitary. But now generalize what you have there. Here, you put a minus a dagger minus a, because that was anti-Hermitian, and you put the real constant. Now this alpha will belong to the complex numbers. Quantum mechanics is all about complex numbers. You've got complex vector spaces, complex numbers. It's all over the place. So how do we do this? We do this exponential of alpha a dagger. And I want it to be anti-Hermitian. So I should put minus alpha star a on the vacuum. This thing for alpha equals-- this real number reduces to that. But now, with alpha complex, it's a little more complicated operator. And it's more general. But it's still unitary. And it preserves a norm. And its most of what you want from these states. So the first thing you do to figure out what this operator does is to calculate something that maybe you would not expect it to be simple. But it's worth doing. What is a acting on the alpha state? Well, I would have to do a acting on this exponential of alpha a dagger minus alpha star a on the vacuum. Now, a kills the vacuum. So maybe you're accustomed already to the next step. I can replace that product by a commutator. Because the other ordering is 0. So this is equal to the commutator. Because the other term when a is on the other side is 0 anyway. And now I have to compute the commutator of a with an exponential. Again, it's a little scary. But A with an exponential e to the B-- it's in the formula sheet, this Campbell-Baker-Hausdorff again-- is A, B e to the B if A, B commutes with B. Well, this is A. This is B. A with B is-- A with B, the exponent-- just alpha times 1 because of this. So it's a number. So this is safe. So you get alpha times the same exponential. But the same exponential means the state alpha-- a little quick, isn't it? OK, A with B, this factor, was alpha. And e to the B anyway on the vacuum is the state alpha. So there you go. You have achieved the impossible. You've diagonalized a non-Hermitian operator. This is not Hermitian, and you found its eigenvalues. How could that happen? Well, it can happen. But then, all of the theorems that you like about Hermitian operators don't hold. So it's a fluke. This can be done. But then, states that correspond to different eigenvalues will not be orthogonal, and they will not form a complete set of states, and nothing will be quite as you may think it would be. But still, it's quite remarkable that this can be done. So this characterizes the coherent state in a nice way. They're eigenstates of the destruction operator. And they're the most general exponentials of creation and annihilation operators acting on the vacuum. Now, we knew that when alpha is real, it has to do with x0. So we've put a complex alpha. What will it do? A complex alpha, what it does is gives the original coherent state some momentum. Remember, the original state that we had was an x0. And how did it move? x0 cosine of omega t. So at time equals 0, it had 0 momentum. This creates a coherent state at x0, and it gives it a momentum controlled by the imaginary part of this thing. In fact, we can do this as follows. You can ask, what is the expectation value of x in this state? Well, x is written here. It's d over square root of 2 alpha a plus a dagger alpha. And look, these are easy to compute. a gives an alpha, gives you alpha. a dagger and alpha, you don't know what it is. But a dagger on bra alpha is alpha star. So this one you know on the right. This one you know on the left. It gives you the over square root of 2 alpha plus alpha star. So it's square root of 2d real of alpha. So the real part of alpha is the expectation value of x. So I'll go here. I'm almost done-- waiting for the punch line. Similarly, you can calculate the expectation value of the momentum. It will be alpha p alpha. And p is a minus a dagger. So you're going to get alpha star minus alpha, so the imaginary part. So p is actually square root of 2 h bar over d imaginary part of alpha. So the physics is clear. Maybe the formulas are a little messy. But when you have an alpha, the real part of alpha is telling you where you're positioning the coherent state. The imaginary part of alpha is telling what kick are you giving to it. And you now have produced the most general coherent state. So how do we describe that geometrically? We imagine it, the alpha plane. And here it is. The alpha plane, here is the vector, the complex number alpha that you've chosen maybe for your state, some particular complex value alpha. On the x-axis, the real part of alpha is the expectation value of x over square root of 2d. The real part of alpha is the expectation value of x over square root of 2d. And the imaginary part of alpha is the expectation value of p over square root of 2 h bar d. And there it is, your state at time equals 0. What is it going to do a little later? Well, that will be the last thing I want to calculate for you. It's a nice answer. And you should see it. It's going to take me two minutes. And what is it? Well, alpha at time t, you have to evolve the state-- e to the minus iHt over h bar on the state, which is e to the alpha a dagger minus alpha star a e to the output and into the iHt over h bar, and an e to the minus iHt over h bar on the vacuum. So I put the one here. I evolve with this. But I take the state and put this and that. This is simple. It's e to the minus iH omega over 2. That's the energy of the ground state. But what is this part? It's pretty much the Heisenberg operator. But the sign came out wrong. Well, it didn't come out wrong. It's what it is. It just means that what I have to put here is the Heisenberg operator at minus t. Because I have t for minus t. So this is e to the alpha a Heisenberg at minus t dagger minus alpha star. I'm sorry, I have too many parentheses here. That's it, much better-- minus alpha star a Heisenberg of minus t acting on this thing. And what is this? Well, we have the formula for the Heisenberg states here. So you've got e to the alpha H a dagger of minus t would be e to the minus I omega t a dagger. And here, you have minus alpha star e to the i omega t a on e to the minus i omega h bar over 2 times the vacuum. And look what has happened. Alpha has become alpha times e to the minus i omega t. Because the star is here. It's minus the star one. So the only thing that has changed is that this state, alpha at time t, is e to the minus ih bar omega over 2. I'm sorry, I'm missing a t here. AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, I dropped-- yeah, minus i omega t over 2, minus i omega t over 2, minus i omega t over 2, times the coherent state, the time independent coherent state of value e to the minus i omega t alpha. That's a new complex number. That's what has happened. The number alpha has become e to the minus i omega t. Now, this is a face for the whole state multiplicative. It's irrelevant. So what has this alpha done? It has been rotated by e to the minus i omega t. So this at the time t is the state alpha t. Here is the state alpha. And it has rotated by omega t. So the coherent state can be visualized as a complex number in this complex plane. This real part is the expectation value of x at time equals 0 whose imaginary part is the expectation value of the momentum at time equals 0. And how does it evolve? This state just rotates with frequency omega all along and forever. All right, that's it for today. See you on Wednesday. [APPLAUSE] PROFESSOR: Thank you, thank you.

MIT_805_Quantum_Physics_II_Fall_2013

2_Wave_Mechanics_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Very good. So today, we'll begin with a study of one dimensional potentials and the energy eigenstates or properties. And we'll discuss this for about half of the lecture. And then go into the variational principle. So let me go back then to our discussion of last time, where we were talking about energy eigenstates and the Schrodinger equation in one dimension. So there's a term that we use all the time. This term is called the bound state. And bound state seems something sort of very non-trivial. But mathematically, we can say quite clearly what it is. A bound state is something that is not spread all over space, basically. And the way we therefore use the terminology for bound state it that we only speak of bound states when there are energy eigenstates. So an energy eigenstate may or may not be a bound state, but any bound state is an energy eigenstate. So an energy eigenstate is a bound state if the wave function goes to zero when you go sufficiently far away. So it's a simple definition, but it helps us understand that the basic idea is at the state is just not spread all over the world. Now remember we're trying to find energy eigenstates, and that is to find wave functions, time independent wave functions, that solve the time independent Schrodinger equation. Which I have written there for convenience. This is the same equation we wrote last time. For a time independent potential, so that the full wave function can be written as a phase that contains the information of the energy times a function of x, sine of x. That still may be complex, but it doesn't have to be complex. As you can see, this is a real equation. Now in order to clean up some of the constants in this equation, it's-- yes? AUDIENCE: Why is it not V x minus e? PROFESSOR: Well you have to look back at the Schrodinger equation, and bring the things to the other side, and take care of the signs. It is correct as stated. Check it. Now we have, in order to do this and just to write things with a little more clarity, we scale the energy. So we define a calligraphic energy, which is 2m over h squared times energy E. And a calligraphic V with the same scaling factor. So that the Schrodinger equation then takes the form-- this same equation-- sine. And I'll use the double prime notation for two derivatives. Plus now E minus cal of x sine equals zero. So this is how our equation looks. It's relatively simple. And can be treated without carrying constants all over the place. Now we're going to discuss a few results that are extremely important that you keep in mind for the future. It helps you think clearly. So these results are little theorems that I'm going to state. And the proofs are given in the notes. I will skip some of the proofs, although I will tell you what the strategy is for the proofs and then show how the theorem implies, as a corollary, interesting things that could easily be misconstrued. So let me give you the first theorem. Theorem one. This is just the statement that if you are talking about bound states in one dimensional potentials, there are no degeneracies. So let's write it. There is no degeneracy for bound states of one dimensional potentials. So it's a fundamental result. We'll see some way to understand it a little later. And you have seen this in 804, I believe. So what is a strategy to a proof? So we want a rigorous proof. So what is a strategy? I will not go through the proof. The strategy is to assume there is a degeneracy. So strategy, assume a degeneracy. So sine 1 and sine 2 different with the same energy. And then you write this equation for sine 1 and sine 2, and they have the same energy. Then you take the two equations, and you multiply the first by something, the second by another thing. Subtract it. Do a bit of algebra. And you suddenly can prove that one solution is equal to a constant times the other solution. So you do the work, and that's what comes out. Once you've shown that one is proportional to the other, we've declared in quantum mechanics that two wave functions that differ by a constant have exactly the same physics. You can normalize it, and normalize them. And the constant can be just a phase, but the phase doesn't matter. So those two wave functions are really the same. So you get a contradiction. You thought they were different. But the difference is just too trivial, and therefore they are really the same wave function. Because we've said that two wave functions that differ by a constant should be identified. So that's this theorem. And it's a key theorem. Let's go to the next theorem that is also important. So theorem two. Energy eigenstates sine f of x can be chosen to be real. So here it is. We mentioned that this differential equation allows for real solutions. There's no contradiction, because there's no i in there. But now we say more. That even if you find a complex solution, you can work with real solutions. So what is the strategy here? Again strategy. To have you get a complex solution. Sine f of x. And then you go about to prove that actually this complex solution implies the existence of two real solutions. So this complex solution implies existence of two real solutions. And moreover, two real-- I should say it better here-- degenerate solutions. How could you prove that? Well, it probably would occur to you that you would take a solution and say, oh, this solves it. Then you would show that sine star also solves this equation. And then by linearity of the Schrodinger equation, you could form what we would call the real solution formed by taking sine plus sine star and adding them. Or the real solution that comes from the imaginary part. You see the real and imaginary parts of a complex number are real. So is the case here as well. That they can take sine and sine star that are different. And the sum is real by construction. You put the star, and recalling that star of star gives nothing, you see that this function is equal to its star. So it's real. So is this one. This function, which is the imaginary part of the solution that you had, is also real. So you get two real solutions. Moreover, you can show that sine and sine star have the same energy. Therefore sine r and sine m have the same energy. So I'm really skipping very little. But I don't want to clutter the blackboard with the whole derivation at this moment. So here is the strategy. A complex solution implies two real solutions. And therefore, since you have real solutions, you can always choose them to be real. No loss of generality. You had a complex solution. You say, I like just real solutions. And you chose your two real solutions. You give them. You've given the whole thing. But then there is a nice corollary for the case of one dimension bound states. So this is the part that is perhaps somewhat interesting. And the corollary is that if you're talking bound states of one dimension, any solution is equal to a real solution up to a constant. You may say, well what's the difference? Let's write it down and look at it. Corollary for bound states of one dimensional potentials. Any solution is, up to a phase, equal to a real solution. So how does that go? Maybe we can use more of this blackboard. Why would any solution be, up to a phase, equal to a real solution? OK. Suppose you have these two solutions here. Well this is real, so that's trivially true. And this is real. But what if I put a linear combination of them? Or take the original complex solution? Why would it be, up to a phase, equal to a real solution? The answer is the following. Because of theorem and, that there's no degeneracy, you've got two solutions that claim to be different and degenerate. But that can't be. So by theorem one, you must have that-- in the notations, I have here-- sine imaginary of x must be equal to a constant times sine real of x. Now if these function's anyway are real, the constant here must be real. Therefore, even if you form the original complex solution, sine, which is sine real plus i sine imaginary. You can check that. It's clear by definition that that's the way it should be. If you think this, this is equal to 1 plus i times c times sine real of x. Therefore this is equal to a constant times the phase of-- this is a complex number. So this is yet another constant that we could say, the norm of 1 plus i c times some phase times sine r of x. So you see that any solution, any linear combination, in fact, with any numbers of sine r and sine imaginary, will be a constant times a phase times this thing. So it is really just real anyway. So you can just have one solution. And any solution that you may write, that represents a bound state, is, up to a phase, equal to a real solution. OK. So that's our theorem two. Theorem three. It's also very famous. If the potential V of x is even-- that is, V of minus x is equal to V of x-- the eigenstate can be chosen to be even or odd under x2 minus x. OK. So here is another claim. So the potential is even. There's no theorem for all the potentials. No clear theorems, no simple theorems for all the-- but if the potential is even, the claim is that you can choose eigenstates to be even or odd. The word chosen is very important here. Otherwise, this would not be a precise statement. You cannot say the eigenstates are even or odd. You can choose them to be. So how does the proof go? Strategy begin with a wave function, sine of x, that is neither even nor odd. And then you do a little work with the Schrodinger equation. Take the Schrodinger equation, change all x's to minus x's and show that, in fact, not only is sine of x a solution, but sine of minus x is also a solution. With the same energy. So prove that sine of minus x is a solution with the same energy. And in this case, of course, we can already have shown that we can choose these wave functions to be real. So we can choose all of these wave functions to be real. And what do we do next? If we have these two solutions with the same energy, then you can build of sine s, which is 1/2 of sine of x plus sine of minus x. And of sine a. s for symmetric, and a for anti-symmetric. And of sine a, that is 1/2 of sine of x minus sine of minus x. And this tool would be this even, under the exchange of x for minus x. This one odd, under the exchange of x for minus x. And both would be solutions by superposition. And both would have the same energy. So that's the end of the theorem because then these things are even or odd and have the same energy. So the solutions can be chosen to be even or odd under x. So if you've proven this, you've got it already. But now we get the corollary. For bound states in one dimension, the solutions not anymore the word chosen. We can delete the word chosen. The solutions are either odd or even. So it's a lot stronger. It's not anymore, you can choose them to be, but a general one is neither odd nor even. No. You try to find the solution that is neither odd nor even, and you can't find it. So it's very strong. Yes? AUDIENCE: Is this for even potentials? PROFESSOR: Even potentials. Yes. For bound states in one dimension with even potentials. V of x. V of minus x equal V of x. Yes. So how do we show that? Well, again, you've got two solutions here that are degenerate that have the same energy. Sine of x and sine of minus x. So given that there's no degeneracy in one dimensional problems, this thing that they have the same energy, the only thing that can be happening is that sine of minus x is equal to a constant times sine of x. Where this constant is real. Why real? Because we said already by the previously theorem, wave functions can be chosen to be real. So you've got this thing already that this is true. Sine of minus x is equal to c times sine of x. Which, if you use this property again by saying, oh, but sine of x is equal to c times sine of minus x. Basically, what this says is that you can change the sign of the argument by putting an extra c. So you do it again, sine of x is equal to this. So this is c squared times sine of minus x. So c squared must be equal to 1. And therefore c is equal to either plus or minus 1. No other option. So the functions are either even or odd, but they can't be arbitrary. This point is sufficiently settled that part two general exam at MIT 10 years ago had a question like that. And the person that invented the problem claimed that there would be a solution that could be neither even nor odd. So even faculty members at MIT sometimes get this wrong. It's not as weak as this, that can be chosen. But it's really either or other in the case you have one dimension. OK. So these are our main theorems. And we're going to proceed now by clarifying a little more the nature of the spectrum. So are there questions? Yes? AUDIENCE: Can you give an example of state that's not bound? PROFESSOR: OK. The question is can I give an example of a state that is not bound? Yes. We can give such state. You have a potential like this. And you have an energy like that. And then the wave function could look like this. Then do something like that. And then look like that. It just doesn't vanish when you go to infinity. AUDIENCE: So that can't be normalized? PROFESSOR: Can't be normalized. That's right. If it's not bound, it can't be normalized. Other questions? Yes? AUDIENCE: So you can't really represent-- it's doesn't really represent single particles, more like a stream of particles? PROFESSOR: Yes. So it doesn't represent a single particle. Now trying to interpret it as a stream of particles is a little delicate. So what we usually do is we build superpositions of those states that can represent a localized thing. But it's true that, morally speaking, it seems to represent more than one particle. OK. So now we talk a little about the nature of the spectrum. So what do we want to say here? We want to go back to the Schrodinger equation here. And just move one term to the right hand side. And just see what can happen in terms of singularities and discontinuities. So first of all, we always begin with the idea that sine must be continuous. And the reason sine must be continuous is that we don't want singularities worse than delta functions in potentials. If a function is continuous, the derivative might be discontinuous, and the second derivative would have a delta function. So the second derivative would have a delta function. But if the function is not even continuous, the second derivative would have derivatives of delta functions. So somehow the potential would have to have that. And we don't want it. So to simplify our life, we say that sine must be continuous. Now if sine is continuous, we will consider several possibilities, possibilities for V, possibilities for V of x. So first possibility-- one, V is continuous. So psi is continuous, and V is continuous. If psi is continuous and V is continuous, this product is continuous. Psi double prime is continuous. And psi prime is continuous. Two, V has finite jumps. Well, if V has finite jumps, and psi is continuous, this product has finite jumps. So psi double prime has finite jumps. If psi prime has finite jumps, the worst is that psi prime still must be continuous. But it changes. Psi prime could look like that, could have a corner. But it cannot be worse than that. Because if V has finite jumps, if psi double prime has finite jumps, and if psi prime is not continuous, it would have delta functions. So for these two conditions, continuous or even finite jumps, psi prime is still continuous. Things change qualitatively if three, V has delta functions. If V has a delta function, then psi double prime has a delta function. And psi prime therefore jumps. Psi prime is not continuous. Psi prime jumps. This may be reminiscent to you whenever you had to solve the problem of a bound state of a delta function, you got a wave function that looked like this in which psi prime jumps. And it has to jump. Because psi double prime has a delta function. Another case in which psi prime jumps is for if V has a hard wall. That is, the potential suddenly at one point becomes infinite, and it prevents the particle from moving across. A hard wall is a place, as you remember in the infinite square well, in which the wave function vanishes, but the derivative doesn't vanish. So you could say that psi is 0 here outside. Psi prime, then, jumps in that it's non-0 here and it's 0 afterwards. So now, you could object to what I'm saying by explaining that, well, we shouldn't talk about the wave function beyond the hard wall. And in some sense, you're right. But suppose we do and we say the wave function is just 0 everywhere, we see that psi prime jumps. So really, psi prime jumps. So this is as bad as things can get. So we can summarize this in just one sentence. And the sentence reads, psi and psi prime are continuous unless V has delta functions or hard walls, in which case psi prime can have finite jumps. So basically, psi and psi prime, continuous. Exceptions-- delta functions and hard walls, and psi prime can change. So questions. There was a question before. Maybe it's still there. Yes. AUDIENCE: Do you absolutely need that function [INAUDIBLE]? Or do we just assume that [INAUDIBLE]? PROFESSOR: We assume it-- so the question is, do I need that in absolute generality to do quantum mechanics? I don't think so. I presume you could discuss some potentials that lead to psi that are not continuous and still make sense. But we will not discuss them. And actually, I don't think I've encountered any of those. Yes. AUDIENCE: Can you give an example of a physical system whose potential is well approximated by a delta function [INAUDIBLE]? PROFESSOR: Yes, there are systems like that. For example, this one that is somewhat well approximated by a delta function. For example, a nucleus sometimes is considered to be like a spherical cavity in which particles are bound by a deep potential and don't escape. So the nuclei are moving like that and don't escape. In that case, it would be a three dimensional delta function that vanishes at the origin. I presume there are many examples. Any potential that sort of begins like a finite square well that is sufficiently deep will start to look like a delta function after awhile. AUDIENCE: [INAUDIBLE] PROFESSOR: Well, it's again neither-- yeah, I guess so. But it depends how big is this psi. So yeah, probably you're right. This looks a little more like an analog of a hard wall. But if a hard wall is very narrow and very deep, it looks like a delta function. So it's idealizations for sure. But I'm sure we could get a better example. And I'll try to find one. Now, the next thing we want to do is give you intuition for this incredible result that there's no degeneratives in one dimensional potentials. That is not to say that the proof is not good enough. It is just to say that we can illustrate that without going into a mathematical proof that is more complicated. So how do we do that? We'll consider the following case, a simple case, an example of a potential of this form. V of x-- this is x. And here is V of x. And we will try to find a solution with some energy that is like that, an energy that is right there below the barrier. So this would be a bound state. Why? Because solutions here are exponentials that decay, exponentials that decay. And here, the wave function would be oscillating presumably. So the wave functions go to 0 and infinity. You could get a bound state. So let's see how we get a bound state. Now, the argument I'm going to follow is just an elaboration of something you can read in Shankar. And it's a nice and simple argument. So we want to understand why we would get here no degeneracies. Or even more-- in fact not just no degeneracies, but the spectrum is quantized. That is, you find one energy, and then another energy maybe, and another energy. So how do we see that? Well, you look at the way you can write solutions and count the parameters of the solutions and try to see how many conditions you have to satisfy. So here, the wave function would be a decay in exponential. A decay in exponential is of the form alpha e to the K, kappa, x. Because x here is negative. So this decays as x goes to minus infinity if kappa is positive. And that's how a solution looks. You need one coefficient here to determine this solutions. So I'll put a 1 here. Now, in here, the solution is oscillatory. So it's a sine plus cosine. So you need two coefficients. In here, the solution must again be decaying. And therefore, you just need one coefficient. Again, this time it would be a beta e to the minus Kx. The fact that this potential looks symmetric-- I'm not assuming it is. Yes. AUDIENCE: Won't one of the coefficients be unconstrained by normalization? Isn't one just the normalization factor? PROFESSOR: OK, how about normalization? Indeed, we have one, two, and two and one, so a total of four coefficients, four parameters. But indeed, suppose you wrote your whole solution. You could say, look, let me divide this solution by 3. That's an equivalent solution. I'm just checking if it solves the Schrodinger equation. That's all I have to check. I don't have to check normalization. Normalization, in fact, is sort of irrelevant here. You just need to know if a bound state exists. So indeed, even though you have these four parameters, given that you can multiply the solution by a constant, there's just three constants to fix. Four parameters via normalization or the multiplication by any constant-- just three constants to fix. But this potential is nice enough that psi and psi prime must be continuous. So you get two conditions here and two conditions here. So four conditions-- continuity of psi and psi prime, continuity of psi and psi prime, four conditions. So what did we get? We got in trouble. We've shown that this is unsolvable in general. Because there are more conditions than parameters. Now, this equation could to be a little peculiar. Maybe this is not completely general. But you seem to have more conditions than parameters. But here comes the catch. The solution really is determined. Like kappa-- do you know kappa? Well, you know kappa if you know the energy of the solution. Kappa is determined by the energy of the solution. So some parameters in the solution depend on the energy. So the way we have to think of this is that in fact three constants to fix, but four conditions. So we really need four constants to fix. And the fourth constant is the energy. So the energy is the fourth constant to fix. And with four conditions, these three constants that we had there and the energy, they can just be fixed. So the solution should fix the energy and should fix this coefficient. So the solution exists for some energy, or possibly some values of the energies, but not all values of the energy. So this shows, or at least very clearly illustrates, that you are going to find sets of energies for which you have solutions depending on how the equations look, and one solution each time. So you get what is called a discrete non-degenerate spectrum. Now, there are more cases to discuss, the case in which you have just the step, or the case in which you have three bound states. And I will not do them but state the results. Again, all that I don't do explicitly can be found in the notes. So you would look at them later. And so here is the second case, a potential like this and an energy like that, energy level. And what you get here is that in fact, doing the counting and analyzing the boundary conditions, you should do it by yourselves. But you will see the answers in the notes. You get here continuous spectrum, non-degenerate. So you will get a solution for every value of the energy-- that's to mean, continuous spectrum-- and one solution each time. Finally, this case-- if you have an energy like this, e, you get continuous spectrum and doubly degenerate, so two solutions. Now, after this, there's one more result that qualifies as a theorem. And it's hard to prove rigorously. I will not attempt to prove it here nor even in the notes. It's hard enough. So this theorem has to do with nodes. Theorem-- so if you have the discrete bound state spectrum of a one dimensional potential, and you list the energies E1 less than E2 less than E3 like that, E1 is the ground state energy. Remember, this spectrum is discrete. So this is less than E2, less than E3, and it goes on like that, the ground state energy. Then, you have associated wave functions, energy eigenstates psi 1 of x, psi 2 of x, psi 3 of x. Here is the theorem. The theorem tells you something about the vanishing of the wave function. It says that psi 1 has no nodes. Psi 2 has one node. Psi 3 has two nodes. And so it goes so that psi n has n minus one node. So psi n is greater than-- well, it's correct. Any n greater or equal to 1 has n minus 1 nodes. Now, there are several ways people show this. Mathematicians show it in a rather delicate analysis. Physicists have an argument as well for this, which is based on approximating any potential by infinite square wells to begin with. So suppose you have a potential like that. Well, think of it as first being a little potential like that, an infinite square well. And you start making the window of the square well bigger. The argument-- it's a neat argument. Maybe you can discuss it in recitation. I would suggest that. It's a good argument. But it's not rigorous. But still, one can do something like that. You make it grow. And what you know is that the infinite square well, the first wave function has no node. And as you change the screen to make the potential really what it's supposed to be and not just that of a square well, the wave function cannot gain a node. On the other hand, what you will show in the homework is something that is a partial result which says that the solution with n plus 1 has at least one more node than the solution with n. So it's part of what you can show. And it doesn't take too much effort. And you can prove it rigorously. So we will assign that eventually for a homework to do. In the homework that you have for the first homework, you also have a problem with delta functions. And I suggest that you read the notes that will be posted today. Because there's an example there with delta functions. If you study that example, you'll find the problem quite easy to solve. You may have solved already. Some of you are very eager to get going with the homework. So it's something you can study first, and then make your life a little easier. So what we're going to do now for the rest of the lecture is consider the variational problem, which is something you probably haven't seen before, the variational problem. This problem has to do with calculus of variations. Now, calculus of variations is something considered fairly advanced. And therefore, as you will see, we will avoid some of the major difficulties of calculus of variations in our discussion of the variational principle. But I wanted to mention a little story about this problem. So this calculus of variations is a more complicated version of maxima and minima in which in maxima and minima of functions you look at the function. And if you could plot it, you could say, here's a maximum, here's a minimum. If you want to figure out where they are, you know. You take a derivative, set it equal to 0, you find the maxima and minima. So the typical calculus problem is one in which you have a function, and you want the maxima and minima. The variational problem is a problem in which you want to maximize or minimize something. But what you don't know is not where the maximum or minimum occurs, but which kind of function will give you this maximum or minimum. So your unknown is not a point where there's a maximum or a minimum but a function where there is a maximum and a minimum. So it's slightly more complicated. So this is the calculus of variations. And people wonder when did it start. It actually seems to have first been discussed by Newton. And it's quite an interesting story. Newton was trying to understand apparently the following problem. If you would have a cross sectional area like this, he asked the question, how should you make a solid out of this by tapering it and ending with this, tapering it, in such a way that as it moves in a viscous fluid, the resistance is the minimum possible-- very complicated problem. And as you can imagine, this is a complicated problem because you're trying to find a shape-- not just a maximum or a minimum of a function but what shape maximizes or minimizes this. So apparently, he solved the problem and wrote it in Principia but didn't explain his solution. And people for years were trying to figure it out. And nobody could figure out how he did it. Then, the story goes that this mathematician Johann Bernoulli in 1696 came up with a challenge to all mathematicians. At that time, people would announce a problem and challenge to see who's smart, who can solve this problem. So Johann Bernoulli in around 1696 poses a problem of, you're given two points in the plane, in the vertical plane like this blackboard, point this, A, and point B in here. You must design the curve of shortest time for fall, so some curve here. If you put an object and let it fall, it will get the fastest to that point, so maybe something that looks like this. It's a complicated curve, or at least not all that simple. And he asked all the people to solve it, gave them one year to solve it. So who was around at that time? Well, one person that got the letter was Leibniz. He got it on the 9th of June of that year, 1696. And he answered it, sent an email back, by the 16th of June with a letter with a complete solution. It's a funny thing that actually apparently Newton was very busy and didn't receive this letter. Or something happened, and he got it one day, and he actually solved the problem in one night. It took him one full night to solve it. Now, you say, well, how brilliant. And true, but given that he had solved this problem, he was criticized as being really slow and-- how come you took 12 hours to solve this problem? So it's quite amazing. There's a lot of Bernoullis. And apparently, this question by Jacob Bernoulli, the main purpose of this question was to demonstrate to everybody that his older brother, Jacob Bernoulli, who had invented the Bernoulli numbers, was actually an incompetent person that could not solve this problem. So that was apparently what he wanted to do. It's a rather famous family. But they obviously didn't get along. But apparently, Jacob did manage to solve the problem. So Jacob Bernoulli, Leibniz, and Newton all solved the problem. Johann Bernoulli, the one that started this problem-- and I think it's maybe with a double N, I'm sorry-- his son is Daniel Bernoulli. And engineers know him, because that's the Bernoulli of the Bernoulli fluid dynamics stuff. So the problem is not all that easy. And calculus of variations determines this shape. So the calculus of variation applied to quantum mechanics asks, here, this function is determined by the principle that it minimizes time. So you have the Schrodinger Equation. And you could ask, you have all these Eigenfunctions. What do they minimize? Is there something they're minimize? And the answer is yes. And this is what you'll understand in the next few minutes. So what is the problem we want to solve? We want to solve the problem h psi equal e psi. So some Hamiltonian. Now my notation will be such that it applies to three dimensions as well. So I'll put just arrows on top of it, and you would have to write the proper Hamiltonian. It will not be necessary for you to know the Hamiltonian. So I'll put psi of x here, meaning that this is equally valid for more than one dimension. Now we want to find solutions of this equation. And you can say, what do they maximize or minimize? Well we won't get to it until 15 minutes. First let's try something simpler. How about, can we learn something about the ground state energy of the system? So let's try to think about the ground state energy. State energy. Now consider ground state energy and we'll consider an arbitrary-- arbitrary is the most important word here-- psi of x that is normalized. Is normalized. So integral the x of psi squared is equal to 1. And here comes the claim. The first claim that we can make. You see, this wave function doesn't solve the Schrodinger equation. That's what we mean by arbitrary. It's just any function of space that is normalizable. Doesn't solve the Schrodinger equation. Never the less, you are commanded to compute the following quantity. This quantity is also by definition what we call the expectation value of the Hamiltonian in the state psi. I love the fact, the remarkable fact that we're going to show now, is that this thing provides an upper bound for the ground state energy for all psi. So let me try to make sure we understand what's happening here. Here it says you don't know the ground state energy but you're going to learn something about it. Something that's interesting is if you know that it has an upper bound, so the ground state energy is definitely not higher than this one, so you learn something. Would be ideal if you had also lower bound so you knew it's in this range. But an upper bound is a nice thing to have. And the claim here is that each time you try an arbitrary function, you put anything here, you ever write, you've got an upper bound. So how is this used? You try arbitrary functions that you think look possibly like the wave function of a bound state. And you get numbers and you already know that the ground state energy smaller than some number. So it's a rather nice way of getting some information about the ground state energy. So this psi effects is called a trial wave function. Is a trial wave function. So is the statement of this theorem clear? Not the proof, but the statement. Do we have questions? Yes. AUDIENCE: Is there any statement about how using the wave function will give us how accurate an estimate [INAUDIBLE] PROFESSOR: No. We're going to become good and figure out some nice way of choosing wave functions, but no. Once you tried you got some information. And you may not know so easily whether you could go much lower. You can try a little, but there's no clear way to know. This is just some partial information. OK. So let me first prove this. We'll prove it and then explain a little more what it all means. So the proof. Now it's a proof under quotation marks. I will make a few assumptions. Basically, that I don't have a continuous spectrum. Now that assumption is done for me to write a simpler proof, not because the result doesn't hold. So the proof is good, but I will just consider for notational purposes no continuous spectrum. So we'll have a ground state energy which is e1 that is maybe less than or equal to e2 less than or equal e3 like that. So you even may consider the [? genorisies. ?] And we have h psi n is equal to en psi n. So what do we have? We have a trial wave function. So your trial wave function since it's an arbitrary function of x should be expandable by completeness as a serious or a superb position of the energy eigenstates. Let me clarify this point. This is a trial wave function. Doesn't solve the Schrodinger equation. So this Doesn't solve this energy eigenstate equation. So in fact, it doesn't solve it because this is a superb position of many in here. So that's consistent with this, and the fact that this wave function as given in here just can be represented using the energy eigenstates. But being a superb position, it's not an energy eigenstate which is true because a trial wave function is something that you invent out of your head. It's not a solution. If you had a solution, you wouldn't need this. So you don't have a solution. You invent the trial wave function, and you have this. A couple of things. The psi squared integral being one. You can do that integral, that condition, is the sum of their bn-2nd is equal to 1. This I use the orthonormality. You can elevate this. It's sort of the kinds of things we're doing last time. Please make sure you know how to do that. Then there's the other computation that we also have sketched last time, which is that the integral of psi star h psi which is what we want to compute, h hat psi is actually bn-2nd en. So that was something we're doing towards the end of last lecture. And this computation takes a few lines, but it was there. It's in the notes. And now comes the kind of thing we want to say. Look at this sum. It has b1, e1, b2, e2, b3, e3, but e2, e3, e4, all those are bigger, or at most, equal to e1. So if I did, here, the following bad joke of substituting en for e1, which is not the same, if I put here bn squared n equals 1 to infinity. I put here e1, well this is bigger than that because e2 is possibly bigger than e1, e3 is bigger than e1. But it may be equal. But at this moment, e1 can go out of the sum. So this is e1 times this sum which is 1 because bn is equal to 1. And e1 is the ground state energy by definition. So the ground state energy is less than this which is the expectation value of the Hamiltonian. Pretty simple, in fact, the proof is really a little too simple. Where do we go from now? Well let's make a more general statement of the variational principal. Again, sometimes it's not all that convenient to have normalized wave functions. So recall that if psi of x is not normalized, psi of x over the square root of integral of psi-2nd dx is. Therefore if you hand me an arbitrary psi of x that is really arbitrary. You don't even bother to normalize it. Then when I plug here in this formula it is supposed to be normalized. So I plug the second expression there. So therefore I get that egs less than or equal to the integral of psi star h psi the x over integral of psi star psi the x. This is actually nicer in one way because you don't have to work with normalized wave functions. And that result must be true still. Yes? AUDIENCE: [INAUDIBLE] PROFESSOR: Sure. It cannot be completely arbitrary, the function should we normalizable. Doesn't have to be normalized but normalizable. So here you got. And here let me introduce just a little name. f of psi. f of psi is what is called a functional. f of is a functional. What is a functional? A functional is a machine or an expression whose input is a function and whose output is a number. So here f of psi is a functional. And maybe I should use brackets. Many times people with brackets denote that, watch out, this is not a function, it's a functional. And here it is. No dash there. You give me a psi of x, which is a function, and then this is a number because you've done the integrals. So that is like the Brachistochrone problem, that a funny name for it. Here it is. There is a function now which is the time that it takes to go here. You give me a path, a function, and I can calculate the time it will take the mass to get here. So this was the issue of finding a critical point of a functional. So actually we start to see, it seems that the ground state energy is the minimum of this functional. And what this interesting as well is that when you get the minimum you will have gotten a ground state wave function. So the ground state wave function actually is the thing that minimizes this functional and gives you some value, the ground state energy. A little more in a couple of minutes. Let's do an example. How do we use this so now if you think about this carefully it's kind of dizzying because what is sell functional, really? It's, in some sense, a function in an infinite dimensional space because a function itself is specified by infinitely many numbers that you can change. So how many dimensions you have is how many ways you can move your hands, but they are linearly independent. But if you have a function, and you can change it, you can change it here, you can change it there, or you can change it there, and those are all orthogonal directions. You're finding a critical point. When you find the critical point, you should imagine that you're plotting a function that is not one dimensional function or two dimensional function, but it's infinitely dimensional function. Direction one, direction two, infinitely many directions. And suddenly in this infinitely many directions you find the critical point. It's really incredible that one can do these things. So you're at that critical point, and you can deform the energy eigenstate by making it a little fatter here or thinner here or up there. And those are the infinite directions in any direction that you will the energy goes up because you're at the global minimum it's pretty amazing. Something that you will prove in the homework is that actually it's even more. Every single eigenstate is a critical point of this functional. So you've got the lowest energy state and in that infinite dimensional space in every direction that you move you go up. The first excited state is another critical point. But it will not be an absolute minimum. It will be a [? saddle ?] an infinite dimensional [? saddle ?] which are infinitely many directions in which you go up. There's one direction in which you go down because you could flow towards the ground state. So the first excited state is the [? saddle ?] but these are all stationary points of this functional. So we'll conclude by doing this example. Sorry, what is the question? AUDIENCE: [INAUDIBLE] PROFESSOR: I didn't assume it's non-degenerate. So if you have two things that have the same ground state, the functional will have, in fact, the degeneracy there will be two solutions that have the same energy. And any linear combination of them will have the same energy. The proof that I did here doesn't assume non-degeneracy, it's even true with degenerate things. So the example is an example for illustration, not for solving something that you can't do otherwise. So it's a delta function potential. v of x is minus alpha delta of x with alpha positive. And the ground state energy is well known. It's minus m alpha squared over 2h-2nd. You've solved this problem many times in 804. So trial wave function. Well you know how it should look, but let's assume you don't. And you say, it's some sort of [INAUDIBLE]. So trial. It would be psi of x equals e to the minus x squared. While this would do, you're going to work hard and you're not going to reap all the benefits of this calculation. So what you should do at this moment is put a constant here. Minus beta squared x squared and I'll put the minus one half. This is our trial wave function. You see, by this, you're going to get an expression. You calculate this number, and you're going to get the function of beta. Beta is not going to disappear. And therefore, you're going to know that the ground state energy is less than this function of beta. And then you can adjust beta to get the best bound. So beta is put as a parameter to begin with, and we hope to use it. So note that integral of psi squared dx in this case is equal to square root of pi over beta. So we have to calculate this whole functional. So this integral of psi star-- well I don't have to bother with psi star because it's real. h psi over psi psi, and what do we get? Well the denominator is easy. So we beta over square root of pi, and let me write the whole thing here. dx the psi would have e to the minus one half beta squared x squared minus h squared over 2m d-2nd dx2nd minus alpha delta of x. And another wave function, e to the minus one half beta squared x-2nd. OK. So you have to evaluate that. And that's the part that is not so much fun. For any integral that you have in 805, we believe that you can use Mathematica or Maple or MATLAB or whatever and do it. The only reason not to use any of these things is if you think you could not do the integral. But once you realize, oh, this is an integral, I know how to do, don't waste time. Use any of those programs. Now, this part of the integral is kind of easy because the delta function just picks the value of 0. So this part of the integral gives you minus beta alpha over square root of pi. The other part of the integral, however, is a little more complicated. Believe it or not, it makes a big difference whether you take the two derivatives of this function or you integrate by parts. If you integrate by parts, you save a lot of work. So let me integrate by parts. This becomes, plus beta over square root of h, h-2nd over 2m, integral dx of ddx of e to the minus one half beta squared x-2nd squared. So you integrate by parts one of the ddx and then you have the other ddx, so it's the thing squared. And that's an easier integral to do. We don't want to bother with this, but this whole thing then becomes minus beta over square root of pi, alpha plus beta squared h squared over 4m. That's the whole answer And that's the valuation of this whole thing. So look what you get. You got a function of beta indeed. So how does that function of beta look? It's 0 for beta equals 0 is 0 for some other possible beta, it's going to look like this. So there's a point at which this function is going to have a minimum, and we should choose that point to get the best upper bound on the function. Our claim is that following from the variational theorem that we've proven is that the e ground state is less than or equal than the minimum value over beta of this beta squared h squared over 4m minus beta square root of pie alpha. So you minimize over beta, and yet still the ground state energy must be a little smaller than that. Well what do you get? You do this, this minimization gives beta equal 2m alpha over h squared square root of pi. It's a little messy but not terrible. And you substitute, and you get that e ground state is less than or equal than m alpha squared over pi h squared. And to better write it as 2 over pi times minus m alpha squared over 2h-2nd which is the true ground state energy. So let's just make sure we understand what has happened. Here is the energy. Here is zero. Energy as just a vertical plot here is zero. The true ground state energy of this problem is this one, let's call it egs, is negative. And we go 2 over pi times that. 2 over pi is about 0.64. So the bound says that here is 0.64 egs. So that's what the bound told you. The bound state energy must be lower than this quantity. We got close. Not impressively close, but the work functional was not all that bad either. Question? AUDIENCE: [INAUDIBLE] always going to be a constant times incorrect energy value or is it just the closest approximation? PROFESSOR: Is it going to be what? AUDIENCE: Is it always going to be a constant times the correct energy value or is it just-- PROFESSOR: Well it typically is like that because of dimensional units. You're looking for a constant because you're not looking for the function. So you will get a number times a correct value, yes, indeed. That's an illustration of the problem of wave functions. You know the variational principle tells you things about the ground state but allows you to find the first excited state as well if the potential is symmetric, will allow you to prove that any attractive potential has a bound state. And you will prove in the homework that stationary points of these things are the eigenfunctions. See you next Monday.

MIT_805_Quantum_Physics_II_Fall_2013

3_Wave_Mechanics_continued_and_SternGerlach_Experiment.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. So today, we'll continue our kind of review that included, of course, the last lecture, the variational principle that's supposed to be new stuff you didn't see in 804. And today, as we continue, we'll talk about position and momentum for about 30 minutes or 40 minutes, and then begin the study of spin. That will be spin-1/2 with a Stern-Gerlach experiment and the mathematics that comes out of it. Now, we will talk about the Stern-Gerlach experiment in quite some detail so that you can appreciate what was going on there. And then we will extract a few of the mathematical lessons that this experiment tells us about quantum mechanics. Immediately after that, which will be probably middle of next lecture, we will pivot. And as we learn this mathematics that the Stern-Gerlach experiment is telling us or asking us for, we will go in some detail on the necessary mathematics for quantum mechanics. We'll talk about vector spaces, linear operators, Hermitian operators, unitary operators, diagonalisation, matrix representations, all kinds of things. That probably will be about two weeks, three lectures at least. So it will be a nice study. And in that way, people that don't have a background in linear algebra will feel more comfortable with what we're going to be doing. And I think even for the people that have a background in linear algebra, you will gain a new appreciation about the concepts that we meet here. So today, we begin, therefore, with position and momentum, and these are operators in quantum mechanics. And they have letters to denote them. x, we put a hat with it, that's a position operator. p, we put a hat on it. And the position and momentum operators don't commute. And the commutator is given by ih bar. Now, we have been dealing so far with wave functions. Our wave functions, where these functions of x and t, they represent the dynamics of your system, the dynamics of your particle as it moves in time. But time, as you are seeing in quantum mechanics, is a little bit of a spectator. It's an arena where things happen. But the operators, and most of the interesting things, are going on without reference to time. Time evolution, you have an expansion of a wave function in terms of energy, eigenstates, at a given time. And then you can evolve it easily with the way we've learned, adding e to the minus i et over h bar for each energy eigenstate. So time will play no role here. So when I talk about the wave function, at this moment you could put the time, but we will talk about the wave functions that have no time dependence. So, say, a psi of x wave function. So this psi of x may be the true wave function at time equals 0, or you could just simply think of it as the psi of x. Now, this wave function means that we're treating x in a particular way, and we say that we're working in the x representation, the position representation. Now, this means that we have an easy way to figure out what this operator does when it acts on this function. So what it acts on this function, it will give you another function, and the definition of this is that the position operator acting on the function psi of x is defined to be another function, which is the function x times psi of x. Well, we're talking about these wave functions and operators on wave functions. And a recurrent theme in quantum mechanics is that we will think of wave functions, sometimes we call them states. Sometimes we call them vectors. And we basically think of wave functions as vectors. And things that act on wave functions are the things that act on vectors. And the things that act on vectors, as you know in mathematics, is matrices. So we're compelled, even at this early stage, to get a picture of how that language would go if we're talking about these things. So how do we think of a wave function as a vector? And how do we think of x as a matrix? So there's a way to do that. It will not be totally precise, but it's clear enough. So suppose you have a wave function, and we're interested in its values from 0 up to a. This wave function is a function of x between 0 and a. So it's the psi of x for x between a and 0. That's all the information. What we're going to do is we're going to divide this thing, this line, this segment, into a lot of pieces. And we're going to say, look, instead of writing a function like sine of x or cosine of x, let's just give the values and organize them as if this will be a vector of many components. So let's divide this in sizes epsilon, such that N times epsilon is equal to a. So there are N of these intervals. So we think of psi as a vector whose first component is psi at 0. The second is psi at epsilon. The third is psi at 2 epsilon. And the last one is psi at N epsilon. And depending on how much accuracy you want to work with, you take epsilon smaller and larger, keeping a constant. And this would be like summarizing all the information of a function in a vector. Now, that's intuitively a nice way to think of it. May look, with your background in classical physics, a little strange that we sort of put the value at 0 along the x-axis, first component, the value at epsilon along the y, the value of 2 epsilon along the z. But we need more axes. So you need many axes here. In this case, this is a N plus 1 column vector. It has N plus 1 entries, because 0 up to N, that's N plus 1 entries. But that's a fine way of thinking of it. Not exact because we have an epsilon. In this way of thinking about the wave function, we can then ask, what does the matrix x hat look like? So x hat is an operator, and it acts this way. So here is how it looks like. We would think of x hat as an N plus 1 times N plus 1 matrix. And its entries are 0 everywhere, except in the diagonal, where they are 0 epsilon, 2 epsilon, up to N epsilon. And here is a big 0 and a big 0. This, I claim, is the way you should think of the x operator if you thought of the wave function the way we wrote it. And how do we check that? Well, x operator acting on psi should be this acting on that. And then, indeed, we see that if x hat is acting on psi of x, what do we get? Well, it's easy to multiply a diagonal matrix times a vector. Here you get 0 times psi of 0. You get a vector, so let me make this thinner. Then I get epsilon times psi of epsilon, 2 epsilon times psi of 2 epsilon, up to N epsilon times psi of N epsilon. And indeed, that matrix looks like the matrix associated with this wave function because here is the value at 0 of this wave function. Here is the value at epsilon of this wave function, and so on. So this has worked out all right. We can think of the wave function as a column vector, and then the position operator as this vector as well. Now, given that we know how the x operator is defined, we can also think easily about what is the expectation value of x on a wave function. Something that you really know, but now maybe becomes a little clearer. Here you're supposed to do psi star of x times the x operator acting on psi of x. But we have the definition of this, so this is, as you imagine, dx-- and I should put primes maybe, well, I don't have to put primes-- dx psi star of x x psi of x, which is what you would have done anyway. Well, given that we've started with this, we can ask also, is there eigenstates of the x operator? Yes, there are. but then fortunately, are a bit singular. So what should be an eigenstate of x? It's some sort of state. Intuitively, it has a definite value of the position. So it just exists for some value of x. So it's naturally thought as a delta function. So let me define a function, psi sub x0 of x. So it's a function of x labeled by x0, and define it to be delta of x minus x0. So I claim that is an eigenstate of x hat. x hat on psi x0 of x is equal, by definition, to x times psi x0 of x, which is x times delta of x minus x0. And when you multiply a function of x times a delta function in x, it is possible to evaluate the function that is being multiplied by the delta function at the place where the delta function fires. It has the same effect on integrals or anything that you would do. So here, this is equal to x0 times delta x minus x0. You evaluate the x at x0. And finally, this is x0 times that function psi x0 of x. And therefore, you've shown that this operator acting on this function reproduces the function-- that's the definition of eigenstate as an operator-- and the eigenvalue is the number that appears here, and it's x0. So this function is an eigenstate of x hat with eigenvalue, e.v., x0. The only complication with this eigenfunction is that it's not normalizable. So it doesn't represent the particle. It can be used to represent the particle, but it's a useful function. You can think of it as something that can help you do physics, and don't insist that it represents a particle. So this is the story for position. And the position gets actually more interesting as soon as you introduce the dual quantity, momentum. So what is momentum here? So momentum is an operator, and this operator must be defined. Now, you had a shorthand for it in 804, which is p hat equal h bar over i d dx. And this shorthand means actually that, in what we call the position representation where we're using wave functions that depend on x, well, the momentum is given by this operator. And the story of why this was the case was sort of something that was elaborated on in 804, the work of de Broglie, that saw that the wavelength of the wave has to do with the momentum of a wave. And finally, people understood that this would measure the momentum of the wave. So this is the operator. And therefore, in the representation that we're working-- representation is a word that has a lot of precise meaning, but now I'm just using it in the sense that, well, we're working either with x's or with p's. And we're working with x's. That's why p looks like something to do with x. So what is p hat on a wave function? Well, that's what this means. It's another wave function obtained by taking the x derivative. So that's the definition of it acting on a wave function. The one thing that must be verified, of course, is that this definition is consistent or implies this commutation relation. So you've defined it as an operator. x, we've defined it as an operator. But most of us think that it doesn't look like an operator is multiplying. But it is an operator. So this one does look like an operator. It's a differential operator. And you can try to see if this equation is true. And the way to test these commutators is something that, again, I don't think is unfamiliar to you, but let's go through it, is that you try to evaluate this product of operators acting on a wave function. And if things work out well, we'll see you should get ih bar times that wave function. If that is the case, you say, OK, I've proven that equation, because it's an operator equation. The left-hand side of that equation is the product in different orders of two operators, therefore it's an operator. The right-hand side is another operator. It's the operator multiplied by ih, anything that you'll get. Well, if this is an operator identity, the operator on the left must be equal to the operator on the right, which just means that, acting on anything, they must give the same answer. So if I managed to prove that this is equal to this, I've proven that for anything that is the answer. And therefore, I can write the top one. And let me just do it, even though this may be kind of familiar to many of you. It's good to do this slowly once in your life. So let's go through this. So this says x operator p operator on psi minus p operator of x operator on psi. When you have several operators, like ABC acting on psi, this really means let C act on psi, and then let B act on C psi, and then let A act on that. The operators act one by one. The closest one acts first. So here I'm supposed to let B act on psi, but that means that thing. So now x is acting on h over i d psi dx. On this one, I have p acting on x psi, because that's what x hat psi is. Here, this is multiplication by x of a function of x. So this is just h over i x d psi dx. And here, I have h over i d dx of this whole thing x psi. And you can see that when you act here, you act first on the x, and you get something. And then you act on the psi, and you get this same term. So the only contribution here is equal to minus h over i, the d dx on x times psi, which is ih bar psi, which is what I wanted to show. So this is true. And therefore, you could say that this definition is consistent with your definition of x, and they represent this operator. One more thing you could try to do, and it's fun to do it, is we had a matrix representation for x. Can I think of p as a matrix? How would you do it? What kind of matrix would p look like? Well, yes? AUDIENCE: You just generate a finite difference equation. PROFESSOR: You could do it, exactly, with taking finite differences. So for example, if you think that you want to produce the wave function psi prime at 0, psi prime at epsilon, psi prime, that's what the derivative gives you, you'll write this as 1 over epsilon, say, psi at epsilon minus psi at 0. That's the derivative at 0 roughly. It would be psi at 2 epsilon minus psi at 0 over 2. And you could build it. You could build it. I'm not going to do it. You may want to do it and try and see how the derivative operator looks as a matrix. And then if you really want to spend some time thinking about it, you could try to see if this matrix and this matrix commute to give the right answer. And as you try it, you will figure out all kinds of funny things that we will talk about later in the course. So you can represent the momentum operator as a matrix indeed, and there are interesting things to say about it, and it's a good subject. So let's continue with the momentum and ask for eigenstates of the momentum. So eigenstates of p, you know them. They're e to the ipx things. So let's write them with some convenient normalization. This is an [INAUDIBLE] wave function that depends on x with momentum p. And we'll write it, as a definition, as e to the ipx over h bar, and I'll put it a 2 pi h bar here. It's kind of a useful normalization. Then p hat on psi p of x, well, p hat is supposed to take h over i d dx, and take h over i d dx of psi p. And h over i cancels the i over h. When you take the d dx, you get p out, and you get the same wave function. So indeed, you get p times psi p of x. So indeed, this is the eigenstate of the momentum operator, and it has momentum p. Well, what is the use of this? Well, say you have a representation, what we call the position representation, of the wave function and operators. Let us think now of the momentum representation. So what does all that mean? Well, there is the Fourier transform operation in which we have psi of p. Well, let me write it this way, actually. I'll write any psi of x physically can be represented as a sum of momentum eigenstates. Therefore, that's Fourier's theorem, minus infinity to infinity dp e to the ipx over h bar square root of 2 pi h psi tilde of p. That's Fourier transformation, defines psi tilde of p. And Fourier's theorem is the fact that not only you can do that, but you can invert it so that psi tilde of p can also be written as an integral, this time over x from minus infinity to infinity e to the minus ipx over h bar, also 2 pi h bar psi of x. So let's ponder this equation for a couple of minutes. Well, as a physicist, you think of this, well, this is telling me that any wave function could be written as a superposition of momentum eigenstates. Here are the momentum eigenstates. And for each value of momentum, you have some coefficient here that tells me how much of that momentum eigenstate I have. Now, here is the opposite one. Psi tilde of p and psi of x are related in this way. So these coefficients, if you want to calculate them, you calculate them this way. But now let's think of it as a change of representation. The physics is contained in psi of x. All what you wish to know about this physical system in quantum mechanics is there in psi of x. But it's also there in psi of p, because they contain the same information. So there are different ways of encoding the same information. What is the relation between them? This, we thought of it as a vector, vector in position space, an infinite dimensional space that is talking about positions. This is another vector in momentum space. Think of it now the infinite line. So this is an infinite vector with all those points little by little, from minus infinity to plus infinity, all of them there, gigantic vector. And here is another gigantic vector with p from minus infinity to infinity. And in between, there's an integral. But now, with your picture of quantum mechanics, you see an integral, but you also see a matrix. And what is this matrix? Think of this as some sort of psi sub p. And this as some sort of matrix, px psi x, in which if you have a product-- you'll remember when you multiply matrices, a matrix on a vector, you sum over the second index. That's the product for matrix. And then the first index is the index here. So here is what it, more or less, is like. Psi tilde of p [? subtend ?] by this, and this matrix depends on two labels, p and x, and it's that. So it's a matrix full of phases. So how do you pass from the coordinate representation of the information, a vector of all values of the wave function in all positions? By multiplying with this matrix of phases that is here, and it gives you this representation. So different representations means using different vectors to represent the physics. And this vector is a very nice one. And because of these properties of the momentum operator and all these things, this vector is also a very nice one. And there's an integral transform or some sort of infinite matrix product that relates them. And we shouldn't be uncomfortable about it. That's all fine. So we say that we have, for example, psi of x as one representation of the state and psi tilde of p as another representation of the same physics. We can do one more thing here, If I continue. We can take that boxed equation on the blackboard up there and act with h bar over i d dx on psi of x. So that is equal to h i d dx, and I'll write what psi of x is, is minus infinity to infinity dp e to the ipx over h bar square root of 2 pi h bar psi tilde of p. Now, when we act on this, as you know, h bar over i d dx just acts on this and produces the factor of p. So this is equal to minus infinity to infinity dp e to the ipx over h bar over square root of 2 pi h bar p times psi tilde of p. So look at this equation again. This double arrow is to mean that there are equivalent physics in them. They have the same information. It's the same data encoded in a different way. And that different way, this arrow is Fourier transformation. And this Fourier transformation is explained here. So now you have Fourier transformation the same way. So here we have-- what we've learned is that h over i d dx of psi is represented in momentum space by p psi tilde of p. And this was p hat acting on psi of x. So the corresponding thing in momentum space of p hat acting on psi of x is p multiplying psi tilde of p, which is to say that we can think of the abstract operator p hat acting on psi tilde of p as just p psi tilde of p. So in momentum space, the operator p hat acts in a very easy way. In coordinate space, it takes derivatives. In momentum space, it's multiplicative. So in position space, x is multiplicative. But in momentum space, x would not be multiplicative. x would also be a derivative. So I leave it for you as an exercise to show that or convince yourself in several ways, that x hat is really i h bar d dp in p space, in i h bar d dp. All right. So that's really all I wanted to say about position and momentum operators at this moment. They will come back when we'll introduce bra-ket notation in detail. We'll revisit this a little. But the main concepts have been illustrated. Are there questions? We're about to leave this, so if you have any questions at this moment. Yes? AUDIENCE: Could you explain again how you used this [INAUDIBLE] h bar over i d dx assign to [INAUDIBLE]? PROFESSOR: Right. So the question was, why did I associate these things? So it really goes back here to what the meaning of this arrow is. The meaning of this arrow is Fourier transformation. So this psi tilde and psi of x are related in this way. That's Fourier transformation, and that's what we mean by this arrow. It also means that whatever physics you have here, you have it there. So really, when you have something acting on a state, for example, if you have some operator acting in here, well, you get a new wave function. And there should be one on the right that corresponds to it, that has the same information as the one in which you've acted with something. So what we claim here is that, also in the sense of Fourier transformation or having the same information, h bar over i, the derivative of psi, is encoded by this. So we say, thinking abstractly, what is this? This is the momentum operator. Therefore, I'm going to say that the momentum operator really is the same momentum operator, whether it acts on wave functions that you show them to mean this way or wave functions that, because you're in another mood, you decide to give them to me in momentum space. So as you change your mood, the operator takes different forms but is doing the same thing. It's totally reversible. It's acting on that-- you see, the operator is always the same, but you give me the data in two different ways, then the operator has to do the thing in a different way. So that's what it means that the operator has different representations. In the [INAUDIBLE] representation, it looks like a derivative. In the momentum representation, it looks like multiplying. Other questions? Yes? AUDIENCE: So by saying that they sort of represent [INAUDIBLE] to the same positions, does that mean that h bar over i p e to the xi and p psi p are like the same [INAUDIBLE]? PROFESSOR: That h bar over d dx psi and p-- yeah. They are the same data, the same state represented in different ways. Yeah. All right. So time for a change. We're going to talk about Stern-Gerlach and spin. Now, spin will keep us busy the biggest chunk of this semester. So it will be spin-1/2, and we're really going to go into enormous detail on it. So this is just the beginning of the story that will be elaborated at various stages. So at this moment, I will talk about this experiment that led to the discovery of spin, and if you try to invent the theory that describes this experiment, what you would possibly begin doing. And then we go through the mathematics, as I mentioned to you, for maybe a week and a half or two weeks, and then return to the spin with more tools to understand it well. So the subject is the Stern-Gerlach experiment, Stern-Gerlach experiment. So the Stern-Gerlach experiment was done in Frankfurt, 1922. It was an experiment that, in fact, people were extraordinarily confused. It was not clear why they were doing it. And for quite a while, people didn't understand what they were getting, what was happening with it. In fact, Pauli had thought that the electron has like two degrees of freedom and didn't know what it was, those two degrees of freedom. Kronig suggested that it had to do somehow with the rotation of the electron. Now, Pauli said that's nonsense. How can an electron rotate and have angular momentum because it has a rotation? It would have to rotate so fast, even faster than the speed of light to have the angular momentum, and then this little ball that would be the electron would disintegrate. And it made no sense to him that there would be such a thing. So Kronig didn't publish this. Then there were another two people, Uhlenbeck and Goudsmit, at the same time, around 1925, had the same idea, angular momentum of this particle. And their advisor was Ehrenfest, and said it doesn't make too much sense, but you should publish it. [LAUGHTER] And thanks to their publishing, they are given credit for discovering the spin of the electron. And Pauli, a couple of years later, decided, after all, I was wrong. Yes, it is spin, and it's all working out. And 1927, five years after the experiment basically, people understood what was going on. So what were these people trying to do? First, Stern and Gerlach were atomic physicists, and they were just interested in measuring speeds of thermal motion of ions. So they would send beams of these ions and put magnetic fields and deflect them and measure their velocities. And eventually, they were experts doing this kind of thing. And they heard of Bohr, that said that the electron has angular momentum and is going around the proton in circles, so it might have angular momentum. They said, oh, if it has angular momentum because it's going around the proton, maybe we can detect it. And when they did the experiment, they got something. And they said, well, we're seeing it. But it was not that. They were not seeing the orbital angular momentum of the electron because that electron in these silver atoms actually has no angular momentum, as we will see, no orbital angular momentum. It only has spin. So they were actually seeing the spin. So it was a big confusion. It took some time. Basically, they took the beam, and they split it with a magnetic field, and the clean split was something nobody understood. So they called it space quantization, as of it's separated in space. Space is quantized. A pretty awful name, of course. There's nothing quantized about space here. But it reflects that when you don't know what's really happening, your names don't come out too well. So what we have to understand here, our goal today is to just see what's happening in that experiment, quantify a bit the results, and then extract the quantum mechanical lessons from it. So let us begin with the important thing. You don't see the spin directly. What you see is magnetic moments. So what's that? So what are magnetic moments? Magnetic moments, mu, is the analog, the magnetic analog of an electric dipole. A mu is called a magnetic dipole. You say it has a magnetic moment. And the magnetic moment is given by I times the area. What does that mean? Well, a precise discussion would take some time. But roughly, you can simplify when you think of a loop that is in a plane, in which case there's an area associated to it. And if the loop is this one, the area is defined as the normal vector to the oriented loop. So an oriented loop has an area vector. And the orientation could be focused the direction of the current. There is some area. And the magnetic moment is given by this thing. It points up in the circumstances when this current goes like that. So that's a magnetic moment. A little bit of units. The way units work out is that mu B-- magnetic moments and magnetic fields have units of energy. So magnetic moments you could define as energy, which is joules, divided by tesla, or ergs divided by gauss, because mu B has units of energy. So how do magnetic moments originate in a charge configuration? Well, you can simply have a little current like that. But let's consider a different situation in which you have a ring of charge, a ring of charge of some radius R. It has a total charge Q, and it has a linear charge density lambda. It's uniform, and it's rotating with some velocity v. If you wish, it also has a mass M. There are all kinds of [? parameters. ?] How many? Mass, charge, radius, and velocity. Here we go. We have our solid ring of charge rotating, and we want to figure out something quite fundamental, which is the origin of this principle. We said, you really never see spins directly. You never see this intrinsic angular momentum directly. You see magnetic moments. But then actually what happens is that there's a universal relation between magnetic moments and angular momentum. This is a key concept in physics. Maybe you've seen it before. Maybe you haven't. Probably you might have seen that in 802. So how does that go? Let's calculate the magnetic moment. So the current is the linear charge density times the velocity. The linear charge density is the total charge divided by 2 pi R times the velocity. Now the area, to give the magnetic moment, we'll have mu is equal to I times the area. So it would be this Q times 2 pi R v times the area, which would be pi R squared. So the pi's cancel, and we get 1/2 QvR. OK. 1/2 QvR, and that's fine and interesting. But OK, depends on the radius, depends on the velocity. So here is the magnetic moment is supposed to be going up. But what else is going up? The angular momentum of this thing is also going up. So what is the magnitude of the angular momentum L? L is angular momentum. Well, it's the mass times the momentum-- it's the mass momentum cross R, so MvR. The momentum of R cross p for each piece, contributes the same, so you just take the total momentum. This really is 0, but add them up little by little, and you've got your MvR. So here you have vR, so here you put 1/2 Q over M MvR. And you discover that mu is equal to 1/2 Q over M L. So maybe write it better-- Q over 2M L. I'm sorry, this is the normal. The M shouldn't change, M. And I box this relation because an interesting thing has happened. All kinds of incidentals have dropped out. Like the velocity has dropped out. The radius has dropped out as well. So if I have one ring with this radius and another ring with a bigger radius, the relation between mu and L is the same, as long as it's rotating with the same speed. So this is actually a universal relation. It is not just true for a little ring. It's true for a solid sphere or any solid object axially symmetric. It would be true. You could consider any object that is axially symmetric, and then you start considering all the little rings that can be built. And for every ring, mu over L is the same, and they all point in the same direction. Therefore, it's true under very general grounds. And that is a very famous relation. So now you could speculate that, indeed, the reason that a particle may have a magnetic moment if it's made by a little ball of charge that is rotating. But that was exactly what Pauli didn't like, of course. And you would like to see what's really happening with particles. So when you think of a true quantum mechanical particle, let's think of a particle in general, a solid particle rotating. We'll change the name to S for spin angular momentum. Because that little part, this is just one particle. We're not thinking of that little particle going around a nucleus. We're thinking of that little particle rotating. So this is a little piece of that little particle that is rotating. So you could ask, if, for the electron, for example, is it true that mu is equal to e over 2 mass of the electron times its spin? So this would be a vindication of this classical analysis. It might be that it's related in this way. So actually, it's not quite true, but let's still improve this a little bit. In terms of units, we like to put an h bar here and a 2Me. And put spin here, angular momentum, divided by h. Because this has no units, h bar has the units of angular momentum, x times p. It's the same units, so units of angular momentum. So h bar would be convenient. So that over here, you would have units of a dipole moment, or magnetic moment, magnetic moment units. So what does happen for the electron? Well, it's almost true, but not quite. In fact, what you get is that you need a fudge factor. The fudge factor is that, actually, for elementary particles, you have a g, which is a constant, which is the fudge factor, e h bar 2 over M of the particle S over h bar. Sometimes called the Lande factor. You must put a number there. Now, the good thing is that the number sometimes can be calculated and predicted. So when people did this, they figured out that for the electron the number is actually a 2. So for the electron, g of the electron is equal to 2. Now that, you would say, cannot be an accident. It's twice what you would predict sort of classically. And the Dirac equation, the relativistic equation of the electron that you have not studied yet but you will study soon, predicts this g equal to 2. It was considered a great success that that equation gave the right answer, that people understood that this number was going to be 2. So for the electron, this is 2. So this quantity is called-- it's a magnetic dipole moment-- is called mu B for Bohr magneton. So how big is a mu B? It's about 9.3 times 10 to the minus 24 joules per tesla. AUDIENCE: Professor. PROFESSOR: Yes? AUDIENCE: [INAUDIBLE]. So where exactly does the fudge factor come in? Is it just merely because [INAUDIBLE]?? PROFESSOR: Right. So the classical analysis is not valid. So it's pretty invalid, in fact. You see, the picture of an electron, as of today, is that it's a point particle. And a point particle literally means no size. The electron is not a little ball of charge. Otherwise, it would have parts. So an electron is a point particle. Therefore, a point particle cannot be rotating and have a spin. So how does the electron manage to have spin? That you can't answer in physics. It just has it. Just like a point particle that has no size can have mass. How do you have mass if you have no size? You get accustomed to the idea. The mathematics says it's possible. You don't run into trouble. So this particle has no size, but it has an angular spin, angular momentum, as if it would be rotating. But it's definitely not the case that it's rotating. And therefore, this 2 confirms that it was a pointless idea to believe that it would be true. Nevertheless, kind of unit analyses or maybe some truth to the fact that quantum mechanics changes classical mechanics. Turns out that it's closely related. For the proton, for example, the magnetic moment of the proton is quite complicated as well because the proton is made out of quarks that are rotating inside. And how do you get the spin of the proton and the magnetic moment of the proton? It's complicated. The neutron, that has no charge, has a magnetic moment, because somehow the quarks inside arrange in a way that their angular momentum doesn't quite cancel. So for example, the value for a neutron, I believe, is minus 2.78 or something like that. It's a strange number. Another thing that is sort of interesting that is also true is that this mass is the mass of a particle. So if you're talking about the magnetic moment of the proton or the neutron, it's suppressed with respect to the one of the electron. The electron one is much bigger because, actually, the mass shows up here. So for a neutron or a proton, the magnetic moment is much, much smaller. So, in fact, for an electron then, you would have the following. Mu is equal to minus g, which is 2, mu B S bar over h. And actually, we put the minus sign because the electron has negative charge. So the magnetic moment actually points opposite. If you rotate this way, the angular momentum is always up. But if you rotate this way and you're negative, it's as if the current goes in the other direction. So this is due to the fact that the electron is negatively charged. And that's the final expression. So OK, so that's the general story with magnetic moments. So the next thing is, how do magnetic moments react when you have magnetic fields? So that is something that you can calculate, or you can decide if you have a picture. For example, if you have a loop of charge like this, and you have magnetic field lines that go like this, they diverge a bit. Let me see you use your right-hand rule and tell me whether that loop of current will feel a force up or down. I'll give you 30 seconds, and I take a vote. Let's see how we're doing with that. And I'll prepare these blackboards in the meantime. All right. Who votes up? Nobody. Who votes down? Yeah, [INAUDIBLE]. Down, exactly. How do you see down? Well, one way to see this, look at the cross-section. You would have this wire here like that. The current is coming in on this side and going out this way. Here you have the field lines that go through those two edges, and the magnetic field is like that. And the force goes like I cross B. So I goes in, B goes out. The force must be like that, a little bit of force. In this one, I cross B would be like that, a little bit of force. Yep. Has a component down because the field lines are diverging. So what is the force really given by? The force is given by the gradient of mu dot B. This is derived in E&M. I will not derive it here. This is not really the point of this course. But you can see that it's consistent. This is saying that the force goes in the direction that makes mu dot B grow the fastest. Now mu, in this case, is up. So mu dot B is positive, because mu and the magnetic field go in the same direction. So mu dot b is positive. So the force will be towards the direction-- that's what the gradient is-- that this becomes bigger. So it becomes bigger here, because as the field lines come together, that means stronger magnetic field. And therefore, mu dot B would be larger, so it's pointing down. If you have a magnetic field that is roughly in the z direction, there will be a simplification, as we will see very soon. So what did Stern and Gerlach do? Well, they were working with silver atoms. And silver atoms have 47 electrons, out of which 46 fill up the levels and equal 1, 2, 3, and 4. Just one lone electron, a 5s electron, the 47th electron, it's a lonely electron that is out in a spherical shell, we know now with zero orbital angular momentum. It's an S state. And therefore, throwing silver atoms through your apparatus was pretty much the same thing as throwing electrons, because all these other electrons are tied up with each other. We know now one has spin up, one spin down. Nothing contributes, no angular momentum as a whole. And then you have this last electron unpaired. It has a spin. So it's like throwing spins. So moreover, throwing spins, as far as we're concerned, Stern and Gerlach wouldn't care. Because of these relations, it's throwing in dipole moments. And they would care about that because magnetic fields push dipole moments up or down. Therefore, what is the apparatus these people had? It was sort of like this, with an oven, and you produce some silver atoms that come out as a gas, a collimating slit. Then you put axes here-- we put axes just to know the components we're talking about. And then there's magnets, some sort of magnet like this, and the screen over there. So basically, this form of this magnet that I've tried to draw there, although it's not so easy, if I would take a cross-section it would look like this. So the magnetic field has a gradient. The lines bend a bit, so there's a gradient of the magnetic field. And it's mostly in the z direction, so z direction being pointed out here. So there's the magnetic field. The beam then comes here. And the question is, what do you get on this screen? Now, I have it a little too low. The beam comes there and goes through there. So the analysis that we would have to do is basically an analysis of the forces. And relatively, we don't care too much. The fact is that there's basically, because the magnetic field is mostly in the z direction and varies in z direction, there will be a force basically in the z direction. Why is that? Because you take this, and you say, well, that's roughly mu z Bz, because it's mostly a magnetic field in the z direction. And mu is a constant, so it's basically gradient of Bz. Now, that's a vector. But we're saying also most of the gradient of Bz is in the z direction, so it's basically dBz dz. Now, there is some bending of the lines, so there's a little bit of gradient in other directions. But people have gone through the analysis, and they don't matter for any calculation that you do. They actually average out. So roughly, this gradient is in the z direction. I'm sorry, the gradient is supposed to be a vector. So you get a force in the z direction. And therefore, the thing that people expected was the following. You know, here comes one atom, and it has its magnetic moment. Well, they've all been boiling in this oven for a while. They're very disordered. Some have a z component of magnetic-- the magnetic moment is pointing like that, so they have some component, some down. Some are here. They have no component. It's all Boltzmann distributed all over the directions. Therefore, you're going to get a smudge like this. Some ones are going to be deflected a lot because they have lots of z component of angular momentum or z magnetic moment. Others are going to be deflected little. So this was the classical expectation. And the shock was that you got, actually, one peak here, an empty space, and another peak there. That was called space quantization. Stern and Gerlach worked with a magnetic field that was of about 0.1 tesla, a tenth of a tesla. And in their experiment, the space quantization, this difference, was 1/5 of a millimeter. So not that big, but it was a clear thing. It was there. So everybody was confused. They thought it was the orbital spin, angular momentum that somehow had been measured. At the end of the day, that was wrong. It couldn't have been that. People understood the Bohr atom, realized, no, there's no angular momentum there. The idea of the spin came back, and you would have to do a calculation to determine what is the value of the spin. So the exact factor took a while to get it right. But with the idea that mu z is equal to minus 2 Bohr magenton Sz over h bar, which we wrote before. Well, mu z, if you know the strength of your magnetic field, you can calculate the deflections. You know what mu B is. So therefore, you get the value for Sz over h. And experiments suggested that Sz over h was either plus or minus 1/2. And this kind of particle, it has Sz over h bar equal plus or minus 1/2, is called the spin-1/2 particle. So again, from this equation, this can be measured. And you then use this, and you get this value. So the experiment is a little confusing. Why did this happen? And how do we think of it quantum mechanically? Now 804 sort of began with these kind of things. And you know by now that what's happening is the following, that somehow, mathematically, every state is a superposition of a spin up and a spin down. So every particle that goes there has half of its brain in the spin up and half of its brain in the spin down. And then as it goes through the magnetic field, this thing splits, but each particle is in both beams still. And they just have this dual existence until there's a screen and there's detectors. So they have to decide what happens, and then either collapses in the top beam or lower beam. Nothing happens until you put the screen. That's what we think now is the interpretation of this experiment. But let's use the last few minutes to just write this in terms of boxes and get the right ideas. So instead of drawing all that stuff, we'll draw a little box called a z hat box, a Stern-Gerlach apparatus. In comes a beam, out would come two beams, Sz equal h bar over 2 and Sz equal minus h bar over 2. And the convention is that the plus goes up and the minus goes down, which I think is probably consistent with that drawing. And that's the Stern-Gerlach apparatus. It measures Sz, and it splits the beam. Each particle goes into both beams until there's a device that measures and decides where you go. So you can do the following arrangements. So here's arrangement number 1, a Stern-Gerlach device with z. Then you block the lower one and let the top one go as Sz equal h bar over 2. And then you put another Stern-Gerlach machine, z hat, that has two outputs. And then you ask, what's going to happen? And the experiment can be done and, actually, there's nothing here coming out, and all the particles come out here with Sz equal h bar over 2. What are we going to learn from this? In our picture of quantum mechanics, we're going to think of this as there are states of the electron that have-- and I will write them with respect to z-- they have plus h bar over 2 and states that have minus h bar over 2. And what we will think is that these are really old basis states, that any other state, even one that points along x, is a superposition of those two. This is a very incredible physical assumption. It's saying this system is a 2-dimensional complex vector space, two vectors, two unit, two basis vectors. And from those two, all linear combinations that are infinite represent all possible spin configurations. And what is this saying? Well, as we will translate it into algebra, we will say that, look, here is a state plus. And when you try to measure, if it had any minus component, it had nothing. So we will state that as saying that these states are orthogonal. The minus state and the plus state have zero overlap. They are orthogonal basis states. And, for example, well, you could also do it this way. That would also be 0. And you could also say that z plus and z plus is 1, because every state that came in as a plus came out as a plus. They had perfect overlap. So these are two orthonormal basis vectors. That's what this seems to suggest. And it's a little strange, if you think, because there's a clash between arrows and the notion of orthonormality. In 3-dimensional vectors, you think of this vector being orthogonal to this. But you wouldn't think of this vector as being orthogonal to that one. And here is the spin is up, and this is the spin down. And those two are orthogonal. You say, no, they're anti-parallel. They're not orthogonal. No, they are orthogonal. And that's the endlessly confusing thing about spin-1/2. So these states, their pictures of the spins are arrows. But don't think that those arrows and the dot product give you the orthogonality, because this is up and down. If you would be doing the dot product of an up and down vector, you would not get 0. But this is 0. Then you do the following experiment. So let's do the next one. And the next one is, again, the z filter. Take this one, block it. Then you put an x filter. And what actually happens is that you would get states with Sx, now, h bar over 2 and Sx equal minus h bar over 2, because it's an x filter. The magnetic field is a line in the x direction. Now, all these things have Sz equal h bar over 2. And what happens in the experiment is that 50% of the particles come out here and 50% come out there. So a spin state along the x direction has some overlap with a spin state along the z direction. Normal vectors, a z vector and an x vector, are orthogonal. Not here for spins. The spin pointing in the z and the spin pointing in the x are not orthogonal states. They have overlaps. So this means that, for example, the x plus state and the z plus state have an overlap. This is notations that-- we're going to be precise later. But the same thing with the x minus state, it has an overlap, and somehow they're about the same. Finally, the last experiment is this, z hat, block again, x hat, but this time block one. So here is a state with Sx equals minus h bar over 2. Here is a state with Sz equal h bar over 2. And now you put the z machine again. And what happens? Well, there's two options. People who were inventing quantum mechanics no wonder thought about them. Here they could say, look, I filtered this thing, and now all these electrons have Sz equal h bar over 2. And now all these electrons have Sx equal minus h bar over 2. Maybe, actually, all these electrons have both Sz equal h over 2 and that because I filtered it twice. So it maybe satisfies both. So if all these electrons would have Sz equals h over 2 and this, then you would only get something from the top one. But no, that's not what happens. You get in both. So somehow, the memory of these states coming from Sz equals h over 2 has been destroyed by the time it turned into a state with Sx equal minus h over 2. And a state cannot have simultaneously this and that. That's two properties, because you get 50% here and 50% there. So we'll discuss next time a little more about these relations and how can the states be related, the ones that we use as the basis vectors and all the others along x and others that we could build some other way. All right. See you next week. There's office hours today, 5:00 to 6:00, Monday, 4:30 to 5:30.

MIT_805_Quantum_Physics_II_Fall_2013

11_Uncertainty_Principle_and_Compatible_Observables_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Going to get started right away. I want to make a comment about energy time uncertainty relations. And we talked last time about the fact that the energy time uncertainty relation really tells you something about how fast a state can change. So an interesting way to try to evaluate that is to consider a system that has to state that time equals 0 and compute the overlap with a state at time equals t. Now this overlap is a very schematic thing. It's a bracket. It's a number. And you know what that means. You can keep in mind what that is. It's an integral over space possibly of psi star at t equals 0 x psi t x. It's a complete integral, its inner product. So you may want to understand this, because, in a sense, this is telling you how quickly a state can change. At time equals 0, this overlap is just 1. A little later, this overlap is going to change and perhaps after some time the overlap is going to be 0. And we're going to say that we actually have changed a lot. So this number is very interesting to compute. And in fact, we might as well square it, because it's a complex number. So to understand better what it is we'll square it. And we'll have to evaluate this. Now how could you evaluate this? Well, we'll assume that this system that governs this time evolution has a time independent Hamiltonian. Once this evolution is done by a time independent Hamiltonian, you can wonder what it is. Now it's quite interesting, and you will have to discuss that in the homework because it will actually help you prove that version of the time energy uncertainty relationship that says that the quickest time state can turn orthogonal to itself is bounded by some amount. Cannot do it infinitely fast. So you want to know how fast this can change. Now it's very surprising what it depends on, this thing. Because suppose you had an energy eigenstate, suppose psi at time equals 0 is an energy eigenstate. What would happen later on? Well, you know that energy eigenstates evolve with a phase, an exponential of e to the minus iht over h bar. So actually if you had an energy eigenstate, this thing would remain equal to 1 for all times. So if this is going to be non-zero, it's because it's going to have-- you have to have a state is not an energy eigenstate. That you're going to have an uncertainty in the energy and energy uncertainty. So the curious thing is that you can evaluate this, and expand it as a power in t, and go, say, to quadratic ordering in t evaluating what this is. And this only depends on the uncertainty of h, and t, and things like that. So only the uncertainty of h matters at this moment. So this would be quite interested, I think, for you to figure out and to explore in detail. That kind of analysis has been the center of attention recently, having to do with quantum computation. Because in a sense, in a quantum computer, you want to change states quickly and do operations. So how quickly you can change a state is crucial. So in fact, the people that proved these inequalities that you're going to find say that this eventually will limit the speed of a quantum computer, and more slow that computers become twice as fast every year or so, and double the speed. So this limit apparently, it's claimed, will allow 80 duplications of the speed until you hit this limit in a quantum computer due to quantum mechanics. So you will be investigating this in next week's homework. Today we want to begin quickly with an application of the uncertainty principal to find exact bounds of energies for quantum systems, for ground states of quantum systems. So this will be a very precise application of the uncertainty principle. Then we'll turn to a completion of these things that we've been talking about having to do with linear algebra. We'll get to the main theorems of the subject. In a sense, the most important theorems of the subject. These are the spectral theorem that tells you about what operators can be diagonalized. And then a theorem that leads to the concept of a complete set of commuting observables. So really pretty key mathematical ideas. And the way we're good to do it, I think you will see, that we have gained a lot by learning the linear algebra concepts in a slightly abstract way. I do remember doing this proof that we're going today in previous years, and it was always considered the most complicated lecture of the course. Just taking the [? indices ?] went crazy, and lots of formulas, and the notation was funny. And now we will do the proof, and we'll write a few little things. And we'll just try to imagine what's going on. And will be, I think, easier. I hope you will agree. So let's begin with an example of a use of the uncertainty principle. So example. So this will be maybe for 20 minutes. Consider this Hamiltonian for a one dimensional particle that, in fact, you've considered before, alpha x to the fourth, for which you did some approximations. You know that the expectation value of the energy in the ground state. You've done it numerically. You've done it variationally. And variationally you knew that the energy at every stage was smaller at a given bound. The uncertainty principle is not going to give us an upper bound. It's going to give us a lower bound. So it's a really nice thing, because between the variational principal and the uncertainty principle, we can narrow the energy of this ground state to a window. In one of the problems that you're doing for this week-- and I'm sorry only today you really have all the tools after you hear this discussion-- you do the same for the harmonic oscillator. You do a variational estimate for the ground state energy. You do the uncertainty principle bound for ground state energy. And you will see these two bounds meet. And therefore after you've done this to bounds, you found the ground state energy of the harmonic oscillator, so it's kind of a neat thing. So we want to estimate the ground state energy. So we first write some words. We just say H in the ground state will be given by the expectation value of p squared in the ground state plus alpha times the expectation value of x to the fourth in the ground state. Haven't made much progress, but you have, because you're starting to talk about the right variables. Now this thing to that you have to know is that you have a potential that is like this, sort of a little flatter than x squared potential. And what can we say about the expectation value of the momentum on the ground state and the expectation value of x in the ground state? Well, the expectation value of x should be no big problem. This is a symmetric potential, therefore wave functions in a one dimensional quantum mechanics problems are either symmetric or anti symmetric. It could not be anti symmetric because it's a ground state and kind of have a 0. So it's asymmetric. Has no nodes. So there's the wave function of the ground state, the symmetric, and the expectation value of x in the ground state is 0. Similarly, the expectation value of the momentum in the ground state, what is it? Is? 0 too. And you can imagine just computing it. It would be the integral of psi. Psi is going to be a real d dx h bar over i psi. This is a total derivative. If it's a bound state, it's 0 at the ends. This is 0. So actually, we have a little advantage here. We have some control over what p squared is, because the uncertainty in p in the ground state-- well, the uncertainty in p squared is the expectation value of p squared minus the expectation value of p squared. So in the ground state, this is 0. So delta p squared in the ground state is just p squared on the ground state. Similarly, because the expectation value of x is equal to 0, delta x squared in the ground state is equal to expectation value of x squared in the ground state. So actually, this expectation of p squared is delta p. And we want to use the uncertainty principle, so that's progress. We've related something we want to estimate to an uncertainty. Small complication is that we have an expectation value of x to the fourth. Now we learned-- maybe I can continue here. We learned that the expectations value for an operator squared is bigger than or equal to the expectation value of the operator squared. So the expectation value of x to the fourth is definitely bigger than the expectation value of x squared squared. And this is true on any state. This was derived when we did uncertainty. We proved that the uncertainty squared is positive, because the norm of a vector, and that gave you this thing. So here you think of the operator as x squared. So the operator squared is x to the fourth. And here's the operator expectation value squared. So this is true for the ground state. It's also here true for any state, so is the ground state. And this x squared now is delta x. So this is delta x on the ground state to the fourth. So look what we have. We have that the expectation value of H on the ground state is strictly equal to delta p on the ground state squared over 2m plus alpha. And we cannot do a Priorean equality here, so we have this. This is so far an equality. But because of this, this thing is bigger than that. Well, alpha is supposed to be positive. So this is bigger than delta p ground state squared over 2m plus alpha delta x on the ground state to the fourth. OK, so far so good. We have a strict thing, this. And the order of the inequality is already showing up. We're going to get, if anything, a lower bound. You're going to be bigger than or equal to something. So what is next? Next is the uncertainty principle. We know that delta p delta x is greater than or equal to h bar over 2 in any state. So the delta p ground state and delta x on the ground state still should be equal to that. Therefore delta p ground state is bigger than or equal than h over 2 delta x in the ground state like that. So this inequality still the right direction. So we can replace this by something that is bigger than this quantity day without disturbing the logic. So we have H ground state now is greater than or equal to replace the delta p by this thing here, h squared over 8, because this is squared and there's another 2m delta x ground state squared plus alpha delta x ground state to the fourth. And that's it. We've obtained this inequality. So here you say, well, this is good but how can I use it? I don't know what delta x is in the ground state, so what have I gained? Well, let me do a way of thinking about this that can help you. Plot the right hand side as a function of delta x on the ground. So you don't know how much it is, delta x on the ground state, so just plot it. So if you plot this function, there will be a divergence as this delta x goes to 0, then it will be a minimum. It will be a positive minimum, because this is all positive. And then it will go up again. So the right hand side as a function of delta x is this. So here it comes. You see I don't know what delta x is. Suppose delta x happens to be this. Well, then I know that the ground state energy is bigger than that value. But maybe that's not delta x. Delta x may be is this on the ground state. And then if it's that, well, the ground state energy is bigger than this value over here. Well, since I just don't know what it is, the worst situation is if delta x is here, and therefore definitely H must be bigger than the lowest value that this can take. So the claim is that H of gs, therefore is greater than or equal than the minimum of this function h squared over 8m, and I'll just write here delta x squared plus alpha delta x to the fourth over delta x. The minimum of this function over that space is the bound. So I just have to do a calculus problem here. This is the minimum. I should take the derivative with respect to delta x. Find delta x and substitute. Of course, I'm not going to do that here, but I'll tell you a formula that will do that for you. A over x squared plus Bx to the fourth is minimized for x squared is equal to 1 over 2-- it's pretty awful numbers. 2 to the 1/3 A over B to the 1/3. And its value at that point is 2 to the 1/3 times 3/2 times A to the 2/3 times B to the 1/3. A little bit of arithmetic. So for this function, it turns out that A is whatever coefficient is here. B is whatever coefficient is there, so this is supposed to be the answer. And you get H on the ground state is greater than or equal to 2 to the 1/3 3/8 h squared square root of alpha over m to the 2/3, which is about 0.4724 times h squared square root of alpha over m to the 2/3. And that's our bound. How good or how bad is the bound? It's OK. It's not fabulous. The real answer is done numerically is 0.668. I think I remember variational principal gave you something like 0.68 or 0.69. And this one says it's bigger than 0.47. It gives you something. So the important thing is that it's completely rigorous. Many times people use the uncertainty principle to estimate ground state energies. Those estimates are very hand wavy. You might as well just do dimensional analysis. You don't gain anything. You don't know the factors. But this is completely rigorous. I never made an approximation or anything here. Every step was logical. Every inequality was exact. And therefore, this is a solid result. This is definitely true. It doesn't tell you an estimate of the answer. If you dimensional analysis, you say the answer is this times 1, and that's as good as you can do with dimensional analysis. It's not that bad. The answer turns out to be 0.7. But the uncertainty principle really, if you're careful, sometimes, not for every problem, you can do a rigorous thing and find the rigorous answer. OK, so are there any questions? Your problem in the homework will be to do this for the harmonic oscillator and find the two bounds. Yes? AUDIENCE: How does the answer change if we don't look at the ground state? PROFESSOR: How do they what? PROFESSOR: How does the answer change if we look at a state different from the ground state? PROFESSOR: Different from the ground state? So the question was how would this change if I would try to do something different from the ground state. I think for any state, you would still be able to say that the expectation value of the momentum is 0. Now the expectation value of x still would be 0. So you can go through some steps here. The problem here being that I don't have a way to treat any other state differently. So I would go ahead, and I would have said for any stationary state, or for any energy eigenstate, all of what I said is true. So I don't get a new one. These things people actually keep working and writing papers on this stuff. People sometimes find bounds that are a little original. Yes? AUDIENCE: How do you know the momentum expectation is 0 again? PROFESSOR: The momentum expectation for a bound state goes like this. So you want to figure out what is psi p psi. And you do the following. That's integral. Now psi in these problems can be chosen to be real, so I won't bother. It's psi of x h bar over i d dx of psi. So this is equal to h bar over 2i the integral dx of d dx of psi squared. So at this moment, you say well that's just h bar over 2i, the value of psi squared at infinity and at minus infinity. And since it's a bound state, it's 0 here, 0 there, and it's equal to 0. A state that would have expectation value of momentum you would expect it to be moving. So this state is there is static. It's stationary It doesn't have expectation value of momentum. Yes? AUDIENCE: Is the reason that you can't get a better estimate for the things that are on the ground state, because if you consider the harmonic oscillator, the uncertainty delta x [? delta p ?] from the ground state [INAUDIBLE] you go up to higher states. PROFESSOR: Right, I think that's another way. AUDIENCE: [INAUDIBLE] higher state using the absolute. PROFESSOR: Yeah. That's a [INAUDIBLE]. So the ground state of the harmonic oscillator saturates the uncertainty principal and the others don't. So this argument, I think, is just good for ground state energies. One more question. AUDIENCE: It appears that this method really works. Doesn't particularly work well if we have a potential that has an odd power, because we can't use [? the packet ?], like x [INAUDIBLE] expectation value x to the fourth is something, some power, expectation. PROFESSOR: Right, if it's an odd power, the method doesn't work well. But actually for an odd power, the physics doesn't work well either, because the system doesn't have ground states. And so let's say that if you had x to the fourth plus some x cubed, the physics could still make sense. But then it's not clear I can do the same. Actually you can do the same for x to the eighth and any sort of powers of this type. But I don't think it works for x to the sixth. You can try a few things. OK, so we leave the uncertainty principle and begin to understand more formerly the operators for which there's no uncertainty and you can simultaneously diagonalize them. So we're going to find operators like A and B, that they commute. And then sometimes you can simultaneously diagonalize them. Yes? You have a question. AUDIENCE: So part of [INAUDIBLE] we use here is [INAUDIBLE], right? PROFESSOR: Right. AUDIENCE: If we get an asset-- is there any way that we can better our [INAUDIBLE] principle based on the wave function with non saturated? Can we get an upper bound for just [INAUDIBLE] principle with an h bar over it? PROFESSOR: Can I get an upper bound? I'm not sure I understand your question. AUDIENCE: [INAUDIBLE] the fact that the [INAUDIBLE] principle will not be saturated. Can you put the bound for just taking [INAUDIBLE]? PROFESSOR: Yeah, certainly. You might have some systems in which you know that this uncertainty might be bigger than the one warranted by the uncertainty principles. And you use that information. But on general grounds, it's hard to know that a system might come very close to satisfy the uncertainty principle in its ground state. We don't know. There are systems that come very close in the ground state to satisfy this and some that are far. If they are far, you must have some reason to understand that to use it. So I don't know. So let me turn now to this issue of operators and diagonalization of operators. Now you might be irritated a little even by the title. Diagonalization of operation. You'll be talking about diagonalization of matrices. Well, there's a way to state what we mean by diagonalizing an operator in such a way that we can talk later about the matrix. So what is the point here? You have an operator, and it's presumably an important operator in your theory. You want to understand this operator better. So you really are faced with a dilemma. How do I get some insight into this operator? Perhaps the simplest thing you could do is to say, OK let me choose some ideal basis of the vector space, such as that operator is as simple as possible in that basis. So that's the origin of this thing. Find a basis in the state space, so the operator looks as simple as possible. So you say that you can diagonalize an operator if you can find the basis such that the operator has just diagonal entries. So let me just write it like this. So if you can find a basis in V where the matrix representing the operator is diagonal, the operator is said to be diagonalizable. So to be diagonalizable is just a statement that there is some basis where you look at the matrix representation operator, and you find that it takes form as a diagonal. So let's try to understand this conceptually and see what actually it's telling us. It tells us actually a lot. Suppose t is diagonal in some basis u1 up to un. So what does it mean for it to be diagonal? Well, you may remember all these definitions we had about matrix action. If T acting on a ui is supposed to be Tki uk in some basis sum over k. You act on ui, and you get a lot of u's. And these are the matrix elements of the operator. Now the fact that it's diagonalizable means that in some basis, the u basis, this is diagonal. So ki in this sum only happens to work out when k is equal to i. And that's one number and you get back to the vector ui. So if it's diagonal in this basis, you have the T on u1 is lambda a number times u1 T on u2 is lambda 2 in u2. And Tun equal lambda n un. So what you learn is that this basis vector-- so you learn something that maybe you thought it's tautological. It's not tautological. You learn that if you have a set of basis vectors in which the operator is diagonal, these basis vectors are eigenvectors of the operator. And then you learn something that is quite important, that an operator is diagonalizable if, and only if, it has a set of eigenvectors that span the space. So the statement is very important. An operator T is that diagonalizable if it has a set of eigenvectors that span the space. Span V. If and only if. If this double f. If and only if. So here it's diagonalizable, and we have a basis, and it has a set of these are eigenvectors. So diagonalizable realizable really means that it has a set of eigenvectors that span the space. On the other hand, if you have the set of eigenvectors that span the space, you have a set of u's that satisfy this, and then you read that, oh yeah, this matrix is diagonal, so it's diagonalizable. So a simple statement, but an important one, because there are examples of matrices that immediately you know you're never going to succeed to diagonalize. So here is one matrix, 0 0 1 0. This matrix has eigenvalues, so you do the characteristic equation lambda squared equals 0. So the only eigenvalue is lambda equals 0. And let's see how many eigenvectors you would have for lambda equals 0. Well, you would have if this is T, T on some vector a b must be equal to 0. So this is 0 1 0 0 on a b, which is b and 0, must be zero. So b is equal to 0. So the only eigenvector here-- I'll just write it here and then move to the other side. The only eigenvector for lambda equals 0, the only eigenvector is with b equals 0. So it's 1 0. One eigenvector only. No more eigenvectors. By the theorem, or by this claim, you know it's a two dimensional vector space you just can't diagonalize this matrix. It's impossible. Can't be done. OK, a couple more things that I wish to say about this process of diagonalization. Well, the statement that an operator is diagonal is a statement about the existence of some basis. Now you can try to figure out what that basis is, so typically what is the problem that you face? Typically you have a vector spaces V. Sorry? AUDIENCE: I have a question. PROFESSOR: Yes? If you had an infinite dimensional space and you had an operator whose eigenvectors do not span the space, can it still have eigenvectors, or does it not have any then? PROFESSOR: No. You said it has some eigenvectors, but they don't span the space. So it does have some eigenvectors. AUDIENCE: So my question is was what I just said a logical contradiction in an infinite dimensional space? PROFESSOR: To have just some eigenvectors? I think-- PROFESSOR: I'm looking more specifically at a dagger for instance. PROFESSOR: Yes. AUDIENCE: In the harmonic oscillator, you I think mentioned at some point that it does not have-- PROFESSOR: So the fact that you can' diagonalize this thing already implies that it's even worse in higher dimensions. So some operator may be pretty nice, and you might still be able to diagonalize it, so you're going to lack eigenvectors in general. You're going to lack lots of them. And there are going to be blocks of Jordan. Blocks are called things that are above the diagonal, things that you can't do much about. Let me then think concretely now that you have a vector space, and you've chosen some basis v1 vn. And then you look at this operator T, and of course, you chose an arbitrary basis. There's no reason why its matrix representation would be diagonal. So T on the basis v-- Tij. Sometimes to be very explicit we write Tij like that-- is not diagonal. Now if it's not diagonal, the question is whether you can find a basis where it is diagonal. And then you try, of course, changing basis. And you change basis-- you've discussed that in the homework-- with a linear operator. So you use a linear operator to produce another basis, an invertible in your operator. So that you get these vectors uk being equal to some operator A times vk. So this is going to be the u1's up to un's are going to be another basis. The n vector here is the operator acting with the n vector on this thing. And then you prove, in the homework, a relationship between these matrix elements of T in the new basis, in the u basis. And the matrix elements of T in the v basis. You have a relationship like this, or you have more explicitly Tij in the basis u is equal to A minus 1 ik Tkp of v Apj. So this is what happens. This is the new operator in this basis. And typically what you're trying to do is find this matrix A that makes this thing into a diagonal matrix. Because we say in the u basis the operator is diagonal. I want to emphasize that there's a couple of ways in which you can think of diagonalization. Sort of a passive and an active way. You can imagine the operator, and you say look, this operator I just need to find some basis in which it is diagonal. So I'm looking for a basis. The other way of thinking of this operator is to think that A minus 1 TA is another operator, and it's diagonal in original basis. So it might have seem funny to you, but let's stop again and say this again. You have an operator, and the question of diagonalization is whether there is some basis in which it looks diagonal, its matrix is diagonal. But the equivalent question is whether there is an operator A such that this is diagonal in the original basis. To make sure that you see that, consider the following. So this is diagonal in the original basis. So in order to see that, think of Tui is equal to lambda i ui. We know that the u's are supposed to be this basis of eigenvectors where the matrix is diagonal, so here you got it. Here the i not summed. It's pretty important. There's a problem with this eigenvalue notation. I don't know how to do it better. If you have several eigenvalues, you want to write this, but you don't want this to think that you're acting on u1 and you get lambda 1 u1. Not the sum right here. OK, but they ui is equal to A on vi. So therefore this is lambda i A on vi. And then you'll act with A minus 1. Act with A minus 1 from the left with the operator. So you get A minus 1 TA vi is equal to lambda i vi. So what do you see? You see an operator that is actually diagonal in the v basis. So this operator is diagonal in the original basis. That's another way of thinking of the process of diagonalization. There's one last remark, which is that the columns of A are the eigenvectors, in fact. Columns of A are the eigenvectors. Well, how do you see that? It's really very simple. You can convince yourself in many ways, but the uk are the eigenvectors. But what are uk's? I have it somewhere. There it is. A on vk. And A on vk is this matrix representation is sum over i Aik vi. So now if this is the original basis, the vi's are your original basis, then you have the following, that the vi's can be thought as the basis vectors and represented by columns with a 1 in the ith entry. So this equation is saying nothing more, or nothing less than uk, in terms of matrices or columns, is equal to A1k v1 plus Ank vn, which is just A1k A2k Ank. Because vi is the ith basis vector. So 1 0's only in the ith position. So these are the eigenvectors. And they're thought as linear combinations of the vi's. The vi's are the original basis vectors. So the eigenvectors are these numbers. OK, we've talked about diagonlization, but then there's a term that is more crucial for our operators that we're interested in. We're talking about Hermitian operators. So the term that is going to be important for us is unitarily diagonalizable. What is a unitarily diagonalizable operator? Two ways again of thinking about this. And perhaps the first way is the best. And I will say it. A matrix is set to be unitarily diagonalizable if you have an orthonormal basis of eigenvectors. Remember diagonalizable meant a basis of eigenvectors. That's all it means. Unitarily diagonalizable means orthonormal basis of eigenvectors. So T has an orthonormal basis of eigenvectors. Now that's a very clear statement. And it's a fabulous thing if you can achieve, because you basically have broken down the space into basis spaces, each one of them with a simple thing before your operators. And they're orthonormal, so it's the simplest possible calculational tool. So it's ideal if you can have this. Now the way we think of this is that you start with-- concretely, you start with a T of some basis v that is an orthonormal basis. Start with an orthonormal basis, and then pass to another orthonormal basis u. So you're going to pass to another orthonormal basis u with some operator. But what you have learned is that if you want to pass from v orthonormal to another basis u, a vector that is also orthonormal, the way to go from one to the other is through a unitary operator. Only unitary operators pass you from orthonormal to orthonormal basis. Therefore really, when you start with your matrix in an orthonormal basis that is not diagonal, the only thing you can hope is that T of u will be equal to sum u dagger, or u minus 1, T of v u. Remember, for a unitary operator, where u is unitary, the inverse is the dagger. So you're doing a unitary transformation, and you find the matrix that is presumably then diagonal. So basically, unitarily diagonalizable is the statement that if you start with the operator in an arbitrary orthonormal basis, then there's some unitary operator that takes you to the privilege basis in which your operator is diagonal, is still orthonormal. But maybe in a more simple way, unitarily diagonalizable is just a statement that you can find an orthonormal basis of eigenvectors. Now the main theorem of this subject, perhaps one of the most important theorems of linear algebra, is the characterization of which operators have such a wonderful representation. What is the most general operator T that will have an orthonormal basis of eigenvectors? Now we probably have heard that Hermitian operators do the job. Hermitian operators have that. But that's not the most general ones. And given that you want the complete result, let's give you the complete result. The operators that have this wonderful properties are called normal operators, and they satisfy the following property. M is normal if M dagger, the adjoint of it, commutes with M. So Hermitian operators are normal, because M dagger is equal to M, and they commute. Anti Hermitian mission operators are also normal, because anti Hermitian means that dagger is equal to minus M, and it still commutes with M. Unitary operators have U dagger U equal to U U dagger equals to 1. So U and U dagger actually commute as well. So Hermitian, anti Hermitian, unitary, they're all normal operators. What do we know about normal operators? There's one important result about normal operators, a lemma. If M is normal and W is an eigenvector, such that MW is equal to lambda W. Now normal operators need not have real eigenvalues, because they include unitary operators. So here I should write Hermitian, anti Hermitian, and unitary are normal. So here is what a normal operator is doing. You have a normal operator. It has an eigenvector with some eigenvalue. Lambda is a complex number in principle. Then the following result is true, then M dagger omega is also an eigenvector of M dagger. And it has eigenvalue lambda star. Now this is not all that easy to show. It's a few lines, and it's done in the notes. I ask you to see. It's actually very elegant. What is the usual strategy to prove things like that Is to say oh, I want to show this is equal to that, so I want to show that this binds that is 0. So I have a vector that is zero what is the easiest way to show that it's 0? If I can show it's norm is 0. So that's a typical strategy that you use to prove equalities. You say, oh, it's a vector that must be 0 as my equality. Let's see if it's 0. Let's find its norm, and you get it. So that's a result. So with this stated, we finally have the main result that we wanted to get to. And I will be very sketchy on this. The notes are complete, but I will be sketchy here. It's called the spectral theorem. Let M be an operator in a complex vector space. The vector space has an orthonormal basis of eigenvectors of M if and only if M is normal. So the normal operators are it. You want to have a complete set of orthonormal eigenvectors. Well, this will only happen if your operator is normal, end of story. Now there's two things about this theorem is to show that if it's diagonalizable, it is normal, and the other thing is to show, that if it's normal, it can be diagonalized. Of course, you can imagine the second one is harder than the first. Let me do the first one for a few minutes. And then say a couple of words about the second. And you may discuss this in recitation. It's a little mathematical, but it's all within the kind of things that we do. And really it's fairly physical in a sense. We're accustomed to do such kinds of arguments. So suppose it's unitarily diagonalizable, which means that M-- so if you have U dagger, MU is equal to a diagonal matrix, DM. I'm talking now matrices. So these are all matrices, a diagonal matrix. There's no basis to the notion of a diagonal operator, because if you have a diagonal operator, it may not look diagonal in another basis. Only the identity operator is diagonal in all basis, but not the typical diagonal operator. So unitarily diagonalizable, as we said, you make it-- it's gone somewhere. Here. You act with an inverse matrices, and you get the diagonal matrix. So from this, you find that M is equal to U DM U dagger by acting with U on the left and with U dagger from the right, you solve for M, and it's this. And then M dagger is the dagger of these things. So it's U to DM dagger U dagger. The U's sort of remain the same way, but the diagonal matrix is not necessarily real, so you must put the dagger in there. And now M dagger M. To check that the matrix is normal that commutator should be 0. So M dagger M. You do this times that. You get U DM dagger. U dagger U. That's one. DM U dagger. And M M dagger you multiply the other direction you get U DM DM dagger U dagger. So the commutator of M dagger M is equal to U DM dagger DM minus DM Dm dagger U dagger. But any two diagonal matrices commute. They may not be that simple. Diagonal matrices are not the identity matrices, but for sure they commute. You multiply elements along with diagonal so this is 0. So certainly any unitarily diagonalizable matrix is normal. Now the other part of the proof, which I'm not going to speak about, it's actually quite simple. And it's based on the fact that any matrix in a complex vector space has at least one eigenvalue. So what you do is you pick out that eigenvalue and it's eigenvector, and change the basis to use that eigenvector instead of your other vectors. And then you look at the matrix. And after you use that eigenvector, the matrix has a lot of 0's here and a lot of 0's here. And then the matrix has been reduced in dimension mansion, and then you go step by step. So basically, it's the fact that any operator has at least one eigenvalue and at least one eigenvector. It allows you to go down. And normality is analogous to Hermiticity in some sense. And the statement that you have an eigenvector generally tells you that this thing is full of 0's, but then you don't know that there are 0's here. And either normality or Hermiticity shows that there are 0's here, and then you can proceed at lower dimensions. So you should look at the proof because it will make clear to you that you understand what's going on. OK but let's take it for granted now you have these operators and can be diagonalized. Then we have the next thing, which is simultaneous diagonalization. What is simultaneous diagonalization? It's an awfully important thing. So we will now focus on simultaneous diagonalization of Hermitian operators. So simultaneous diagonalization of Hermitian ops. Now as we will emphasize towards the end, this is perhaps one of the most important ideas in quantum mechanics. It's this stuff that allows you to label and understand your state system. Basically you need to diagonalize more than one operator most of the time. You can say OK, you found the energy eigenstates. You're done. But if you find your energy eigenstates and you think you're done, maybe you are if you have all these energy eigenstates tabulated. But if you have a degeneracy, you have a lot of states that have the same energy. And what's different about them? They're certainly different because you've got several states, but what's different about them? You may not know, unless you figure out that they have different physical properties. If they're different, something must be different about them. So you need more than one operator, and your facing the problem of simultaneously diagonalizing things, because states cannot be characterized just by one property, one observable. Would be simple if you could, but life is not that simple. So you need more than one observable, and then you ask when can they be simultaneously diagonalizable. Well, the statement is clear. If you have two operators, S and T that belong to the linear operators in a vector space, they can be simultaneously diagonalized if there is a basis for which every basis vector is eigenstate of this and an eigenstate of that. Common set of eigenstates. So they can be simultaneously diagonalized. Diagonalizable is that there is a basis where this basis is comprised of the eigenvectors of the operator. So this time you require more, that that basis be at the same time a basis set of eigenvectors of this and a set of eigenvectors of the second one. So a necessary condition for simultaneous diagonalization is that they commute. Why is that? The fact that two operators commute or they don't commute is an issue that is basis independent. If they don't commute, the order gives something different, and that you can see in every basis. So if they don't commute and they're simultaneously diagonalizable, there would be a basis in which both are diagonal and they still wouldn't commute. But you know that diagonal matrices commute. So if two operators don't commute, they must not commute in any base, therefore there can't be a basis in which both are at the same time diagonal. So you need, for simultaneous diagonalizable, you need that S and P commute. Now that may not be enough, because not all operators can be diagonalized. So the fact that they commute is necessary, but not everything can be diagonalizable. Well, you've learned that every normal operator, every Hermitian operator is diagonalizable. And then you got now a claim of something that could possibly be true. Is the fact that whenever you have two Hermitian operators, each one can be diagonalized by themselves. And they commute. There is a simultaneous set of eigenvectors of 1 that are eigenvectors of the first and eigenvectors of the second. So the statement is that if S and T are commuting Hermitian operators, they can be simultaneously diagonalized. So this theorem would be quite easy to show if there would be no degeneracies, and that's what we're going to do first. But then we'll consider the case of degeneracies. So I'm going to consider the following possibilities. Perhaps neither one has a degenerate spectrum. What does it mean a degenerate spectrum? Same eigenvalue repeated many times. But that is a wishful thinking situation. So either both are non degenerate, either one is non degenerate and the other is degenerate, or both are degenerate. And that causes a very interesting complication. So let's say there's going to be two cases. It will suffice. In fact, it seems that there are three, but two is enough to consider. There is no degeneracy in T. So suppose one operator has no degeneracy, and let's call it T. So that's one possibility. And then S may be degenerate, or it may not be degenerate. And the second possibility is that both S and T are degenerate. So I'll take care of case one first. And then we'll discuss case two, and that will complete our discussion. So suppose there's no the degeneracy in the spectrum of T. So case one. So what does that mean? It means that T is non degenerate. There's a basis U1 can be diagonalized to UM, orthonormal by the spectral theorem. And there's eigenvectors T U-- these are eigenvectors. Lambda I Ui. And lambda I is different to lambda j for i different from j. So all the eigenvalues, again, it's not summed here. All the eigenvalues are different. So what do we have? Well, each of those eigenvectors, each of the Ui's that are eigenvectors, generate invariant subspaces. There are T invariant subspaces. So each one, each vector U1 you can imagine multiplying by all possible numbers, positive and negative. And that's an invariant one dimensional subspace, because if you act with T, it's a T invariant space, you get the number times a vector there. So the question that you must ask now is you want to know if these are simultaneous eigenvectors. So you want to figure out what about S. How does S work with this thing? So you can act with S from the left. So you get STUi is equal to lambda i SUi. So here each Ui generates an invariant subspace Ui T invariant. But S and T commute, so you have T SUi is equal to lambda i SUi. And look at that equation again. This says that this vector belongs to the invariant subspace Ui, because it satisfies exactly the property that T acting on it is equal to lambda i Ui. And it couldn't belong to any other of the subspaces, because all the eigenvalues are different. So spaces that are in Ui are the spaces-- vectors that are in Ui are precisely all those vectors that are left invariant by the action of T. They're scaled only. So this vector is also in Ui. If this vector is in Ui, SUi must be some number Wi times Ui. And therefore you've shown that Ui is also an eigenvector of S, possibly with a different eigenvalue of course. Because the only thing that you know is that SUi is in this space. You don't know how big it is. So then you've shown that, indeed, these Ui's that were eigenstates of T are also eigenstates of S. And therefore that's the statement of simultaneously diagonalizable. They have the common set of eigenvectors. So that's this part. And it's relatively straight forward. Now we have to do case two. Case two is the interesting one. This time you're going to have degeneracy. We have to have a notation that is good for degeneracy. So if S is degeneracies, has degeneracies, what happens with this operator? It will have-- remember, a degenerate operator has eigenstates that form higher than one dimensional spaces. If you have different eigenvalues, each one generates a one dimensional operator invariant subspace. But if you have degeneracies, there are operators-- there are spaces of higher dimensions that are left invariant. So for example, let Uk denote the S invariant subspace of some dimension Dk, which is greater or equal than 1. I will go here first. We're going to define Uk to be the set of all vectors so that SU is equal to lambda k U. And this will have dimension of Uk is going to be Dk. So look what's happening. Basically the fact is that for some eigenvalues, say the kth eigenvalue, you just get several eigenvectors. So if you get several eigenvectors not just scaled off each other, these eigenvectors correspond to that eigenvalue span of space. It's a degenerate subspace. So you must imagine that as having a subspace of some dimensionality with some basis vectors that span this thing. And they all have the same eigenvector. Now you should really have visualized this in a simple way. You have this subspace like a cone or something like that, in which every vector is an eigenvector. So every vector, when it's acted by S is just scaled up. And all of them are scaled by the same amount. That is what this statement says. And corresponding to this thing, you have a basis of vectors, of these eigenvectors, and we'll call them Uk1, the first one, the second, up to U Dk1, because it's a subspace that we say it has the dimensionality Dk. So look at this thing. Somebody tells you there's an operator. It has degenerate spectrum. You should start imagining all kind of invariant subspaces of some dimensionality. If it has degeneracy, it's a degeneracy each time the Dk is greater than 1, because if it's one dimensional, it's just one basis vector one eigenvector, end of the story. Now this thing, by the spectral theorem, this is an orthonormal basis. There's no problem, when you have a degenerate subspace, to find an orthonormal basis. The theorem guarantees it, so these are all orthonormal. So at the end of the day, you have a decomposition of the vector space, V as U1 plus U2 plus maybe up to UM. And all of these vector spaces like U's here, they may have some with just no degeneracy, and some with degeneracy 2, degeneracy 3, degeneracy 4. I don't know how much degeneracy, but they might have different degeneracy. Now what do we say next? Well, the fact that S is a Hermitian operator says it just can be diagonalized, and we're can find all these spaces, and the basis for the whole thing. So the basis would look U1 of the first base up to U d1 of the first base. These are the basis vectors of the first plus the basis vectors of the second. All the basis vectors U1 up to Udm of the mth space. All this is the list. This is the basis of V. So I've listed the basis of V, which a basis for U1, all these vectors. U2, all of this. So you see, we're not calculating anything. We're just trying to understand the picture. And why is this operator, S, diagonal in this basis? It's clear. Because every vector here, every vector is an eigenvector of S. So when you act with S on any vector, you get that vector times a number. But that vector is orthogonal to all the rest. So when you have some U and S and another U, this gives you a vector proportional to U. And this is another vector. The matrix element is 0, because they're all orthogonal. So it should be obvious why this list produces something that is completely orthogonal-- a diagonal matrix. So S, in this basis, looks like the diagonal matrix in which you have lambda 1 d1 times up to lambda m dm times. Now I'll have to go until 2:00 to get the punchline. I apologize, but we can't stop right now. We're almost there, believe it or not. Two more things. This basis is good, but actually another basis would also be good. I'll write this other basis would be a V1 acting on the U1 up to V1 acting on that U1 up to a Vm acting on this U1 up to Vm acting on that U1. This is m. m. This is dm. And here it's not U1. It's Ud1. You see, in the first collection of vectors, I act with an operator V1 up to here with an operator Vm. All of them with Vk being a unitary operator in Uk. In every subspace, there are unitary operators. So you can have these bases and act with a unitary operator of the space U1 here. A unitary operator with a space U2 here. A unitary operator of the space Un here. Hope you're following. And what happens if this operator is unitary, this is still an orthonormal basis in U1. These are still orthonormal basis in Um. And therefore this is an orthonormal basis for the whole thing, because anyway those different spaces are orthogonal to each other. It's an orthogonal decomposition. Everything is orthogonal to everything. So this basis would be equally good to represent the operator. Yes? AUDIENCE: [INAUDIBLE] arbitrary unitary operators? PROFESSOR: Arbitrary unitary operators at this moment. Arbitrary. So here comes the catch as to the main property that now you want to establish is that the spaces Uk are also T invariant. You see, the spaces Uk were defined to be S invariant subspaces. And now the main important thing is that they are also T invariant because they commute with that. So let's see why that is the case. Suppose U belongs to Uk. And then let's look at the vector-- examine the vector Tu. What happens to Tu? Well, you want to act on S on Tu to understand it. But S and T commute, so this is T SU. But since U belongs to Uk, that's the space with eigenvalue lambda k. So this is lambda k times u, so you have Tu here. So Tu acted with S gives you lambda k Tu. So Tu is in the invariant subspace Uk. What's happening here is now something very straightforward. You try to imagine how does the matrix T look in the basis that we have here. Here is this basis. how does this matrix T look? Well, this matrix keeps the invariant subspaces. So you have to think of it blocked diagonally. If it acts on it-- here are the first vectors that you're considering, the U1. Well if you act on it with T of the U1 subspace, you stay in the U1 subspace. So you don't get anything else. So you must have 0's all over here. And you can have a matrix here. And if you act on the second Uk U2, you get a vector in U2, so it's orthogonal to all the other vectors. So you get a matrix here. And you get a matrix here. So actually you get a blocked diagonal matrix in which the blocks correspond to the degeneracy. So if there's a degeneracy d1 here, it's a d1 times d1. And d2 times d2. So actually you haven't simultaneously diagonalized them. That's the problem of degeneracy. You haven't, but you now have the tools, because this operator is Hermitian, therefore it's Hermitian here, and here, and here, and here. So you can diagonalize here. But what do you need for diagonalizing here? You need a unitary matrix. Call it V1. For here you need another unitary matrix. Call it V2. Vn. And then this matrix becomes diagonal. But then what about the old matrix? Well, we just explained here that if you change the basis by unitary matrices, you don't change the first matrix. So actually you succeeded. You now can diagonalize this without destroying your earlier result. And you managed to diagonalize the whole thing. So this is for two operators in the notes. You'll see why it simply extends for three, four, and five, or arbitrary number of operators. See you next time.

MIT_805_Quantum_Physics_II_Fall_2013

26_Addition_of_Angular_Momentum_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK so we're going to do this thing of the hydrogen atom and the algebraic solution. And I think it's not that long stuff so we can take it easy as we go along. I want to remind you of a couple of facts that will play a role. One result that is very general about the addition of angular momentum that you should of course know is that if you have a j1 times j2. What does this mean? You have some states of-- first angular momentum J1 that so you have a whole multiplet with J1 equals little j1. Which means the states in that multiplet have J1 squared, giving you h squared. Little j1 times little j1 plus 1. That's having a j1 multiplet. You have a j2 multiplet. And these are two independent commuting angular momenta acting on different degrees of freedom of the same particle or different particles. And what you've learned is that this can be written as a sum of representations. As a direct sum of angular momenta, which goes from j1 plus j2 plus j1 plus j2 minus 1 all the way up to the representation with j1 minus j2. And these are all representations or multiplets that live in the tense or product, but they are multiplets of J equals j1 plus j2. These states here can be reorganized into these multiplets, and that's our main result for the addition of angular momentum. Mathematically, this formula summarizes it all. These states, when you write them as a basis here-- you take a basis state here times a basis state here-- these are called the coupled bases. And then you reorganize, you form linear combinations that you have been playing with, and then they get reorganized into these states. So these are called the coupled bases in which we're talking about states of the sum of angular momentum. So that's one fact we've learned about. Now as far as hydrogen is concerned, we're going to try today to understand the spectrum. And for that let me remind you what the spectrum was. The way we organized it was with an L versus energy levels. And we would put an L equals 0 state here-- well maybe-- there's color, so why not using color. Let's see if it works. Yeah, it's OK. L equals 0. And this was called n equals 1. There's an n equals 2 that has an L equals 0 state and an L equals 1 state. There's an n equals 3 state, set of states that come within L equals 0 and L equals 1 and an L equals 2. And it just goes on and on. With the energy levels En equals minus e squared over 2a0. That combination is familiar for energy, Bohr radius, charge of electron, with a 1 over n squared. And the fact is that for any level, for each n, L goes from 0, 1, 2, up to n minus 1. And for each n there's a total of n squared states. And you see it here, you have n equals 2, n equals 1, one state. n equals 2, you have L equals 0, one state. L equals 1 is 3 states. So it's 4. Here we'll have 4 plus 5. So 9. And maybe you can do it, it's a famous thing, there's n squared states at every level. So this pattern that of course continues and-- it's a little difficult to do a nice diagram of the hydrogen atom in scale because it's all pushed towards the zero energy with 1 over n squared, but that's how it goes. For n equals 4, you have 1, 2, 3, 4 for example. And this is what we want to understand. So in order to do that, let's return to this Hamiltonian, which is p squared over 2m minus e squared over r. And to the Runge-Lenz vector that we talked about in lecture and you've been playing with. So this Runge-Lenz vector, r, is defined to be 1 over 2me squared p cross L minus L cross p minus r over r. And it has no units. It's a vector that you've learned has interpretation of a constant vector that points in this direction, r. And it just stays fixed wherever the particle is going. Classically this is a constant vector that points in the direction of the major axis of the ellipse. With respect to this vector, this vector is Hermitian. And you may recall that when we did the classical vector, you had just p cross L and no 2 here. There are now two terms here. And they are necessary because we want to have a Hermitian operator, and this is the simplest way to construct the Hermitian operator, r. And the way is that you add to this this term, that if L and p commuted as they do in classical mechanics, that the term is identical to this. And you get back to the conventional thing that you had in classical mechanics. But in quantum mechanics, of course, they don't commute, so it's a little bit different. And moreover this thing, r, is Hermitian. L and p are Hermitian but when you take the Hermitian conjugate, L goes to the other side of p. And since they don't commute, that's not the same thing. So actually the Hermitian conjugate of this term is this. There's an extra minus sign in hermiticity when you have a cross product. So this is the Hermitian conjugate of this, this is the Hermitian conjugate of this second term, here's the first and therefore this is actually a Hermitian operator. And you can work with it. Moreover, in the case of classical mechanics, it was conserved. In the case of quantum mechanics this statement of conservation quantum mechanics is something that in one of the exercises that you were asked to try to do this computation so these computations are challenging. They're not all that trivial and are good exercises. So this is one of them. This is practice. OK this is the vector r. What about it? A few more things about it that are interesting. Because of the hermiticity condition or in-- a way you can check this directly in fact was one of the exercises for you to do, was p cross L-- you did it long time ago, I think-- is equal to minus L cross b plus 2ih bar p. This is an identity. And this identity helps you write this kind of term in a way in which you have just one order of products and a little extra term, rather than having two complicated terms. So the r can be written as 1 over me squared alone, p cross L minus ihp minus r over r. For example. By writing this term as another p cross L minus that thing gives you that expression for r. You have an alternative expression in which you solve for the other one. So it's 1 over me squared minus L cross p plus ih bar p. Now, r-- we need to understand r better. That's really the challenge of this whole derivation. So we have one thing that is conserved. Angular momentum is conserved. It commutes with the Hamiltonian. We have another thing that is conserved, this r. But we have to understand better what it is. So one thing that you can ask is, well, r is conserved, so r squared is conserved as well. So r squared, if I can simplify it-- if I can do the algebra and simplify it-- it should not be that complicated. So again a practice problem was given to do that computation. And I think these forms are useful for that, to work less. And the computation gives a very nice result, where r squared is equal to 1 plus 2 H over me to the fourth L squared plus h bar squared. Kind of a strange result if you think about it. People that want to do this classically first would find that there's no h squared. And here, this h, is that whole h that we have here. It's a complicated thing. So this right hand side is quite substantial. You don't have to worry that h is in this side or whether it's on the other side because h commutes with L. L is conserved. So h appears like that. And this, again, is the result of another computation. So we've learned something. r-- oh, I'm sorry, this is r squared. Apologies. r is conserved. r squared must be conserved because if h commutes with r it commutes with r squared as well. And therefore whatever you see on the right hand side, the whole thing must be conserved. And h is conserved, of course. And L squared is conserved. Now we need one more property of a relation-- you see, you have to do these things. Even if you probably don't have an inspiration at this moment how you're going to try to understand this, there are things that just curiosity should tell that you should do. We have L, we do L squared. It's an important operator. OK. We had r, we did r squared, which is an important operator. But one thing we can do is L dot r. It's a good question what L dot r is. So what is L dot r? So-- or r dot L. What is it? Well a few things that are important to note are that you did show before that you know that r dot L, little r dot L, is 0. And little p dot L is 0. These are obvious classically, because L is perpendicular to both r and p. But quantum mechanically they take a little more work. They're not complicated, but you've shown those two. So if you have r dot L, you would have, for example, here-- r dot L, you would have to do and think of this whole r and put an L on the right. Well this little r dotted with the L on the right would be 0. That p dotted with L on the right would be 0. And we're almost there, but p cross L dot L, well, what is that? Let me talk about it here. P cross L dot L-- so this is part of the computation of this r dot L. We've already seen this term will give nothing, this term will give nothing. But this term could give something. So when you face something like that, maybe you say, well, I don't know any identities I should be using here. So you just do it. Then you say, this is i-th component of this vector times the i-th of that. So it's epsilon ijkpjLkLi. And then you say look this looks a little-- you could say many things that are wrong and get the right answer. So you could say, oh, ki symmetric and ki anti-symmetric. But that's wrong, because these k and i are not symmetric really because these operators don't commute. So the answer will be zero, but for a more complicated reason. So what do you have in here? ki. Let's move the i to the end of the epsilon, so jkipjLkLi. And now you see this part is pj L cross cross L j. Is the cross product of this. But what is L cross L? You probably remember. This is ih bar L L cross L, that's the computation in relation of angular momentum. In case you kind of don't remember it was ih bar L. Like that. So now p dot L is anyway 0. So this is 0. So it's kind of-- it's a little delicate to do these computations. But so since that term is zero, this thing is zero. Now you may as well-- r dot L is 0. Is L dot r also 0? It's not all that obvious you can even do that. Well in a second we'll see that that's true as well. L dot r and r dot L, capital R, are 0. Let's remember-- let's continue-- let's see, I wanted to number some of these equations. We're going to need them. So this will be equation one. This will be equation two, it's an important one. Now what was-- let me remind you of a notation we also had about vectors and their rotations. Vector under rotations. So what was a vector, Vi, under rotations was something that you had LiVj equals ih bar epsilon ijkvk. So there is a way to write this with cross products that is useful in some cases. So I will do it. You probably have seen that in the notes, but let me remind you. Consider this product, L cross V plus V cross L and the i-th component of it. i-th component of this product. So this is epsilon ijk and you have LjVk plus VjLk. Now in this term you can do something nice. If you think of it like expanded out, you have the second term has epsilon ijkVjk. Change j for k. If you change j for k, this will be VkLj. And these would have the opposite order. But this order can be changed up to the cost of a minus sign. So I claim this is ijk-- first term is the same-- minus VkLj. So in the second term, for this term alone, we've done for this term, multiplied with this of course, we've done j and k relabeling. But this is nothing else than the commutator of L with V. So this is epsilon ijkLjVk. That's epsilon ijk, and this is epsilon jkp or LVL. Now, 2 epsilons with 2 common indices is something that it simple. It's a commutator dealt on the other indices. Now it's better if they are sort of all aligned in the same way, but they kind of are because this L, without paying a price, can be put as the first index. So you have jk as the second and third and-- jk as the second and third-- once L has been moved to the first position. So this thing is 2 times delta iL. And there's an h bar, ih bar I forgot here. ih bar. 2 delta ik ih bar iL ih bar VL. So this is 2 ih bar Vi. So this whole thing the i-th component of this thing, using this commutation relation is this. So what we've learned is that L cross V plus V cross L you see go to 2 ih bar V. And that's a statement as a vector relation of the fact that V is a vector in the rotations. So for V to be a vector in the rotations means this. And if you wish, it means this thing as well. It's just another thing of what it means. Now R is a vector in the rotations. This capital R. Why? You've shown that if you have a vector in the rotations and you multiply it by another vector in the rotations under the cross product, it's still a vector in the rotations. So this is a vector in the rotations, this is, and this is a vector in the rotations. R is a vector in the rotations. So this capital R is a vector on the rotations, which means two things. It means it satisfies this kind of equation. So r cross-- or L cross R plus R cross L is equal to ih bar R. So R is a vector in the rotation. It's a fact beyond doubt. And that means that we now know the commutation relations between L and R. So we're starting to put together this picture in which we get familiar with R and the commutators that are possible. So I can summarize it here. L dot R LiRj is ih bar epsilon ijkRk. That's the same statement as this one but in components. And now you see why R dot L is equal to L dot R. Because actually if you put the same two indices here, i and i, you get zero. So when you have R dot L you have R1L1 plus R2L2 plus R3L3. And each of these two commute when the two indices are the same. Because of the epsilon. So R dot L is 0. And now you also appreciate that L dot R is also 0, too. OK. Now comes, in a sense, the most difficult of all calculations. Even if this seemed a little easy. But you can get quite far with it. So what do you do with Ls? You computed L commutators and you got the algebra of angular momentum. Over here. This is the algebra for angular momentum. And this kind of nontrivial calculation, you did it by building results. You knew how R was a vector in the rotation or how p was a vector in the rotation. You multiplied the two of them, and it was not so difficult. But the calculation that you really need to do now is the calculation of the commutator say of Ri with Rj. And that looks like a little bit of a nightmare. You have to commute this whole thing with itself. Lots of p's, L's, R's. 1 over R's, those don't commute with p. You remember that. So this kind of calculation done by brute force. You're talking a day, probably. I think so. And probably it becomes a mess, but. You'll find a little trick helps to organize it better. It's less of a mess, but still you don't get it and-- try several times. So what we're going to do is try to think of what the answer could be by some arguments. And then once we know what the answer can be, there's still one calculation to be done. That I will probably put in the notes, but it's not a difficult one. And the answer just pops out. So the question is what is R cross R. R cross R is really what we have when we have this commutator. So we need to know what R cross R is, just like L cross L. Now R is not likely to be an angular momentum. It's a vector but it's not an angular momentum. Has nothing to do with it. It's more complicated. So what is R cross R quantum-mechanically? Classically, of course, it would be zero. So first thing is you think of what this should be. We have a vector, because the cross product of two vectors. Now I want to emphasize one other thing, that it should be this thing-- R cross R-- is tantamount to this thing. What is this thing? It should be actually proportional to some conserved quantity. And the reason is quite interesting. So this is a small aside here. If some operator is conserved, it commutes with the Hamiltonian. Say if S1 and S2 are symmetries, that means that S1 with h is equal to S2 with h is equal to zero. Then the claim is that the commutator of this S1 and S2 claim S1 commutator with S2 is also a symmetry. So the reason is because commutator of S1 S2 commutator with h is equal actually to zero. And why would it be equal to zero? It's because of the so-called Jacobi identity for commutators. You'll remember when you have three things like that, this term is equal to 1-- this term plus 1, in which you cycle them. And plus another one where you cycle them again is equal to zero. That's a Jacobi identity. And in those cyclings you get an h with S2, for example, that is zero. And then an h with S1, which is zero. So you use these things here and you prove that. So I write here, by Jacobi. So if you have a conserved-- this is the great thing about conserved quantities, if you have one conserved quantity, it's OK. But if you have two, you're in business. Because you can then take the commutator of these two and you get another conserved quantity. And then more commutators and you keep taking commutators and if you're lucky you get all of the conserved quantities. So here R cross R refers to this commutator. So whatever is on the right should be a vector and should be conserved. And what are our conserved vectors? Well our conserved vectors-- candidates here-- are L, R itself, and L cross R. That's pretty much it. L and R are our only conserved things, so it better be that. Still this is far too much. So there could be a term proportional to L, a term proportional to R, a term proportional to L dot R. So this kind of analysis is based by something that Julian Schwinger did. This same guy that actually did quantum electrodynamics along with Feynman and Tomonaga. And he's the one of those who invented the trick of using three-dimensional angular momentum for the two-dimensional oscillator. And had lots of bags of tricks. So actually this whole discussion of the hydrogen atom-- most books just say, well, these calculations are hopeless. Let me give you the answers. Schwinger, on the other hand, in his book on quantum mechanics-- which is kind of interesting but very idiosyncratic-- finds a trick to do every calculation. So you never get into a big mess. He's absolutely elegant and keeps pulling tricks from the bag. So this is one of those tricks. Basically he goes through the following analysis now and says, look, suppose I have the vector R and I do a parity transformation. I change it for minus R. What happens under those circumstances? Well the momentum is the rate of change of R, should also change sign. Quantum mechanically this is consistent, because a commutation between R and p should give you h bar. And if R changes, p should change sign. But now when you do this, L, which is R cross p, just goes to L. And R, however, changes sign because L doesn't change sign but p does and R does. So under these changes-- so this is the originator, the troublemaker and then everybody else follows-- R also changes sign. So this is extremely powerful because if you imagine this being equal to something, well it should be consistent with the symmetries. So as I change R to minus R, capital R changes sign but the left hand side doesn't change sign. Therefore the right hand side should not change sign. And R changes sign and L cross R changes sign. So computation kind of finished because the only thing you can get on the right is L. This is the kind of thing that you do and probably if you were writing a paper on that you would anyway do the calculation. The silly way, the- the right way. But this is quite save of times. So actually what you have learned is that R cross R is equal to some scalar conserved quantity, which is something that is conserved that could be like an h, for example, here. But it's a scalar. And, L. Well once you know that much, it doesn't take much work to do this and to calculate what it is. But I will skip that calculation. This is the sort of thoughtful part of it. And R cross R turns out to be ih bar minus 2 h again. h shows up in several places, like here, so it tends-- it has a tendency to show up. me to the fourth L. So this is our equation for-- and in a sense, all the hard work has been done. Because now you have a complete understanding of these two vectors, L and R. You know what L squared is, what R squared is, what L dot R is. And you know all the commutators, you know the commutation of L with L, L with R, and R with R. You've done all the algebraic work. And the question is, how do we proceed from now to solve the hydrogen atom. So the way we proceed is kind of interesting. We're going to try to build from this L that is an angular momentum. And this R that is not an angular momentum. Two sets of angular momenta. You have two vectors. So somehow we want to try to combine them in such a way that we can invent two angular momenta. Just like the angular momentum in the two-dimensional harmonic oscillator. It was not directly through angular momentum, but was mathematical angular momentum. These two angular momenta we're going to build, one of them is going to be recognizable. The other one is going to be a little unfamiliar. But now I have to do something that-- it may sound a little unusual, but is necessary to simplify our life. I want to say some words that will allow me to think of this h here as a number. And would allow me to think of this h as a number. So here's what we're going to say. It's an assumption-- it's no assumption, but it sounds like an assumption. But there's no assumption whatsoever. We say the following: this hydrogen atom is going to have some states. So let's assume there is one state, and it has some energy. If I have that state with some energy, well, that would be the end of the story. But in fact, the thing that they want to allow the possibility for is that at that the energy there are more states. One state would be OK, maybe sometimes it happens. But in general there are more states at that energy. So I don't-- I'm not making any physical assumption to state that there is a subspace of degenerate states. And in that subspace of degenerate states, there may be just one state, there are two states, there are three states, but there's subspace of degenerate states that have some energy. And I'm going to work in that subspace. And all the operators that I have are going to be acting in that subspace. And I'm going to analyze subspace by subspace of different energies. So we're going to work with one subspace of degenerate energies. And if I have, for example, the operator R squared acting on any state of that subspace, since h commutes with L squared, h can go here, acts on this thing, becomes a number. So you might as well put a number here. You might as well put a number here as well. It has to be stated like that. Carefully. We're going to work on a degenerate subspace of some energy. But then we can treat the h as a number. So let me say it here. We'll work in a degenerate subspace with eigenvalues of h equal to h prime, for h prime. Now I want to write some numbers here to simplify my algebra. So without loss of generality we put what this dimensionless-- this is dimensionless. I'm sorry, this is not dimensionless. This one has units of energy. This is roughly the right energy, with this one would be the right energy for the ground state. Now we don't know the energies and this is going to give us the energies as well. So without solving the differential equation, we're going to get the energies. So if I say, well that's the energies you would say, come on, you're cheating. So I'll put one over nu squared where nu can be anything. Nu is real. And that's just a way to write things in order to simplify the algebra. I don't know what nu is. How you say-- you don't know, but you have this in mind and it's going to be an integer, sure. That's what good notation is all about. You write things and then, you know, it's nu. You don't call it N. Because you don't know it's an integer. You call it nu, and you proceed. So once you have called it nu, you see here that, well, that's what we call h really. h will be-- this h prime is kind of not necessary. This is what-- where h becomes in every formula. So from here you get that minus 2h over me to the fourth is 1 over h squared nu squared. I have a minus here, I'm sorry. 2h minus me to the fourth down is h squared nu squared. So we can substitute that in our nice formulas that hme to the fourth so our formulas four and five have become-- I'm going to use this blackboard. Any blackboard where I don't have a formula boxed can be erased. So I will continue here. And so what do we have? R cross R, from that formula, well this thing is over there minus 2h over me to the fourth, you substitute it in here. So it's i over h bar, one over nu squared L. Doesn't look that bad. And, R squared is equal to 1 minus 1 over h bar nu squared. Like this. L squared plus h squared. 2h, that's minus h squared nu squared. Yeah. So these are nice formulas. These are already quite clean. We'll call them five, equation five. I still want to rewrite them in a way that perhaps is a little more understandable or suggestive. I will put an h bar nu together with each R. So h nu R cross h nu R is equal to ih bar L. Makes it look nice. Then for this one you'll put h squared nu squared R squared is equal to h squared nu squared minus 1 minus L squared. It's sort of trivial algebra. You multiply by h squared nu squared, you get this. You get h squared nu squared minus L squared because it's all multiplied minus h squared. So these two equations, five, have become six. So five and six are really the same equations. Nothing much has been done. And if you wish, in terms of commutators this equation says that the commutator h nu Ri with h nu Rj is equal to ih bar epsilon ijkLk. H cross this thing h nu R, h nu R cross equal iHL in components means this. That is not totally obvious. It requires a small computation, but is the same computation that shows that this thing is really LiLj equal ih bar epsilon ijkLk. In which these L's have now become R's. OK so, we've cleaned up everything. We've made great progress even though at this moment it still looks like we haven't solved the problem at all. But we're very close. So are there any questions about what we've done so far? Have I lost you in the algebra, or any goals here? Yes. AUDIENCE: Why is R cross R not a commutation? Why would we expect that to not be a commutation? PROFESSOR: In general, it's the same thing as here. L cross L is this. The commutator of two Hermitian operators is anti-Hermitian. So there's always an i over there. Other questions? It's good, you have to-- you should worry about those things. Are the units right, or the right number of i's on the right hand side. That's a great way to catch mistakes. OK so we're there. And now it should really almost look reasonable to do what we're going to do. h nu R with h nu R gives you like L. So you have L with L, form angular momentum. L and R are vectors in their angular momentum. Now R cross R is L. And with these units, h nu R and h nu R looks like it has the units of angular momentum. So h nu R can be added to angular momentum to form more angular momentum. So that's exactly what we're going to do. So here it comes. Key step. J1-- I'm going to define two angular momenta. Well, we hope that they are angular momenta. L plus h nu R. And J2, one half L minus h nu R. These are definitions. It's just defining two operators. We hope something good happens with these operators, but at this moment you don't know. It's a good suggestion because of the units match and all that stuff. So this is going to be our definitions, seven. And from these of course follows that L, the quantity we know, is J1 plus J2. And R, or h nu R, is J1 minus J2. You solve in the other way. Now my first claim is that J1 and J2 commute. Commute with each other. So these are nice, commuting angular momenta. Now this computation has to be done-- let me-- yeah, we can do it. J1i with J2J. It's one half and one half gives you one quarter of Li plus h nu Ri with LJ minus h nu RJ. Now the question is where do I-- I think I can erase most of this blackboard. I can leave this formula. It's kind of the only very much needed one. So I'll continue with this computation here. This gives me one quarter-- and we have a big parentheses-- ih bar epsilon iJkLk. For the commutator of these two. And then you have the commutator of the cross terms. So what do they look like? They look like minus h nu Li with RJ, and minus h nu Ri with-- no. So I have minus h nu Li with RJ, and now I have a plus of this term. But I will write this as a minus h nu of LJ with Ri. Those are the two cross products. And then finally we have this thing, the h nu with h nu Rijk. So I have minus h nu squared, and you have then RiRJk. No, I'll do it this way. I'm sorry. You have minus over there, and I have this thing so it's minus ih bar epsilon iJkLk from the last two commutators. So this one you use essentially equation six. Now look. This thing and this thing cancels. And these two terms, they actually cancel as well. Because here you get an epsilon iJR. And here there's an epsilon Ji something. So these two terms actually add up to zero. And this is zero. So indeed, J1i and J2i-- 2J-- is zero. And these are commuting things. I wanted to say commuting angular momentum, but not quite yet. Haven't shown their angular momenta. So how do we show their angular momenta? We have to try it and see if they really do form an algebra of angular momentum. So again, for saving room, I'm going to erase this formula. It will reappear in lecture notes. But now it should go. So the next computation is something that I want to do. J1 cross J1 or the J2 cross J2, to see if they form angular momenta. And I want to do them simultaneously, so I will do one quarter of J1 cross J2 would be L plus minus h nu R cross L plus minus h nu R. OK that doesn't look bad at all, especially because we have all these formulas for products. So look, you have L cross L, which we know. Then you have L cross R plus R cross L that is conveniently here. And finally, you have R cross R which is here. So it's all sort of done in a way that the composition should be easy. So indeed 1 over 4 L cross L gives you an ih bar L. From L cross L. From these ones, you get plus minus with plus minus. It's always plus but you get another ihL. So you get another ihL. And then you get plus minus L cross h nu R plus h nu R cross L. So here you get one quarter of 2 ihL. And look at this formula, just put an h nu here and h nu here and an h nu here. So you get plus minus 2 ih from here and an h nu R. OK so the twos and the fours and the iH's go out and then you get ih times one half times L plus minus h nu R, which is either J1 or J2. So, very nicely, we've shown that J1 cross J1 is ih bar J1 and J2 cross J2 is ih bar J2. And now finally you can say that you've discovered two independent angular momenta in the hydrogen atom. You did have an angular momentum on an R vector, and all of our work has gone into showing now that you have two angular momenta. Pretty much we're at the end of this because, after we do one more little thing, we're there. So let me do it here. I will not need these equations anymore. Except this one I will need. So L dot R is zero. So from L dot R equals zero, this time you get J1 plus J2 is equal to-- no, times-- J1 minus J2 is equal to zero. Now J1 and J2 commute. So the cross terms vanish. J1 and J2 commute. So this implies that J1 squared is equal to J2 squared. Now this is a very surprising thing. These two angular momenta have the same length squared. Let's look a little more at the length squared of it. So let's, for example, square J1. Well, if I square J1, I have one fourth L squared plus h squared nu squared R squared. No L dot R term, because L dot R is 0. And h squared nu squared R squared is here. So this is good news. This is one fourth L squared plus h squared nu squared minus 1 minus L squared. The L squared cancels. And you've got that J1 equals to J2 squared. And it's equal to one fourth of h squared nu squared minus 1. OK. Well the problem has been solved, even if you don't notice at this moment. It's all solved. Why? You've been talking a degenerate subspace with angular momentum with equal energies. And there's two angular momenta there. And their squares equal to the same thing. So these two angular momenta, our squares are the same and the square is precisely what we call h squared J times J plus 1, where j J is quantized. It can be zero, one half, one, all of this. So here comes a quantization. J squared being nu squared, we didn't know what nu squared is, but it's now equal to these things. So at this moment, things have been quantized. And let's look into a little more detail what has happened and confirm that we got everything we wanted. So let me write that equation again here. J1 squared is equal J2 squared is equal to one quarter h squared nu squared minus 1, which is h squared J times J plus 1. So cancel the h squares and solve for nu squared. Nu squared would be 1 plus 4J times J plus 1, which is 4J squared plus 4J plus 1, which is 2J plus 1 squared. That's pretty neat. Why is it so neat? Because as J is equal to zero, all the possible values of angular momentum-- three halves, all these things-- nu, which is 2J plus 1, will be equal to 1, 2, 3, 4-- all the integers. And what was nu? It was the values of the energies. So actually you've proven the spectrum. Nu has come out to be either 1, 2, 3, but you have all representations of angular momentum. You have the singlet, the spin one half-- where are the spins here? Nowhere. There was an electron, a proton, we never put spin for the hydrogen atom. But it all shows up as these representations in which they come along. Even more is true, as we will see right away and confirm that everything really shows up the right way. So what happened now? We have two independent, equal angular momentum. So what is this degenerate subspace we were inventing? Is the space J, which is J1 and m1 tensor product with J, which is J2 but has the same value because the squares are the same, m2. So this is an uncoupled basis. Uncoupled basis of states in the degenerate subspace. And now, you know, it's all a little surreal because these don't look like our states at all. But this is the way algebraically they show up. We choose our value of J, we have then that nu is equal to this and for that value of J there's some values of m's. And therefore, this must be the degenerate subspace. So this is nothing but the tensor product of a J multiplet with a J multiplet. Where J is that integer here. And what is the tensor product of a J multiplet? First, J is for J1. The second J is for J2. So at this moment of course we're calling this N for the quantum number. But what is this thing? This is 2J plus 2J minus 1 plus-- all the way up to the singlet. But what are these representations of? Well here we have J1 and here is J2. These must be the ones of the sum. But who is the sum, L? So these are the L representations that you get. L is your angular momentum. L representations. And if 2J plus 1 is N, you got a representation with L equals N minus 1, because 2J plus 1 is N, L equals N minus 2, all the way up to L equals 0. Therefore, you get precisely this whole structure. So, just in time as we get to 2 o'clock, we've finished the quantization of the hydrogen atom. We've finished 805. I hope you enjoyed. I did a lot. [INAUDIBLE] and Will did, too. Good luck and we'll see you soon.

MIT_805_Quantum_Physics_II_Fall_2013

9_Diracs_Bra_and_Ket_Notation.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Now, a theorem that was quite powerful and applied to complex vector spaces for old V longing to V complex vector space. This implied that the operator was zero, and it's not true for real vector spaces. And we gave a proof that made it clear that indeed, a proof wouldn't work for a real vector space. I will ask you in the homework to do something that presumably would be the first thing you would do if you had to try to understand why this is true-- take two by two matrices, and just see why it has to be 0 in one case, and it doesn't have to be 0 in the other case. And I think that will give you a better perspective on why this happens. And then once you do it for a two by two, and you see how it works, you can do it for n by n matrices, and it will be clear as well. So we'll use this theorem. Our immediate application of this theorem was a well-known result that we could prove now rigorously-- that t is equal to t dagger, which is to say the operator is equal to the adjoined. And I think I will call this-- and in the notes you will see always the adjoint, as opposed to Hermitian congregate. And I will say whenever the operator is equal to the adjoint, that it is Hermitian. So a Hermitian operator is equivalent to saying that v Tv is a real number for all v. And we proved that. In other words, physically, Hermitian operators have real expectation values. This is an expectation value, because-- as you remember in the bracket notation-- v Tv, you can write it v T v-- the same thing. So it is an expectation value, so it's an important thing, because we usually deal with Hermitian operators, and we want expectation values of Hermitian operators to be real. Now that we're talking about Hermitian operators, I delay a complete discussion of diagonalization, and diagonalization of several operators simultaneously, for the next lecture. Today, I want to move forward a bit and do some other things. And this way we spread out a little more the math, and we can begin to look more at physical issues, and how they apply here. But at any rate, we're just here already, and we can prove two basic things that are the kinds of things that an 805 student should be able to prove at any time. They're really very simple. And it's a kind of proof that is very short-- couple of lines-- something you should be able to reproduce at any time. So the first theorem says the eigenvalues of Hermitian operators-- H is for Hermitian-- are real. And I will do a little bit of notation, in which I will start with an expression, and evaluate it sort of to the left and to the right. So when you have an equation, you start here, and you start evaluating there. So I will start with this-- consider v Tv. And I will box it as being the origin, and I will start evaluating it. Now if v is an eigenvector-- so let v be an eigenvector so that Tv is equal to lambda v. And now we say consider that expression in the box. And you try to evaluate it. So one way to evaluate it-- I evaluate it to the left, and then evaluate to the right. Is the naive evaluation-- t on v is lambda v, so substitute it there. v, lambda v-- we know it. And then by homogeneity, this lambda goes out, and therefore it's lambda v v. On the other hand, we have that we can go to the right. And what is the way you move an operator to the first position? By putting a dagger. That's a definition. So this is by definition. Now, we use that the operator is Hermitian. So this is equal to Tv v. And this is by T Hermitian. Then you can apply again the equation of the eigenvalues, so this is lambda v v. And by conjugate homogeneity of the first input, this is lambda star v v. So at the end of the day, you have something on the extreme left, and something on the extreme right. v-- if there is an eigenvector, v can be assumed to be non-zero. The way we are saying things in a sense, 0-- we also think of it as an eigenvector, but it's a trivial one. But the fact that there's an eigenvalue means there's a non-zero v that solves this equation. So we're using that non-zero v. And therefore, this is a number that is non-zero. You bring one to the other side, and you have lambda minus lambda star times v v equals 0. This is different from 0. This is different from 0. And therefore, lambda is equal to lambda star. So it's a classic proof-- relatively straightforward. The second theorem is as simple to prove. And it's already interesting. And it states that different eigenvectors of a Hermitian operator-- well, different eigenvalues of Hermitian operators correspond to orthogonal eigenfunctions, or eigenvectors. So different eigenvalues of Hermitian ops correspond to orthogonal eigenfunctions-- eigenvectors, I'm sorry. So what are we saying here? We're saying that suppose you have a v1 that [INAUDIBLE] T gives you a lambda1 v1. That's one eigenvalue. You have another one-- v2 is equal to lambda 2 v2, and lambda 1 is different from lambda 2. Now, just focusing on a fact that is going to show up later, is going to make life interesting, is that some eigenvalues may have a multiplicity of eigenvectors. In other words, If a vector v is an eigenvector, minus v is an eigenvector, square root of three v is an eigenvector, but that's a one-dimensional subspace. But sometimes for a given eigenvalue, there may be a higher dimensional subspace of eigenvectors. That's a problem of degeneracy, and it's very interesting-- makes life really interesting in quantum mechanics. So if you have degeneracy, and that set of eigenvectors form a subspace, and you can choose a basis, and you could have several vectors here. Now what do you do in that case? The theorem doesn't say much, so it means choose any one. If you had the bases there, choose any one. The fact remains that if these two eigenvalues are different, then you will be able to show that the eigenvectors are orthogonal. So if you have some space of eigenvectors-- a degenerate higher-dimensional space of eigenvectors, one eigenvalue, and another space with another-- any vector here is orthogonal to any vector there. So how do you show this? How do you show this property? Well you have to involve v1 and v2, so you're never going to be using the property that gives Hermitian, unless you have an inner product. So if you don't have any idea how to prove that, you presumably at some stage realize that you probably have to use an inner product. And we should mix the vectors, so maybe a V2 inner product with this. So we'll take a v2 inner product with T v1. And this is interesting, because we can use it, that T v1 is lambda 1 v1 to show that this is just lambda 1 v2 v1. And that already brings all kinds of good things. You're interested in this inner product. You want to show it's 0, so it shows up. So it's a good idea. So we have evaluated this, and now you have to think of evaluating it in a different way. Again, the operator is Hermitian, so it's asking you to move it to the other side and exploit to that. So we'll move it to the other side a little quicker this time. It goes as T dagger, but T dagger is equal to T, because it's Hermitian. So this is the center of the equation. We go one way. We go the other way-- this time down. So we'll put T v2 v1, and this is equal to lambda-- let me go a little slow here-- lambda 2 v2 v1. Your impulse should be it goes out as lambda 2 star, but the eigenvalues are already real, so it goes out as lambda 2 v2 v1, because the operator is Hermitian. So at this moment, you have these two equations. You bring, say, this to the right-hand side, and you get lambda 1 minus lambda 2 v1 v2 is equal to 0. And since the eigenvalues are supposed to be different, you conclude that v1 inner product with v2 is 0. So that's the end of the proof. And those are the two properties that are very quickly proven with rather little effort. So where do we go from now? Well there's one more class of operators that are crucial in the physics. They are perhaps as important as the Hermitian operators, if not more. They are some operators that are called unitary operators, and the way I will introduce them is as follows-- so I will say-- it's an economical way to introduce them-- so we'll talk about unitary operators. If S is unitary, and mathematicians call it anisometry-- if you find that S acting on any vector-- if you take the norm, it's equal to the norm of the vector for all u in the vector space. So let's follow this, and make a couple of comments. An example-- a trivial example-- this operator lambda times the identity. Lambda times the identity acts on vectors. What does it do, lambda times identity? The identity does nothing on the vector, and lambda stretches it. So lambda, in order not to change the length of any vector should be kind of 1. Well, in fact, it suffices-- it's unitary-- if the absolute value of lambda is equal to 1. Because then lambda is a phase, and it just rotates the vector. Or in other words, you know that the norm of av is equal to the absolute value of a times the norm of v, where this is a number. And remember these two norms are different. This is the norm of a vector. This is the normal of a complex number. And therefore, if you take lambda i u-- norm-- is lambda u is equal absolute value of lambda u, and absolute value of lambda is equal to 1 is the answer. So that's a simple unitary operator, but an important one. Another observation-- what are the vectors annihilated by this operator u? Zero-- it's the only vector, because any other vector that's nonzero has some length, so it's not killed. So it kills only zero. So the null space of S is equal to the 0 vector. So this operator has no kernel, nothing nontrivial is put to zero. It's an invertible operator. So s is invertible. So that's a few things that you get very cheaply. Now from this equation, S u equals u-- if you square that equation, you would have S u S u is equal to u u. Maybe I should probably call it v. I don't know why I called it u, but let's stick to u. Now, remember that we can move operators from one side to the other. So I'll move this one to that side. If you move an S here, you would put an S dagger. But since the dagger of an S dagger is S, you can move also the S to that side as S dagger. So u S dagger, S u-- you see that. If you want to move this one, you can move it by putting another dagger, and you get that one. And this is u u, and therefore you get u S dagger, S minus the identity acting on u is equal to 0 for all u. So for every vector, this is true, because this is true. We just squared it. And now you have our favorite theorem, that says if this is true in a complex vector space, this is 0, and therefore, you've shown that S dagger S is equal to 1. So that's another property of unitary operators. In fact that's the way it's many times defined. Unitary operators sometimes are said to be operators whose inverse is S dagger. I will not go into the subtleties of what steps in all these things I'm saying are true or not true for infinite dimensional operators-- infinite dimensional vector spaces. So I will assume, and it will be true in our examples, that if S dagger is an inverse from the left, it's also an inverse from the right. And perhaps everything is true for infinite dimensional vector spaces, but I'm not 100% positive. So S dagger is the inverse of S. And that's a pretty important thing. So one last comment on unitary operators has to do with basis. So suppose you have an orthonormal basis, e1 up to en. Now you can define another basis. fi equal-- I'll change to a letter U-- U e i where U is unitary, so it's like the S. In fact, most books in physics call it U for unitary. So maybe I should have changed that letter in there, too, as well. So suppose you change basis. You put-- oh, there was something else I wanted to say before. Thanks to this equation, consider now the following thing-- S U Sv. SUSv-- you can move this S, for example, to the other side-- S dagger S U v, and S dagger S is equal to 1, and it's Uv. So this is a pretty nice property. We started from the fact that it preserved the norm of a single vector, of all vectors, and now you see that in fact, it preserved the inner product. So if you have two vectors, to compare their inner product, compute them after action with U or before action with U, and it doesn't make a difference. So suppose you define a second basis here. You have one orthonormal basis. You define basis vectors like this. Then the claim is that the f1 up to fn is orthonormal. And for that you simply do the following-- you just check f i f j is equal to U e i, U e j. By this property, you can delete both U's, rules, and therefore this is e i, e j. And that's delta i j. So the new basis is orthonormal. If you play with these things, it's easy to get some extra curious fact here. Let's think of the matrix representation of the operator U. Well, we know how these things are, and let's think of this in the basis e basis. So U k i is equal to ek U e i. That's the definition of U in the basis e-- the matrix elements of U. You can try to figure out what is Uki in the f basis. How does operator U look in the f basis? Well, let's just do it without thinking. So in the f basis, I would put fk U fi. Well, but fk is U ek, so I'll put Uek Ufi. Now we can delete both U's, and it's ek fi. And I can remember what fi was, which is ek U ei. And it's just the same as the one we had there. So the operator, unitary operator, looks the same in both bases. That might seem strange or a coincidence, but it's not. So I leave it to you to think about, and visualize why did that happen. What's the reason? So the bracket notation-- we've been using it here and there-- and I will ask you to please read the notes. The notes will be posted this afternoon, and they will have-- not maybe all we've done today, but they will have some of what we'll do today, and all of what we've been doing. And the way it's done-- it's first done in this sort of inner product language, and then things are done in the bracket language. And it's a little repetitious, and I'm trying to take out some things here and there, so it's less repetitious. But at this moment it's probably worth reading it, and reading it again. Yes. AUDIENCE: [INAUDIBLE] if you have two orthonormal bases, is the transformation between them necessarily unitary? PROFESSOR: Yes, yes. All right. So as I was saying we're going to go into the Dirac notation again. And here's an example of a place where everybody, I think, tends to use Dirac notation. And the reason is a little curious, and you will appreciate it quite fast. So this will be the case of where we return to x and p operators, on a non-denumerable basis. So we're going to try to do x and p. now this is the classic of Dirac notation. It's probably-- as I said-- the place where everybody likes to use Dirac notation. And the reason it's efficient is because it prevents you from confusing two things. So I've written in the notes, and we have all these v's that belong to the vector space. And then we put this, and we still say it belongs to the vector space. And this is just a decoration that doesn't do much. And we can play with this. Now, in the non-denumerable basis, the catch-- and the possible confusion-- is that the label is not quite a vector in the vector space. So that is the reason why the notation is helpful, because it helps you distinguish two things that you could confuse. So here we go. We're going to talk about coordinate x, and the x operator, and the states. Well, this is a state space. So what kind of states do we have here? Well, we've talked about wave functions, and we could give the value of the wave function of different places. We're going to go for a more intrinsic definition. We're going to try to introduce position states. And position states will be called this-- x. Now, what is the meaning of this position state? We should think of this intuitively as a particle at x. Now here's how you can go wrong with this thing, if you stop thinking for a second. What is, then, ax? Is it ax, a being a number. Is it the same thing? No, not at all. This is a particle at the coordinate ax, and this is a particle at x with some different amplitude-- very different. So this is not true-- typical mistake. This is not minus x. That's totally different. So there's no such thing as this, either. It doesn't mean anything. And the reason is that these things are not our vectors. Our vector is this whole thing that says a particle at x. Maybe to make a clearer impression, imagine you're in three dimensions, and you have an x vector. So then you have to ket this. This is the ket particle at x. x is now a vector. It's a three-dimensional vector. This is a vector, but it's a vector in an infinite dimensional space, because the particle can be anywhere. So this is a vector in quantum mechanics. This is a complex vector space. This is a real vector space, and it's the label here. So again, minus x is not minus x vector. It's not the vector. The addition of the bra has moved you from vectors that you're familiar with, to states that are a little more abstract. So the reason this notation is quite good is because this is the number, but this i--- or this is a coordinate, and this is a vector already. So these are going to be our basis states, and they are non-denumerable. And here you can have that all x must belong to the real numbers, because we have particles in a line, while this thing can be changed by real numbers. The states can be multiplied by complex numbers, because we're doing quantum mechanics. So if you want to define a vector space-- now, this is all infinite dimension. It's a little worse in this sense the basis is non-denumerable. If I use this basis, I cannot make a list of all the basis vectors. So for an inner product, we will take the following-- we will take x with y to be delta of x minus y. That will be our inner product. And it has all the properties of the inner product that we may want. And what else? Well at this moment, we can try to-- this is physically sensible, let me say, because if you have a particle at one point and a particle at another point, the amplitude that this particle at one point is at this other point is 0. And these states are not normalizable. They correspond to a particle at the point, so once you try to normalize them, you get infinity, and you can't do much. But what you can do here is state more of the properties, and learn how to manipulate this. So remember we had one was the sum of all e i e i. The unit operator was that. Well, let's try to write a similar one. The unit operator will be the sum over all x's. And you could say, well, looks reasonable, but maybe there's a 1/2 in here, or some factor. Well, no factor is needed. You can check that-- that you've defined this thing properly. So let me do it. So act to on this so-called resolution of the identity with the vector y, so 1 on y is equal to y. And now let's add on the right xxy. This is delta of x minus y. And then when you integrate, you get y. So we're fine. So this looks a little too abstract, but it's not the abstract if you now introduce wave functions. So let's do wave functions. So you have a particle, a state of the particle psi. Time would be irrelevant, so I will put just this psi like that without the bottom line. And let's look at it. Oh, I want to say one more thing. The x operator acts on the x states to give x x. So these are eigenstates of the x operator. We declare them to be eigenstates of the x operator with eigenvalue x. That's their physical interpretation. I probably should have said before. Now, if we have a psi as a state or a vector, how do we get the wave function? Well, in this language the wave function, which we call psi of x, is defined to be the overlap of x with psi. And that makes sense, because this overlap is a function of this label here, where the particle is. And therefore, the result is a complex number that is dependent on x. So this belongs to the complex numbers, because inner products can have complex numbers. Now, I didn't put any complex number here, but when you form states, you can superpose states with complex numbers. So this psi of x will come out this way. And now that you are armed with that, you can even think of this in a nicer way. The state psi is equal to 1 times psi. And then use the rest of this formula, so this is integral-- dx x x psi. And again, the bracket notation is quite nice, because the bra already meets the ket. This is a number, and this is dx x psi of x. This equation has a nice interpretation. It says that the state is a superposition of the basis states, the position states, and the component of your original state along the basis state x is precisely the value of the wave function at x. So the wave function at x is giving you the weight of the state x as it enters into the sum. So one can compute more things. You will get practice in this type of computations. There are just a limited type of variations that you can do, so it's not that complicated. Basically, you can introduce resolutions of the identity wherever you need them. And if you introduce too many, you waste time, but you typically get the answer anyway. So it's not too serious. So suppose you want to understand what is the inner product of two states. Put the resolution of the identity in between. So put phi, and then put the integral dx x x psi. Well, the integral goes out, and you get phi x x psi. And remember, if x psi is psi of x, phi x is the complex conjugate, so it's phi star of x. And you knew that. If you have two wave functions, and you want to compute the overlap, you integrate the complex conjugate of one against the other. So this notation is doing all what you want from this. You want to compute a matrix element of x. Well, put another resolution of the identity here. So this would be integral dx phi-- the x hat is here. And then you put x x psi. The x hat on x is x. That's what this operator does, so you get integral dx of-- I'll put x phi x x psi, which is what you expect it to be-- integral of x phi star of x, psi of x. Now we can do exactly the same thing with momentum states. So I don't want to bore you, so I just list the properties-- basis states are momenta where the momenta is real. p prime p is equal delta of p minus p prime. One is the integral dp of p p. And p hat p is equal to p p. So these are the momentum bases. They're exactly analogous. So all what we've done for x is true. The completeness and normalization work well together, like we checked there, and everything is true. The only thing that you need to make this more interesting is a relation between the x basis and the p basis. And that's where physics comes in. Anybody can define these two, but then a physical assumption as to what you really mean by momentum is necessary. And what we've said is that the wave function of a particle with momentum p is e to the i px over h bar over square root of 2 pi h-- convenient normalization, but that was it. That was our physical interpretation of the wave function of a particle with some momentum. And therefore, if this is a wave function, that's xp. A state of momentum p has this wave function. So we write this. OK, there are tricks you can do, and please read the notes. But let's do a little computation. Suppose you want to compute what is p on psi. You could say, well, I don't know why would I want to do something with that? Looks simple enough. Well, it's simple enough, but you could say I want to see that in terms of wave functions, coordinate space wave functions. Well, if you want to see them in terms of coordinate space wave functions, you have to introduce a complete set of states. So introduce p x x psi. Then you have this wave function, and oh, this is sort of known, because it's the complex conjugate of this, so it's integral dx px over h bar, square root of 2 pi h bar times psi of x. And this was the Fourier transform-- what we call the Fourier transform of the wave function. So we can call it psi tilde of p, just to distinguish it, because we called psi with x, psi of x. So if I didn't put a tilde, you might think it's the same functional form, but it's the momentum space wave function. So here is the wave function in the p basis. It's the Fourier transform of the wave function in the x basis. One last computation, and then we change subjects again. It's the classic computation that you have now a mixed situation, in which you have the momentum operator states and the coordinate bra. So what is the following expression-- X p hat psi? OK. What is your temptation? Your temptation is to say, look, this is like the momentum operator acting on the wave function in the x basis. It can only be h bar over i d dx of psi of x. That's probably what it means. But the notation is clear enough, so we can check if that is exactly what it is. We can manipulate things already. So let's do it. So for that, I first have to try to get rid of this operator. Now the only way I know how to get rid of this operator p is because it has eigenstates. So it suggests very strongly that we should introduce momentum states, complete them. So I'll put v p x p hat p p psi. And now I can evaluate the little-- because p hat and p is little p, or p without the hat. So this is p xp p psi. Now you can look at that, and think carefully what should you do. And there's one thing that you can do is look at the equation on top. And this is a way to avoid working very hard. So look at the equation on top-- x p is equal to that. How do I get a p to multiply this? I can get a p to multiply this xp by doing h bar over i d dx of x p. Because if I see it there, I see that differentiating by d dx brings down an ip over h bar. So if I multiply by h bar over i, I get that. So let's do this. Now I claim we can take the h over i d dx out of this integral. And the reason is that first, it's not an x integral. It's a p integral, and nothing else except this factor depends on x. So I take it out and I want to bring it back, it will only act on this, because this is not x dependent. So you should think of psi, psi doesn't have an x dependence. Psi is a state, and here is p-- doesn't have an x dependence? You say no, it does, it looks here. No, but it doesn't have it, because it's been integrated. It really doesn't have x dependence. So we can take this out. We'll have h over i d dx. And now we have vp x p p psi. And now by completeness, this is just 1. So this becomes x psi. So h bar over i d dx of x psi, which is what we claimed it would be. So this is rigorous-- a rigorous derivation. There's no guessing. We've introduced complete states until you can see how things act. But the moral is here that you shouldn't have to go through this more than once in your life, or practice it. But once you see something like that, you think. You're using x representation, and you're talking about the operator p. It cannot be anything like that. If you want to practice something different, show that the analogue p x hat psi is equal i h bar d dp of psi tilde. So it's the opposite relation. All right. Questions? Yes. AUDIENCE: So how's one supposed to-- so what it appears is happening is you're basically taking some state like psi, and you're basically writing in terms of some basis. And then you're basically using the [INAUDIBLE] coordinates of this thing. But the question is, what does this basis actually look like? Like, what do these vectors-- because if you put them in their own coordinates, they're just infinite. PROFESSOR: Yup. AUDIENCE: They're not even delta-- I mean-- PROFESSOR: They are delta functions. AUDIENCE: [INAUDIBLE] PROFESSOR: These vectors are delta functions because if you have a state that has this as the position state of a particle, you find the wave function by doing x on it. That's our definition of a wave function. And its infinite. So there's is not too much one can say about this. If people want to work more mathematically, the more comfortable way, what you do is, instead of taking infinite things, you put everything on a big circle. And then you have a Fourier series and they transform as sums, and everything goes into sums. But there's no real need. These operations are safe. And we managed to do them, and we're OK with them. Other questions? Yes. AUDIENCE: [INAUDIBLE] PROFESSOR: Probably not. You know, infinite bases are delicate. Hilbert spaces are infinite dimensional vector spaces, and they-- not every infinite dimensional space is a Hilbert space. The most important thing of a Hilbert space is this norm, this inner product. But the other important thing is some convergence facts about sequences of vectors that converge to points that are on the space. So it's delicate. Infinite dimensional spaces can be pretty bad. A Banach space is not a Hilbert space. It's more com-- AUDIENCE: [INAUDIBLE] PROFESSOR: Only for Hilbert spaces, and basically, this problem of a particle in a line, or a particle in three space is sufficiently well known that we're totally comfortable with this somewhat singular operation. So the operator x or the operator p may not be what mathematicians like them to be-- bounded operators in Hilbert spaces. But we know how not to make mistakes with them. And if you have a very subtle problem, one day you probably have to be more careful. But for the problems we're interested in now, we don't. So our last topic today is uncertainties and uncertainty relations. I probably won't get through all of it, but we'll get started. And so we'll have uncertainties. And we will talk about operators, and Hermitian operators. So here is the question, basically-- if you have a state, we know the result of a measurement of an observable is the eigenvalue of a Hermitian operator. Now, if the state is an eigenstate of the Hermitian operator, you measure the observable, and out comes eigenvalue. And there's no uncertainty in the measured observable, because the measured observable is an eigenvalue and its state is an eigenstate. The problem arises when the state that you're trying to measure this property is not an eigenstate of the observable. So you know that the interpretation of quantum mechanics is a probabilistic distribution. You sometimes get one thing, sometimes get another thing, depending on the amplitudes of the states to be in those particular eigenstates. But there's an uncertainty. At this time, you don't know what the measured value will be. So we'll define the uncertainty associated to a Hermitian operator, and we want to define this uncertainty. So A will be a Hermitian operator. And you were talking about the uncertainty. Now the uncertainty of that operator-- the first thing that you should remember is you can't talk about the uncertainty of the operator unless you give me a state. So all the formulas we're going to write for uncertainties are uncertainties of operators in some state. So let's call the state psi. And time will not be relevant, so maybe I should delete the-- well, I'll leave that bar there, just in case. So we're going to try to define uncertainty. But before we do that, let's try to define another thing-- the expectation value. Well, the expectation value-- you know it. The expectation value of A, and you could put a psi here if you wish, to remind you that it depends on the state-- is, well, psi A psi. That's what we call expectation value. In the inner product notation would be psi A psi. And one thing you know-- that this thing is real, because the expectation values of Hermitian operators is real. That's something we reviewed at the beginning of the lecture today. So now comes the question, what can I do to define an uncertainty of an operator? And an uncertainty-- now we've said already something. I wish to define an uncertainty that is such that the uncertainty is 0 if the state is an eigenstate, and the uncertainty is different from 0 if it's not an eigenstate. In fact, I wish that the uncertainty is 0 if and only if the state is an eigenstate. So actually, we can achieve that. And in some sense, I think, the most intuitive definition is the one that I will show here. It's that we define the uncertainty, delta A, and I'll put the psi here. So this is called the uncertainty of A in the state psi. So we'll define it a simple way. What else do we want? We said this should be 0 if and only if the state is an eigenstate. Second, I want this thing to be a real number-- in fact, a positive number. What function do we know in quantum mechanics that can do that magic? Well, it's the norm. The norm function is always real and positive. So this-- we'll try to set it equal to a norm. So it's the norm of the state A minus the expectation value of A times 1 acting on psi. This will be our definition of the uncertainty. So it's the norm of this vector. Now let's look at this. Suppose the norm uncertainty is 0. And if the uncertainty is 0, this vector must be 0. So A minus expectation value of A on psi is 0. Or A psi is equal to expectation value of A on psi. The 1 doesn't do much. Many people don't write the 1. I could get tired and stop writing it. You should-- probably it's good manners to write the i, but it's not all that necessary. You don't get that confused. If there's an operator and a number here, it must be an identity matrix. So the uncertainty is 0, the vector is 0, then this is true. Now, you say, well, this equation looks kind of funny, but it says that psi is an eigenstate of A, because this is a number. It looks a little funny, because we're accustomed to A psi lambda psi, but this is a number. And in fact, let me show you one thing. If you have A psi equal lambda psi-- oh, I should say here that psi is normalized. If psi would not be normalized, you change the normalization. You change the uncertainty. So it should be normalized. And look at this-- if you have a psi equal lambda psi, do the inner product with psi. Psi comma A psi would be equal to lambda, because psi inner product with psi is 1. But what is this? This is the expectation value of A. So actually, given our definition, the eigenvalue of some operator on this state is the expectation value of the operator in the state. So back to the argument-- if the uncertainty is 0, the state is an eigenstate. And the eigenvalue happens to be the expectation value-- that is, if the uncertainty is 0. On the other hand, if you are in an eigenstate, you're here. Then lambda is A, and this equation shows that this vector is 0, and therefore you get 0. So you've shown that this norm or this uncertainty is 0, if and only if the state is an eigenstate. And that's a very powerful statement. The statement that's always known by everybody is that if you have an eigenstate-- yes-- no uncertainty. But if there's no uncertainty, you must have an eigenstate. That's the second part, and uses the fact that the only vector with 0 norm is the zero vector-- a thing that we use over and over again. So let me make a couple more comments on how you compute this. So that's the uncertainty so far. So the uncertainty vanishes in that case. Now, we can square this equation to find a formula that is perhaps more familiar-- not necessarily more useful, but also good. For computations, it's pretty good-- delta A of psi, which is real-- we square it. Well, the norm square is the inner product of this A minus A psi A minus A psi. Norm squared is the inner product of these two vectors. Now, the thing that we like to do is to move this factor to that side. How do you move a factor on the first input to the other input? You take the adjoint. So I should move it with an adjoint. So what do I get? Psi, and then I get the adjoint and this factor again. Now, I should put a dagger here, but let me not put it, because A is Hermitian. And moreover, expectation value of A is real. Remember-- so no need for the dagger, so you can put the dagger, and then explain that this is Hermitian and this is real-- or just not put it. And now look at this. This is a typical calculation. You'll do it many, many times. You just spread out the things. So let me just do it once. Here you get A squared minus A expectation value of A minus expectation value of A A plus expectation value of A squared psi. So I multiplied everything, but you shouldn't be all that-- I should put a 1 here, probably-- shouldn't worry about this much. This is just a number and an A, a number and an A. The order doesn't matter. These two terms are really the same. Well, let me go slowly on this once. What is the first term? It's psi A squared psi, so it's the expectation value of A squared. Now, what is this term? Well, you have a number here, which is real. It goes out of whatever you're doing, and you have psi A psi. So this is expectation value of A. And from the leftover psi A psi, you get another expectation value of A. So this is A A. Here the same thing-- the number goes out, and you're left with a psi A psi, which is another expectation value of A, so you get minus A A. And you have a plus expectation value of A squared. And I don't need the i anymore, because the expectation values have been taken. And this always happens. It's a minus here, a minus here, and a plus here, so there's just one minus at the end of the day. One minus at the end of the day, and a familiar, or famous formula comes out that delta of A on psi squared is equal to the expectation value of A squared minus expectation value of A squared. Which shows something quite powerful. This has connections, of course, with statistical mechanics and standard deviations. It's a probabilistic interpretation of this formula, but one fact that this has allowed us to prove is that the expectation value of A squared is always greater or equal than that, because this number is positive, because it is the square of a real positive number. So that's a slightly non-trivial thing, and it's good to know it. And this formula, of course, is very well known. Now, I'm going to leave a funny geometrical interpretation of the uncertainty. Maybe you will find it illuminating, in some ways turning into pictures all these calculations we've done. I think it actually adds value to it, and I don't think it's very well known, or it's kind of funny, because it must not be very well known. But maybe people don't find it that suggestive. I kind of find it suggestive. So here's what I want to say geometrically. You have this vector space, and you have a vector psi. Then you come along, and you add with the operator A. Now the fact that this thing is not and eigenstate means that after you add with A, you don't keep in the same direction. You go in different directions. So here is A psi. So what can we say here? Well, actually here is this thing. Think of this vector space spanned by psi. Let's call it U psi. So it's that line there. You can project this in here, orthogonally. Here is the first claim-- the vector that you get up to here-- this vector-- is nothing else but expectation value of A times psi. And that makes sense, because it's a number times psi. But precisely the orthogonal projection is this. And here, you get an orthogonal vector. We'll call it psi perp. And the funny thing about this psi perp is that its length is precisely the uncertainty. So all this, but you could prove-- I'm going to do it. I'm going to show you all these things are true, but it gives you a bit of an insight. you have a vector. A moves you out. What is the uncertainty is this vertical projection-- vertical thing is the uncertainty. If you're down there, you get nothing. So how do we prove that? Well, let's construct a projector down to the space U psi, which is psi psi. This is a projector, just like any e1. e1 is a projection into the direction of 1. Well, take your first basis vector to be psi, and that's a projection to psi. So let's see what it-- so the projection to psi. So now let's see what it gives you when it acts on A psi-- this project acting on A psi is equal to psi psi A psi. And again, the usefulness of bracket notation is kind of nice here. So what is this? The expectation value of A. So indeed psi expectation value of A is what you get when you project this down. So then, the rest is sort of simple. If you take psi, and subtract from psi-- well, I'll subtract from psi, psi times expectation value of A. I'm sorry, I was saying it wrong. If you think the original vector-- A psi, and subtract from it what we took out, which is psi times expectation value of A, the projected thing-- this is some vector. But the main thing is that this vector is orthogonal to psi. Why? If you take a psi on the left, this is orthogonal to psi. And how do you see it? Put the psi from the left. And what do you get here? Psi A psi, which is expectation value of A, psi psi, which is 1, and expectation value A is 0. So this is a vector psi perp. And this is, of course, A minus expectation value of A acting on the state psi. Well, precisely the norm of psi perp is the norm of this, but that's what we defined to be the uncertainty. So indeed, the norm of psi perp is delta A of psi. So our ideas of projectors and orthogonal projectors allow you to understand better what is the uncertainty-- more pictorially. You have pictures of vectors, and orthogonal projections, and you want to make the uncertainty 0, you have to push the A psi into psi. You have to be an eigenstate, and you're there. Now, the last thing of-- I'll use the last five minutes to motivate the uncertainty, the famous uncertainty theorem. And typically, the uncertainly theorem is useful for A and B-- two Hermitian operators. And it relates the uncertainty in A on the state psi to the uncertainty in B of psi, saying it must be greater than or equal than some number. Now, if you look at that, and you think of all the math we've been talking about, you maybe know exactly how you're supposed to prove the uncertainty theorem. Well, what does this remind you of? Cauchy-Schwarz-- Schwarz inequality, I'm sorry-- not Cauchy-Schwarz. Why? Because for Schwarz inequality, you have norm of u, norm of v is greater than or equal than the norm of the inner product of u and v-- absolute value of the inner product of u and v. Remember, in this thing, this is norm of a vector, this is norm of a vector, and this is value of a scalar. And our uncertainties are norms. So it better be that. That inequality is the only inequality that can possibly give you the answer. So how would you set this up? You would say define-- as we'll say f equal A minus A acting on psi, and g is equal to B minus B acting on psi. And then f f, or f f is delta A squared. f g g is delta B squared. And you just need to compute the inner product of f g, because you need the mixed one. So if you want to have fun, try it. We'll do it next time anyway. All right that's it for today.

MIT_805_Quantum_Physics_II_Fall_2013

19_Multiparticle_States_and_Tensor_Products_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, let me get going. Last time we were talking about multi-particle states and tensor products. And for that, we explained that if we have a system, a quantum mechanical system of one particle described by a vector space V, and the quantum mechanical system of another particle described with a vector space W, the quantum mechanics of the total system composed by the two particles is defined on a new vector space called the space V tensor W. And that was a construction that showed that in particular it was not true to say that, oh if you want to know the system of particle 1 and 2, you just tell me what state particle 1 is and what state particle 2 is, and that's the end of the story. No, the story is really more sophisticated than that. So the typical elements on this space were of the form aij vi cross wj. And it's a sum over i and j numbers times these vectors. So you pick a vector in the first vector space, a vector in the second vector space, you put them in here and take linear combinations of them. So that's the general state in the system. Now we said a few things about this. One thing I didn't say too much about was the issue of the vector 0 in this tensor space. And well, vector 0 is some element of any vector space is an important element. And we could get a little confused about how it looks. And here's for example, the vector 0 in v cross w. An example of the vector 0 is the vector 0 tensor wi. If you put in the first input, the vector 0, that's it. That is also the vector 0 in here. Vi tensor the vector 0 in w. Here is 0 in w. Here is 0 in v. This is also 0. It's maybe a little surprising. Now how do we see that? Well we had a property. For example, this one. av tensor w is equal to av tensor w, where a is a number. So pick a equals 0. Well 0 times any vector is the 0 vector. 0 cross w. But 0 times any vector is also the vector 0. So this is the 0 in v cross w. So 0 cross w is the vector 0. Once you put 0 in one of the two inputs, you're there. You're at 0. You don't have more. So that's just a comment on the vector 0. Now we did a few things. And one thing I didn't do last time was to define an inner product on the new vector space. So let's define a way to get numbers from one vector in the tensor space and another vector in the tensor space. So inner product. And again, here you're supposed to define it to your best understanding and the hope that, once you make the right definitions, it has all that axiomatic properties it should have. So let me take the following thing. The inner product with this thing aij vi omega j with bpq vp wq. So I will define this by assuming the linearity in the inputs on the right inputs and the anti-linearity here on the left input. So this would be the sum over inj here. So I'll put sum over inj aij star sum over pq bpq and then vi wj comma vp wq. So by declaring that this is the case, I'm saying that the inner product in the tensor space has the-- I'm demanding it has the properties that we expect. If you have a vector plus another vector here, well you get this times the first plus this times the second. So you can take the sums out and arrange it this way. But I still haven't got a number, and the inner product is supposed to be a number. So how do we get a number at this stage? I have this thing, and nobody has told me what this is supposed to be. At this stage, the only thing you can say is, well, you know I suspect that, if I had an inner product in V and I had an inner product in w, I must have an inner product here, and somehow I should use that. So they still define to be ij pq aij bpq. And then what you do is use the inner product in v to get a number from these two vectors. This is going the v inner product. And use the inner product on w to get a number from the two w vectors. And that's it. The end of the definition. Now here maybe this is the sort of most interesting step, where this part was set equal to this. And consistent with what I was telling you about 0, suppose any of this vi was 0. If this vi was 0, we would have 0 with vp. That would be 0, so this whole number is 0. So the way this can happen is one of the vectors must be 0 here. And well, you have the 0 vector here, and the zero vector inner product with anything is 0. So it's, again, consistent to think that, once you put one of these entries to 0, you've got the 0 vector. So where are we going today? Well, we have now the inner product, and I want to go back to a state we had last time. What we're going to do today is define what we called an entangled state. Then we will consider basis of entangled states, and we will be able to discuss this sort of nice example of teleportation, quantum teleportation. So that's where we're going today. I wanted to remind you of a calculation we were doing last time. We had established that there was a state in the tensor product of 2 spin 1/2 particles. And the state was alpha plus tensor minus minus minus tensor plus. Now you can sometimes-- this is an example of a superposition of vectors of the from in the v cross w. So here is a vector of that form. There is a vector of this form. Sometimes we put here 1 and 2. And sometimes it will be useful to put those labels. Because if you don't put the labels, you better make sure that you're always talking that the first ket, is the one for the first vector space, and the second ket is the one for the second vector space. There's nothing really non-commutative here. So if somebody would write for you 1 minus 2, or they would write minus 2 1, both of you would be talking about the same state. But if you don't put the labels, you know you're not something about the same state, because you assume always the first one goes to the first Hilbert space. The second one goes with the second vector space. So we considered an entangled state of two spin 1/2 particles. I'm not using-- it's not fair to use the word entangled yet, but we'll be able to say this very soon. So the one thing we can do now given the inner product is try to normalize this state. So how do we normalize this state? Well, we must take the inner product of this state with itself. So phi phi. So then what do we do? Well, given these rules, we're supposed to take all this vector here, all that vector there, 1 alpha-- the alpha that is on the left goes out as an alpha star. The alpha that this on the right goes out as an alpha. And we have plus minus minus minus plus inner product with plus minus minus minus plus. Now this is easier than what it seems from what I'm writing. You will be able to do these things, I think. Or you can already maybe do them by inspection. Basically at this stage, you have to do each one with each one here. And let's see what we get. Well, what is the inner product of this with this? This works, because the inner product of plus with plus is 1 and minus with minus is 1. This on the other hand, doesn't give any contribution, because the first one is a plus has 0 inner product with a minus. A minus has 0 with a plus. That doesn't matter. It's an overkill. So this one couples with this, and this one couples with that. Another way people would do this is to say oh don't worry just take the bra here. So it's plus minus. Here is one. I'll put the labels too. Minus the bra of the minus is the minus like that. 1 plus 2. And now you do this with this ket, the plus minus 1 2 minus the minus plus 1 2. And bras and kets, you know that this one goes with this one. Plus plus, minus minus, this one goes with this one. And here I put the labels, because when I form the bra, it's not obvious which one you would put first, but it doesn't really matter. So back here, we have norm of alpha squared. And this with this is 1. And minus is one, this is another one. So this is 2 alpha squared. So if I want it to be normalized, I take alpha 1 over square root of 2. And this is the well normalized state. So this is the unit normalized state. So we have this state. This state is something you've played with over last week. Is that state that we started very fine in lecture that had 0 z component of angular momentum, 0 x component of angular momentum, and 0 y component of angular momentum. Total angular momentum as we defined it. And this has a state with absolutely no angular momentum. And what you verified in the homework was that that state, in fact, is rotational invariant. You apply a rotation operator to that state by rotating in both spaces, and out comes the same state. The state is not changed. So it's a very interesting state that will be important for us later. All right, so having taken care of inner products and normalizations, let's talk a little about entangled states. So entangled states. So these are precisely those states in which you cannot say, or describe them by saying particle one is doing this, particle two is doing that. You've learned that v cross w includes a state superpositions alpha ij vi cross omega j. The question is, if somebody hands you a state like this, maybe you could do some algebra or some trickery. And is it equal, you ask, to some sort of vector u star tensor v star times some vector w star. Is it equal? Is there vectors v star and omega star belonging to v and belonging to w in such a way that this thing, the sum, can be written as a product of something and that. If you would have that, then you would be able to say look, yes, this is an interesting state, but actually it's all simple here. Particle one is to state v star. Particle two is in state w star. If this has happened, if so, this state of the two particles is not entangled. So if you can really factor it, it's not entangled. If there are no such vectors v star and w star, then it is entangled. So you can say, well, it's a complicated factorization problem. And indeed, it might take a little work to figure out if a state is entangled or not. It's not a basis dependence problem. It's not like it's entangled in one basis or not. Here is a state, and you find any two things that tensor this way give you the state. So the simplest example to illustrate this is two dimensional vector spaces, v and w. Two dimensional complex. So v will have a basis e1 and e2. w will have a basis f1 and f2. And the most general state you could write is a state, general state, is a number a11 e1 f1 plus a2 e1 f2 plus a21 e2 f1 plus a22 e2 f2. That's it. There's two basis states in v, two basis state in w. v cross w is dimension for product of dimensions for basis states, the products of the e's with the f's. So that's it. That's the general vector. The question is if this is it equal to something like a1 e1 plus a2 e2. Some general vector, you write the most general vector in v, and you write the most general vector b1 f1 plus b2 f2 in w. And you ask is it equal to a product, tensor product, of some vector in v with some vector in w. So the question is really are their numbers a1, a2, b1, and b2 so that this whole thing gets factorized. So that's happily not a complicated problem. We could see if those number exist, if a1, a2, b1, b2 exist, then the state is not entangled. You've managed to factor it out. So let's see. Well, we know the distributive laws apply. So actually e1 f1 can only arise from this product. So to have a solution you must have that a11 is equal to a1 b1. a12 can only appear from the product of e1 with f2. So a12 must be equal to a1 b2. a21 must be equal to a2 b2. And a22 must be equal to a2 b2. And we must try to solve for these quantities. Actually, there is a consistency condition. You see these quantities repeat here in a funny way. If this holds from this, a11 a22 minus a12 a21 is equal to what? a11 a22 would be a1 b1 a2 b2. And a12 a21 also have the same things. a1 b2 a2 b1. Well, both terms have both a's and both b's, so this system only has a solution if this product is 0. So if you give me four numbers, if you hope to factorize it, you must have the determinant of this matrix-- if you collapse it into a matrix, a11 a12 a21 a22, if you encode the information about this state in a matrix, it's necessary that the determinant of the matrix a be equal 0. So the determinant of a is equal to 0 is certainly necessary for the factorization to take place. But a very small argument that will be in the notes, or you can try to complete it, shows that the determinant equal to 0, in fact, guarantees that then you can solve this system. There's a solution. And this is not complicated. So determinant equals 0 is actually the same as not entangled. We've done not entangled. So there's a solution implies determinant a equals 0, but determinant of a equals 0 also implies not entangled. You do that by solving this. Let's not spend time doing that. The basic way to do it is to assume-- consider, say, a11 equals 0 and solve it. Then a11 different from 0, and then you can show that you can choose these quantities. So it can be factored. And you have that, if these numbers are such that the determinant is 0, then the state is entangled. And it's very easy to have a determinant of this non-zero. For example, you could have these two 0 and these two non-zero. That will be entangled because the determinant is non-zero. You can have this two that will be entangled. There are many ways of getting entangled states. So in fact, there's enough ways to get entangled states that we can construct a basis. We had a basis here of e1 f1 e2 f2. This thing. This four vector basis. We can construct a basis that is all the states, all the basis vectors are entangled states. That's what we're going to do next. But maybe it's about time for questions, things that have become a little unclear as I went along. Yes? AUDIENCE: So what exactly does an entangled state mean? What are the [INAUDIBLE] to give me an entangled state. PROFESSOR: Well, the main thing that it happens is that there will be interesting correlations when you have an entangled state. If you have an entangled state and you find a state that is not entangled, you can say particle one is doing this and particle two is doing that. And particle two is doing this independent of what particle one is doing. But when a state is entangled, whatever is happening with particle one is correlated with what is happening in particle two in a strange way. So if particle one is doing something, then particle two is doing another thing. But if particle one is doing another thing, then particle two is doing something. And these particles can be very far apart, and that's when it gets really interesting. So we're going to do a lot of things with entangled states. Today we're doing this teleportation using entangled state, and you will see how subtle it is. Next time we do EPR, these Einstein Podolsky Rosen arguments and the Bell inequalities that answered that with entangled states. There's a couple of problems in the homework set also developing entangled states in different directions. And I think by the time we're done, you'll feel very comfortable with this. So a basis of entangled states. Here it goes. We're going to use spins. So we're going to use v is the state space of spin 1/2. And we're going to consider a v tensor v where this refers to the first particle and this this to the second particle. So let's take one state, phi 0, defined to be 1 over square root of 2, and I don't put indices. And probably at some stage, you also tend to drop the tensor product. I don't know if it's early enough to drop it. Probably we could drop it. We'll put plus plus minus minus. Of course, people eventually drop even the other ket and put it plus plus. So those are the evolutions of notation. As you get to more and more calculations, you write less, but hopefully, it's still clear. But I will not do this one. I will still keep that because many times, I will want to keep labels. Otherwise, it's a little more cumbersome. So this state is normalized. Phi0 phi0 is equal to 1. It's the state we built. Oh, in fact, I want it with a plus. Sorry. It's similar to the state we had there. And by now, you say, look, yes, it's normalized. Let's take the dual. Plus plus with plus plus will give me 1. The minus minus with minus minus will give me 1. This is 2. 1 over square root of 2 squared, 1. It should become sort of easy by inspection that this is normalized. And this is entangled state because in the matrix representation, it's a 1 here and a 1 there. You have the 1 1 product and the 2 2 product. So 1 1, the determinant is non-zero. There's no way, we've proven, you can find how to factor this. There's no alpha. There's no way to write this as an alpha plus, plus beta minus, times a gamma plus, plus delta minus. Just impossible. We've proven it. It's entangled. So this is an entangled state, but the state space is four dimensional. So if it's four dimensional, we need three more basis states. So here they are. I'm going to write a formula for them. Phi i for i equals 1, 2, and 3 will be defined to be the following thing. You will act with the operator 1 tensor sigma i on phi 0. So three ways of doing. Let's do 1, for example, phi 1. What is it? Well, you would have 1 times sigma 1 acting on the state phi 0, which is 1 over square root of 2 plus, plus, plus minus, minus. Well, the 1 acts on the first ket, the sigma acts on the second ket. So what do we get here? 1 over square root of 2-- let me go a little slow-- plus sigma 1 plus, plus, minus sigma 1 minus. And this is phi 1 equals sigma 1 plus is the minus state, and sigma 1 minus is the plus state. 1 over square root of 2. Those are things that you may just remember sigma 1 is this matrix. So you get 1 over square root of 2 plus, minus, plus, minus, plus. So that's phi 1. And phi 1 is orthogonal to phi 0. You can see that because plus minus cannot have an overlap with plus plus, nor with minus minus. Here minus plus, no. In order to get something, you would have to have the same label here and the same label here so that something matches. Well, we can do the other ones as well. I will not bother you too much writing them out. So what do they look like? Well, you have phi 2 would be 1 tensor sigma 2 on phi 0. And that would give you-- I will just copy it-- an i because sigma 2 has i's there. So i over square root of 2 plus, minus, minus, minus, plus. Finally, phi 3 is 1 tensor sigma 3 phi 0. And it's 1 over square root of 2 plus, plus, minus, minus, minus. We got the states here. Let's just check they're orthonormal. Well, here's one thing. If you take phi 0 with 1 tensor sigma i phi 0, which is phi 0 with phi i. Well, this is 0. You could say, well, how do you know? How do you prove it easily? Well, I think the best way is just inspection, so let's look at that. Phi 1, we said, is orthogonal to phi 0 because it has plus minus and minus plus, and that can never do anything with that. Phi 2 also has plus minuses and minus pluses, so we can never have anything to do with phi 0. The only one that has a chance to have an inner product with phi 0 is phi 2 because it has a plus plus and a minus minus. On the other hand, when you flip them, this term with a plus plus of phi 0 will give you 1, but here's a difference of sign. So this with the second term of phi 00 will give you a minus, and therefore, it will be 0. So these things are all 0 by inspection. You don't really have to do a calculation there. The one that takes a little more work is to try to understand what is the inner product of phi i with phi j. Now, you could say, OK, I'm going to do them by inspection. After all, there's just six things to check. But let's just do it a little more intelligently. Let's try to calculate this by saying, well, this is phi 0. Since the Pauli matrices are Hermitian, this phi i is also 1 tensor sigma i. They're Hermitian, so acting on the left, they're doing the right thing. Given our definition, here is a definition as well. So you take the bra and that's what it is. It would have been dagger here but it's not necessary. And then you have the phi j, which is 1 tensor sigma j. And that's phi 0 here. That sounds like the kind of thing that we can make progress using our Pauli identities. Indeed, first thing is that the product of operators, they multiply just in that order in the tensor product. So phi 0, you have 1 times 1, which is 1 tensor sigma i sigma j phi 0. And this is equal to phi 0 1 tensor. Now, the product of two Pauli matrices gives you an identity plus a Pauli matrix. You may or may not remember this formula, but it's 1 times delta ij plus i epsilon ijk sigma k phi 0. Now, what do we get? Look, the second term has a sigma k on phi 0, so it's some number with a phi k here, while the first term is very simple. What do we get from the first term? From the first term, we get-- well, 1 tensor 1 between any two things is nothing because the 1 acting on things and the 1 acting on another thing is 0. So the unit operator in the tensor product is 1 tensor 1. That's nothing whatsoever. So what do you get here? Delta ij times phi 0 phi 0 plus i epsilon ijk phi 0 phi k. But that is 0. We already showed that any phi i with phi 0 is 0. And this is 1. So what have we learned? That this whole thing is delta ij. And therefore, the basis is orthonormal. So we've got a basis of orthonormal states in the tensor product of two spin 1/2 particles. And the nice thing about this basis is that all of these basis states are entangled states. They're entangled because they fill different parts of the matrix. Here you have 1 and 1 and minus 1 here. This would be plus minus, would be an i here and a minus i there. The determinants are non-zero for all of them, and therefore, they can't be factored, and therefore, they're entangled. So the last thing I want to do with this is to record a formula for you, which is a formula of the basis states in the conventional way, written as superposition of entangled states. So for example, you say, what is plus plus? Well, plus plus, looking there, how would you solve it? You would solve it from phi 0 and phi 3. You would take the sum so that the minus minus states cancel. Phi 0 and phi 3, and therefore, this state must be 1 over square root of 2, phi 0 plus phi 3. A useful relation. Then we have plus minus. Then we have minus plus. And finally, minus minus. Well, minus minus would be done by 1 over square root of 2 phi 0 minus phi 3. The other ones, well, they just leave complex numbers. Phi 1 has this plus minus, and this has a plus minus in phi 2. The only problem is it has an i, so you must take this state minus i times this state will produce this state twice and will cancel this term. That's what you want. So phi 1, this should be 1 over square root of 2 phi 1 minus i phi 2. And this one should be phi 1 plus i phi 2. And if this was a little quick, it's just algebra, one more line. You do it with patience in private. So here it is. It's the normal product, simple product basis expressed as a superposition of entangled states. This is called the bell basis, this phi 1 up to phi 4, the bell basis. And now, I have to say a couple more things and we're on our way to begin the teleportation thing. Are there questions? Any questions about bell basis or the basis we've introduced? Any confusion? Errors on the blackboard? So we have a basis, and I want to make two remarks before we get started with the teleportation. It's one remark about measurement and one remark about evolution of states. Two facts. The first fact has to do with measurement in orthonormal basis. If you have an orthonormal basis, the postulate of measurement of quantum mechanics can be stated as saying that you can do an experiment in which you find the probability of your state being along any of these basis states of the orthonormal basis. So you can do an experiment to detect in which of the basis states the state is. Now, the state, of course, is in a superposition of basis states, but it will collapse into one of them with some probability. So the Stern-Gerlach experiment was an example in which you pick two basis states, orthogonal, and there was a device that allowed you to collapse the state into one or the other. So this is a little more general, not just for two state systems. If there would be a particle with three states, well, orthonormal states, then there is in principle an operator in quantum mechanics that allows it to measure which of these basis states you go into. So let me state this as saying, given an orthonormal basis, e1 up to en, we can measure a state, phi, and we get that the probability to be in ei is, as you know, ei overlapped with a state squared. And if you measure that this probability, the state will collapse into one of these states. So after the measurement, the state goes into some ek. There are different probabilities to be in each one of those basis states, but the particle will choose one. Now, the other thing I want to mention is that a fact that has seemed always a gift, the Pauli matrices are not only Hermitian, but they square to one, and therefore they're also unitary. So the Pauli matrices are unitary. So actually, they can be realized as time evolution. So you have a state and you want to multiply it by sigma 1. You say, OK, well, that's a very mathematical thing. Not so mathematical because it's a unitary operator, so it could respond to some time evolution. So we claim there is a Hamiltonian that you can construct that will evolve the state and multiply it by sigma 1. So all these Pauli matrices, sigma 1, sigma 2, and sigma 3 are unitary as operators. They can be realized by time evolution with a suitable Hamiltonian. So if you're talking spin states, some magnetic field that lifts for some few picoseconds according to the dipole, and that's it. It will implement sigma one. Just in fact, you can check, for example, that e to the i pi over 2 minus 1 plus sigma i. This is i this, and this is Hermitian. Well, this is 1 and sigma i. 1 and sigma i commute, so this is equal to e to the minus i pi over 2 times e to the i pi sigma 1 over 2. The first factor is a minus i, and the second factor is 1 times cosine of pi over 2 plus i sigma 1 sine of pi over 2. So this is minus i times-- this is 0-- times i sigma 1. So this is sigma 1. So we've written sigma 1 as the exponential of i times the Hermitian operator. And therefore, you could say that this must be equal to some time times some Hamiltonian over h bar. And you decide, you put the magnetic field in the x, y, z direction. You realize it. So sigmas can be realized by a machine. We're all done with our preliminary remarks, and it's now time to do the teleportation stuff. Quantum teleportation. So we all know this teleportation is the stuff of science fiction and movies and kind of stuff like that, and it's pretty much something that was, classically, essentially impossible. You have an object, you sort of dematerialize it and create it somewhere else. No basis for doing that. The interesting thing is that quantum mechanically, you seem to be able to do much better, and that's the idea that we want to explain now. So this is also not something that has been known for a long time. The big discovery that this could be done is from 1993. So it's just 20 years ago people realized finally that you could do something like that. In that way, quantum mechanics is, in a sense, having a renaissance because there's all kinds of marvelous experiments-- teleportation, entanglement, ideas that you could build one day a quantum computer. It's all stimulating thinking better about quantum mechanics more precisely, and the experiments are just amazing. This thing was done by the following people. We should mention them. Bennett at IBM, Brassard, Crepeau-- can't pronounce that-- Jozsa, all these people in Montreal. Peres, at Technion, and Wootters at Williams College. 1993. So big collaboration all over the world. So what is the question that we want to discuss? In this game, always there's two people involved, and the canonical names are Alice and Bob. Everybody calls Alice and Bob. It's been lots of years that people talk about Alice and Bob. They use it also for black hole experiments. Depending on your taste, Alice stays out and Bob is sucked into the black hole, or Bob stays out, Alice goes down. But it's Alice and Bob all the time. So this time, the way we're going to do it, Alice has a quantum state. It has been handed to her, and it's a state of a spin 1/2 particle. Spin 1/2 is nice because you have discrete labels. So she has this state. It's alpha plus beta minus. And she has it carefully there in a box, just hoping that the state doesn't get entangled with anything and disappear, or doesn't get measured. And her goal is to send this state to Bob, who's far away. So Alice is sitting here and has this state, and Bob is sitting somewhere here and has no state. And she wants to send this state. This is the state to be teleported. Now, there's a couple of things you could try to do before even trying to teleport this. Why teleport it? Why don't you create a copy of this state and just put it in FedEx and send it to Bob, and he gets it? The problem is that there's something in quantum mechanics, something called no cloning, that you can't create a copy of a state, actually, with a quantum mechanical process. It's really a funny thing. You've got a qubit-- this is called a qubit-- a quantum bit. Bit is something that can be 0 or 1. Quantum, it can be two things. So instead of calling it a spin state, sometimes people call it a qubit. For us, it's a spin state. It has two numbers. And there's no cloning. We will not discuss it here. It's a nice topic for a recitation. It's a simple matter. You can't make a copy. So given that you can't make a copy, let's avoid that idea, save ourselves $15 of FedEx and just try to do something else. So the one thing Alice could do is that she could say, all right. Well here is alpha and beta. Let me measure the state. Find alpha and beta. And then I'll of send that information to Bob. OK. But she has one copy of the state. How is she going to measure alpha and beta with one copy of the state. She puts it through a Stern-Gerlach experiment, and the particle comes out the plus side. Now what? The probability that it went to the plus. You've got some information about the alpha squared. Not even because you just did the experiment once and your cubit is gone. So Alice actually can't figure out alpha and beta. So if she's handed the qubit, she better not measure it. Because if she measures it, she destroys the state, goes into a plus or a minus, and it's all over. The state is gone before she could do anything. So that doesn't work either. Now there's the third option. Maybe Alice cannot talk to Bob, and Alice created that state with some Hamiltonian. And she knows because she created it what alpha and beta is. So she could in principle tell Bob, OK. Here is alpha and here is beta. Create it again. That would be a fine strategy, but actually there's even plausibly a problem with that. Because maybe she knows this state, but alpha is a number. It is 0.53782106, never ends. Doesn't repeat. And she has to send that infinite string of information to Bob, which is not a good idea either. She's not going to manage to send the right state. So these are the things we speculate about because it's a natural thing to one wonder. So what we're going to try to do is somehow produce an experiment in which she'll take this state, get it in, and somehow Bob is going to create that state on his other side. That's the teleportation thing that we'll try to do. So let's do a little diagram of how we're going to do this. So here is going to be the state that is going to be teleported. We'll call it the state C. So I'll write it as phi alpha plus in the state space C sub particle plus beta minus in this state space C. And C is the state she is going to try to teleport. But now they're not going to be able to do it unless they use something different. They try something different. And the whole idea is going to be to use an entangled state. So basically what we're going to do is we're going to put the source here, entangled state source. And we're going to produce and an entangled state of two particles. And one particle is going to be given to A, to Alice. And one particle is going to be given to Bob. So particle B for Bob is going to be given to Bob. And particle A is going to be given to Alice. And this is an entangled pair. So there it is. Now what's going to happen? What are we going to do? Entanglement really correlates what goes here with what goes in there. Now entanglement happens instantaneously, and we can discuss this. You have no way of sending information through entanglement in general. There's no such thing as learning something about A when B doesn't measure, learning anything nontrivial about A. So the entangled state is there, and that's what we're going to try to use in order to do the teleporting. Now morally speaking, suppose I wanted to teleport myself from one place in this room to another. What I would have to do is create an enormous reservoir of entangled states. So here's my generator, and I create billions of entangled pairs. And I put them all here, all the ones here and all the corresponding pairs over there. And then I sort of-- somebody takes me and these billions of entangled pairs, one side of the pair, and does a measurement in which every atom or every quantum state in my body is measured with some entangled state. They've done the measurement, and boom. I reappear on the other side. That's what's going to happen. So we're going to do this. We're going to have this state, and now we're going to a measurement between this state and this state. Alice is going to do a measurement. That's going to force this particle to actually pretty much become the state you wanted to teleport. So that's the goal. So let me say a couple more things. Alice will have to send some information actually. Because she is going to have to do a measurement, and she has a console with four lights, zero, one, two, and three. Four lights. And when she will do her measurement, one of the lights will blink. And she will have to tell Bob which one blinked. So she will have to send the number and information of two bits. Because with two bits, you can represent any of four numbers, binary code. So she will send information of which clicked. And then Bob will have a machine with four entries here. And according to the information that he gets, he will make the state to go through one of those machines, the zero, the one, the two, or the three. So he will push B into one of them out, we claim, will come this teleported state. So that's the set up. You have to get a feel for the set up. So are there questions on what we're doing? AUDIENCE: So after teleportation would have some kind of copy [INAUDIBLE]? PROFESSOR: No. After the replication, this state will be destroyed beyond repair as you will see. So there will not be a copy created by this procedure. You destroy. It's really what teleportation was supposed to be. Not to create another copy of you there, but to take you there. Destroy you here and recreate you there. So no other copy. Other questions? Yes. AUDIENCE: Does this also work if C is an entangled state? PROFESSOR: If what? AUDIENCE: If C say itself contains different parts which are entangled with each other? PROFESSOR: Well, it's a more complicated thing. I'm pretty sure it would work. Maybe you would need more than one entangled pair here. You would need a source that is more complicated. More questions. AUDIENCE: What do you mean about pushes the state into one of the [INAUDIBLE]? PROFESSOR: What do I mean by pushes it through one of them? Well you know, Hamiltonians. You get your state. You can put them in a magnetic field. Let them evolve a little bit. Those are machines. So any of these machines are some unitary time evolution. It does something to the state. AUDIENCE: But one [INAUDIBLE] PROFESSOR: Sorry. AUDIENCE: Are there Hamiltonians that are based off of what Alice measures? PROFESSOR: Yes. So they will be correlated as you will see. So if Alice measures that the light zero beeps, the instruction for Bob is to send the state through the zero Hamiltonian, and one, two, and three Hamiltonian. More questions? It's good to really have a good feeling of this or what we're trying to do and why it's nontrivial. Yes. AUDIENCE: This might be a little too intuitive, but in a state which-- Can a Hamiltonian which Bob needs to send B through in order to yield the same state that Alice had, can that also be transmitted quantumly through qubits? Or would you just get like an infinite line of qubits needing to-- PROFESSOR: No no. You know, this is a device that they can build by themselves. As you will see once we do the calculation, Alice will construct a device that has these four lights and she knows what they mean. And Bob will construct a device that has these things, and they can use it to transport any state. So these machines are independent of the state you want to teleport. You teleported this, you want to teleport another state with alpha prime and beta prime? Sure. Use exactly the same machines, give me another entangled pair, and do it. AUDIENCE: Well, I think what I meant is that the information between the two machines, does that have to be transmitted classically, or is there some way to transmit-- PROFESSOR: There's no real information. The machines were built, say, in the same laboratory of IBM. And then they're built, and we will tell you how to build each of these machines. And then just put aside, taken away by these two people, and then we'll do it. There's no mystery of sending information about it. That probably will become clear with the computation, which I better start doing soon. Yes. AUDIENCE: The difference-- PROFESSOR: Louder. AUDIENCE: Just a question about the first part on the left side of the board. So, when we first do a measurement, does that mean it's something that's like a microscopic quantity, like an energy or something? Or does it just refer to any? PROFESSOR: When we refer to measurements and quantum mechanics, we talk-- Let me give you just a little bit of intuition here. We typically talk about measuring hermitian operators, because they have eigenvalues. So we don't have to say what they are-- energy, momentum. It's a hermitian operator you measure. And projector operators into basis states of hermitian operators. So you could imagine that's one way of thinking about these measurements. OK. So let's do this. All right. The state to be teleported is this one, and the A B pair is an entangled state. So it will be one of the bell states, phi zero AB 1 over square root of 2 plus A plus b plus minus A plus minus B. So this is the state they share. Of course, Alice only has a handle on particle A, and Bob only has a handle on particle B. Nevertheless the state is entangled even though this could be 200 kilometers apart. So the total state-- well, we've been tensoring two things. Well, tensoring three is three particles. So I don't think you will be too unhappy to just tensor the whole thing. So phi zero AB tensor alpha plus C plus beta minus C. So here comes the interesting point. Alice has available the state A. The particle A is not the state A because A is in a funny thing. It's entangled. But it has a particle A available, and it has a particle C available. So Alice is going to do a measurement, and it's going to be a sneaky measurement. It's going to use a bases. Since she has two particles, she can choose a basis of two particle states. Any orthonormal basis will do well by the idea that we can measure with any orthonormal basis. So what she's going to try to do is use the bell basis for A and C. So let's try to think of what that means. That requires a small calculation here. So this is equal to 1 over square root of 2 plus-- so I anticipate that this will become clear in a second, what that measurement means-- plus minus A minus b Tensor alpha plus plus beta minus C. So I just wrote what this is. OK. Some algebra. This is the total state, phi total. Let's multiply these things out, and I will keep the labels all the time because I don't want there to be any confusion about what's happening. So what do we get first? Alpha multiplying plus of A. I should write in plus of B, but the order doesn't really matter if I keep the labels. So I'll put plus of C times plus of B. Then keep multiplying. So we have plus beta, from this with that. So I'll have plus of A minus of C and plus of B. Maybe it's easier to read if I use another line. So I now must multiply the second state times this. So I get plus alpha minus of A with plus of C and minus of B. So this is this times that, minus of A plus of C minus of B plus beta minus of A minus of C minus of B. OK. So there here my state. But now I have written it in a way that I have here A and C A and C A and C and A and C. So I could decide to measure in this basis. This is an orthonormal basis for A and C. But it's not a very smart basis because it's not entangled. So let's go to the entangled basis. So let's rewrite this state, this total state. Nothing has been done yet to the state. We're just mathematically rewriting it, nothing else. We have this, this, this, and that. And I want you now to use these formulas to do this. So I'll do this on this blackboard. We'll have to erase those important names. So what do we get? Well a little of algebra. Let's do it. A with C plus plus would be that. So I'll write it with one over square root of 2 becomes one half. A with C would be phi zero AC plus phi three AC multiplying alpha plus on B. So I took care of the first term. The alpha is there. The B is there. And AC is there, in which, you know, you can put any labels you want to here. AB, this is the AB state. The entangled AB state. We used AC. Second term plus one half. Now we have plus A minus C. So it's the second line in there. So it would be phi one AC minus I phi 2 AC beta plus B. Next line, I'll just copy it, one half. Well not. Alpha minus B and here you'll have the minus plus which is the same thing, phi 1 AC plus I phi 2 AC. And the last term is plus one half phi 0 AC minus phi 3 AC. And we get beta minus B. OK, almost there. Let's rewrite this as-- let's collect the phi zeroes, phi 0 and phi 0. You see we're do nothing yet. We're just mathematically rewriting the states in a different basis, the total states. So it is equal to one half phi 0 AC. and look what you get here, very curiously. You get alpha plus B plus beta minus B. Very curious, that was precisely the state we wanted to teleport. Alpha plus plus beta minus. All right. Let's see what else happens. Here we get plus one half phi-- which other one do I want to copy? Phi 1 AC. You see this is the state we wanted to teleport. It's here. And it sort of has appeared in the B space. Phi 1 AC, well this time I have this term and this term. So actually it seems a little different. Now we get beta plus B plus alpha minus B. Then we go to the next. One half of phi 2 AC. So phi 2 is here. So you get I alpha minus B minus I beta plus B. OK. Finally linear combinations. And finally phi 3. What is phi 3? Well two terms also for phi 3. This one and this one. So you get alpha plus B minus beta minus B. Kind of the end of math by now. You've proven a funny identity actually in doing this. And maybe this blackboard should-- to make sure you understand. This is the calculation of total state. And here we go. So let me show you one thing. This is actually the state we wanted. So this will be called phi in the B basis, in the B space. The state that you wanted to teleport that was phi in the C basis, now it's phi in the B basis. Those ones look a little funny, but this one actually looks like this thing, looks like sigma 3 times phi. Because if you have sigma 3 on this state, it gives you a plus 1 here and a minus eigenvalue. So that's sigma 3 phi. This actually has flipped the plus and the minus. So that actually is sigma 1 phi. And this state is actually sigma 2 phi. OK everything is in place now. We've just done math, but now comes the physics. Alice is going to measure in the bell space of A and C. So these are the four bases states. So she's going to measure in one of these bases states. And as see measures, she falls and the wave function of her collapses into one of them. So when she gets the zero basis state, this light blanks. If doing the measurement on AC, because she has both particles A and C, she gets this basis state-- recall the postulate of measurement-- light one blinks. If she gets the third like 2 and the fourth here. Suppose the state light zero shines. Well the state collapsed into this. She is now sitting with phi 0 AC that has no memory whatsoever of the original state C, but B is sitting with this state, the state we wanted to teleport. So if light zero shines, she tells Bob, let it go to machine zero where there's no magnetic field, nothing. So actually the same state goes out. If she gets phi 1 as the measured state, again no memory in this state about alpha and beta. But Bob gets sigma 1 phi 1. So he puts it into the first Hamiltonian for a picosecond, produces a sigma 1. This Hamiltonian, this box I takes a state into sigma I state. It's a unitary operation. So puts a sigma 1 and gets phi. If light two shines, goes to the machine two, which produces a sigma 2, and so he gets the state. Light four shines, the third Hamiltonian, he gets the state. Any of the four options, he gets the precise state. The state has been teleported. You needed to send only the information of which light shone, and the state is on the other side of the ocean. All right. That's it for today.

MIT_805_Quantum_Physics_II_Fall_2013

8_Linear_Algebra_Vector_Spaces_and_Operators_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: It's good to be back. I really want to thank both Aaron and Will who took my teaching duties over last week. You've been receiving updates of the lecture notes, and, in particular, as I don't want to go back over some things, I would like you to read some of the material you have there. In particular, the part on projectors has been developed further. We will meet projectors a lot in this space, because in quantum mechanics, whenever you do a measurement, the effect of a measurement is to act on a stage with a projector. So projectors are absolutely important. And orthogonal projectors are the ones that we're going to use-- are the ones that are relevant in quantum mechanics. There's a property, for example, of projectors that is quite neat that is used in maximization and fitting problems. And you will see that in the PSET. In the PSET, the last problem has to do with using a projector to find best approximations to some functions using polynomials. So there's lots of things to say about projectors, and we'll find them along when we go and do later stuff in the course. So please read that part on projectors. The other thing is that much of what we're going to do uses the notation that we have for describing inner products-- for example, u, v. And then, as we've mentioned, and this is in the notes-- this, in the bracket notation, becomes something like this. And the bracket notation of quantum mechanics is fairly nice for many things, and it's used sometimes for some applications. Everybody uses the bracket notation for some applications. I hope to get to one of those today. So much of the theory we've developed is done with this as the inner product. Nevertheless, the translation to the language of bras and kets is very quick. So the way the notes are going to be structured-- and we're still working on the notes, and they're going to chang a bit-- is that everything regarding the math is being developed more in this notation, but then we turn into bras and kets and just go quickly over all you've seen, just how it looks with bras and kets so that you're familiar. Then, in the later part of the course, we'll use sometimes bras and kets, and sometimes this. And sometimes some physicisists use this notation with parentheses. So for example, Weinberg's recent book on quantum mechanics uses this notation. It doesn't use bras and kets I think at all. So you have to be ready to work with any notation. The bra and ket notation has some nice properties that make it very fast to do things with it. It is very efficient. Nevertheless, in some ways this notation is a little clearer. So many of the things we'll develop is with this notation. So today I'm going to develop the idea of the Hermitian conjugator for an operator, or the adjoint of an operator. And this idea is generally a little subtle, a little hard to understand. But we'll just go at it slowly and try to make it very clear. So adjoints or Hermitian operators, or Hermitian conjugates-- adjoints or Hermitian conjugates. So the idea of Adjoints, or Hermition conjugates, really begins with some necessary background on what they're called-- linear functionals. It sounds complicated, but it's not. What is a linear functional? A linear functional on V-- on a vector space V-- is a linear map from V to the numbers F. We've always been calling F the numbers. So it's just that, something that, once you have a vector, you get a number and it's linear. So a linear function of Phi, if it's a linear functional, Phi on v belongs to F. Phi acts on a vector, v, that belongs to the vector space and gives you a number. So "linear" means Phi of v1 plus v2 is Phi of v1 plus Phi of v2. And Phi of av, for a is number, is a Phi of v. So seems simple, and indeed it is. And we can construct examples of linear functionals, some trivial ones, for example. Let Phi be a map that takes the vector space, reals in three dimensions, to the real numbers. So how does it act? Phi acts on a vector, which is x1, x2, and x3-- three components. And it must give a numbers, so it could be 3x1 minus x2 plus 7x3, as simple as that. It's linear. x1, x2, and x3 are the coordinates of a single vector. And whenever you have this vector, that is, this triplet-- now, I could have written it like this-- Phi of x1, x2, and x3, as a vector. It looks like that. But it's easier to use horizontal notation, so we'll write it like that. And, if you have an inner product on this space-- on this three dimensional vector space-- there's something you can say. Actually this Phi is equal-- and this we call the vector V-- is actually equal to u, inner product with v, where u is the vector that has components 3, minus 1, and 7, because if you take the inner product of this vector with this vector, in three dimensions real vector spaces-- inner product is a dot product. And then we make the dot product of u with the vector V. Maybe I should have called it v1, v2, v3. I'll change that-- v1, v2, v3 here are components of the vector-- v1, v2, and v3, not to be confused with three vectors. This whole thing is a vector V. So this linear functional, that, given a vector gives me a number. The clever thing is that the inner product is this thing that gives you numbers out of vectors. So you've reconstructed this linear functional as the inner product of some vector with the vector you're acting on, so, where u is given by that. The most important result about linear functionals is that this is not an accident. This kind be that very generally. So any time you give me a linear functional, I can find a vector that, using the inner product, acts on the vector you're acting on the same way as the linear function of thus. The most general linear functional is just some most general vector acting this way. So let's state that and prove it. So this is a theorem, it's not a definition or anything like that. Let Phi be a linear functional on v. Then there is a unique vector u belonging to the vector space such that Phi acting on v is equal to u, v. Since this is such a canonical thing, you could even invent a notation. Call this the linear functional created by u, acting on v. Everybody doesn't use this, but you could call it like that. This is a linear functional acting on v, but it's labeled by u, which is the vector that you've use there. This is important enough that we better understand why it works. So I'll prove it. We're going to use an orthonormal basis, say e1 up to en is an orthonormal, O-N, basis. AUDIENCE: That means we're assuming v is finite dimensional here? PROFESSOR: Sorry? AUDIENCE: We're assuming V is finite dimensional, correct? PROFESSOR: Yeah, it's finite dimensional I'm going to prove it using a finite basis like that. Is true finite dimensional? I presume yes. AUDIENCE: If it's not [INAUDIBLE]. PROFESSOR: What hypothesis? AUDIENCE: You say continuous when you're talking [INAUDIBLE]. PROFESSOR: OK, I'll check. But let's just prove this one finite dimensional like this. Let's take that. And now write the vector as a superposition of these vectors. Now we know how to do that. We just have the components of v along each basis vector. For example, the component of v along e1 is precisely e1, v. So then you go on like that until you go en, v, en. I think you've derived this a couple of times already, but this is a statement you can review, and let's take it to be correct. Now let's consider what is Phi acting on a v like that. Well, it's a linear map, so it takes on a sum of vectors by acting on the vectors, each one. So it should act on from this plus that, plus that, plus that. Now, it acts on this vector. Well, this is a number. The number goes out. It's a linear function. So this is e1, v, Phi of e1, all the way up to en, v Phi of en. Now this is a number, so let's bring it into the inner product. Now, if you brought it in on the side of V as a number it would go in just like the number. If you bring it into the left side, remember it's conjugate homogeneous, so this enters as a complex number. So this would be e1, Phi of e1 star times V plus en, Phi of en star, v. And then we have our result that this Phi of v has been written now. The left input is different on each of these terms, but the right input is the same. So at this moment linearity on the first input says that you can put here e1, Phi of e1 star plus up to en, Phi of en star, v. And this is the vector you were looking for, the vector U. Kind of simple, at the end of the day you just used the basis and made it clearer. It can always be constructed. Basically, the vector you want is e1 times Phi of u1 star plus en up to Phi of en star. So if you know what the linear map does to the basis vectors, you construct the vector this way. Vector is done. The only thing to be proven is that it's unique. Uniqueness is rather easy to prove at this stage. Suppose you know that u with v works and gives you the right answer. Well, you ask, is there a u prime that also gives the right answer for all v? Well, pass it to the other side, and you would have u minus u prime, would have zero inner product with v for all v. Pass to the other side, take the difference, and it's that. So u minus u prime is a vector that has zero inner product with any vector. And any such thing as always zero. And perhaps the easiest way to show that, in case you haven't seen that before, if x with v equals 0 for all for all v. What can you say about x? Well, take v is the value for any v. So take v equal x. So you take x, x is equal to 0. And by the axioms of the inner product, if a vector has 0 inner product with itself, it's 0. So at this stage, you go u minus u prime equals 0, and u is equal to u prime. So it's definitely unique, you can't find another one that works. So we have this thing. This theorem is proven. And now let's use to define this the adjoint, which is a very interesting thing. So the adjoing, or Hermitian conjugate, sometimes called adjoint-- physicists use the name Hermitian conjugate, which is more appropriate. Well, I don't know if it's more appropriate. It's more pictorial if you have a complex vector space. And if you're accustomed with linear algebra about Hermition matrices, and what they are, and that will show up a little later, although with a very curious twist. So given an operator T belonging to the set of linear operators on a vector space, you can define T dagger, also belonging to l of v. So this is the aim-- constructing an operator called the Hermitian conjugate. Now the way we're going to do it is going to be defining something that is a T star. Well, I said "T star" because mathematicians in fact call it star. And most mathematicians, they complex conjugate if a number is not z star but z bar. So that's why we call it T star and I may make this mistake a few times today. We're going to use dagger. And so I will make a definition that will tell you what T dagger is supposed to be, acting on things. But it might not be obvious, at least at first sight, that it's a linear operator. So let's see how does this go. Here is the claim. Consider the following thing-- u, T, v-- this inner product of u with T, v. And think of it as a linear functional. Well, it's certainly a linear functional of v. It's a linear functional because if you put a times v the a goes out. And if you put v1 plus v2 you get it's linear. So it's linear, but it's not the usual one's that we've been building, in which the linear functional looks like u with v. I just put an operator there. So by this theorem, there must be some vector that this can be represented as this acting with that vector inside here, because any linear operator is some vector acting on the vector-- on the vector v. Any linear functional, I'm sorry-- not linear operator. Any linear functional-- this is a linear functional. And every linear function can be written as some vector acting on v. So there must be a vector here. Now this vector surely will depend on what u is. So we'll give it a name. It's a vector that depends on U. I'll write it as T dagger u. At this moment, T dagger is just a map from v to v. We said that this thing that we must put here depends on u, and it must be a vector. So it's some thing that takes u and produces another vector called T dagger on u. But we don't know what T dagger is, and we don't even know that it's linear. So at this moment it's just a map, and it's a definition. This defines what T dagger u is, because some vector-- it could be calculated exactly the same way we calculated the other ones. So let's try to see why it is linear. Claim T dagger belongs to the linear operators in v. So how do we do that? Well, we can say the following. Consider u1 plus u1 acting on Tv. Well, by definition, this would be the T dagger of u1 plus u2, some function on u1 plus u2, because whatever is here gets acted by T dagger times v. On the other hand, this thing is equal to u1, Tv plus u2, Tv, which is equal to T dagger u1, v plus T dagger u2, v. And, by linearity, here you get equal to T dagger u1 plus T dagger on u2. And then comparing this too-- and this is true for arbitrary v-- you find that T dagger, acting on this sum of vectors, is the same as this thing. And similarly, how about au, Tv? Well, this is equal to T dagger on au, v. Now, T dagger on au, do you think the a goes out as a or as a bar? Sorry? a or a-bar? What do you think T dagger and au is supposed to be? a, because it's supposed to be a linear operator, so no dagger here. You see-- well, I didn't show it here. Any linear operator, T on av, is supposed to be a T of v. And we're saying T dagger is also a linear operator in the vector space. So this should be with an a. We'll see what we get. Well, the a can go out here, and it becomes a star u1, Tv, which is equal. I'm going through the left side. By definition, a bar T dagger of u, v. And now the constant can go in, and it goes back as a, T dagger u, v. So this must be equal to that, and you get what we're claiming here, which is T dagger on au, is equal to a T dagger of u. So the operator is linear. So we've defined something this way, and it's linear, and it's doing all the right things. Now, you really feel proud at this stage. This is still not all that intuitive. What does this all do? So we're going to do an example, and we're going to do one more property. Let me do one more property and then stop for a second. So here is one property-- ST dagger is supposed to be T dagger S dagger. So how do you get that? Not hard-- u, STv. Well, STv is really the same as S acting on Tv. Now the first S can be brought to the other side by the definition that you can bring something to the other side. Put in a dagger. So the S is brought there, and you get S dagger on u, T on v. And then the T can be brought here and act on this one, and you get T dagger S dagger u, v. So this thing is the dagger of this thing, and that's the statement here. There's yet one more simple property, that the dagger of S dagger is S. You take dagger twice and you're back to the same operator. Nothing has changed. So how do you do that? Take, for example, this-- take u, put S dagger here, and put v. Now, by definition, this is equal to-- you put the operator on the other side, adding a dagger. So that's why we put that one like this. The operator gets daggers, so now you've got the double dagger. So at this moment, however, you have to do something to simplify this. The easiest thing to do is probably the following-- to just flip these two, which you can do the order by putting a star. So this is equal. The left hand side is equal to this. And now this S dagger can be moved here and becomes an S. So this is u, Sv, and you still have the star. And now reverse this by eliminating the star, so you have S-- I'm sorry, I have this notation completely wrong. Sv-- this is u. The u's v's are easily confused. So this is v, and this is u. I move the S, and then finally I have Su, v without a star. I flipped it again. So then you compare these two, and you get the desired result. OK, so we've gone through this thing, which is the main result of daggers, and I would like to see if there are questions. Anything that has been unclear as we've gone along here? And question? OK. No questions. So let's do a simple example, and it's good because it's useful to practice with explicit things. So here's an example. There's a vector space V, which is three complex numbers, three component vectors-- complex vectors. So a v is equal to v1, v2, v3-- three numbers are all the vi. Each one belongs to the complex number. So three complex numbers makes a vector space like this. So somebody comes along and gives you the following linear map-- T on a vector, v1, v1, v3, gives you another vector. It's a linear map. So what is it? It's 0 times v1 plus 2v2 plus iv3 for the first component. The first component of the new vector-- I put the 0v1 just so you see that it just depends on v2 and v3. The second component is v1 minus iv2 plus 0v3. Those are not vectors. These are components. These are numbers. So this is just a complex number. This is another complex number, as it should be. Acting on three complex numbers gives you, linearly, three other ones. And then the third component-- they don't have space there, so I'll put it here-- 3iv1 plus v2 plus 7v3. And the question is two questions. Find T dagger, and write the matrix representations of T and T dagger. Write the matrices T and T dagger using the standard basis in which the three basis vectors are 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, and 0, 0, 1. These are the three basis vectors-- e1, e2, and e3. You know, to write the matrix you need the basis vectors. So that's a problem. It's a good problem in order to practice, to see that you understand how to turn an operator into a matrix. And you don't get confused. Is it a row? Is it a column? How does it go? So let's do this. So first we're going to try to find the rules for T dagger. So we have the following. You see, you use the basic property. u on Tv is equal to T dagger u on v. So let's try to compute the left hand side, and then look at it and try to see if we could derive the right hand side. So what is u supposed to be a three component vector? So for that use, u equals u1, u2, u3. OK, now implicit in all that is that when somebody tells you-- OK, you've got a three dimensional complex vector space what is the inner product? The inner product is complex conjugate of the first component. That's first component of the second, plus complex conjugate of the second times star, times star. So it's just a generalization of the dot product, but you complex conjugate the first entries. So what is this? I should take the complex conjugate of the first term here-- u1-- times the first one. So I have 2v2 plus iv3. This is the left hand side, plus the complex conjugate of the second component-- there's the second component-- so u2 times v1 minus iv2 plus-- well, 0v3-- his time I won't write it-- plus u3 bar times the last vector, which is 3iv1 plus v2 plus 7v3. OK, that's the left hand side. I think I'm going to use this blackboard here, because otherwise the numbers are going to be hard to see from one side to the other. So this information, those two little proofs, are to be deleted. And now we have this left hand side. Now, somehow when you say, OK, now I'm going to try to figure out this right hand side your head goes and looks in there and says well, in the left hand side the u's are sort of the ones that are alone, and the v's are acted upon. Here the v's must be alone. So what I should do is collect along v. So let's collect along v. So let's put "something" times v1 plus "something" like v2 plus "something" like v3. And then I will know what is the vector T star this. So let's do that. So v1, let's collect. So you get u2 bar for this v1, and 3iu3 bar. v2 will have 2u1 bar minus iu2 bar plus u3 bar. I think I got them right. OK. And then v3, let's collect-- iu1 bar, nothing here, and v3 7u3 bar. OK, and now I must say, OK, this is the inner product of T dagger u times v3. So actually, T dagger on u, which is u1, u2, u3, must be this vector with three components for which this thing is the inner product of this vector with the vector V. So I look at this I say, well, what was the formula for the inner product? Well, you complex conjugate the first entry of this and multiply by the first entry of that. Complex conjugate the second entry. So here I should put u2 minus 3iu3, because the complex conjugate of that is that as multiplied by v1. So here I continue-- 2u1 plus iu2 plus u3. And, finally, minus iu1 plus 7u3. And that's the answer for this operator. So the operator is there for you. The only thing we haven't done is the matrices. Let me do a little piece of one, and you try to compute the rest. Make sure you understand it. So suppose you get T on the basis vector e1. It's easier than what it looks. I'm going to have to write some things in order to give you a few components, but then once you get a little practice, or you look what it means, it will become clear. So what is T on e1? Well, it's T on the vector 1, 0, 0. T on the vector 1, 0, 0-- look at the top formula ther3-- is equal to 0, 1, and 3i. Top formula-- the v1 is 1, and all others are 0. And this is e2 plus 3ie3. So how do you read, now, matrix elements? You remember the formula that T on ei is supposed to be Tkiek-- sum over k. So this thing is supposed to be equal to T11e1 plus T21e2 plus T31e3. Your sum over the first index, T of e1, is there for that. So then I read this, and I see that T21 is equal to 1. This is equal to 3i. And this is equal to 0. So you've got a piece of the matrix, and the rest I will just tell you how you see it. But you should check it. You don't have to write that much after you have a little practice with this. But, the matrix T-- what you've learned is that you have 0, 1, and 3i. So 0, 1, and 3i are these numbers, in fact-- 0, 1, and 3i. And they go vertical. So 2, minus i, and 1 is the next column. 2, minus i, and 1 is the next column, and the third one would be i-- look at the v3 there. It has an i for the first entry, a 0 for the second, and a 7. So this is the matrix. How about the matrix T dagger? Same thing-- once you've done one, don't worry. Don't do the one. So this you look for the first column. It's going to be a 0-- no u1 here-- a 2, and a minus i. 0, 2, and a minus i, then 1, i, and 0, minus 3i, 1, and 7. And those are it. And look how nice. The second one is in fact the Hermitian conjugate of the other. Transpose and complex conjugate gives it to you. So that example suggests that that, of course, is not an accident. So what do you need for that to happen? Nobody said that what you're supposed to do to find T dagger is transpose some complex conjugate, but somehow that's what you do once you have the matrix, or at least what it seems that you do when you have the matrix. So let's see if we can get that more generally. So end of example. Look at T dagger u, v is equal to u, Tv. We know this is the key equation. Everything comes from this. Now take u and v to be orthonormal vectors, so u equal ei, and v equal ej. And these are orthonormal. The e's are going to be orthonormal each time we say basis vectors-- e, orthonormal. So put them here, so you get T dagger on ei times ej is equal to ei, Tej. Now use the matrix action on these operators. So T dagger on ei is supposed to be T dagger kiek. The equation is something worth knowing by heart. What is the matrix representation? If the index of the vector goes here, the sum index goes like that. So then you have ej here, and here you have ei, and you have Tkjek. So now this basis orthonormal. This is a number, and this is the basis. The number goes out. T dagger ki-- remember, it's on the left side, so it should go out with a star. And then you have ekej. That's orthonormal, so it's delta kej. The number here goes out as well, and the inner product gives delta ik. So what do we get? T dagger ji star is equal to Tij. First, change i for j, so it looks more familiar. So then you have T dagger ij star is equal to Tji. And then take complex conjugate, so that finally you have T dagger ij is equal to Tji star. And that shows that, as long as you have an orthonormal basis you can see the Hermitian conjugate of the operator by taking the matrix, and then what you usually call the Hermitian conjugate of the matrix. But I want to emphasize that, if you didn't have an orthonormal basis-- if you have your operator, and you want to calculate the dagger of it, and you find its matrix representation. You take the Hermitian conjugate of the matrix. It would be wrong if your basis vectors are not orthonormal. It just fails. So what would happen if the basis vectors are not orthonormal? Instead of having ei with ej giving you delta iej, you have that ei with ej is some number. And you can call it aij, or alpha iej, or gij, I think, is maybe a better name. So if the basis is not orthonormal, then ei with ej is some sort of gij. And then you go back here. And, instead of having deltas here, you would have g's. So you would have the T dagger star ki with gkj is equal to Tkj, gik. And there's no such simple thing as saying, oh, well you just take the matrix and complex conjugate and transpose. That's not the dagger. It's more complicated than that. If this matrix should be invertible, you could pass this to the other side using the inverse of this matrix. And you can find a formula for the dagger in terms of the g matrix, its inverses and multiplications. So what do you learn from here? You learn a fundamental fact, that the statement that an operator-- for example, you have T. And you can find T dagger as the adjoint. The adjoint operator, or the Hermitian conjugate operator, has a basis independent definition. It just needs that statement that we've written many times now, that T dagger u, v is defined via this relation. And it has nothing to do with a basis. It's true for arbitrary vectors. Nevertheless, how you construct T dagger, if you have a basis-- well, sometimes it's a Hermitian conjugate matrix, if your basis is orthonormal. But that statement, that the dagger is the Hermitian conjugate basis, is a little basis dependent, is not a universal fact about the adjoint. It's not always constructed that way. And there will be examples where you will see that. Questions? No questions? Well, let's do brackets for a few minutes so that you see a few properties of them. With the same language, I'll write formulas that we've-- OK, I wrote a formula here, in fact. So for example, this formula-- if I want to write it with bras and kets, I would write u Tv. And I could also write it as u T v, because remember this means-- the bra and the ket-- just says a way to make clear that this object is a vector. But this vector is obtained by acting T on the vector v. So it's T on the vector v, because a vector v is just something, and when you put it like that that's still the vector v. The kit doesn't do much to it. It's almost like putting an arrow, so that's why this thing is really this thing as well. Now, on the other hand, this thing-- let's say that this is equal to v, T dagger u star. So then you would put here that this is v T dagger u star. So this formula is something that most people remember in physics, written perhaps a little differently. Change v and u so that this left hand side now reads u T dagger v. And it has a star, and the right hand side would become v T u. And just complex conjugated it. So u T dagger v is equal to v T u star-- a nice formula that says how do you get to understand what T dagger is. Well, if you know T dagger's value in between any set of states, then you know-- well, if you know T between any set of states u and v, then you can figure out what T dagger is between any same two states by using this formula. What you have to do is that this thing is equal to the reverse thing. So you go from right to left and reverse it here. So you go v, then T, then u, and you put a star, and that gives you that object. Another thing that we've been doing all the time when we calculate, for example, ei, T on ej. What is this? Well, you know what this is. Let's write it like that-- ei. Now T on ej is the matrix T kjek. If this is an orthonormal basis, here is a delta iek. So this is nothing else but Tij. So another way of writing that matrix element, ij, of a matrix is to put an ei, an ej here, and a T here. So people write it like that-- Tij is ei comma Tej. Or, in bracket language, they put ei T ej. So I need it to be flexible and just be able to pass from one notation to the other, because it helps you. One of the most helpful things in this object is to understand, for example, in bra and ket notation, what is the following object? What is ei ei? This seems like the wrong kind of thing, because you were supposed to have bras acting on vectors. So this would be on the left of that, but otherwise it would be too trivial. If it would be on the left of it, it would give you a number. But think of this thing as a object that stands there. And it's repeated endlessly, so it's summed. So what is this object? Well, this object is a sum of things like that, so this is really e1 e1 plus e2 e2, and it goes on like that. Well, let it act on a vector. This kind of object is an operator. Whenever you have the bra and the ket sort of in this wrong position-- the ket first, and the bra afterwards-- this is, in Dirac's notation, an operator, a particular operator. And you will see in general how it is the general operator very soon. So look at this. You have something like that, and why do we call it an operator? We call it an operator business if it acts on a vector-- you put a vector here, a bra-- this becomes a number, and there's still a vector left. So this kind of structure, acting on something like that, gives a vector, because this thing goes in here, produces a number, and the vector is left there. So for example, if you act with this thing on the vector a-- an arbitrary vector a-- what do you get? Whatever this operator is is acted on a. Well, you remember that these thing are the components of a, and these are the basis vectors. So this is nothing else but the vector a again. You see, you can start with a equals some alpha i's with ei's, and then you calculate what are the alpha i's. You put an ej a, and this ej on that gives you alpha j. So alpha j-- these numbers are nothing else but these things, these numbers. So here you have the number times the vector. The only difference is that this is like ei alpha i. The number has been to the right. So this thing acting on any vector is the vector itself. So this is perhaps the most fundamental relation in bracket notation, is that the identity operator is this. Yes. AUDIENCE: Is that just 1 e of i, or sum over all e of i? PROFESSOR: It's sum of over all. So here implicit sum is the sum of all up to en en. You will see, if you take just one of them, you will get what is an orthogonal projector. Now this allows you to do another piece of very nice Dirac notation. So let's do that. Suppose you have an operator T. You put a 1 in front of it-- a T and a 1 in front of it. And then you say, OK, this 1, I'll put ei ei. Then comes the T, and then comes the ej ej-- another 1. And then you look at that and you suddenly see a number lying there. Why? Because this thing is some number. So this is the magic of the Dirac notation. You write all this thing, and suddenly you see numbers have been created in between. This number is nothing else but this matrix representation of the operator. T, between this, is Tij. So this is ei Tij ej. So this formula is very fundamental. It shows that the most general operator that you can ever invent is some sort of ket before a bra, and then you superimpose them with these numbers which actually happen to be the matrix representation of the operator. So the operator can be written as a sum of, if this is an n by n matrix n squared thinks of this form-- 1 with 1, 1 with 2, 1 with 3, and all of them. Bu then, you know this formula is so important that people make sure that you realize that you're summing over i and j. So just put it there. Given an operator, these are its matrix elements. And this is the operator written back in abstract notation. The whole operator is back there for you. I want to use the last part of the lecture to discuss a theorem that is pretty interesting, that allows you to understand things about all these Hermitian operators and unitary operators much more clearly. And it's a little mysterious, this theorem, and let's see how it goes. So any questions about this Dirac notation at this moment, anything that I wrote there? It takes a while to get accustomed to the Dirac notation. But once you get the hang of it, it's sort of fun and easy to manipulate. No questions? Can't be. You can prove all kinds of things with this matrix representation of the identity. For example, you can prove easily something you proved already, that when you multiply two operators the matrices multiply. You can prove all kinds of things. Pretty much everything we've done can also be proven this way. OK, so here comes the theorem I want to ask you about. Suppose somebody comes along, and they tell you, well, you know, here's a vector v, and I'm going to have a linear operator acting on this space. So the operator's going to be T, and I'm going act with the vector v. And moreover, I find that this is 0 for all vectors v belonging to the vector space. And the question is-- what can we say about this operator? From all vectors it's just 0. So is this operator 0, maybe? Does it have to be 0? Can it be something else? OK, we've been talking about real and complex vector spaces. And we've seen that it's different. The inner product is a little different. But let's think about this. Take two dimensions, real vector space. The operator that takes any vector and rotates it by 90 degrees, that's a linear operator. And that is a non-trivial linear operator, and it gives you 0. So case settled-- there's no theorem here, nothing you can say about this operator. It may be non-zero. But here comes the catch. If you're talking complex vector spaces, T is 0. It just is 0, can't be anything else. Complex vector spaces are different. You can't quite do that thing-- rotate all vectors by something and do things. So that's a theorem we want to understand. Theorem-- let v be a complex inner product space. By that is a complex vector space with an inner product. Then v, Tv equals 0 for all v implies that the operator is just 0. I traced a lot of my confusions in quantum mechanics to not knowing about this theorem, that somehow it must be true. I don't know why it should be true, but somehow it's not, because it really has exceptions. So here it is. We tried to prove that. It's so important, I think, that it should be proven. And how could you prove that? And at first sight it seems it's going to be difficult, because, if I do just a formal proof, how is it going to know that I'm not talking real or complex vector spaces. So it must make a crucial difference in the proof whether it's real or complex. So this property really sets the complex vector spaces quite apart from the real ones. So let's see what you would need to do. Well, here's a strategy-- if I could prove that u, Tv is equal to 0 for all u and all v. You see, the problem here is that these two are the same vector. They're all vectors, but they're the same vector. If I could prove that this is 0 for all u and v, then what would I say? I would say, oh, if this is 0 for all u and v, then pick u equal to Tv. And then you find that Tv, Tv is 0, therefore Tv is the 0 vector. By the axiom of the inner product, for all v is a 0 vector, so T kills all vectors, therefore T is 0. So if I could prove this is true, I would be done. Now, of course, that's the difficulty. Well, I wouldn't say of course. This takes a leap of faith to believe that this is the way you're going to prove that. You could try to prove this, and then it would follow. But maybe that's difficult to prove. But actually that's possible to prove. But how could you ever prove that this is true? You could prove it if you could somehow rewrite u and Tv as some sort of something with a T and something plus some other thing with a T, and that other thing plus some-- all kinds of things like that. Because the things in which this is the same as that are 0. So if you can do that-- if you could re-express this left hand side as a sum of things of that kind-- that would be 0. So let's try. So what can you try? You can put u plus v here, and T of u plus v. That would be 0, because that's a vector, same vector here. But that's not equal to this, because it has the u, Tu, and it has the v Tv. And it has this in a different order. So maybe we can subtract u minus v, T of u minus v. Well, we're getting there, but all this is question marks-- u, Tu, v, Tv-- these cancel-- u, Tu, v, Tv. But, the cross-products, what are they? Well here you have a u, Tv. And here you have a v, Tu. And do they cancel? No. Let's see. u, Tv, and up here is u minus Tv about. But there's another minus, so there's another one there. And v, Tu has a minus, minus is a plus. So actually this gives me two of this plus two of that. OK, it shouldn't have been so easy anyway. So here is where you have to have the small inspiration. Somehow it shouldn't have worked, you know. If this had worked, the theorem would read different. You could use a real vector space. Nothing is imaginary there. So the fact that you have a complex vector space might help. So somehow you have to put i's there. So let's try i's here. So you put u plus iv and T of u plus iv. Well, then you probably have to subtract things as well, so u minus iv, T of u minus iv. These things will be 0 because of the general structure-- the same operator here as here. And let's see what they are. Well, there's u, Tu, and here's minus u, Tu, so the diagonal things go away-- the minus iv, minus iv, iv, and a T. You have minus iv, minus iv subtracted, so that also cancels. So there's the cross-products. Now you will say, well, just like the minus signs, you're not going to get anything because you're going to get 2 and 2. Let's see. Let's see what we get with this one. You get u with Tiv, so you get i u, Tv. But look, this i on the left, however, when you take it out, becomes a minus i, so you get minus i v, Tu. And the other products [INAUDIBLE]. So let's look what you get here-- a u with a minus iv and a minus here gives you a 2 here. And the other term, v, Tu-- well, this goes out as a plus i. But with a minus, it becomes a minus i, so v, Tu is this. So there's a 2 here. So that's what these terms give you. And now you've succeeded. Why? Because the relative sign is negative. So who cares? You can divide by i, and divide this by i. You are constructing something. So let me put here what you get. I can erase this blackboard. So what do we get? I claim that if you put one quarter of u plus v, T u plus v minus u minus v, T of u minus v, then, let's see, what do we need to keep? We need to keep u and Tv. So divide this by i plus 1 over i u plus iv, T of u plus iv minus 1 over i, u minus iv, T of u minus iv. And close it. You've divided by i. You get here four of these ones, zero of these ones, and you got the answer you wanted. So this whole thing is written like that, and now, since this is equal to u with Tv, by the conditions of the theorem, any vector-- any vector here-- these are all 0. You've shown that this is 0, and therefore the operator is 0. And you should be very satisfied, because the proof made use of the fact that it was a complex vector space. Otherwise you could not add vectors with an imaginary number. And the imaginary number made it all work. So the theorem is there. It's a pretty useful theorem, so let's use it for the most obvious application. People say that, whenever you find that v, Tv is real for all v, then this operator is Hermitian, or self-adjoint. That is, then, it implies T dagger equals T. So let's show that. So let's take v, Tv. Proof. You take v, Tv, and now this thing is real. So since this is real, you can say it's equal to v, Tv star. Now, because it's real-- that's the assumption. The number is real. Now, the star off an inner product is Tv, v. But on the other hand, this operator, by the definition of adjoint, can be moved here. And this is equal to T dagger v, v. So now you have done this is equal to this. So if you put it to one side, you get that T dagger minus T on v times v is equal to 0. Or, since any inner product that is 00-- it's complex conjugate is 0-- you can write it as v, T dagger minus v is 0 for all v. And so this is an actually well known statement, that any operator that gives you real things must be Hermitian. But it's not obvious, because that theorem is not obvious. And now you can use a theorem and say, well, since this is true for all v, T dagger minus T is 0, and T dagger is equal to T. Then you can also show, of course, if T dagger is equal to T, this thing is real. So in fact, this arrow is both ways. And this way is very easy, but this way uses this theorem. There's another kind of operators that are called unitary operators. We'll talk a little more about them next time. And they preserve the norm of vectors. People define them from you, and you see that they preserve the norm of vectors. On the other hand, you sometimes find an operator that preserves every norm. Is it unitary? You will say, yes, must be. How do you prove it? You need again that theorem. So this theorem is really quite fundamental to understand the properties of operators. And we'll continue that next time. All right.

MIT_805_Quantum_Physics_II_Fall_2013

13_Quantum_Dynamics_continued_Heisenberg_Picture.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, so let me go back to what we were doing. The plan for today is as follows. We're going to look at this unitary time evolution and calculate this operator u, given the Hamiltonian. That will be the first order of business today. Then we will look at the Heisenberg picture of quantum mechanics. And the Heisenberg picture of quantum mechanics is one where the operators, the Schrodinger operators, acquire time dependence. And it's a pretty useful way of seeing things, a pretty useful way of calculating things as well, and makes the relation between classical mechanics and quantum mechanics more obvious. So it's a very important tool. So we'll discuss that. We'll find the Heisenberg equations of motion and solve them for a particular case today. All this material is not so to be covered in the test. The only part-- of course, the first few things I will say today about solving for the unitary operator you've done in other ways, and I will do it again this time. So going back to what we were saying last time, we postulated unitary time evolution. We said that psi at t was given by some operator U of t t0 psi t0. And then we found that this equation implied the Schrodinger equation with a Hamiltonian given by the following expression. ih dU dt of t t0 u dagger of t t0. So that was our derivation of the Schrodinger equation. We start with the time evolution. We found that, whenever we declare that states evolve in time in that way, they satisfy a first order time differential equation of the Schrodinger form in which the Hamiltonian is given in terms of U by this equation. And we talked about this operator. First we showed that it doesn't depend really on t0. Then we showed that it's Hermitian. It has units of energy. And as you may have seen already in the notes, there is a very clear correspondence between this operator and the way the dynamics follows with the ideas of Poisson brackets that are the precursors of commutators from classical mechanics. So that's in the notes. I will not go in detail in this. Many of you may have not heard of Poisson brackets. It's an interesting thing, and really that will be good enough. So our goal today is to find U given H, because as we mentioned last time, for physics it is typically more easy to invent a quantum system by postulating a Hamiltonian and then solving it than postulating a time evolution operator. So our goal in general is to find U of t t0 given H of t. That's what we're supposed to do. So the first thing I'm going to do is multiply this equation by u. By multiplying this equation by a u from the right, I will write first this term. ih dU dt of t t0 is equal to H of t U of t t0. So I multiplied this equation by u from the right. This operator is unitary, so u dagger u is one. That's why this equation cleaned up to this. Now there's no confusion really here with derivatives, so I might this well write them with normal derivatives. So I'll write this equation as d dt of U t t0 is equal to H of t U of t t0. You should be able to look at that equation and say I see the Schrodinger equation there. How? Imagine that you have a psi of t0 here, and you put it in. Then the right hand side becomes h and t acting on psi of t. And on the left hand side, this psi of t0 can be put inside the derivative because it doesn't depend on t. Therefore this becomes ih bar d dt of psi of t. So the Schrodinger equation is there. OK so now let's solve this. We'll go through three cases. Case one, h is time independent. So we're doing this sort of quickly. So H of t is really H like that. No explicit time dependence there. So what do we have? ih bar. Let's write dU dt is equal H times U. And we tried to write a solution of the form U use equal to e to the minus iHt over h bar times U0. Does that work? Well, we can think du dt and ih. So we get ih. When I take dU dt, I have to differentiate this exponential. And now in this exponential, this full operator H is there. But we are differentiating with respect to time. And H doesn't depend on time, so this is not a very difficult situation. You could imagine the power series expansion. And H, as far as this derivative goes, is like if it would be even a number. It wouldn't make any difference if it's an operator. So the derivative with respect to time of this thing is minus iH over h times the same exponential. Moreover, the position of this h could be here, or it could be to the right. It cannot be to the right of U0 though, because this is a matrix, a constant matrix that we've put in here as a possible thing for boundary condition. So so far we've taken this derivative, and then i's cancel, the h bar cancels, and you get H. But this whole thing is, again, U. So the equation has been solved. So try this. And it works. So having this solution we can write, for example, that U of t t0 is going to be e to the minus iHt over h bar, some constant matrix. When t is equal to t0, this matrix becomes the unit matrix. So this is e to the minus iHt0 over h bar times U0. And therefore from here, U0 is the inverse of this matrix, which is nothing else but e to the iHt0 over h bar. So I can substitute back here what U0 is and finally obtain U of t t0 is e to the minus iH over h bar t minus t0. And this is for h time independent. And that's our solution. There's very little to add to this. We discussed that in recitation on Thursday. This unitary operator you've been seeing that from the beginning of the course in some sense, that you evolve energy eigenstate. If this acts on any energy eigenstate, h is an energy-- if you act here on an energy eigenstate, the energy eigenstate is an eigenstate precisely for H, you can put just the number here. That is e to the, say, alpha h on a state psi n is equal to e to the alpha en psi n if h on psi n is equal to en on psi n. So the function of an operator acting on an eigenstate is just the function evaluated at the eigenvalue. So this is a rule that you've been using a really long time. OK, so when h is time independent, that's what it is. How about when h has a little time dependence? What do I call a little time dependence? A little time dependence is an idea, the sign to make it possible for you to solve the equation, even though it has some time dependence. So you could have Hamiltonians that are time dependent, but still have a simplifying virtue. So H of t is time dependent. But assume that H at t1 and h at t2 commute for all t1 and t2. So what could that be? For example, you know that the particle in a magnetic field, the spin in a magnetic field is minus gamma B dot the spin. And you could have a time dependent magnetic field, B of t times the spin. I'm not sure this is the constant gamma that they usually call gamma, but it may be. Now then if the magnetic field is time dependent, but imagine its direction is not time dependent. So if its direction is not time dependent, then, for example, you would have here minus gamma Bz of t times Sz. And the Hamiltonian at different times commute because Sz commutes with itself, and the fact that it's time independent doesn't make it fail to commute. So if you have a magnetic field that is fixed in one direction but change in time, you can have a situation where your Hamiltonian is time dependent, but still at different times it commutes. And you will discuss such case because it's interesting. But later on as we do nuclear magnetic resonance, we will have the more interesting case in which a magnetic field rotates and therefore it's not that simple. So what happens if you have a time dependent Hamiltonian that actually commutes? Well, the claim is that U of t t0 is given by a natural extension of what we had before. You would want to put exponential of minus iHt, but the reason this worked was because the derivative with respect to time brought down an iH over h bar. So one way to fix this is to put t t0 H of t prime dt prime. So this is an answer to try this. Look at this. If the Hamiltonian were to be time independent, you could take it out. And then you would get t minus t0. That brings you back to this case, so this looks reasonable. So let me call this quantity R of t. And then you notice that R dot of t, the derivative of this quantity with respect to time. Well, when you differentiate an integral the upper argument, you get just the integrand evaluated at the time represented by the upper argument of the upper limit of integration. So this is H of t. And now here comes a crucial point. You're trying to differentiate. This U is really e to the R. And you're trying to differentiate to see if the equation holds dU dt. So what is the dU dt? Would be d dt of 1 plus R plus RR plus 1 3 factor RRR. And now what happens? You differentiate here, and the first term is R dot. Here You, would have one half R dot R plus R R dot. And then 1 over 3 factorial, but three factors. R dot RR plus R R dot R plus RR R dot. But here is the claim R dot commutes with R. Claim R dot and R commute. Why is that? Well, R dot depends on H. And R is an integral of H as well, but the H at different times commute anyway, so this must be true. There's no place where you can get a contribution, because R dot is like an H, and here's an integral of H. So since the Hamiltonians are assumed to commute, R dot commutes with R. And this becomes like a normal derivative of an exponential in which you can move the R dot to the left everywhere. And you're differentiating the usual thing. So this is R dot and times the exponential of R. So actually that means that we've got pretty much our answer, because R dot is minus i over h bar H of t. And e to the R is U, so we got dU dt equals this, which is the same as this equation. The only reason a derivative with respect to time will not give the usual thing is if R and R dot fail to commute, and they don't. So you could put the R dot here. You can put R dot on the other side, because it commutes with R, but it's better here. And therefore you've got this very nice solution. So the solution is not that bad. Now finally, I want to discuss for a second the general case. So that's case-- there was a 1, a 2, a 3 H of t general. What can you do? Well, if H of t is general, there's not too much you can do. You can write something that will get you started doing things, but it's not obviously terribly useful. But it's interesting anyway that there's a way to write something that makes sense. So here it is. U of t and t0. I'll write the answer and explain how it looks, and then you will see that it's OK. It's interesting. But it probably is not the most practical way you can solve this problem. So here it is. There's an acronym for this thing. T it's called the time ordered exponential. This operator does something to the exponential function. So it's a definition. So I have to say what this time ordered exponential is, and it's the following. You take the exponential and just begin to expand. So 1 minus i over h bar-- or I'll put like this, plus minus i over h bar integral from t0 to t of dt1 H of t1. So far, so good. I've just expanded this. Now if I would continue expanding, I would get something that doesn't provide the solution. You see, this thing is the solution when the Hamiltonian at different times commute. So it's unlikely to be the solution when they don't commute. In fact, it's not the solution. So what is the next term here? The next term is you think of the exponential as you would expand as usual. So you will have here plus one half of this thing squared. So I will put something and then erase it, so maybe don't copy. One half minus i over h bar squared. And you would say, well, t0 to t dt prime H of t prime. t0 to t dt double prime H of double prime. Well, that would be just an exponential. So what is a time ordered exponential? You erase the one half. And then for notation call this t1 and t1. And then the next integral do it only up to time t1, and call this t2. So t1 will always be greater than t2, because t2 is integrated from t0 to t1. And as you integrate here over the various t1's, you just integrate up to that value. So you're doing less of the full integral then you should be doing, and that's why the factor of one half has disappeared. This can be continued. I can write the next one would be minus i over h bar cubed integral t0 to t H of t1 integral t0 to t1 dt2 H of t2. And then they next integral goes up to t2. So t0 to t2 dt3 H of t3. Anyway, that's a time ordered exponential. And I leave it to you to take the time derivative, at least to see that the first few terms are working exactly the way they should. That is, if you take a time derivative of this, you will get H times that thing. So since it's a power series, you will differentiate the first term, and you will get the right thing. Then the second term and you will start getting everything that you need. So it's a funny object. It's reassuring that something like this success, but in general, you would want to be able to do all these integrals and to sum them up. And in general, it's not that easy. So it's of limited usefulness. It's a nice thing that you can write it, and you can prove things about it and manipulate it. But when you have a practical problem, generally that's not the way you solve it. In fact, when we will discuss the rotating magnetic fields for magnetic resonance, we will not solve it in this way. We will try to figure out the solution some other way. But in terms of completeness, it's kind of pretty in that you go from the exponential to the time ordered exponential. And I think you'll see more of this in 806. So that's basically our solution for H and for the unitary operator U in terms of H. And what we're going to do now is turn to the Heisenberg picture of quantum mechanics. Yes, questions? AUDIENCE: Why does R dot [INAUDIBLE]? PROFESSOR: Because that's really a property of integrals. d dx integral up to x from x0 g of x prime dx prime is just equal to g of x. This is a constant here, so you're not varying the integral over in this limit. So if this limit would also be x dependent, you would get another contribution, but we only get the contribution from here. What's really happening is you're integrating up to x, then up to x plus epsilon subtracting, so you pick up the value of the function of the upper limit. Yes? AUDIENCE: So what happens to the T that was pre factor? PROFESSOR: What happens to this T? AUDIENCE: Yeah, what happens? PROFESSOR: That's just a symbol. It says time order the following exponential. So at this stage, this is a definition of what t on an exponential means. AUDIENCE: OK. PROFESSOR: It's not-- let me say T is not an operator in the usual sense of quantum mechanics or anything like that. It's an instruction. Whenever you have an exponential of this form, the time ordered exponential is this series that we've written down. It's just a definition. Yes? AUDIENCE: So when we have operators in differential equations, do we still get [INAUDIBLE]? PROFESSOR: If we have what? AUDIENCE: If we have operators in differential equations do we still get unique [INAUDIBLE] solutions? PROFESSOR: Yes, pretty much. Because at the end of the day, this is a first order matrix differential equation. So it's a collection of first order differential equations for every element of a matrix. It's pretty much the same as you have before. If you know the operator at any time, initial time, with the differential equation you know the operator at a little bit time later. So the operator is completely determined if you know it initially and the differential equation. So I think it's completely analogous. It's just that it's harder to solve. Nothing else. One last question. AUDIENCE: So let's say that we can somehow fly in this unitary operator, and then we have a differential equation, and we somehow, let's say, get a wave function out of it. What is the interpretation of that wave function? PROFESSOR: Well, it's not that we get the wave function out of this. What really is happening is that you have learned how to calculate this operator given H. And therefore now you're able to evolve any wave function. So you have solved the dynamical system. If somebody tells you a time equals 0, your system is here, you can now calculate where it's going to be at the later time. So that's really all you have achieved. You now know the solution. When you're doing mechanics and they ask you for an orbit problem, they say at this time the planet is here. What are you supposed to find? x is a function of time. You now know how it's going to develop. You've solved equations of motion. Here it's the same. You know the wave function of time equals. If you know it at any time, you've solved problem. OK, so Heisenberg picture of quantum mechanics. Heisenberg picture. So basically the Heisenberg picture exists thanks to the existence of the Schrodinger picture. Heisenberg picture of quantum mechanics is not something that you necessarily invent from the beginning. The way we think of it is we assume there is a Schrodinger picture that we've developed in which we have operators like x, p, spin, Hamiltonians, and wave functions. And then we are going to define a new way of thinking about this, which is called the Heisenberg picture of the quantum mechanics. So it all begins by considering a Schrodinger operator As hat, which is s is for Schrodinger. And the motivation comes from expectation values. Suppose you have time dependent states, in fact, matrix elements. One time dependent state alpha of t, one time dependent state beta of t. Two independent time dependent states. So you could ask what is the matrix element of A between these two time dependent states, a matrix element. But then, armed with our unitary operator, we know that As is here, and this state beta at time t is equal to U of t comma 0 beta at time 0. And alpha t is equal to alpha at 0 U dagger of t0. So the states have time dependence. But the time dependence has already been found, say, in principle, if you know U dagger. And then you can speak about the time dependent matrix elements of the operator As or the matrix element of this time dependent operator between the time equals 0 states. And this operator is sufficiently important that this operator is called the Heisenberg version of the operator s. Has time dependence, and it's defined by this equation. So whenever you have Schrodinger operator, whether it be time dependent or time independent, whatever the Schrodinger operator is, I have now a definition of what I will call the Heisenberg operator. And it is obtained by acting with a unitary operator, U. And operators always act on operators from the left and from the right. That's something that operators act on states from the left. They act on the state. But operators act on operator from the left and from the right, as you see them here, is the natural, ideal thing to happen. If you have an operator that's on another from the right only or from the left only, I think you have grounds to be suspicious that maybe you're not doing things right. So this is the Heisenberg operator. And as you can imagine, there's a lot of things to be said about this operator. So let's begin with a remark. Are there questions about this Heisenberg operator. Yes? AUDIENCE: Do we know anything about the Schrodinger operator? PROFESSOR: You have to speak louder. AUDIENCE: Is the Schrodinger operator related to the Hamiltonian [INAUDIBLE]? PROFESSOR: Any Schrodinger operator, this could be the Hamiltonian, this could be x hat, it could be Sz, could be any of the operators you know. All the operators you know are Schrodinger operators. So remarks, comments. OK, comments. One, at t equals 0 A Heisenberg becomes identical to A Schrodinger at t equals 0. So look why. Because when t is equal to 0, U of t-- of 0 0 is the operator propagates no state, so it's equal to the identity. So this is a wonderful relation that tell us you that time equals 0 the two operators are really the same. And another simple remark. If you have the unit operator in the Schrodinger picture, what is the unit operator in the Heisenberg picture? Well, it would be U t 0 dagger 1 U t 0. But 1 doesn't matter. U dagger with U is 1. This is a 1 Schrodinger, and therefore it's the same operator. So the unit operator is the same. It just doesn't change whatsoever. OK, so that's good. But now this is something interesting also happens. Suppose you have Schrodinger operator C that is equal to the product of A with B, two Schrodingers. If I try to figure out what is CH, I would put U dagger-- avoid all the letters, the t 0. It's supposed to be t 0. Cs U. But that's equal U dagger As Bs U. But now, in between the two operators, you can put a U U dagger, which is equal to 1. So As U U dagger Bs U. And then you see why this is really nice. Because what do you get is that C Heisenberg is just A Heisenberg times B Heisenberg. So if you have C Schrodinger equals A Schrodinger, B Schrodinger, C Heisenberg is A Heisenberg B Heisenberg. So there's a nice correspondence between those operators. Also you can do is for commutators. So you don't have to worry about this thing. So for example, if A Schrodinger with B Schrodinger is equal to C Schrodinger, then by doing exactly the same things, you see that A Heisenberg with B Heisenberg would be the commutator equal to C Heisenberg. Yes? AUDIENCE: That argument for the identity operators being the same in both pictures. If the Hamiltonian is time independent, does that work for any operator that commutes with the Hamiltonian? PROFESSOR: Hamiltonian is [INAUDIBLE]. AUDIENCE: Because then you can push the operator just through the exponential of the Hamiltonian. PROFESSOR: Yeah, we'll see things like that. We could discuss that maybe a little later. But there are some cases, as we will see immediately, in which some operators are the same in the two pictures. So basically operators that commute with the Hamiltonian as you say, since U involves the Hamiltonian, and this is the Hamiltonian, if the operator commutes with the Hamiltonian and you can move them across, then they are the same. So I think it's definitely true. So we will have an interesting question, in fact, whether the Heisenberg Hamiltonian is equal to the Schrodinger Hamiltonian, and we'll answer that very soon. So the one example that here I think you should keep in mind is this one. You know this is true. So what do you knowing the Heisenberg picture? That X Heisenberg of t times P Heisenberg of t commutator is equal to the Heisenberg version of this. But here was the unit operator. And therefore this is just ih bar times the unit operator again, because the units operator is the same in all pictures. So these commutation relation is true for any Heisenberg operator. Whatever commutation relation you have of Schrodinger, it's true for Heisenberg as well. OK, so then let's talk about Hamiltonians. Three, Hamiltonians. So Heisenberg Hamiltonian by definition would be equal to U dagger t 0 Schrodinger Hamiltonian times U of t 0. So if the Schrodinger Hamiltonian-- actually, if Hs at t1 commutes units with Hs at t2, the Schrodinger Hamiltonian is such that for all t1 and t2 they commute with each other. Remember, if that is the case, the unitary operator is any way built by an exponential. It's this one. And the Schrodinger Hamiltonians commute. So as was asked in the question before, this thing commutes with that, and you get that they are the same. So if this is happening, the two Hamiltonians are identical. And we'll have the chance to check this today in a nice example. So I will write in this as saying the Heisenberg Hamiltonian as a function of time then is equal to the Schrodinger Hamiltonian as a function of time. And this goes Hs of t1 and Hs of t2 commute. OK, now I want you to notice this thing. Suppose the Hs of t is some Hs of x,p, and t, for example. OK, now you come and turn it into Heisenberg by putting a U dagger from the left and a U from the right. What will that do? It will put U dagger from the left, U dagger on the right. And then it will start working it's way inside, and any x that it will find will turn into a Heisenberg x. Any p will turn into Heisenberg p. Imagine, for example, any Hamiltonian is some function of x. It has an x squared. Well the U dagger and U come and turn this into x Heisenberg squared. So what I claim here happens is that H Heisenberg of t is equal to U dagger H Schrodinger of x, p, t, U. And therefore this becomes H Schrodinger of x Heisenberg of t, P Heisenberg of t, and t. So here is what the Heisenberg Hamiltonian is. It's the Schrodinger Hamiltonian where X's, and P's, or spins and everything has become Heisenberg. So the equality of the two Hamiltonians is a very funny condition on the Schrodinger Hamiltonian, because this is supposed to be equal to the Schrodinger Hamiltonian, which is of x, p, and t. So you have a function of x, p, and t. And you put X Heisenberg P Heisenberg, and somehow the whole thing is the same. So this is something very useful and we'll need it. One more comment, expectation values. So this is three. Comment number four on expectation values, which is something you've already-- it's sort of the way we began the discussion and wanted to make sure it's clear. So four, expectation values. So we started with this with alpha and beta, two arbitrary states, matrix elements. Take them equal and to be equal to psi of t. So you would have psi t As psi t is, in fact, equal to psi 0 A Heisenberg psi 0. Now that is a key equation. You know you're doing expectation value at any given time of a Schrodinger operator, turn it into Heisenberg and work at time equals 0. It simplifies life tremendously. Now this is the key identity. It's the way we motivated everything in a way. And it's written in a way that maybe it's a little too schematic, but we write it this way. We just say the expectation value of As is equal to the expectation value of AH. And this, well, we save time like that, but you have to know what you mean. When you're computing the expectation value for a Schrodinger operator, you're using time dependent states. When you're computing the expectation value of the Heisenberg operator, you're using the time equals 0 version of the states, but they are the same. So we say that the Schrodinger expectation value is equal to the Heisenberg expectation value. We right it in the bottom, but we mean the top equation. And we use it that way. So the Heisenberg operators, at this moment, are a little mysterious. They're supposed to be given by this formula, but we've seen that calculating U can be difficult. So calculating the Heisenberg operator can be difficult sometimes. So what we try to do in order to simplify that is find an equation that is satisfied by the Heisenberg operator, a time derivative equation. So let's try to find an equation that is satisfied by the Heisenberg operator rather than a formula. You'll say, well, this is better. But the fact is that seldom you know U. And even if you know U, you have to do this simplification, which is hard. So finding a differential equation for the operator is useful. So differential equation for Heisenberg operators. So what do we want to do? We want to calculate ih bar d dt of the Heisenberg operator. And so what do we get? Well, we have several things. Remember, the Schrodinger operator can have a bit of time dependence. The time dependence would be an explicit time dependence. So let's take the time derivative of all this. So you would have three terms. ih bar dU dagger dt As U plus U dagger As dU dt plus-- with an ih bar-- U dagger ih bar dAs minus dt. dAs dt and U. Well, you have these equations. Those were the Schrodinger equations we started with today. The derivatives of U, or the derivatives of U dagger. so what did we have? Well, we have that ih bar dU dt was HU-- H Schrodinger times U. And therefore ih bar dU dagger dt. I take the dagger of this. I would get a minus sign. I would put it on the other side. Is equal to U dagger Hs with a minus here. And all the U's are U's of t and t0. I ran out of this thick chalk. So we'll continue with thin chalk. All right, so we are here. We wrote the time derivative, and we have three terms to work out. So what are they? Well we have this thing, ih bar this. So let's write it. ih bar d d dt of As-- of A Heisenberg, I'm sorry-- Is equal to that term is minus U dagger Hs A Schrodinger U. The next term plus ih bar dU dt on the right. So we have plus U dagger As Hs dU dt, so U. Well, that's not bad. It's actually quite nice. And then the last term, which I have very little to say, because in general, this is a derivative of a time dependent operator. Partial with respect to time, it would be 0 if As depends, just say, on X, on P, on Sx, or any of those things, has to have a particular t. So I will just leave this as plus ih bar dAs dt Heisenberg. The Heisenberg version of this operator using the definition that anything, any operator that we have a U dagger in front, a U to the right, is the Heisenberg version of the operator. So I think I'm doing all right with this equation. So what did we have? Here it is. ih bar d dt of A Heisenberg of t. And now comes the nice thing, of course. This thing, look at it. U dagger U. This turns everything here into Heisenberg. H Heisenberg, A Heisenberg. Here you have A Heisenberg H Heisenberg, and what you got is the commutator between them. So this thing is A Heisenberg commutator with H Heisenberg. That whole thing. And then you have plus ih bar dAs dt Heisenberg. So that is the Heisenberg equation of motion. That is how you can calculate a Heisenberg operator if you want. You tried to solve this differential equation, and many times that's the simplest way to calculate the Heisenberg operator. So there you go. It's a pretty important equation. So let's consider particular cases immediately to just get some intuition. So remarks. Suppose As has no explicit time dependence. So basically, there's no explicit t, and therefore this derivative goes away. So the equation becomes ih bar dAh, of course, dt is equal to Ah Heisenberg sub h of t. And you know the Heisenberg operator is supposed to be simpler. Simple. If the Schrodinger operator is time independent, it's identical to the Schrodinger Hamiltonian. Even if the Schrodinger operator has time dependence, but they commute, this will become the Schrodinger Hamiltonian. But we can leave it like that. It's a nice thing anyway. Time dependence of expectation value. So let me do a little remark on time dependence of expectation values. So suppose you have the usual thing that you want to compute. How does the expectation value of a Schrodinger operator depend on time? You're faced with that expectation value of As, and it changes in time, and you want to know how you can compute that. Well, you first say, OK, ih bar d dt. But this thing is nothing but psi 0 A Heisenberg of t psi 0. Now I can let the derivative go in. So this becomes psi 0 ih bar dAh dt psi 0. And using this, assuming that A is still no time dependence, A has no explicit time dependence, then you can use just this equation, which give you psi 0 Ah Hh psi 0. So all in all, what have you gotten? You've gotten a rather simple thing, the time derivative of the expectation values. So ih bar d dt. And now I write the left hand side as just expectation value of H Heisenberg of t. And on the left hand side has to the A Schrodinger expectation value, but we call those expectation values the same thing as a Heisenberg expectation value. So this thing becomes the right hand side is the expectation value of A Heisenberg H Heisenberg like that. And just the way we say that Heisenberg expectation values are the same as Schrodinger expectation values, you could as well write, if you prefer, as d dt of A Schrodinger is equal to the expectation value of A Schrodinger with H Schrodinger. It's really the same equation. This equation we derived a couple of lectures ago. And now we know that the expectation values of Schrodinger operators are the same as the expectation value of their Heisenberg counterparts, except that the states are taking up time equals 0. So you can use either form of this equation. The bottom one is one that you've already seen. The top one now looks almost obvious from the bottom one, but it really took quite a bit to get it. One last comment on these operators. How about conserved operators? What are those things? A time independent As is set to be conserved if it commutes with a Schrodinger Hamiltonian. If As commutes with As equals 0. Now you know that if As with Hs is 0, Ah with Hh is 0, because the map between Heisenberg and Schrodinger pictures is a commutator that is valued at the Schrodinger picture is valued in the Heisenberg picture by putting H's. So what you realize from this is that this thing, this implies Ah commutes with Hh. And therefore by point 1, by 1, you have to dAh dt is equal to 0. And this is nice, actually. The Heisenberg operator is actually time independent. It just doesn't depend on time. So a Schrodinger operator, it's a funny operator. It doesn't have time in there. It has X's, P's, spins, and you don't know in general, if it's time independent in the sense of conserve of expectation values. But whenever As commutes with Hs, well, the expectation values don't change in time. But as you know, this d dt can be brought in, because the states are not time dependent. So the fact that this is 0 means the operator, Heisenberg operator, is really time independent. Whenever you have a Schrodinger operator, has no t, the Heisenberg one can have a lot of t. But if the operator is conserved, then the Heisenberg operator will have no t's after all. It will really be conserved. So let's use our last 10 minutes to do an example and illustrate much of this. In the notes, there will be three examples. I will do just one in lecture. You can do the other ones in recitation next week. There's no need really that you study these things at this moment. Just try to get whatever you can now from the lecture, and next week you'll go back to this. So the example is the harmonic oscillator. And it will illustrate the ideas very nicely, I think. The Schrodinger Hamiltonian is p squared over 2m plus 1/2 m omega squared x hat squared. OK, I could put x Schrodinger and p Schrodinger, but that would be just far too much. x and p are the operators you've always known. They are Schrodinger operators. So we leave them like that. Now I have to write the Heisenberg Hamiltonian. Well, what is the Heisenberg Hamiltonian? yes? AUDIENCE: It's identical. PROFESSOR: Sorry? AUDIENCE: It's identical. PROFESSOR: Identical, yes. But I will leave that for a little later. I will just assume, well, I'm supposed to do U dagger U. As you said, this is a time independent Hamiltonian. It better be the same, but it will be clearer if we now write what it should be in general. Have a U dagger and a U from the right. They come here, and they turn this into P Heisenberg over 2m plus 1/2 m omega squared x Heisenberg. OK, that's your Heisenberg Hamiltonian. And we will check, in fact, that it's time independent. So how about the operators X Heisenberg and P Heisenberg. What are they? Well, I don't know how to get them, unless I do this sort of U thing. That doesn't look too bad but certainly would be messy. You would have to do an exponential of e to the minus iHt over t with the x operator and another exponential. Sounds a little complicated. So let's do it the way the equations of the Heisenberg operators tell you. Well, X and P are time independent Schrodinger operators, so that equation that I boxed holds. So ih dx Heisenberg dt is nothing else than X Heisenberg commuted with H Heisenberg. OK, can we do that commutator? Yes. Because X Heisenberg, as you remember, just commutes with P Heisenberg. So instead of the Hamiltonian, you can put this. This is X Heisenberg P Heisenberg squared over 2m. OK well, X Heisenberg P Heisenberg is like you had X and P. So what is this commutator? You probably know it by now. You act with this and these two p. So it acts on one, acts on the other, gives the same on each. So you get P Heisenberg times the commutator of X and P, which is ih bar times a factor of 2. So we could put hats. Better maybe. And then what do we get? The ih there and ih cancels. And we get some nice equation that says dX Heisenberg dt is 1 over m P Heisenberg. Well, it actually looks like an equation in classical mechanics. dx dt is P over m. So that's a good thing about the Heisenberg equations of motion. They look like ordinary equations for dynamical variables. Well, we've got this one. Let's get P. Well, we didn't get the operator still, but we got an equation. So how about P dP dt. So ih dP Heisenberg dt would be P Heisenberg with H Heisenberg. And this time only the potential term in here matters. So it's P Heisenberg with 1/2 m omega squared X Heisenberg squared. So what do we? We get 1/2 m omega squared. Then we get again a factor of 2. Then we get one left over Xh. And then a P with Xh, which is a minus ih bar. So the ih bars cancel, and we get dPh dt is equal to-- the h bar cancelled-- m omega squared Xh. Minus m. All right, so these are our Heisenberg equations of motion. So how do we solve for them now? Well, you sort of have to try the kind of things that you would do classically. Take a second derivative of this equation. d second Xh dt squared would be 1 over m dPh dt. And the dPh dt would be [INAUDIBLE] 1 over m times minus m omega squared Xh. So d second Xh dt squared is equal to minus omega squared Xh, exactly the equation of motion of a harmonic oscillator. It's really absolutely nice that you recover those equations that you had before, except that now you're talking operators. And it's going to simplify your life quite dramatically when you try to use these operators, because, in a sense, solving for the time dependent Heisenberg operators is the same as finding the time evolution of all states. This time the operators change, and you will know what they change like. So you have this. And then you write Xh is equal to A cosine omega t plus B sine omega t where A and B are some time independent operators. So Xh of t, well, that's a solution. How about what is P? Ph of t would be m dX m dx dt. So you get minus m omega sine omega tA plus m omega cosine omega tB. OK, so that's it. That is the most general solution. But it still doesn't look like what you would want, does it? No, because you haven't used the time equals 0 conditions. At time equals 0, the Heisenberg operators are identical they to the Schrodinger operators. So at t equals 0, Xh of t becomes A, but that must be X hat, the Schrodinger operator. And at t equals 0, Ph of t becomes equal to this is 0 m omega B. And that must be equal to the P hat operator. So actually we have already now A and B. So B from here is P hat over m omega. And therefore Xh of t is equal to A, which is X hat cosine omega t plus B, which is P hat over m omega sine omega t. An Ph of t is here. A is-- Ph of t is m omega B, which is [INAUDIBLE] P. So it's P hat cosine omega t minus m omega X hat sine omega t. So let's see. I hope I didn't make mistakes. P hat minus m omega X hat sine omega t. Yep, this is correct. This is your whole solution for the Heisenberg operators. So any expectation value of any power of X and P that you will want to find its time dependence, just put those Heisenberg operators, and you will calculate things with states at time equals 0. It will become very easy. So the last thing I want to do is complete the promise that we had about what is the Heisenberg Hamiltonian. Well, we had the Heisenberg Hamiltonian there. And now we know the Heisenberg operators in terms of the Schrodinger one. So Hh of t is equal to Ph-- 1/2m Ph squared. So I have P hat cosine omega t minus m omega X hat sine omega t squared plus 1/2 m omega squared Xh squared. So X hat cosine omega t plus P hat over m omega sine omega t squared. So that's what the Heisenberg Hamiltonian is. So let's simplify this. Well, let's square these things. You have 1/2m cosine squared omega t P hat squared. Let's do the square of this one. You would have plus 1/2m m squared omega squared sine squared omega t X squared. And then we have the cross product, which would be plus-- or actually minus 1/2m. The product of these two things. m omega sine omega t cosine omega t. And you have Px plus XP. OK, I squared the first terms. Now the second one. Well, let's square the P squared here. What do we have? 1/2 m omega squared over m squared omega squared sine squared of omega t P squared. The x plus 1/2 m omega squared cosine squared omega t X squared. And the cross term. Plus 1/2 m omega squared over m omega times cosine omega t sine omega t XP plus PX. A little bit of work, but what do we get? Well, 1/2 m. And here we must have 1/2 m, correct. 1/2 m. Sine squared omega t P squared. So this is equal 1/2 m P squared. These one's, hall here you have 1/2 m omega squared. So it's 1/2 m omega squared cosine and sine squared is X hat squared. And then here we have all being over 2. And here omega over 2, same factors, same factors, opposite signs. Very good. Schrodinger Hamiltonian. So you confirm that this theoretical expectation is absolutely correct. And what's the meaning? You have the Heisenberg Hamiltonian written in terms of the Heisenberg variables. But by the time you substitute these Heisenberg variables in, it just becomes identical to the Schrodinger Hamiltonian. All right, so that's all for today. I hope to see in office hours in the coming days. Be here Wednesday 12:30, maybe 12:25 would be better, and we'll see you then. [APPLAUSE]

MIT_805_Quantum_Physics_II_Fall_2013

6_Linear_Algebra_Vector_Spaces_and_Operators_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ARAM HARROW: So let's get started. This week Professor Zwiebach is away, and I'll be doing today's lecture. And Will Detmold will do the one on Wednesday. The normal office hours, unfortunately, will not be held today. One of us will cover his hours on Wednesday though. And you should also just email either me or Professor Detmold if you want to set up an appointment to talk to in the next few days. What I'm going to talk about today will be more about the linear algebra that's behind all of quantum mechanics. And, at the end of last time-- last lecture you heard about vector spaces from a more abstract perspective than the usual vectors are columns of numbers perspective. Today we're going to look at operators, which act on vector spaces, which are linear maps from a vector space to itself. And they're, in a sense, equivalent to the familiar idea of matrices, which are squares or rectangles of numbers. But are work in this more abstract setting of vector spaces, which has a number of advantages. For example, of being able to deal with infinite dimensional vector spaces and also of being able to talk about basis independent properties. And so I'll tell you all about that today. So we'll talk about how to define operators, some examples, some of their properties, and then finally how to relate them to the familiar idea of matrices. I'll then talk about eigenvectors and eigenvalues from this operator prospective. And, depending on time today, a little bit about inner products, which you'll hear more about the future. These numbers here correspond to the sections of the notes that these refer to. So let me first-- this is a little bit mathematical and perhaps dry at first. The payoff is more distant than usual for things you'll hear in quantum mechanics. I just want to mention a little bit about the motivation for it. So operators, of course, are how we define observables. And so if we want to know what the properties of observables, of which a key example are of Hamiltonians, then we need to know about operators. They also, as you will see in the future, are useful for talking about states. Right now, states are described as elements of a vector space, but in the future you'll learn a different formalism in which states are also described as dense operators. What are called density operators or density matrices. And finally, operators are also useful in describing symmetries of quantum systems. So already in classical mechanics, symmetries have been very important for understanding things like momentum conservation and energy conservation so on. They'll be even more important in quantum mechanics and will be understood through the formalism of operators. So these are not things that I will talk about today but are sort of the motivation for understanding very well the structure of operators now. So at the end of the last lecture, Professor Zwiebach defined linear maps. So this is the set of linear maps from a vector space, v, to a vector space w. And just to remind you what it means for a map to be linear, so T is linear if for all pairs of vectors in v, the way T acts on their sum is given by just T of u plus T of v. That's the first property. And second, for all vectors u and for all scalars a-- so f is the field that we're working over, it could be reals are complexes-- we have that if T acts on a times u, that's equal to a times t acting on u. So if you put these together what this means is that t essentially looks like multiplication. The way T acts on vectors is precisely what you would expect from the multiplication map, right? It has the distributive property and it commutes with scalars. So this is sort of informal-- I mean, the formal definition is here, but the informal idea is that T acts like multiplication. So if the map that squares every entry of a vector does not act like this, but linear operators do. And for this reason we often neglect the parentheses. So we just write TU to mean T of u, which is justified because of this analogy with multiplication. So an important special case of this is when v is equal to w. And so we just write l of v to denote the maps from v to itself. Which you could also write like this. And these are called operators on v. So when we talk about operators on a vector space, v, we mean linear maps from that vector space to itself. So let me illustrate this with a few examples. Starting with some of the examples of vector spaces that you saw from last time. So one example of a vector space is an example you've seen before but a different notation. This is the vector space of all real polynomials in one variable. So real polynomials over some variable, x. And over-- this is an infinite dimensional vector space-- and we can define various operators over it. For example, we can define one operator, T, to be like differentiation. So what you might write as ddx hat, and it's defined for any polynomial, p, to map p to p prime. So this is certainly a function from polynomials to polynomials. And you can check that it's also linear if you multiply the polynomial by a scalar, then the derivative multiplied by the same scale. If I take the derivative of a sum of two polynomials, then I get the sum of the derivatives of those polynomials. I won't write that down, but you can check that the properties are true. And this is indeed a linear operator. Another operator, which you've seen before, is multiplication by x. So this is defined as the map that simply multiplies the polynomial by x. Of course, this gives you another polynomial. And, again, you can check easily that it satisfies these two conditions. So this gives you a sense of why things that don't appear to be matrix-like can still be viewed in this operator picture. Another example, which you'll see later shows some of the slightly paradoxical features of infinite dimensional vector spaces, comes from the vector space of infinite sequences. So these are all the infinite sequences of reals or complexes or whatever f is. One operator we can define is the left shift operator, which is simply defined by shifting this entire infinite sequence left by one place and throwing away the first position. So you start with x2, x3, and so. Still goes to infinity so it still gives you an infinite sequence. So it is indeed a map-- that's the first thing you should check that this is indeed a map from v to itself-- and you can also check that it's linear, that it satisfies these two products. Another example is right shift. And here-- Yeah? AUDIENCE: So left shift was the first one or-- ARAM HARROW: That's right. So there's no back, really. It's a good point. So you'd like to not throw out the first one, perhaps, but there's no canonical place to put it in. This just goes off to infinity and just falls off the edge. It's a little bit like differentiation. Right? AUDIENCE: Yeah. I guess it loses some information. ARAM HARROW: It loses some information. That's right. It's a little bit weird, right? Because how many numbers do you have before you applied the left shift? Infinity. How many do you have after you applied the left shift? Infinity. But you lost some information. So you have to be a little careful with the infinities. OK The right shift. Here it's not so obvious what to do. We've kind of made space for another number, and so we have to put something in that first position. So this will be question mark x1, x2, dot, dot, dot. Any guesses what should go in the question mark? AUDIENCE: 0? ARAM HARROW: 0. Right. And why should that be 0? AUDIENCE: [INAUDIBLE]. ARAM HARROW: What's that? AUDIENCE: So it's linear. ARAM HARROW: Otherwise it wouldn't be linear. Right. So imagine what happens if you apply the right shift to the all 0 string. If you were to get something non-zero here, then you would map to the 0 vector to a non-zero vector. But, by linearity, that's impossible. Because I could take any vector and multiply it by the scalar 0 and I get the vector 0. And that should be equal to the scalar 0 multiplied by the output of it. And so that means that T should always map 0 to 0. T should always map the vector 0 to the vector 0. And so if we want right shift to be a linear operator, we have to put a 0 in there. And this one is strange also because it creates more space but still preserves all of the information. So two other small examples of linear operators that come up very often. There's, of course, the 0 operator, which takes any vector to the 0 vector. Here I'm not distinguishing between-- here the 0 means an operator, here it means a vector. I guess I can clarify it that way. And this is, of course, linear and sends any vector space to itself. One important thing is that the output doesn't have to be the entire vector space. The fact that it sends a vector space to itself only means that the output is contained within the vector space. It could be something as boring is 0 that just sends all the vectors to a single point. And finally, one other important operator is the identity operator that sends-- actually I won't use the arrows here. We'll get used to the mathematical way of writing it-- that sends any vector to itself. Those are a few examples of operators. I guess you've seen already kind of the more familiar matrix-type of operators, but these show you also the range of what is possible. So the space l of v of all operators-- I want to talk now about its properties. So l of v is the space of all linear maps from v to itself. So this is the space of maps on a vector space, but itself is also a vector space. So the set of operators satisfies all the axioms of a vector space. It contains a 0 operator. That's this one right here. It's closed under a linear combination. If I add together two linear operators, I get another linear operator. It's closed under a scalar multiplication. If I multiply a linear operator by a scalar, I get another linear operator, et cetera. And so everything we can do on a vector space, like finding a basis and so on, we can do for the space of linear operators. However, in addition to having the vector space structure, it has an additional structure, which is multiplication. And here we're finally making use of the fact that we're talking about linear maps from a vector space to itself. If we were talking about maps from v to w, we couldn't necessarily multiply them by other maps from v to w, we could only multiply them by maps from w to something else. Just like how, if you're multiplying rectangular matrices, the multiplication is not always defined if the dimensions don't match up, But since these operators are like square matrices, multiplication is always defined, and this can be used to prove many nice things about them. So this type of structure being a vector space of multiplication makes it, in many ways, like a field-- like real numbers or complexes-- but without all of the properties. So the properties that the multiplication does have first is that it's associative. So let's see what this looks like. So if we have a times bc is equal to ab times c. And the way we can check this is just by verifying the action of this on any vector. So an operator is defined by its action and all of the vectors in a vector space. So the definition of ab can be thought of as asking how does it act on all the possible vectors? And this is defined just in terms of the action of a and b as you first apply b and then you apply A. So this can be thought of as the definition of how to multiply operators. And then from this, you can easily check the associativity property that in both cases, however you write it out, you obtain A of B of C of v. I'm writing out all the parentheses just to sort of emphasize this is C acting on v, and then B acting on C of v, and then A acting on all of this. The fact that this is equal-- that this is the same no matter how A, B, and C are grouped is again part of what let's us justify this right here, where we drop-- we just don't use parentheses when we have operators acting. So, yes, we have the associative property. Another property of multiplication that operators satisfy is the existence of an identity. That's just the identity operator, here, which for any vector space can always be defined. But there are other properties of multiplication that it doesn't have. So inverses are not always defined. They sometimes are. I can't say that a matrix is never invertible, but for things like the reals and the complexes, every nonzero element has an inverse. And for matrices, that's not true. And another property-- a more interesting one that these lack-- is that the multiplication is not commutative. So this is something that you've seen for matrices. If you multiply two matrices, the order matters, and so it's not surprising that same is true for operators. And just to give a quick example of that, let's look at this example one here with polynomials. And let's consider S times T acting on the monomial x to the n. So T is differentiation so it sends this to n times x to the n minus 1. So we get S times n, x to the n minus 1. Linearity means we can move the n past the S. S acting here multiplies by x, and so we get n times x to the n. Whereas if we did the other order, we get T times S acting on x to the n, which is x to the n plus 1. When you differentiate this you get n plus 1 times x to the n. So these numbers are different meaning that S and T do not commute. And it's kind of cute to measure to what extent do they not commute. This is done by the commutator. And what these equations say is that if the commutator acts on x to the n, then you get n plus 1 times x to the n minus n times x to the n, which is just x to the n. And we can write this another way as identity times x to the n. And since this is true for any choice of n, it's true for what turns out to be a basis for the space of polynomials. So 1x, x squared, x cubed, et cetera, these span the space of polynomials. So if you know what an operator does and all of the x to the n's, you know what it does on all the polynomials. And so this means, actually, that the commutator of these two is the identity. And so the significance of this is-- well, I won't dwell on the physical significance of this, but it's related to what you've seen for position and momentum. And essentially the fact that these don't commute is actually an important feature of the theory. So these are some of the key properties of the space of operators. I want to also now tell you about some of the key properties of individual operators. And basically, if you're given an operator and want to know the gross features of it, what should you look at? So one of these things is the null space of an operator. So this is the set of all v, of all vectors, that are killed by the operator. They're sent to 0. In some case-- so this will always include the vector 0. So this always at least includes the vector 0, but in some cases it will be a lot bigger. So for the identity operator, the null space is only the vector 0. The only thing that gets sent to 0 is 0 itself. Whereas, for the 0 operator, everything gets sent to 0. So the null space is the entire vector space. For left shift, the null space is only 0 itself-- sorry, for right shift the null space is only 0 itself. And what about for left shift? What's the null space here? Yeah? AUDIENCE: Some numer with a string of 0s following it. ARAM HARROW: Right. Any sequence where the first number is arbitrary, but everything after the first number is 0. And so from all of these examples you might guess that this is a linear subspace, because in every case it's been a vector space, and, in fact, this is correct. So this is a subspace of v because, if there's a vector that gets sent to 0, any multiple of it also will be sent to 0. And of the two vectors that get sent to 0, their sum will also be sent to 0. So the fact that it's a linear subspace can be a helpful way of understanding this set. And it's related to the properties of T as a function. So for a function we often want to know whether it's 1 to 1, or injective, or whether it's [? onto ?] or surjective. And you can check that if T is injective, meaning that if u is not equal to v, then T of u is not equal to T of v. So this property, that T maps distinct vectors two distinct vectors, turns out to be equivalent to the null space being only the 0 vector. So why is that? This statement here, that whenever u is not equal to v, T of u is not equal to T of v, another way to write that is whenever u is not equal to v, T of u minus v is not equal to 0. And if you look at this statement a little more carefully, you'll realize that all we cared about on both sides was u minus v. Here, obviously, we care about u minus v. Here we only care if u is not equal to v. So that's the same as saying if u minus v is non-zero, then T of u minus v is non-zero. And this in turn is equivalent to saying that the null space of T is only 0. In other words, the set of vectors that get sent to 0 consists only of the 0 vector itself. So the null space for linear operators is how we can characterize whether they're 1 to 1, whether they destroy any information. The other subspace that will be important that we will use is the range of an operator. So the range of an operator, which we can also just write as T of v, is the set of all points that vectors in v get mapped to. So the set of all Tv for some vector, v. So this, too, can be shown to be a subspace. And that's because-- it takes a little more work to show it, but not very much-- if there's something in the output of T, then whatever the corresponding input is we could have multiplied that by a scalar. And then the corresponding output also would get multiplied by a scalar, and so that, too, would be in the range. And so that means that for anything in the range, we can multiply it by any scalar and again get something in the range. Similarly for addition. A similar argument shows that the range is closed under addition. So indeed, it's a linear subspace. Again, since it's a linear subspace, it always contains 0. And depending on the operator, may contain a lot more. So whereas the null space determined whether T was injective, the range determines whether T is surjective. So the range of T equals v if and only if T is surjective. And here this is simply the definition of being surjective. It's not really a theorem like it was in the case of T being injective. Here that's really what it means to be surjective is that your output is the entire space. So one important property of the range of the null space whenever v is finite dimensional is that the dimension of v is equal to the dimension of the null space plus the dimension of the range. And this is actually not trivial to prove. And I'm actually not going to prove it right now. But the intuition of it is as follows. Imagine that v is some n dimensional space and the null space has dimension k. So that means you have input of n degrees of freedom, but T kills k of n. And so k at different degrees of freedom, no matter how you vary them, have no effect on the output. They just get mapped to 0. And so what's left are n minus k degrees of freedom that do affect the outcome. Where, if you vary them, it does change the output in some way. And those correspond to the n minus k dimensions of the range. And if you want to get formal, you have to formalize what I was saying about what's left is n minus k. If you talk about something like the orthogonal complement or completing a basis or in some way formalize that intuition. And, in fact, you can do a little further, and you can decompose the space. So this is just dimensions counting. You can even decompose the space into the null space and the complement of that and show that T is 1 to 1 on the complement of the null space. But for now, I think this is all that we'll need for now. Any questions so far? Yeah? AUDIENCE: Why isn't the null space part of the range? ARAM HARROW: Why isn't it part of the range? AUDIENCE: So you're taking T of v and null space is just the special case when T of v is equal to 0. ARAM HARROW: Right. So the null space are all of the-- This theorem, I guess, would be a little bit more surprising if you realized that it works not only for operators, but for general linear maps. And in that case, the range is a subset space of w. Because the range is about the output. And the null space is a [? subset ?] space of v, which is part of the input. And so in that case, they're not even comparable. The vectors might just have different lengths. And so it can never-- like the null space in a range, in that case, would live in totally different spaces. So let me give you a very simple example. Let's suppose that T is equal to 3, 0, minus 1, 4. So just a diagonal 4 by 4 matrix. Then the null space would be the span of e2, that's the vector with a 1 in the second position. And the range would be the span of e1, e3, in e4. So in fact, usually it's the opposite that happens. The null space in the range are-- in this case they're actually orthogonal subspaces. But this picture is actually a little bit deceptive in how nice it is. So if you look at this, total space is 4, four dimensions, it divides up into one dimension that gets killed, and three dimensions where the output still tells you something about the input, where there's some variation of the output. But this picture makes it seem-- the simplicity of this picture does not always exist. A much more horrible example is this matrix. So what's the null space of this matrix? Yeah? AUDIENCE: You just don't care about the upper [INAUDIBLE]. ARAM HARROW: You don't care about the-- informally, it's everything of this form. Everything with something in the first position, 0 in the second position. In other words, it's the span of e1. What about the range? AUDIENCE: [INAUDIBLE]. ARAM HARROW: What's that? Yeah? AUDIENCE: [INAUDIBLE]. ARAM HARROW: It's actually-- AUDIENCE: Isn't it e1? ARAM HARROW: It's also e1. It's the same thing. So you have this intuition that some degrees of freedom are preserved and some are killed. And here they look totally different. And there they look the same. So you should be a little bit nervous about trying to apply that intuition. You should be reassured that at least the theorem is still true. At least 1 plus 1 is equal to 2. We still have that. But the null space and the range are the same thing here. And the way around that paradox-- Yeah? AUDIENCE: So can you just change the basis-- is there always a way of changing the basis of the matrix? In this case it becomes [INAUDIBLE]? Or not necessarily? ARAM HARROW: No. It turns out that, even with the change of basis, you cannot guarantee that the null space and the range will be perpendicular. Yeah? AUDIENCE: What if you reduce it to only measures on the-- or what if you reduce the matrix of-- [? usability ?] on only [INAUDIBLE] on the diagonal? ARAM HARROW: Right Good. So if you do that, then-- if you do row [? eduction ?] with two different row and column operations, then what you've done is you have a different input and output basis. And so that would-- then once you kind of unpack what's going on in terms of the basis, then it would turn out that you could still have strange behavior like this. What your intuition is based on is that if the matrix is diagonal in some basis, then you don't have this trouble. But the problem is that not all matrices can be diagonalized. Yeah? AUDIENCE: So is it just the trouble that the null is what you're acting on and the range is what results from it? ARAM HARROW: Exactly. And they could even live in different space. And so they really just don't-- to compare them is dangerous. So it turns out that the degrees of freedom corresponding to the range-- what you should think about are the degrees of freedom that get sent to the range. And in this case, that would be e2. And so then you can say that e1 gets sent to 0 and e2 gets sent to the range. And now you really have decomposed the input space into two orthogonal parts. And because we're talking about a single space, the input space, it actually makes sense to break it up into these parts. Whereas here, they look like they're the same, but really input and output spaces you should think of as potentially different. So this is just a mild warning about reading too much into this formula, even though it's the rough idea it counting degrees of freedom is still roughly accurate. So I want to say one more thing about properties of operators, which is about invertibility. And maybe I'll leave this up for now. So we say that a linear operator, T, has a left inverse, S, if multiplying T on the left by s will give you the identity. And T has a right inverse, S prime, you can guess what will happen here if multiplying T on the right by S prime gives you identity. And what if T has both? Then in that next case, it turns out that S and S prime have to be the same. So here's the proof. So if both exist, then S is equal to s times identity-- by the definition of the identity. And then we can replace identity with TS prime. Then we can group these and cancel them and get S prime. So if a matrix has both a left and a right inverse, then it turns out that the left and right inverse are the same. And in this case, we say that T is invertible, and we define T inverse to be S. One question that you often want to ask is when do left to right inverses exist? Actually, maybe I'll write it here. Intuitively, there should exist a left inverse when, after we've applied T, we haven't done irreparable damage. So whatever we're left with, there's still enough information that some linear operator can restore our original vector and give us back the identity. And so that condition is when-- of not doing irreparable damage, of not losing information, is asking essentially whether T is injective. So there exists a left inverse if and only if T is injective. Now for a right inverse the situation is sort of dual to this. And here what we want-- we can multiply on the right by whatever we like, but there won't be anything on the left. So after the action of T, there won't be any further room to explore the whole vector space. So the output of T had better cover all of the possibilities if we want to be able to achieve identity by multiplying T by something on the right. So any guesses for what the condition is for having a right inverse? AUDIENCE: Surjective. ARAM HARROW: Surjective. Right. So there exists a right inverse if and only if T is surjective. Technically, I've only proved one direction. My hand waving just now proved that, if T is not injective, there's no way it will have a left inverse. If it's not surjective, there's no way it'll have a right inverse. I haven't actually proved that, if it is injective, there is such a left inverse. And if it is surjective, there is such a right universe. But those I think are good exercises for you to do to make sure you understand what's going on. This takes us part of the way there. In some cases our lives become much easier. In particular, if v is finite dimensional, it turns out that all of these are equivalent. So T is injective if and only if T is surjective if and only if T is invertible. And why is this? Why should it be true that T is surjective if and only if T is injective? Why should those be equivalent statements? Yeah? AUDIENCE: This isn't really a rigorous statement, but if the intuition of it is a little bit that you're taking vectors in v to vectors in v. ARAM HARROW: Yeah. AUDIENCE: And so your mapping is 1 to 1 if and only if every vector is mapped to, because then you're not leaving anything out. ARAM HARROW: That's right. In failing to be injective and failing to be surjective both look like losing information. Failing to be injective means I'm sending a whole non-zero vector and its multiples to 0, that's a degree of freedom lost. Failing to be surjective means once I look at all the degrees of freedom I reach, I haven't reached everything. So they intuitively look the same. So that's the right intuition. There's a proof, actually, that makes use of something on a current blackboard though. Yeah? AUDIENCE: Well, you need the dimensions of-- so if the [INAUDIBLE] space is 0, you need dimensions of [? the range to p. ?] ARAM HARROW: Right. Right. So from this dimensions formula you immediately get because if this is 0, then this is the whole vector space. And if this is non-zero, this is not the whole vector space. And this proof is sort of non-illuminating if you don't know the proof of that thing-- which I apologize for. But also, you can see immediately from that that we've used the fact that v is finite dimensional. And it turns out this equivalence breaks down if the vector space is infinite dimensional. Which is pretty weird. There's a lot of subtleties of infinite dimensional vector spaces that it's easy to overlook if you build up your intuition from matrices. So does anyone have an idea of a-- so let's think of an example of a vector of something that is on an infinite dimensional space that's surjective but not injective. Any guesses for such an operation? Yeah? AUDIENCE: The left shift. ARAM HARROW: Yes. You'll notice I didn't erase this blackboard strategically. Yes. The left shift operator is surjective. I can prepare any vector here I like just by putting it into the x2, x3, dot, dot, dot parts. So the range is everything, but it's not injective because it throws away the first register. It's maps things with it a non-zero element in the first position and 0's everywhere else to 0. So this is surjective not injective. On the other hand, if you want something that's injective and not surjective, you don't have to look very far, the right shift is injective and not surjective. It's pretty obvious it's not surjective. There's that 0 there which definitely means it cannot achieve any vector. And it's not too hard to see it's injective. It hasn't lost any information. It's like you're in the hotel that's infinitely long and all the rooms are full and the person at the front desk says, no problem. I'll just move everyone down one room to the right, and you can take the first room. So that policy is injective-- you'll always get a room to yourself-- and made possible by having an infinite dimensional vector space. So in infinite dimensions we cannot say this. Instead, we can say that T is invertible if and only if T is injective and surjective. So this statement is true in general for infinite dimensional, whatever, vector spaces. And only in the nice special case of finite dimensions do we get this equivalence. Yeah? AUDIENCE: Can the range and null space of T a [INAUDIBLE] of T the operator again use a vector space [INAUDIBLE]? ARAM HARROW: Yes. the question was do the null space in a range are they properties just of T or also of v? And definitely you also need to know v. The way I've been writing it, T is implicitly defined in terms of v, which in turn is implicitly defined in terms of the field, f. And all these things can make a difference. Yes? AUDIENCE: So do you have to be a bijection for it to be-- ARAM HARROW: That's right. That's right. Invertible is the same a bijection. So let me now try and relate this to matrices. I've been saying that operators are like the fancy mathematician's form of matrices. If you're Arrested Development fans, it's like magic trick versus an illusion. But are they different or not depends on your perspective. There are advantages to seeing it both ways, I think. So let me tell you how you can view an operator in a matrix form. The way to do this-- and the reason why matrices are not universally loved by mathematicians is I haven't specified a basis this whole time. But if I want a matrix, all I needed was a vector space and a function-- a linear function between two vector spaces-- or, sorry, from a vector space to itself. But if I want a matrix, I need additional structure. And mathematicians try to avoid that whenever possible. But if you're willing to take this additional structure-- so if you choose a basis v1 through vn-- it turns out you can get a simpler form of the operator that's useful to compute with. So why is that? Well, the fact that it's a basis that means that any v can be written as linear combinations of these basis elements where a1 though an belong to the field. And since T is linear, if T acts on v, we can rewrite it in this way, and you see that the entire action is determined by T acting on v1 through vn. So think about-- if you wanted to represent an operator in a computer, you'd say, well, there's an infinite number of input vectors. And for each input vector I have to write down the output vector. And this says, no, you don't. You only need to restore on your computer what does T do to v1, what does T do to v2, et cetera. So that's good. Now you only have to write down n vectors, and since these factors in turn can be expressed in terms of the basis, you can express this just in terms of a bunch of numbers. So let's further expand Tvj in this basis. And so there's some coefficient. So it's something times v1 plus something times v2 something times vn. And I'm going to-- these somethings are a function of T so I'm just going to call this T sub 1j, T sub 2j, T sub nj. And this whole thing I can write more succinctly in this way. And now all I need are these T's of ij, and that can completely determine for me the action of T because this Tv here-- so Tv we can write as a sum over j of T times ajvj. And we can move the aj past the T. And then if we expand this out, we get that it's a sum over i from 1 to n, sum over j from 1 to n, of Tijajvi. And so if we act on in general vector, v, and we know the coefficients of v in some basis, then we can re-express it in that basis as follows. And this output in general can always be written in the basis with some coefficients. So we could always write it like this. And this formula tells you what those coefficients should be. They say, if your input vector has coefficients a1 through an, then your output vector has coefficients b1 through bn, where the b sub i are defined by this sum. And of course there's a more-- this formula is one that you've seen before, and it's often written in this more familiar form. So this is now the familiar matrix-vector multiplication. And it says that the b vector is obtained from the a vector by multiplying it by the matrix of these Tij. And so this T is the matrix form-- this is a matrix form of the operator T. And you might find this not very impressive. You say, well, look I already knew how to multiply a matrix by vector. But what I think is nice about this is that the usual way you learn linear algebra if someone says, a vector is a list of numbers. A matrix is a rectangle of numbers. Here's are the rules for what you do with them. If you want to put them together, you do it in this way. Here this was not an axiom of the theory at all. We just started with linear maps from one vector space to another one and the idea of a basis as something that you can prove has to exist and you can derive matrix multiplication. So matrix multiplication emerges-- or matrix-vector multiplication emerges as a consequence of the theory rather than as something that you have to put in. So that, I think, is what's kind of cute about this even if it comes back on the end to something that you had been taught before. Any questions about that? So this is matrix-vector multiplication. You can similarly derive matrix-matrix multiplication. So if we have two operators, T and S, and we act on a vector, v sub k-- and by what I argued before, it's enough just to know how they act on the basis vectors. You don't need to know-- and once you do that, you can figure out how they act on any vector. So if we just expand out what we wrote before, this is equal to T times the sum over j of Sjkvj. So Svk can be re-expressed in terms of the basis with some coefficients. And those coefficients will depend on the vector you start with, k, and the part of the basis that you're using to express it with j. Then we apply the same thing again with T. We get-- this is sum over i, sum over j TijSjkvi. And now, what have we done? TS is an operator and when you act of vk it spat out something that's a linear combination of all the basis states, v sub i, and the coefficient of v sub i is this part in the parentheses. And so this is the matrix element of TS. So the ik matrix element of ts is the sum over j of Tijsjk. And so just like we derived matrix-vector multiplication, here we can derive matrix-matrix multiplication. And so what was originally just sort of an axiom of the theory is now the only possible way it could be if you want to define operator multiplication is first one operator acts, than the other operator acts. So in terms of this-- so this, I think, justifies why you can think of matrices as a faithful representation of operators. And once you've chosen a basis, they can-- the square full of numbers becomes equivalent to the abstract map between vector spaces. And the equivalent-- they're so equivalent that I'm just going to write things like equal signs. Like I'll write identity equals a bunch of 1's down the diagonal, right? And not worry about the fact that technically this is an operator and this is a matrix. And similarly, the 0 matrix equals a matrix full of 0's. Technically, we should write-- if you want to express the basis dependence, you can write things like T parentheses-- sorry, let me write it like this. If you really want to be very explicit about the basis, you could use this to refer to the matrix. Just to really emphasize that the matrix depends not only on the operator, but also on your choice of basis. But we'll almost never bothered to do this. We usually just sort of say it in words what the basis is. So matrices are an important calculational tool, and we ultimately want to compute numbers of physical quantities so we cannot always spend our lives in abstract vector spaces. But the basis dependence is an unfortunate thing. A basis is like a choice of coordinate systems, and you really don't want your physics to depend on it, and you don't want quantity if you compute to be dependent on. And so we often want to formulate-- we're interested in quantities that are basis independent. And in fact, that's a big point of the whole operator picture is that because the quantities we want are ultimately basis independent, it's nice to have language that is itself basis independent. Terminology and theorems that do not refer to a basis. I'll mention a few basis independent quantities, and I won't say too much more about them because you will prove properties [INAUDIBLE] on your p set, but one of them is the trace and another one is the determinant. And when you first look at them-- OK, you can check that each one is basis independent, and it really looks kind of mysterious. I mean, like, who pulled these out of the hat? They look totally different, right? They don't look remotely related to each other. And are these all there is? Are there many more? And it turns out that, at least for matrices with eigenvalues, these can be seen as members of a much larger family. And the reason is that the trace turns out to be the sum of all the eigenvalues and the determinant turns out to be the product of all of the eigenvalues. And in general, we'll see in a minute, that basis independent things-- actually, not in a minute. In a future lecture-- that basis independent things are functions of eigenvalues. And furthermore, that don't care about the ordering of the eigenvalues. So they're symmetric functions of eigenvalues. And then it starts to make a little bit more sense. Because if you talk about symmetric polynomials, those are two of the most important ones where you just add up all the things and when you multiply all the things. And then, if you add this perspective of symmetric polynomial of the eigenvalue, then you can cook up other basis independent quantities. So this is actually not the approach you should take on the p set. The [? p set ?] asks you to prove more directly that the trace is basis independent, but the sort of framework that these fit into is symmetric functions of eigenvalues. So I want to say a little bit about eigenvalues. Any questions about matrices before I do? So eigenvalues-- I guess, these are basis independent quantities. Another important basis independent quantity, or property of a matrix, is its eigenvalue-eigenvector structure. The place where eigenvectors come from is by considering a slightly more general thing, which is the idea of an invariant subspace. So we say that U is a T invariant subspace if T of U-- this is an operator acting on an entire subspace. So what do I mean by that? I mean the set of all TU for vectors in the subspace. If T of U is contained in U. So I take a vector in this subspace, act on it with T, and then I'm still in the subspace no matter which vector I had. So some examples that always work. The 0 subspace is invariant. T always maps it to itself. And the entire space, v, T is a linear operator on v so by definition it maps v to itself. These are called the trivial examples. And usually when people talk about non-trivial invariant subspaces they mean not one of these two. The particular type that we will be interested in are one dimensional ones. So this corresponds to a direction that T fixes. So U-- this vector space now can be written just as the span of a single vector, U, and U being T invariant is equivalent to TU being a mu, because they're just a single vector. So all I have to do is get that single vector right and I'll get the whole subspace right. And that, in turn, is equivalent to TU being some multiple of U. And this equation you've seen before. This is the familiar eigenvector equation. And if it's a very, very important equation it might be named after a mathematician, but this one is so important that two of the pieces of it have their own special name. So these are called-- lambda is called an eigenvalue and U is called an eigenvector. And more or less it's true that all of the solutions to this are called eigenvalues, and all the solutions are called eigenvectors. There's one exception, which is there's one kind of trivial solution to this equation, which is when U is 0 this equation is always true. And that's not very interesting, but it's true for all values of lambda. And so that doesn't count as being an eigenvalue. And you can tell a doesn't correspond to 1D invariant subspace, right? It corresponds to a 0 dimensional subspace, which is the trivial case. So we say that lambda is an eigenvalue of T if Tu equals lambda U for some non-zero vector, U. So the non 0 is crucial. And then the spectrum of T is the collection of all eigenvalues. So there's something a little bit asymmetric about this, which is we still say that 0 vector is an eigenvector with all the various eigenvalues, but we had to put this here or everything would be an eigenvalue and it wouldn't be very interesting. So the-- Oh, also I want to say this term spectrum you'll see it other [INAUDIBLE]. You'll see spectral theory or spectral this or that, and that means essentially making use of the eigenvalues. So people talk about partitioning a graph using eigenvalues of the associated matrix, that's called spectral partitioning. And so throughout math, this term is used a lot. So I have only about three minutes left to tell-- so I think I will not finish the eigenvalue discussion but will just show you a few examples of how it's not always as nice as you might expect. So one example that I'll consider is the vector space will be the reals, 3D real space, and the operator, T, will be rotation about the z-axis by some small angle. Let's call it a theta rotation about the z-axis. Turns out, if you write this in matrix form, it looks like this. Cosine theta minus sine theta 0 sine theta cosine theta 0, 0, 0, 0, 1. That 1 is because it leaves the z-axis alone and then x and y get rotated. You can tell if theta is 0 it does nothing so that's reassuring. And if theta does a little bit, then it starts mixing the x and y components. So that is the rotation matrix. So what is an eigenvalue-- and anyone say what an eigenvalue is of this matrix? AUDIENCE: 1. ARAM HARROW: 1. Good. And what's the eigenvector? AUDIENCE: The z basis vector. ARAM HARROW: The z basis vector. Right. So it fixes a z basis vector so this is an eigenvector with eigenvalue 1. Does it have any other eigenvectors? Yeah? AUDIENCE: If you go to the complex plane, then yes. ARAM HARROW: If you are talking about complex numbers, then yes, it has complex eigenvalues. But if we're talking about a real vector space, then it doesn't. And so this just has one eigenvalue and one eigenvector. And if we were to get rid of the third dimension-- so if we just had T-- and let's be even simpler, let's just take theta to be pi over 2. So let's just take a 90 degree rotation in the plane. Now T has no eigenvalues. There are no vectors other than 0 that it sends to itself. And so this is a slightly unfortunate note to end the lecture on. You think, well, these eigenvalues are great, but maybe they exist, maybe they don't. And you'll see next time part of the reason why we use complex numbers, even though it looks like real space isn't complex, is because any polynomial can be completely factored in complex numbers, and every matrix has a complex egeinvalue. OK, I'll stop here.

MIT_805_Quantum_Physics_II_Fall_2013

24_Addition_of_Angular_Momentum.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocs.mit.edu. PROFESSOR: Today we have plenty to do. We really begin in all generality the addition of angular momentum. But we will do it in the set up of a physical problem. The problem of computing the spin orbit interactions of electrons with the nucleus. So this is a rather interesting and complicated interaction. So we'll spend a little time telling you about the physics of this interaction. And then once the physics is clear, it will become more obvious why we have to do these mathematical contortions of adding angular momentum in order to solve this physical problem. So it's a sophisticated problem that requires several steps. The first step is something that is a result in perturbation theory. Feynman Hellman Theorem of perturbation theory. And that's where we begin. So it's called Feynman Hellman Theorem. It's a very simple result. Theorem. And we'll need it in order to understand how a small perturbation to the Hamiltonian changes the energy spectrum. So we have H of lambda be a Hamiltonian with a parameter in lambda. Lambda. And psi n of lambda being normalized energy eigenstate with energy, En of lambda. So that's the whole assumption of the theorem. We have a Hamiltonian. It depends on some parameter that we're going to vary. And suppose we consider now an eigenstate of this Hamiltonian that depends on lambda, so the eigenstate also depends on lambda. And it has an energy, En of lambda. So the purpose of this theorem is to relate these various quantities. And the claim is that the rate of change of the energy with respect to lambda can be computed pretty much by evaluating the rate of change of the Hamiltonian on the relevant states. So that's the claim. And it's a pretty nice result. It's useful in many circumstances. And for us will be a way to discuss a little perturbation theory. Perturbation theory is the subject of 806 in all details. And it's a very sophisticated subject. Even today we were going to be finding that it's not all that easy to carry it out. So how does this begin? Well, proof. You begin by saying that En of lambda is the energy eigenstate, is nothing else but psi n of lambda. H of lambda. Psi n of lambda. And the reason is, of course, that H and psi n is En of lambda times psi n of lambda. And this is a number goes out. And the inner product of this things is 1, because the state is normalized. So this is a good starting point. And the funny thing that you see already is that, in some sense, you just get the middle term when you take the derivative with respect to lambda. You don't get anything from these two. And it's simple in fact. So let me just do it. V, En, V lambda would be the term that Feynman and Hellman gave. V, H, V lambda, psi n of lambda. Plus one term in which we differentiate this one. V, d lambda of the state psi n of lambda. Times H of lambda, psi n of lambda. Plus the other term in which you differentiate the ket. So psi n of lambda, H of lambda, d, d lambda of psi n of lambda. And the reason these terms are going to vanish is that you can now act with H again on psi n. H is supposed to be Hermitian, so it can act on the left. And therefore, these two terms give you En of lambda, times d d lambda of psi n of lambda-- psi n of lambda-- plus the other term, which would be psi n of lambda, the bra times the derivative of the ket. But this is nothing else than the derivative of the inner product. In the inner product to differentiate-- the inner product differentiates the bra, it differentiates the ket. And do it. And this thing is equal to 1, because it's normalized. So this is 0. End of proof. These two terms vanish. And the result holds. Yes? AUDIENCE: How do you know it stays normalized when you vary lambda? PROFESSOR: It's an assumption. The state is normalized for all values of n. So if you have a state that you've constructed, that is normalized, you can have this result. So it's an assumption. You have to keep the state normalized. Now this is a baby version of perturbation theory. It's a result I think that Feynman did as an undergrad. And as you can see, it's very simple. Calling it a theorem is a little too much. But still, the fact is that it's useful. And so we'll just go ahead and use it. Now I want to rewrite it in another way. So, suppose you have a Hamiltonian, H, which has a term H0, plus lambda, H1. So, the parameter lambda, H of lambda, is given in this way. And that's a reasonable H of lambda. Sometimes, this could be written as H0 plus something that we will call the change in the Hamiltonian. And we usually think of it as a small thing. So what do we have from this theorem? From this here we would have the d, En, d lambda is equal to psi n of lambda, H1, psi n of lambda. Now, we can be particularly interested in the evaluation of this thing at lambda equals 0. So what is d En of lambda? d lambda at lambda equals 0 would be psi n at zero, H1, psi n at 0. And therefore, you would say that the En of lambda energies would be the energies at 0, plus lambda, d En of lambda, d lambda at lambda equals 0, plus order lambda squared. I'm doing just the Taylor expansion of En's of lambda from lambda equals 0. So this thing tells you that En of lambda is equal En of 0, plus-- this derivative you can write it as psi n, lambda, H1, psi n, all at 0. Like that. Plus order lambda squared. So in this step, I just use the evaluation that we did here. I substituted that and put the lambda in. So that I recognize now that En of lambda is equal to En of 0, plus psi n of 0-- and I can write this as delta H-- psi n of 0, plus order delta H squared. It's nice to write it this way, because you appreciate more the power of the theorem. The theorem here doesn't assume which value of lambda you have. And you have to have normalized eigenstates. And you wonder what is it helping you with, if finding the states for every value of lambda is complicated. Well, it certainly helps you to figure out how the energy of the state varies by a simple calculation. Suppose you know the states of the simple Hamiltonian. Those are the psi's, n, 0. So if you have the psi n 0 over here, you can do the following step. If you want to figure out how it's energy has varied, use this formula in which you compute the expectation value of the change in the Hamiltonian on that state. And that is the first correction to the energy of the state. So you have this state. You compute the expectation value of the extra piece in the Hamiltonian. And that's the correction to the energy. It's a little more complicated of course to compute the correction to the state. But that's a subject of perturbation theory. And that's not what we care about right now. So the reason we're doing this is because actually whatever we're going to have with spin orbit coupling represents an addition to the hydrogen Hamiltonian of a new term. Therefore, you want to know what happens to the energy levels. And the best thing to think about them is to-- if you know the energy levels of this one, well, a formula of this type can let you know what happens to the energy levels after the perturbation. There will be an extra complication in that the energy levels that we're going to deal with are going to be degenerate. But let's wait for that complication until it appears. So any questions? Yes? AUDIENCE: So I would imagine that this would work just as well for time. Because time [INAUDIBLE] a parameter in quantum mechanics. So [INAUDIBLE] PROFESSOR: Time dependent perturbation theory is a bit more complicated. I'd rather not get into it now. So let's leave it here, in which we don't have time. And the Schrodinger equation is something like H psi equal [INAUDIBLE] psi, that's all we care. And leave it for that moment. Other questions? OK. So let's proceed with addition of angular momentum. So first, let me give you the fundamental result of addition of angular momentum. It's a little abstract, but it's what we really mean by addition of angular momentum. Of angular momentum. And the main result is the following. Suppose you have a set of operators, J, i, 1, that have the algebra of angular momentum. Of angular momentum. Which is to say Ji1, JJ1, is equal i, h bar, epsilon iJK, JK1. And this algebra is realized on some state space. On some vector space, V1. And suppose you have another operator, J-- set of operators actually. Ji2, which have the algebra of angular momentum. I will not write that. On some V2. OK. Angular momentum, some sets of states. Angular momentum on some other set of states. Here comes the thing. There is a new angular momentum, which is the sum Ji defined as Ji1, added with Ji2. Now, soon enough you will just write Ji1, plus Ji2. But let me be a little more careful now. This sum is Ji1, plus 1, tensor Ji2. So i is the same index. But here, we're having this operator that we're being defined that we call it the sum. Now how do you sum two operators that act in different spaces? Well, the only thing that you can actually do is sum them in the tensor product. So the claim is that this is an angular momentum in V1 tensor V2. That is an operator. You see, you have to sum them. So you have to create a space where both can act, and you can sum them. You cannot sum a thing, an operator that acts on one vector space to an operator that acts on another vector space. You have to create one vector space where both act. And then you can define the sum of the operators. Sum of operators is a simple thing. So you form the tensor product. In here, this operator gets upgraded in this way, in which in the tensor product it has a 1 for the second input. This one gets upgrade to this way. And this is the sum. So this is a claim-- this is a definition. And this is a claim. So this has to be proven. So let me prove it. Ji, JJ. I compute this commutator. So I don't have to do the following. I have to do Ji1, tensor 1, plus 1 tensor Ji2. And then the JJ would be JJ1, tensor 1, plus 1, tensor JJ2. Have to compute this commutator. Now, an important fact about this result that I'm not trying to generalize, if you had put a minus here, it wouldn't work out. If you would have put a 2 here, it wouldn't work out. If you would have put a 1/2 here, it won't work out. This is pretty much the only way you can have two angular momenta, and create a third angular momentum. So look at this. It looks like we're going to have to work hard, but that's not true. Consider this commutator. The commutator of this term with this term. That's 0 actually. Because if you multiply them in this order, this times that, you get Ji1 times Ji2, because the ones do nothing. You multiply them in the reverse order, you get again, Ji1 times Ji2. This is to say that the operators that originally lived in the different vector spaces commute. Yes? AUDIENCE: Since the cross terms between those two are 0-- like you just said, the cross terms are 0. And if you put a minus sign in there, it will cancel. But when you do the multiplications with the second ones, why can't you put a minus sign in there? [INAUDIBLE] PROFESSOR: In the whole thing? In this definition, a minus sign? AUDIENCE: Yeah. PROFESSOR: Well, here if I put a minus-- it's like I'm going to prove that this works. So if-- I'm going to get an angular momentum. If I put a minus sign to angular momentum, I ruin the algebra here. I put a minus minus, it cancels. But then I get a minus sign here. So I cannot really even change a sign. So any way, these are operators acting on different spaces. They commute. It's clear they commute. You just multiply them, and see that. These one's commute as well. The only ones that don't commute are this with this. And that with that. So let me just write them. Ji1, tensor 1, with JJ1, tensor 1. Plus this one, 1 tensor Ji2, 1 tensor JJ2. OK, next step is to realize that actually the 1 is a spectator here. Therefore, this commutator is nothing but the commutator Ji1 with JJ1, tensor 1. You can do it. If you prefer to write it, write it. This product is Ji times JJ, tensor 1. And the other product is JJ, Ji, tensor 1. So the tensor 1 factors out. Here the tensor 1 also factors out. And you get an honest commutator, Ji2, JJ2. So one last step. This is i, h bar, epsilon, iJK. I'll put a big parentheses. JK1, tensor 1, for the first one. Because J1 forms an angular momentum algebra. And here, 1 tensor JK2. And this thing is i, h bar, epsilon, iJK. The total angular momentum, K. And you've shown the algebra works out. Now most people after a little practice, they just say, oh, Ji is J1 plus J2, J1 plus J2. J1 and J2 don't commute. J2 and J1-- I'm sorry. J1 and J2 commute. J2 and J1 commute. Therefore you get this 2, like that. And this gives you-- J1 and J1 gives you J1. J2 and J2 gives you J2, so the sum works out. So most people after a little practice just don't put all these tensor things. But at the beginning it's nice to just make sure that you understand what these tensor things do. All right. So that's our main theorem-- that you start with one angular momentum on a state space. Another angular momentum that has nothing to do perhaps with the first on another vector space. And on the tensor product you have another angular momentum, which is the sum. All right. So now, we do spin orbit coupling to try to apply these ideas. So for spin orbit coupling, we will consider the hydrogen atom coupling. And the new term in the Hamiltonian, mu dot B. The kind of term that we've done so much in this semester. We've looked over magnetic ones. So which magnetic moment at which B? There was no B in the hydrogen atom. Well, there's no B to begin with. But here is one where you can think there is a B. First, this will be the electron dipole moment. Magnetic dipole moment. So we have a formula for it. The formula for it is the mu of the electron is minus E over m, times the spin of the electron. And I actually will use a little different formula that is valued in Gaussian units. ge over mC, S, in Gaussian units. And g is the g factor of the electron, which is 2. I'm sorry. There's a 2 here. OK. So look what I've written. I don't want to distract you with this too much. But you know that the magnetic dipole of the electron is given by this quantity. Now, you could put a 2 up, and a 2 down. And that's why people actually classically there seems to be a 2 down. But there's a 2 up, because it's an effect of the electron. And you have this formula. The only thing I've added in that formula is a factor of C that is because of Gaussian units. And it allows you to estimate terms a little more easily. So that's the mu of the electron. But the electron apparently would feel no magnetic field. You didn't put an external magnetic field. Except that here you go in this way of thinking-- you think suppose you are the electron. You see a proton, which is a nucleus going around you. And a proton going around you is a current going around you. It generates a magnetic field. And therefore, you see a magnetic field created by the proton going around you. So there is a magnetic field. And there's a magnetic field experienced by the electron-- felt by electron. So you can think of this, the electron. Here is the proton with the plus charge, and here's the electron. And the electron is going around the proton. Now, from the viewpoint of the electron, the proton is going around him. So here is the proton. Here is the electron going like that. From the viewpoint of the electron, the proton is going like this. Also, from the viewpoint of the electron, the proton would be going in this direction and creating a magnetic field up. And the magnetic field up corresponds actually to the idea that the angular momentum of the electron is also up-- L of the electron is also up. So the whole point of this thing is that somehow this magnetic field is proportional to the angular momentum. And then, L will come here. And here, you have S. So you have L dot S. That's the fine structure coupling. Now let me do a little of this so that we just get a bit more feeling, although it's unfortunately a somewhat frustrating exercise. So let me tell you what's going on. So consider the electron. At some point, look at it and draw a plane. So the electron-- let's assume it's going down. Here is the proton. It's going around in circles. So here, it's going down. The electron is going down. Electron, its velocity of the electron is going down. The proton is over here. And the electron is going around like that. The proton would produce an electric field of this form. Now, in relativity, the electric and magnetic fields seen by different observers are different. So there is this electric field that we see. We sit here, and we see in our rest frame this proton creates an electric field. And then, from the viewpoint of the electron, the electron is moving. And there is an electric field. But whenever you are moving inside an electric field, you also see a magnetic field generated by the motion, by relativistic effects. The magnetic field that you see is roughly given to first order in relativity by V cross E over c. So V cross E, VE V cross E over c up-- change sign because of this. And the magnetic field consistently, as we would expect, goes in this direction. So it's consistent with the picture that we developed that if you were the electron, the proton, would be going around in circles like that and the magnetic field would be up. Now here I can change the sign by doing E cross V over c. So this is the magnetic field seen by the electron. OK, so we need a little more work on that magnetic field by calculating the electric field. Now, what is the electric field? Well, the scalar potential for the hydrogen atom, we write it as minus e squared over r. It's actually not quite the scalar potential. But it is the potential energy. It has one factor of e more than what the scalar potential is. Remember, the scalar potential in electromagnetism is charge divided by r. So it has one factor of e more. What is the derivative of this potential? With respect to r, it's e squared over r squared. So the electric field goes like e over r squared. So the electric field is equal to dV dr divided by e. That's the magnitude of the electric field. And its direction is radial from the viewpoint of the proton. The electric field is here. So this can be written as r vector divided by r. Therefore, the magnetic field will-- [INAUDIBLE] this. The magnetic field now can be calculated. And we'll see what we claimed was the relation with angular momentum. Because B prime is now E cross V. So you have 1 over ec 1 over r dV dr. I've taken care of this. And now I just have r cross V. Well, r cross V is your angular momentum if you had p here. So we borrow a factor of the mass of the electron, ecm 1 over r dV dr L, L of the electron. p equals mv. So we have a nice formula for B. And then, we can go and calculate delta H. Delta H would then be minus mu dot B. And that would be ge over 2mc spin dot L-- mu was given here-- S dot L 1 over r dV dr. And that is the split spin orbit interaction. Now, the downside of this derivation is that it has a relativistic error. There's a phenomenon called Thomas precession that affects this result. We didn't waste our time. The true result is that you must subtract from this g 1. So g must really be replaced by g minus 1. Since g is approximately 2 for the electron, the true result is really 1/2 of this thing. So this should not be in parentheses, but true result is this. And the mistake that is done in calculating this spin orbit coupling is that this spin orbit coupling affects precession rates. All these interactions of magnetic dipoles with magnetic fields affect precession rates. And you have to be a little more careful here that the system where you've worked, the electron rest frame is not quite an inertial system. Because it's doing circular motion. So there's an extra correction that has to be done. Thomas precession or Thomas correction it's called. And it would be a detour of about one hour in special relativity to do it right. So Griffiths doesn't do it. I don't think Shankar does it. Pretty much graduate books do it. So we will not try to do better. I mentioned that fact that this really should be reduced to one half of its value. And it's an interesting system to analyze. So Thomas precession is that relativistic correction to precession rates when the object that is precessing is in an accelerated frame. And any rotating frame is accelerated. So this result needs correction. OK, but let's take this result as it is-- instead of g, g minus 1. Let's not worry too much about it. And let's just estimate how big this effect is. It's the last thing I want to do as a way of motivating this subject. So delta H is this. Let's estimate it. Now for estimates, a couple of things are useful to remember, that Bohr radius is h squared over me squared. We did that last time. And there's this constant that is very useful, the fine structure constant, which is e squared over hc. And it's about 1 over 137. And it helps you estimate all kinds of things. Because it's a rather complicated number to evaluate, you need all kinds of units and things like that. So the charge of the electron divided by hc being 1 over 137 is quite nice. So let's estimate delta H. Well, g we won't worry-- 2, 1, doesn't matter. e mc-- so far, that is kind of simple. Then we have S dot L. Well, how do I estimate S dot L? I don't do too much. S spin is multiples of h bar. L for an atomic state will be 1, 2, 3, so multiples of h bar. So h bar squared, that's it for S dot L. 1 over r is 1 over r. dV dr is e squared over r squared. And that's it. But here, instead of r, I should put the typical length of the hydrogen atom, which is a0. So what do I get? I'm sorry, I made a mistake here. AUDIENCE: Yeah, it's up there. PROFESSOR: Oh, I made a mistake here in that I didn't put this factor, 1 over ecm. So the e cancels. And this is the result here-- g over 2m squared c squared S dot L 1 over r dV dr. So let me start again. 1 over m squared c squared h bar squared 1 over r dV dr-- that much I got right. So this is roughly 1 over [INAUDIBLE] of the electron squared c squared e squared over a0 cubed h squared-- still quite messy, but not that terrible. So in order to get an idea of how big this is, the ground state energy of the hydrogen atom was e squared over 2a0. So let's divide delta H over the ground state energy. And that's how much? Well, we have all this quantity, 1 over m squared c squared e squared a0 cubed h squared. And now, we must divide by e squared over a0 like this. Well, the e squareds cancel. And we get h squared over m squared c squared a0 squared. You need to know what a0 is. Let's just boil it down to the simplest thing, so h squared m squared c squared. a0 squared would be h to the fourth m squared e to the fourth. So this is actually e to the fourth over h squared c squared, or e squared over hc squared, which is alpha squared. Whew-- lots of work to get something very nice. The ratio of the spin orbit coupling to the ground state energy is 1 over alpha squared. It's alpha squared, which is 1 over 137 squared. So it's a pretty small thing. It's about 1 over 19,000. So when this is called fine structure of the hydrogen atom, it means that it's in the level in your page that you use a few inches to plot the 13.6 electron volts-- well, you're talking about 20,000 times smaller, something that you don't see. But of course, it's a pretty important thing. So all in all, in the conventions of-- this is done in Gaussian units. In SI units, which is what Griffiths uses, delta H is e squared over 8 pi epsilon 0 1 over m squared c squared r cubed S dot L. That's for reference. This is Griffiths. But this is correct as well. This is the correct value. This is the correct value already taking into account the relativistic correction. So here, you're supposed to let g go to g minus 1. So you can put the 1 there, and it's pretty accurate. All right, so what is the physics question we want to answer with this spin orbit coupling? So here it comes. You have the hydrogen atom spectrum. And that spectrum you know. At L equals 0, you have one state here. Then, that's n equals 1, n equals 2. You have one state here and one state here at L equals 1. Then n equals 3, they start getting very close together. n equals 4 is like that. Let's consider if you want to have spin orbit coupling, we must have angular momentum. And that's L. And therefore, let's consider this state here. l equals 1, n equals 1-- n equals 2, I'm sorry. What happens to those states, is the question. First, how many states do you have there and how should you think of them? Well actually, we know that an l equals 1 corresponds to three states. So you'd have lm with l equals 1. And then m can be 1, 0, or minus 1. So you have three states. But there's not really three states. Because the electron can have spin. So here it is, a tensor product that appears in your face because there is more than angular momentum to the electron. There's spin. And it's a totally different vector space, the same particle but another vector space, the spin space. So here, you have the possible spins of the electron. So that's another angular momentum. And well, you could have the plus/minus states, for example. So you have three states here and two states here. So this is really six states, so six states whose fate we would like to understand due to this spin orbit coupling. So to use the language of angular momentum, instead of writing plus/minus, you could write Smz, if you will-- ms I will call, spin of s. You have here spin of 1/2 and states 1/2 or minus 1/2. This is the up. When the z component of the spin that we always call m-- m now corresponds to the z component of angular momentum. So in general, even for spin, we use m. And we have that our two spin states of the electron are spin 1/2 particle with plus spin in the z direction, spin 1/2 particle with minus spin in the z direction. We usually never put this 1/2 here. But now you have here really three states-- 1, 1, 1, 0, 1, minus 1, the first telling you about the total angular momentum. Here, the total spin is 1/2. But it happens to be either up or down. Here, the total angular momentum is 1. But it happens to be plus 1, 0, or minus 1 here. So these are our six states. You can combine this with this, this with that, this with this, this with that. You make all the products. And these are the six states of the hydrogen atom at this level. And we wish to know what happens to them. Now, this correction is small. So it fits our understanding of the perturbation theory of Feynman-Hellman in which we try to find the corrections to these things. Our difficulty now is a little serious, however. It's the fact that Feynman-Hellman assumed that you had a state. And it was an eigenstate of the corrected Hamiltonian as you moved along. And then, you could compute how its energy changes. Here, unfortunately, we have a much more difficult situation. These six states that I'm not listing yet, but I will list very soon, are not obviously eigenstates of delta H. In fact, they are not eigenstates of delta H. They're degenerate states, six degenerate states, that are not eigenstates of delta H. Therefore, I cannot use the Feynman-Hellman theorem until I find what are the combinations that are eigenstates of this perturbation. So we are a little bit in trouble. Because we have a perturbation for which these product states-- we call them uncoupled bases-- are not eigenstates. Now, we've written this operator a little naively. What does this operator really mean, S dot L? In our tensor products, it means S1 tensor L1. Actually, I'll use L dot S. I'll always put the L information first and the S information afterward. So L dot S is clearly an operator that must be thought to act on the tensor product. Because both have to act. S has to act and L has to act. So it only lives in the tensor product. So what does it mean? It means this-- S2 L2 plus S3 L3, or sum over i Si tensor Li. So this is the kind of thing that you need to understand-- how do you find for this operator's eigenstates here? So that is our difficulty. And that's what we have to solve. We're going to solve it in the next half hour. So it's a complicated operator, L dot S. But on the other hand, we have to use our ideas that we've learned already about summing angular momenta. What if I define J to be L plus S, which really means L tensor 1 plus 1 tensor S? So this is what I really mean by this operator. J, as we've demonstrated, will be an angular momentum, because this satisfies the algebra of angular momentum and this satisfies the algebra of angular momentum. So this thing satisfies the algebra of angular momentum. And why do we look at that term? Because of the following reason. We can square it-- JiJi. Now we would have to square this thing. How do you square this thing? Well, there's two ways. Naively-- L squared plus L squared plus 2L dot S-- basically correct. But you can do it a little more slowly. If you square this term, you get L squared tensor 1. If you square this term, you get 1 tensor S squared. But when you do the mixed products, you just must take the i's here and the i's here and multiply them. So actually, you do get two i's, the sum over i Li tensor Si. This is sum over i. This is J squared. So basically, what I'm saying is that J squared naively is L squared plus S squared plus our interaction 2L dot S defined property. So L dot S is equal to 1/2 of J squared minus L squared minus S squared. And that tells you all kinds of interesting things about L dot S. Basically, we can trade L dot S for J squared, L squared, and S squared. L squared is very simple, and S squared is extremely simple as well. Remember, L squared commutes with any Li. So L squared with any Li is equal to 0. S squared with any Si is equal to 0. And Li's and Si's commute. They live in different worlds. So L squared and Si's commute. S squareds and Li's commute. These things are pretty nice and simple. So let's think now of our Hamiltonian and what is happening to it. Whenever we had the hydrogen atom, we had a set of commuting observables H, L squared, and Lz. It's a complete set of commuting observables. Now, in the hydrogen atom, you could add to it S squared and Sz. We didn't talk about spin at the beginning, because we just considered a particle going around the hydrogen atom. But if you have spin, the hydrogen atom Hamiltonian, the original one, doesn't involve spin in any way. So certainly, Hamiltonian commutes with spin, with spin z. L and S don't talk, so this is the complete set of commuting observables. But what happens to this list? This is our problem for H0, the hydrogen atom, plus delta H that has the S dot L. Well, what are complete set of commuting observables? This is a very important question. Because this is what tells you how you're going to try to organize the spectrum. So we could have H, the total, H total. And what else? Well, can I still have L squared here? Can I include L squared and say it commutes with the total H? A little worrisome, but actually, you know that L squared commutes with the original Hamiltonian. Now, the question is whether L squared commutes with this extra piece. Well, but L squared commutes with any Li. And it doesn't even talk to S. So L squared is safe. L squared we can keep. OK, S squared-- can we keep S squared? Well, S squared was here. So it commuted with the Hamiltonian, and that was good. S squared commutes with any Si, and it doesn't talk to L. So S squared can stay. But that's not good enough. We won't be able to solve the problem with this still. We need more. How about Lz? It was here, so let's try our luck. Any opinions on Lz-- can we keep it or not? Yes. AUDIENCE: I don't think so. Because in the J term, we have Lx's and Ly's, which don't commute with Lz. PROFESSOR: Right, it can't be kept. Here, this term has SxLx plus SyLy plus SzLz. And Lz doesn't commute with this one. So no, you can't keep Lz-- no good. On the other hand, let's think about J squared. J squared is here. And J squared commutes with L squared and with S squared. J squared, therefore, is-- well, let me say it this way. Here is L dot S, which is our extra interaction. Here we have this thing. I would like to say on behalf of J squared that we can include it here, J squared, because J squared is really pretty much the same as L dot S up to this L squared and S squared. But J squared commutes with L squared and S squared. I should probably write it there. J squared commutes with L squared. And J squared communicates with S squared that we have here. And moreover, we have over here that J squared therefore will commute, or it's pretty much the same, as L dot S. J squared with L dot S would be J squared times this thing, which is 0. So J squared commutes with this term. And it commutes with the Hamiltonian, your original hydrogen Hamiltonian. So J squared can be added here. J square is a good operator to have. And now we can get one more kind of free from here. It's Jz. Z Because Jz commutes with J squared. Jz commutes with these things. And Jz, which is a symmetry of the original Hamiltonian, also commutes with our new interaction, the L dot S, which is proportional to J squared. So you have to go through this yourselves probably even a little more slowly than I've gone. Just check that everything that I'm saying about whatever commutes commutes. So for example, when I say that J squared commutes with L dot S, it's because I can put instead of L dot S all of this. And go slowly through this. So this is actually the complete set of committing observables. And it's basically saying to us, try to diagonalize this thing with total angular momentum. So it's about time to really do it. We haven't done it yet. But now the part that we have to do now, it's kind of a nice exercise. And it's fun. Now, there's one problem in the homework set that sort of uses this kind of thing. And I will suggest there to Will and Aram that tomorrow, they spend some time discussing it and helping you with it. The last problem in the homework set would've been better if you had a little more time for it and you had more time to digest what I'm doing today. But nevertheless, go to recitation, learn more about the problem. It will not be all that difficult. OK, so we're trying now to finally form another basis of states. We had these six states. And we're going to try to organize them in a better way-- as eigenstates of the total angular momentum L plus S. So I'm going to write them here in this way. Here is one of the states of this L equals 1 electron, the 1, 1 coupled to the 1/2, 1/2. Here are two more states- 1, 0, 1/2, 1/2, 1, 1, 1/2, minus 1/2, so the 1, 0 with the top, the 1, 1 with the bottom. Here are two more states-- 1, 0 with 1/2, minus 1/2 and 1, minus 1 with 1/2, 1/2. And here is the last state-- 1, minus 1 with 1/2, minus 1. These are our six states. And I've organized them in a nice way actually. I've organized them in such a way that you can read what is the value of Jz over h bar. Remember, Jz is 1 over h bar Lz plus Sz. So what is it? These are, I claim, eigenstates of Jz. Why? Because let's act on them. Suppose I act with Jz on this state. The Lz comes here and says, 1. The Sz comes here and says, 1/2. So the sum of them give you Jz over h bar equal to 3/2. And that's why I organized these states in such a way that these second things add up to the same value-- 0 and 1/2, 1 and minus 1/2. So if you act with Jz on this state, it's an eigenstate with Jz. Here, 0 contribution, here 1/2. So this is with plus 1/2. Here, you have 0 and minus 1/2, minus 1, and that is minus 1/2. And here you have minus 3/2. OK, questions. We've written the states. And I'm evaluating the total z component of angular momentum. And these two states are like that. So what does our theorem guarantee for us? Our theorem guarantees that we have-- in this tensor product, there is an algebra of angular momentum of the Jz operators. And the states have to fall into representations of those operators. So you must have angular momentum multiplets. So at this moment, you can figure out what angular momentum you're going to get for the result. Here we obtained a maximum Jz of 3/2. So we must get a J equals 3/2 multiplet. Because a J equaling 3/2 multiplets has Jz 3/2, 1/2, minus 1/2, and 0. So actually, this state must be the top state of the multiplet. This state must be the bottom state of the multiplet. I don't know which one is the middle state of the multiplet and which one is here. But we have four states here, four states. So one linear combination of these two states must be, then, that Jz equals 1/2 state of the multiplet. And one inner combination of these two states must be that Jz equals minus 1/2 state of the multiplet. Which one is it? I don't know. But we can figure it out. We'll figure it out in a second. Once you get this J 3/2 multiplet, there will be one linear combination here left over and one linear combination here left over. Those are two state, one with Jz plus 1/2 and one with Jz equals minus 1/2. So you also get a J equals 1/2 multiplet. So the whole tensor product of six states-- it was the tensor product of a spin 1 with a spin 1/2. So we write it like this. The tensor product of a spin 1 with a spin 1/2 will give you a total spin 3/2 plus total spin 1/2-- funny formula. Here is the tensor product, the tensor product of these three states with these two states. This can be written as 3 times 2 is equal to 4 plus 2 in terms of number of states. The tensor product of this spin 1 and spin 1/2 gives you a spin 3/2 multiplet with four states and a spin 1/2 multiplet with two states. So how do you calculate what are the states themselves? So the states themselves are the following. All right, here I have them. I claim that the J equals 3/2 states, m equals 3/2 states, the top state of that multiplet can only be the state here, the 1, 1 tensor 1/2, 1/2. And there's no way any other state can be put on the right. Because there's no other state with total z component of angular momentum equals 3/2. So that must be the state. Similarly, the J equals 3/2, m equals minus 3/2 state must be the bottom one-- 1, minus 1, 1/2, minus 1/2. The one that we wish to figure out is the next state here, which is the J equals 3/2, m equals 1/2. It's a linear combination of these two. But which one? That is kind of the last thing we want to do. Because it will pretty much solve the rest of the problem. So how do we solve for this? Well, we had this basic relation that we know how to lower or raise states of angular momentum-- m times m plus/minus 1 J-- I should have written it J plus/minus Jm equals h bar square root. More space for everybody to see this-- J times J plus 1 minus m times m plus/minus 1. Close the square root-- Jm plus/minus 1. So what I should try to do is lower this state, try to find this state by acting with J minus. So let me try to lower the state, so J minus on this state, on J equals 3/2, m equals 3/2. I can go to that formula and write it as h bar square root. J is 3/2, so 3/2 times 5/2 minus m, which is 3/2, times m minus 1, 1/2. We're doing the minus-- times the state 3/2, 1/2. So the state we want is here. And it's obtained by doing J minus on that. But we want the number here. So that's why I did all these square roots. And that just gives h bar square root of 3, 3/2, 1/2. Well, that still doesn't calculate it for me. But it comes very close. So you have it there. Now I want to do this but using the right hand side. So look at the right hand side. We want to do J minus, but on 1, 1 tensor 1/2, 1/2. So I applied J minus to the left hand side. Now we have to apply J minus to the right hand side. But J minus is L minus plus S minus on 1, 1 tensor 1/2, 1/2. When this acts, it acts on the first. So you get L minus on 1, 1 tensor 1/2, 1/2. And in the second term, you get plus 1, 1 tensor S minus on 1/2, 1/2. Now, what is L minus on 1, 1? You can use the same formula. It's 1, 1. And it's an angular momentum. So it just goes on and gives you h bar square root of 1 times 2 minus 1 times 0. 1, 0-- it lowers it-- times 1/2, 1/2. Let me go here-- plus 1, 1. And what is S minus on this? Use the formula with J equals 1/2. So this is h bar square root of 1/2 times 3/2 minus 1/2 times minus 1/2 times 1/2 minus 1/2. Whew-- well not too difficult. But this gives you h over square root of 2, 1, 0 tensor 1/2, 1/2 plus just h bar. This whole thing is 1-- 1, 1 tensor 1/2, minus 1/2. OK, stop a second to see what's happened. We had this equality. And we acted with J minus. Acting on the left, it gives us a number times the state we want. Acting on the right, we got this. So actually, equating this to that, or left hand side to right hand side, we finally found the state 3/2, 1/2. So the state 3/2, 1/2 is as follows. 3/2, 1/2 is-- you must divide by that square root. So you get the square root of 3 down. The h bars cancel. So here it is, a very nice little formula-- 2 over 3, 1, 0 tensor 1/2, 1/2 plus 1 over square root of 3, 1, 1 tensor 1/2, minus 1/2. So we have the top state of the multiplet. We have the next state of the multiplet. We have-- I'm sorry, the top state of the multiplet was this. You have the bottom state of the multiplet, the middle state of the multiplet. What you're missing is the bottom and the middle term. And this one can be obtained in many ways. One way would be to raise this state. The minus 3/2 could be raised by one unit and do exactly the same thing. Well, the result is square root of 2 over 3, 1, 0 tensor 1/2, minus 1/2 plus 1 over square root of 3. That square root of 2 doesn't look right to me now. I must have copied it wrong. It's 1 over square root of 3-- 1 over square root of 3, 1, minus 1 tensor 1/2, 1/2. So you've built that whole multiplet. And this state, as we said, was a linear combination of the two possible states. This 3 minus 1/2 was a linear combination of these two possible states. So the other states that are left over, the other linear combinations, form the J equals 1/2 multiplet. So basically, every state must be orthogonal to each other. So the other state, the 1/2, 1/2 and the 1/2, minus 1/2 of the J equals 1/2 multiplet must be this orthogonal to this. And this must be orthogonal to that. So those formulas are easily found by orthogonality. So I'll conclude by writing them-- minus 1 over square root of 3, 1, 0, 1/2, 1/2 plus the square root of 2 over 3, 1, 1, 1/2, minus 1/2. And here, you get 1 over square root of 3, 1, 0, 1/2, minus 1/2 minus 2 over square root of 3, 1, minus 1 tensor 1/2, 1/2. So lots of terms, a little hard to read-- I apologize. Now, the punchline here is that you've found these states. And the claim is that these are states in which L dot S is diagonal. And it's kind of obvious that that should be the case. Because what was L dot S? So one last formula-- L dot S equals 1/2 of J squared minus L squared minus S squared. Now, in terms of eigenvalues, this is 1/2 h squared J times J plus 1 minus L times L plus 1 minus S times S plus 1. Now, all the states that we built have definite values of J squared, definite values of S squared. Because L was 1. And S is 1/2. So here you go h squared over 2 J times J plus 1 minus 1 times 2 is 2 minus 1/2 times 3/2 is 3/4. And that's the whole story. The whole story in a sense has been summarized by this. We have four states with J equals 3/2 and two states with J equals 1/2. So these six states that you have here-- split because of this interaction into four states that have J equal to 3/2 and two states that have J equal to 1/2. And you plug the numbers here. And that gives you the amount of splitting. So actually, this height that this goes up here is h squared over 2. And this is minus h squared by the time you put the numbers J, 3/2, and 1/2. So all our work was because the Hamiltonian at the end was simple in J squared. And therefore, we needed J multiplets. J multiplets are the addition of angular momentum multiplets. In a sense, we don't have to construct these things if you don't want to calculate very explicit details. Once you have that, you have everything. This product of angular momentum 1, angular momentum 1/2 gave you total angular momentum 3/2 and 1/2-- four states, two states. So four states split one way, two states split the other way, and that's the end of the story. So more of this in recitation and more of this all of next week. We'll see you then.

MIT_805_Quantum_Physics_II_Fall_2013

16_Quantum_Dynamics_continued_and_Two_State_Systems.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, today's lecture will begin with photon states, which is a very interesting application of what we've learned about coherent states. And in a sense, it's an advanced topic. Photon states are states of the electromagnetic field. They are quantum states of the electromagnetic field. A photon, this particle, is a quantum of the electromagnetic field. There's a discrete piece of energy and momentum carried by this particle. So when we talk about photon states, we're really doing quantum field theory. So in some sense, this lecture, you will see how quantum field theory works. A first introduction to quantum field theory. And it's interesting that the harmonic oscillator plays such an important role in that. So a key identity that we are going to use, of course, is this coherent states that were defined as displacements of the vacuum. For D, if I remember right, was e to the alpha a dagger minus alpha star a. And it had the property that a acting on alpha was equal to alpha-- alpha, the operator a. So these were the coherent states we've been talking about. And today we're going to talk about photon states. So that will be probably about half of the lecture. And in the second half of the lecture, we will begin a more systematic study of two-state systems. Two-state systems, of course, are our spin states, are the classical two-state system. And we're going to sort of put it all together. We'll understand the general dynamics of a two-spin system, what is the most general Hamiltonian you can have, and therefore the most General Dynamics. And then we'll solve that. And we'll have two physical examples, one having to deal with the ammonia molecule. And another having to do with nuclear magnetic resonance. Both are applications of two-state systems. So till it's the end of the lecture, we'll be doing that. So about photon states. Well, photon states have to do with electromagnetic fields. That's electric and magnetic fields. And one important quantity that you know about the electromagnetic fields is the energy. If you have an electromagnetic field, you have an energy. And remember, energies have to do with Hamiltonians. So we're going to try to do a quantum description of the electromagnetic field. Therefore, knowing the energy would be a good place to start. So the energy, as you know, in an electromagnetic field goes like e squared times some epsilon and b squared. And you add the two of them. So here we go. Let me write a precise formula. The energy is equal to 1/2 the integral over volume, epsilon 0, the electric field squared, plus c squared times the magnetic field squared. So this is our energy. And we're going to try to describe a configuration of electromagnetic fields. We will to focus on one mode of the electromagnetic field. So I will imagine I have some sort of cavity, finite volume. And in there I have one electromagnetic field, what you usually called in 802 or in 8022 or 807 a wave. A single wave with some wavelength, some frequency, and that's all we have. So we're going to simplify to the case where we have a single one consistent with Maxwell's equations and some boundary conditions that we need not worry about. And I will normalize them as follows with a V in here. That this is the volume of the system. So, volume. And that could be the volume of the cavity that has this electromagnetic field. Or some large box. Or you can let it almost be infinite and work with that as well. So we'll have a wave. Omega would be the frequency. K is omega over c for a electromagnetic wave. So we'll have this times omega sine of kz, a spatial distribution. And there will be a function of time, as you know. But this function of time, I want to leave it a little ambiguous at this moment-- or, general. Not ambiguous, general. So I'll call it q of t, some function of time to be determined. There's going to be an electromagnetic and magnetic component to this field, By. c times By will also depend on z and t and will have the same pre-factor. I put the c here. So your c squared b squared also works well. Epsilon 0 v. This time I'll put another function, p of t cosine of kz. It's another function of time and I just call them that way. There is a question there. STUDENT: Why is your frequency outside your function of time? PROFESSOR: It's just another constant here. STUDENT: What would that mean then? PROFESSOR: No particular meaning to it. At this moment, whatever this constant is you would say probably it's useful because you somehow wanted the q here. That has some meaning. So you probably would put the same constants here in first trial. You wouldn't have this omega here. But if you put it, this is just another way of changing their own normalization of q. So it doesn't have a profound meaning so far. Any other questions about this? This is and electromagnetic field configuration. And this q of t and p of t are functions of time. You know your Maxwell's equations. And you will check things related to Maxwell's equations for this configuration in the homework. But at this moment, it's not too crucial. The thing that this important is that we can try to calculate the energy now. And if you do it, well, the squares, the epsilon 0's are going to disappear. And you're going to have to integrate over the box, this integral of sine squared of kz or cosine squared of kz. The functions of time don't matter-- this energy could depend on time. And the way we've prepared is when you integrate over sine squared of kz, if the box is big, it's a good situation where you can replace that for 1/2, which is the average, and 1/2 for the average of this. Or you could define where the box extends, from what values of z to what other values of z's. And so the integral, in fact, is not any complicated integral. And we have immediately the answer that energy is 1/2 p squared of t plus omega squared q squared of t. And that was why this omega was here. There's not really much to this. Except that when you square it and you take the integral over the volume, you replace the sine squared by 1/2 and the cosine squared by 1/2. And that's it. So actually, the labels that we've chosen here are pretty good. This starts to look like a harmonic oscillator. Except that the mass is gone. 1 over 2m p squared should be plus 1/2 m omega squared q squared. So the units are wrong here. p squared over 2m has units of energy. But p squared doesn't have units of energy. And 1/2 m omega squared q squared has units of energy but this one doesn't. So the units are a little off for a harmonic oscillator. So it's interesting to notice now. But you couldn't have done better. Because photons have no mass. And we're trying to describe the electromagnetic field. It has photons. So there's no way this could have showed up a mass there. There's no such thing. And that's why it doesn't show up. On the other hand, you can say, well, OK, let's see if this makes a minimum of sense. How do we deal with this unit? So p has units of square root of energy. And q has units of time times square root of energy. Why is that? Because omega has units of 1 over time. So q over time squared is energy. So q is t times square root of energy. And therefore p doesn't have the right units to deserve the name p. And q doesn't have the right units to deserve the name q. But p times q has the units of time times energy, which are the units of h bar. So that's good. This p and q have the right units in some sense. So this thing could be viewed as an inspiration for you. And you say at this moment, well, I don't know what is a quantum of an electromagnetic field. But here I have a natural correspondence between one mode of vibration, classical, of the electromagnetic field, and an energy functional that looks exactly like a harmonic oscillator. So I will declare these things to be a Hamiltonian and this p of t and q of t to be the Heisenberg operators of the electromagnetic field. So what we're saying now is that I'm going to just call the Hamiltonian 1/2 p hat squared plus omega squared q hat squared. This is a time independent Hamiltonian. If you're doing Heisenberg, it's the same thing as the Hamiltonian that would have p hat square of t plus omega q hat squared of t. Now, at this moment, this might sound to you just too speculative. But you can do a couple of checks that this is reasonable. So one check, remember that the Hamiltonian-- quantum equations of motion, of Heisenberg operators should look like classical equations of motion. So I can now compute what are the Heisenberg equation of motions for the operators. Remember something like v dt of p Heisenberg of t dt is related to h with p Heisenberg. And you can calculate the Heisenberg equations of motion. I may have signs wrong here. Nevertheless, you know those for the harmonic oscillator and you can write them. But you also know Maxwell's equations. And you can plug into Maxwell's equations. And that's one check you will do in homework, in which you will take Maxwell's equations and see what equations you have for q of t p of t. And then they will be exactly the same as the Heisenberg equations of motion of this Hamiltonian, giving you evidence that this is a reasonable thing to do. That we can think of this dynamical system with q and p being quantum operators. So let's accept that this is a Hamiltonian for this quantum system that we want to work with. And therefore, write the operators that we have. And what are they? Well, we had formulas with masses. But now mass goes to 1. So know the units. You cannot let in general in a formula mass going to 1 unless you're going to do something with the units. But we agreed already that these p's and q's have funny units. So those units are in fact consistent with a mass that has no units. And you can set it equal to 1. So I claim that you can take all the formulas we had with m and just put m equals to 1 and nothing would go wrong. Nothing goes funny. So in particular, you had an expression for x that now is called q terms of creation and annihilation operators and now that reads-- And you have an expression for p. And that one reads now a minus a dagger. These formulas used to have m's in there. And I've just set m equals to 1. And that should be the right thing. Unit-wise, indeed h bar omega has units of energy. And we claim that p has units of energy, square root of energy. So this is fine. So what else do we get from this Hamiltonian? Well, we can write it in terms of the number operators. So this Hamiltonian now, it's equal to h bar omega a dagger a plus 1/2. And this is just because this p and q written in this way corresponds to m equals to 1. And m doesn't show up anyway in this formula. So no reason to be worried that anything has gone wrong. And this is H equals to h bar omega, N hat plus 1/2. And this is a number operator. And then you get the interpretation, the physical interpretation that if you have states with some number operator, the energy is the number times h omega, which is exactly what we think about photons. If you have N photons in a given state, you would have an energy N times h bar omega. So it may look a little innocent what we've done here. But this is a dramatic assumption. You've really done something that took physicists 30 years to figure out, how to do quantum field theory. And of course, this is just the very beginning. And there's lots of things to learn about it. But the first thing that is happening is that somehow-- look what's happening. In normal quantum mechanics, x and p became quantum operators. In a sense here, this q and p are like x and p. But they have nothing to do with usual position and momentum. Nothing absolutely. q is like E really. And p is like B. So who has become a quantum operator? Not x and p, in a sense. E and B have become quantum operators. Quantum field theory is the idea that the fields become operators. That's what's really happening. And it seems to be right in the sense that our intuition that the state with N photos would be viewed as a state of a harmonic oscillator, an usual one with mass equals 1. So that this really is not a momentum and this is not a position. But they behave as that. So we can turn now this formula to its Heisenberg form so that q of t is square root of h bar over 2 omega. Remember a as a function of time becomes e to the minus i omega t a hat-- that's the Heisenberg version of a-- plus e to the plus i omega t a hat dagger. So given that, we can substitute back to our electric field that has this omega here, that has this factor in there. So I will write it all together here. Therefore, Ex of z t-- and now I've put a hat here. And z t, the t is the t of a Heisenberg operator now. Is equal to E naught e to the minus i omega t a plus e to the i omega t a hat dagger sine of kz, where this constant E zero is h bar omega over epsilon 0 V. It's just a mnemonic for some constant at this moment. So we plugged-in here, there's all these factors. There's the omega and there's the q. So all these factors together give you this. The factor and sine of kz. And this is the electromagnetic field operator. The electric field is not anymore an electric field. It's an operator. So if we want to get an intuition about this electric field operator, let's try to find its expectation value. It's an operator. The closest thing we can have an intuition about an operator is its expectation value. So very good. Let's take a photon state and energy eigenstate of the harmonic oscillator of occupation number n. And we have a state now of n photons, an energy eigenstate. In fact, with energy n h omega plus this 1/2 h bar omega. And let's figure out what is the expectation value of Ex in that state n. So we go to this formula. And we say, OK, it's E naught e to the minus i omega t n-- and we're all very curious. We want to see how the electromagnetic field of the n-th state of the harmonic oscillator, n photons in an energy eigenstate, how does that wave look? Let's see. n a hat n plus e to the i omega t n a dagger n sine kz. So this is a field operator. So we put it in a state and we want to know how does the field look in that state. And how much do we get? STUDENT: [INAUDIBLE]. PROFESSOR: 0. OK, that seems a little strange. Because indeed, the matrix element of a in an energy eigenstate is 0. This reduces, makes n minus 1 orthogonal to this. So this is 0. And this is n plus 1 n. This is 0. So actually no great illumination has happened. We got 0. So actually this is not too strange. Energy eigenstates are very unintuitive. The energy eigenstate of a harmonic oscillator, the n-th state is some sort of wave that is like that. Nothing changes in time in that wave. Nothing all that interesting happens. So the fact that this electromagnetic field operator has zero expectation value on this n photon state is maybe not too surprising. So let's take a more thoughtful state. We've said so many times that coherent states act like classical states. So let's put a coherent state of photons into this state. So let's see. Now the state will be an alpha state, which is a coherent state. And therefore, the expectation value of Ex on the alpha state will be equal to E naught e to the minus i omega t alpha a alpha plus e to the plus i omega t alpha a hat dagger alpha sine of kz. Well, we're in better shape now. a on alpha is the number alpha, as we reviewed at the beginning. And then alpha with alpha is 1. Remember, it's a unitary transform of the vacuum. Therefore, this whole thing is alpha. So this is E naught alpha being a number e to the minus i omega t plus here is alpha star e to the i omega t sine of kz. And now we're very happy. The coherent state is the state for which the expectation value of the electromagnetic field is precisely the kind of waves you've seen all your life. This wave, travelling waves, stationary waves. All those only appeared because only on coherent states a and a dagger have expectation values. So what we really call a classical wave resonating in a cavity is a coherent state of the electromagnetic field in this sense. The state of photons form a coherent state. They're not an energy eigenstate. They're not positioned for anything. They're not the number eigenstates either, because they're not an energy eigenstates. They have uncertainties. But they have a nice, classical picture. The expectation value of the operator is a real wave. So any time in 802 or in 8022, you have a classical wave to analyze, the quantum description of that wave is a coherent state of the electromagnetic field. Lasers are coherent states of the electromagnetic field. They have these uncertainties that we discussed last time with number and phase that are very strong. If the number goes large, then certainty on the phase is extremely small. So there we go. This is a coherent state. We can do a little more on that, write it more explicitly. This is epsilon 2 E not, the real part of alpha e to the minus i omega t sine of kz. And if we write, for example, alpha to be length of alpha e to the i theta, then this would be 2 E not. Length of alpha would go out, and the i theta to minus i omega t would give you cosine of omega t minus theta sine of kz. And this is something like a standing wave. It just changes in time and with a fixed spatial distribution. So it's a classical wave, and nevertheless, it has a good description classically, a good description quantum mechanically. It's a coherent state. And its energy is the expectation value of the Hamiltonian. The expectation value of the energy-- let me write this expectation value-- of H is H omega. Expectation value of N plus 1/2. And in a coherent state, the expectation value of N is alpha squared. So you have this of the coherent state alpha has alpha squared photons. And that's because it's the number operator, and that's pretty much the end of our story for photon states. There's more that one could do. One could do basically all kinds of things put together different modes. We considered here one mode. You could consider electric fields have superposition of modes and discuss commutation relations for the field operators, and all kinds of things. But that's really a quantum field theory course. At this moment, the main story I wanted to get across is that naturally, the harmonic oscillator has entered here, but in a very funny way. q and p were not positioned in momentum, were basically electric field and magnetic field. And there's an uncertainty between electric and magnetic fields. And the result of all this is that at the end of the day, you have a description by a harmonic oscillator and with energy levels that correspond to different amount of photons in the field. Finally, the classical things, if you want to recover classical waves, you must consider coherence states. These are the states that were classical. When you looked at the harmonic oscillator doing motion and for electromagnetic field, they give you the classical wave picture of an electric and magnetic field oscillating in position and time. So are there any questions? Yes. AUDIENCE: If we're associating h bar with [INAUDIBLE],, what object would you associate the zero point energy with? PROFESSOR: Well, it's a zero point energy of this quantum of vibration. So just like an electromagnetic field, basically, if this is like q and p, there's a minimum energy state in which you're in the ground state of the harmonic oscillator. But E and B cannot be zero, like delta x and delta p cannot be zero. So every mode of the electromagnetic field has a zero point energy. You cannot reduce it. So the vacuum of the electromagnetic field has a lot of zero point energies, one for every mode of radiation. Now, that zero point energies don't get you in trouble unless you're trying to do gravity. Gravity's the universal force and universal interaction that notes every bit of energy. So your zero point energies are quite important if you consider gravity. And you would have encountered here the first complication associated with quantum field theory. Every mode of the electromagnetic field-- a frequency one, a frequency 1.1, a frequency 1.2-- every one of them has a ground state energy of 1/2H bar omega. If you add them all up, you get infinity. So you get an infinity of ground state energies. And people have learned how to work with this infinities. That infinity is not physical. But, if you suitably treat it, you can figure out all kinds of things. And there's several people, I think even some undergraduates, working on this with Professor Kardar and Professor Jaffe, called Casimir energies, in which the zero point energies of the electromagnetic field are treated in a more careful way, and the infinities are seen to be irrelevant, but there are some physical dependence on the parameters that keeps there. So you see the origin of this is because every mode of the electromagnetic field has a zero point energy, just like any quantum oscillator. Yes. AUDIENCE: [INAUDIBLE]. PROFESSOR: Absolutely. AUDIENCE: [INAUDIBLE]. PROFESSOR: Well, uncountable things, we already have seen some. Maybe they didn't look that sophisticated, but we had position states that were uncountable. So the electromagnetic field, yes, it has uncountable things. And there's nothing wrong about it. You just have to work with integrals. AUDIENCE: [INAUDIBLE]. PROFESSOR: Well, no, no. They're not really normalized because just like these states, the position states are not normalized, they're delta function normalized and things like that. So look, if you want to avoid conceptual troubles with that, people and many physicists and textbooks on quantum field theory begin with space, a big, big, box. And then you see that it works for any size box, and then you say, well, it will work if the box is infinite. And we just proceed. All right. So I'll move on now to the second part of the lecture that deals with two-state systems and spin states and goes back and puts together a few of the things we've been doing. AUDIENCE: Professor? PROFESSOR: Yes. AUDIENCE: Could you close the sun shade? I can't really see the board. PROFESSOR: OK, sure. Board. I think maybe we need all the way? No, that won't make a difference. It's the other shades, I think. I'll leave it like that. Maybe I should use another board for the people that watch these movies. That may be better. So let's do this board. OK, so here's what we want to understand. Two-state systems. It's probably going to be about this, and two more lectures on that. And what we want to understand first is spin procession. You say, well, spin procession looks like a very particular kind of problem. When you have spins, you have magnetic fields. But at the end of the day, what we will see is that spin process-- you can view any two-state system as a system in which you've put a spin in a magnetic field. Even though you may be talking about electrons shared between two atoms, or ammonia molecule, or anything like that. Mathematically, you go back always to spins. Because spins are things we have become familiar already. So we exploit that to the maximum. So we do the one thing we haven't done rigorously so far, and then we'll explore this analogy to some point. So what was our discussion of spin? So two-state systems, and we'll begin with spin procession. So the idea of spin procession all arises, as you remember, because if you have a charged particle that has some spin, there's a relation between the particle's magnetic moment and the spin, or the angular momentum, of that particle, of that little ball of material. And we made an argument that this was just q over 2m times the angular momentum. And this will be angular momentum. This was classical. Nevertheless, the fact that we claim is true quantum mechanically is that in fact this idea is roughly right, except that there's two modifications. The true magnetic moment that enters into the Hamiltonian under the particle has is not quite the same as suggested by the classical argument, but it's modified by a g factor. And that modification is important. And this S is not just a plain classical angular momentum of a rotated ball with some mass and some radius, but it's a spin angular momentum and intrinsic angular momentum. A rather abstract thing that in fact should be best viewed as an operator, and that's the way we've thought about it. The magnetic dipole moment now becomes an operator, because it's proportional to the spin operator. So it's an operator. And different values of g apply for different particles. And we saw that g equals 2 applies for the electron. That's a famous value, in fact predicted by Dirac's equation, relativistic equation, for the electron, and observed to great accuracy of course as well. And for other particles, like the proton or the neutron, the quantity g has different values. You might be surprised that the neutron has a dipole moment. Because you would say a neutron is an uncharged particle, so a charge rotating doesn't do anything. Nevertheless, a neutron is uncharged, but it has three quarks, two with some charge, one with an opposite charge to the other two. And if they distribute cleverly, say the negative ones are farther away from the center, and in the center is the positive one, this could have angular magnetic moment. And in fact, it does have magnetic moment. The neutron has a significant magnetic moment. So at the end of the day, we're going to write this as mu equal gamma S. And this constant gamma is going to summarize everything, g, q, m, all these things. And this will be a good notation. Gamma S is brief and simple. And this constant, we're going to use it. So the Hamiltonian minus mu dot B is a quantum Hamiltonian because mu is an operator. B, at this moment, even though we were just talking about photon states, this will be a static magnetic field typically. Can be a to time dependent, but it will not be sufficiently important if it has time dependence, and we have to quantize it and to think of it as a quantum field. But in some problems of radiation of electromagnetic fields by the motion of spins, you would have to quantize the electromagnetic field. But this is not the case now. So this is minus gamma S dot B. And we typically like to write it as minus gamma B dot S. And that means very explicitly minus gamma BxSx operator plus BySy operator plus BzSz operator. So let me remind you of a simple situation when you had a magnetic field in the z direction. B along z if B is B times z hat. Then H is minus gamma. BSz. And the unitary operator that generates time evolution of states, the unitary operator u of t0 is exponential minus i. I'll call it H sub s for spin. H sub s t over H bar. And I'll put it like this exponential of minus i minus gamma B t Sz over H bar. So I substituted what Hs is, moved the t sort of inside the parentheses minus gamma B Sz. I put the Sz out and put this here. So far so good? This is our time development operator. Now, I want you to recall one property that you only justified by checking it in the homework. But in the next few lectures, we will just make sure you understand this why it's true in general. But we talked in the homework about an operator Rn sub alpha, which was exponential of minus i alpha Sn over H bar. Where n was a unit vector, and Sn is defined as n dot S. So nxSx, nySy, nzSz. So this operator that you considered was called the rotation operator, and it did perform rotation of spin states. In fact, what it did was rotate any spin state by an angle, alpha, around the nth direction. So if you had the n direction here, and you had any spin state in some arbitrary direction, it would rotate it by an angle alpha around this. So you have this, it would rotate it to another point over here with an angle alpha in between. So in words, it rotates by an angle alpha, rotates spin states. And when you think of a spin state, you must think of some n vector, n prime vector. So maybe n prime here would be a good notation. So you have a spin state in the n prime direction. Remember your spin states were of the form n plus minus. Well, the state that points in the direction n is n plus, so some n prime direction. This operator rotates those states by an angle alpha. Now, it probably is a little vague in your mind, that idea, because you checked it several weeks ago. And you only checked it by taking some particular states and rotating them. So we will have to elaborate on this, and we will. So this will become clear that this rotates any spin state by an angle alpha and rotates spin states using an axis, with respect to the axis defined by n over here. So that's interpretation of this state, of this operator. That's what it does. And now I want you to look at this operator. Well, it's similar. In fact, this plays the role of alpha, and this plays the role of Sn. So this is the spin in the z direction, and this operator must rotate states by this angle alpha, which is gamma Bt. If what we said is right, that's what this operator must do. Even though I think you've done this calculation as part of tests, problems, or other problems, practice problems, not quite homework., I want to do this calculation again. So let's take an arbitrary spin state, xyz. Now, don't confuse the arbitrary spin states with the n here. The n is here the axis around which this Hamiltonian rotates states. But there's no states here. This is a rotation operator. I'm sorry, I called it a Hamiltonian. It's not precise. This is a unitary operator. It rotates states. And this is the direction, the axis, of rotation. Your spin state is another object. It's a spin that lives in some direction. So here, we're having the magnetic field in the z direction. So the magnetic field is here. And we'll put a spin state over here, an n, a spin state that has some value of phi and some value of theta. And that's the spin state at time equals zero. So psi 0 is the spin state this that with your formula sheet, this cosine theta over 2 plus plus sine theta over 2 e to the i phi, I think with a plus, yes. I'll call it phi not, and maybe theta naught, y naught, and minus. So this is a state, a spin state pointing in this direction, the direction n. That was the general formula for a spin state. Now we are going to apply the operator, the time evolution operator. But let's do a preliminary calculation. HS on plus is minus gamma B Sz on plus minus gamma B H bar over 2 plus, and Hs minus is equal to minus gamma BSz on minus equal plus gamma B H bar over 2 minus. So we want to add with this operator on this state. So here we have it, the state that any time is going to be E to the minus iHst over H bar times this state over here acting on psi 0. So let's do it. Well, on the first term is cosine theta 0 over 2. And you have this exponent acting on plus. But the exponent has Hs that's acting on plus is this. So you can just put that thing on the exponent. So you put e to the minus i, and Hs on plus is this, minus gamma B H bar over 2. Then you have the p and the H bar and the plus. And continue here. So we just need to do the second term, plus sine theta over 2, e to the minus i. And now the same thing, but with a plus sign. Plus gamma B H bar over 2, t over H bar on the minus state. So just in case I got you confused and the small type is a problem here, this operator active on initial state just acts on plus, then acts on minus. On plus, the operator is an eigen state. So you can just put the number in the exponential. So you put the plus eigen value, the minus eigen value. So what do we get? Psi t is equal, cosine theta naught over 2, e to the i gamma B t over 2 plus sine theta naught over 2, e to the minus i gamma B t over 2 minus. Now, this state this is not quite-- I hope I got my signs right. Yes. This state is not quite in readable form. To compare it with a general end state, you need null phase here. So we must factor this phase out. e to the i gamma B t over 2. And it's an irrelevant phase. So then you have cosine theta naught over 2 plus sine theta naught over 2. I'm sorry, I forgot to have the e to the i phi naught here. I didn't copy it. So here, what do we have? e to the i phi naught minus gamma B t minus. Look, when you factor this one out, you get minus the same thing here. So this becomes a minus 1. And then you put the two faces together, and you got that. So now you look at this state, and you say, oh, I know what this is. This is a spin state that has theta as a function of time, just theta naught. But the angle, phi, as a function of time is phi naught minus gamma B t. So this spin will precess and will go like this. Phi naught minus gamma B t is the phi as a function of time. So have the magnetic field. You have a procession of the spin over here. So this is spin procession. And indeed, this is exactly what we're claiming here. If this rotates states by an angle alpha, this operator, this Hamiltonian that we've discussed here, must rotate states by this angle alpha, which is minus gamma Bt, along the z-axis. So you have the z-axis, and you rotate by minus gamma Bt. The sine is the reason the phi decreases in time and goes in this direction, as opposed to going in the other direction. So this is a basic confirmation of what the spin is doing. And I want to give you the general result so that you can really use it more clearly. So I think the lights are gone, so we can go to this blackboard. First of all, classical picture. What is it about spin procession? Is it a quantum phenomenon or a classical phenomenon, or both? Well, it's really both. And this idea of procession, you can get it from the classical picture as well. So what do you have? If you have a mu in a B field, you get a torque. And that you can easily convince yourself. I'm sure you've done the computation in 802. You have a little square wire not aligned with the magnetic field. You calculate the force on one side, the force on the other. You see that there is a torque. And the torque is given by mu cross B. That's E and M. On the other hand, the rate of change of angular momentum is the torque. So this is mu cross B. But mu is gamma S, so this is gamma S cross B. And this is minus gamma B cross S. OK. This equation, which I rewrite it here, ds/dt equals minus gamma B cross S, is a particular case of a very famous equation in classical mechanics, and this equation for a rotating vector. If you have a vector, dx/dt is omega cross x. This is the equation satisfied by a vector x that is rotating with angular frequency omega around the axis defined by the vector omega. A famous equation. OK, so you have here omega vector is omega n. So here is the direction of n, the unit vector. Here's omega. And you have a vector x over here. Then this vector, the solution of this equation, is a vector that is rotating around omega with the angular velocity magnitude of omega. In the notes, I just give a little hint of how you derive that. But truly speaking, you guys should be able to just scribble a few notes if you don't know the situation by heart, and convince yourself this is true. So this equation is of that form in which the omega x is played by S. Omega is minus gamma B. So this defines what is called the Larmor frequency, which is minus gamma B, is the Larmor frequency. Now, this Larmor frequency is precisely that one because was minus gamma B. And here you have minus gamma B times t. Omega times t is the angle. So in fact, this is rotating with a Larmor frequency. And there you go. In the same blackboard, you have a classical mechanics derivation of the Larmor frequency and a quantum mechanical derivation of the Larmor frequency. Again, at the end of the day, this is no coincidence. We've made dynamical classical variables into quantum operators, and we haven't changed the physics. Mu dot B is a classical energy. Well, it became Hamiltonian, and it's doing the right thing. So we can now use the Larmor frequency to rewrite the Hamiltonian, of course. It's here. So a little bit of emphasis is worth it. Hs is minus mu dot B, and it's minus gamma B dot S, and it's finally equal to omega L dot S. So if somebody gives you a Hamiltonian that at the end of the day, you can write it as some vector dot S, you already know that for spins, that is the Larmor frequency of rotation. It's a very simple thing. Hs, something times S, well that's precisely the rotation frequency for the spin states. They will all rotate that way. So we can say that the spin states in this Hamiltonian rotate with omega L frequency. So that's good. That's a general discussion of precession in a magnetic field. But I want to go one more step in generalization. It's a simple step, but let's just take it. So that you see even more generally why any system can be thought of as a spin system. And this is quite practical. In fact, it's probably the best way to imagine physically, the effects of any Hamiltonian. So let's consider time-independent Hamiltonians the most general Hamiltonian for a two-state system. How can it be? Well, a two-state system, remember two-state system is a word. It really means a system with two basis states. Once you have two basis states, a plus and the minus have infinitely many states, of course. But two-state system is two basis states. And therefore, in the Hamiltonian, in this two basis states, is a 2 by 2 matrix. And it's a 2 by 2 Hermitian matrix. So there's not too much it can be. In fact, you can have a constant that I will call maybe not the base notation, g naught and g naught. And that's Hermitian. It's real constant. You can put a g3 and a minus g3. That's still Hermitian. And that's reasonable. There's no reason why this number should be equal to this. So there are two numbers here that are arbitrary, real. And therefore, you can put them wherever you want. And I decided to call one g naught plus g3 and one g naught minus g3. Here, I can put again an arbitrary complex number, as long as I put here the complex conjugate. So I will call this g1 minus ig2, and this g1 plus ig2. And that's the most general 2 by 2 Hamiltonian. Tonya If those would be time-dependent functions, this is the most general Hamiltonian ever for a 2 by 2 system. It doesn't get more complicated. That's a great advantage of this. But I've written it in a way that you can recognize something. You can recognize that this is g naught times the identity plus g1 times sigma 1 plus g2 times sigma 2 plus g3 times sigma 3. And this is because the Pauli matrices are, together with the unit matrix, a basis for all Hermitian 2 by 2 matrices. So the Pauli matrices are Hermitian. The unit matrix is Hermitian. The most general 2 by 2 Hermitian matrix is a number times the one matrix, then number times the first part, then number, second, number, third. OK. So at this moment, we've got the most general Hamiltonian. And I will write it as g naught times 1 plus g vector dot sigma, where g vector is g1, g2, g3. If we write the g vector as length of g, which is just the letter g, shouldn't be confused because we have g not, g1, g2, g3, but we haven't had a g without an index. So g without an index is going to be the magnitude of g vector, and n is going to be a particular vector. So look, you're talking about the most general Hamiltonian, and you're saying it's most easily understood as g naught multiplying the identity, and that g vector multiplying the sigma vector. So on the other hand, g is this. So this is also g naught 1 plus g times n dot sigma. But let's continue here. We know how to solve this problem. And you can say, well, all right. I have to diagonalize this matrix, find the eigen vectors, find the eigenvalues, and all that. But you've done all that work. It's already been done. What were the eigen states? Well, n dot sigma, the eigen states were the end states, the spin states, n plus minus. And they were plus minus n comma plus minus. Remember that S is H over 2 sigma. So this corresponds to n dot S on n plus minus equal plus minus H bar over 2 n plus minus, which might be the form in which you remember it better. But the sigma matrices, n dot sigma is diagonalized precisely by this thing. So in fact, you never have to diagonalize this matrix. It's already been done for you. And these are the eigen states of this Hamiltonian. And what is the value of the energy on n plus minus? Well, energy on n plus minus is g naught times 1 plus g n dot sigma on the n plus minus. And g naught times 1 here on this state is g naught plus g n dot sigma, the thing is plus minus. So plus minus g, n plus minus. So in fact, you have the energies, and you have the eigen vectors. So the eigen states are n plus with energy equal g naught plus g and n minus with energy equal g naught minus g. So what we did by inventing the Pauli matrices and inventing spin states and all that was solve for you the most general 2 by 2 Hamiltonian, Hermitian Hamiltonian. If you have a 2 by 2 Hermitian matrix, you don't have to diagonalize it by hand. You know the answers are this state. And how do you build those states? Well, you know what n is because you know the g's. If you know the three g's, you know what the vector g is. You know what the vector n is. You know what this g is as well. And therefore, with a vector n, you construct this state, as you know already very well. And given that you know g and g not, well the energies are this, at the splitting is 2g between those states. This is the ground state. This is the excited state. Splitting two g's, so you look at the Hamiltonian, and you say, what's the splitting between the two eigen states of this. You just take this numbers, compute g, and multiply by 2. Now, last thing that you would want to do with this Hamiltonian is time evolution. So what do we say about time evolution? Well, we have here H is equal to this. And we also had omega L dot S. So omega L dot S in here should be identified with this. So sigma and S, as you remember, S is equal H bar over 2 sigma. So this term can be written as g vector sigma. In fact, this is better from here. g vector sigma, and sigma is H bar over 2S. I got a 2 over H bar. 2 over H bar S. So from here, g dot sigma is 2g over H bar S. And remember, a Hamiltonian for a spin system, whatever's multiplying the vector that is multiplying S is omega L. So in this system-- I will write it like that-- omega L is 2g over H bar. And this is a great physical help. Because now that you have this, I should remark this part of the Hamiltonian is the one that does procession. A part proportional to the identity cannot do procession, is just a constant term that produces a constant phase, just produces a pure phase. That's a change, an overall phase that doesn't change the state. You would have an extra factor of e to the minus i times that constant, g naught t over H bar, multiplying all the states. Doesn't change the action on plus or minus state. It's an overall phase. This term in the Hamiltonian is almost never very important. It doesn't do anything to the physical states, just gives them pure phases. And this term is the thing that matters. So now with this Hamiltonian, because g dot sigma is the form of the Hamiltonian, and we've identified this physical phenomenon of Larmor frequency, if you know your vector g for any Hamiltonian, this might be the Hamiltonian for ammonia molecule, then you know how the states evolve in time. Because you represent the state. You have one state and a second state. You think of the one state as the plus of a spin, the minus of a spin. And then you know that this is processing with this Larmor frequency. So it may sound a little abstract at this moment, but this gives you the way to evolve any arbitrary state intuitively. You know the vector V where it points. You know where your state points in the configuration space. And you have a physical picture of what it does in time. It's always precessing. Therefore, the dynamics of a two-state system in time is always procession, and that's what we have to learn. So next time will be ammonia molecule, and then NMR.

MIT_805_Quantum_Physics_II_Fall_2013

4_Spin_Onehalf_Bras_Kets_and_Operators.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Last time we spoke about the Stern-Gerlach experiment, and how you could have a sequence of Stern-Gerlach boxes that allow you to understand the type of states and properties of the physics having to do with spin-1/2. So the key thing in the Stern-Gerlach machine was that a beam of silver atoms, each of which is really like an electron with a magnetic moment, was placed in an inhomogeneous strong magnetic field, and that would classically mean that you would get a deflection proportional to the z-component of the magnetic moment. What was a surprise was that by the time you put the screen on the right side, it really split into two different beams, as if the magnetic moments could either be all the way pointing in the z-direction or all the way down, pointing opposite to the z-direction, and nothing in between. A very surprising result. So after looking at a few of those boxes, we decided that we would try to model the spin-1/2 particle as a two-dimensional complex vector space. What is the two-dimensional complex vector space? It's the possible space of states of a spin-1/2 particle. So our task today to go into detail into that, and set up the whole machinery of spin-1/2. So we will do so, even though we haven't quite yet discussed all the important concepts of linear algebra that we're going to need. So today, I'm going to assume that at least you have some vague notions of linear algebra reasonably well understood. And if you don't, well, take them on faith today. We're going to go through them slowly in the next couple of lectures, and then as you will reread this material, it will make more sense. So what did we have? We said that the spin states, or the possible states of this silver atom, that really correspond to an election, could be described by states z comma plus and z colon minus So these are the two states. This state we say corresponds to an angular momentum Sz hat. Sz-- I can put it like that-- of h-bar over 2, and this corresponds to Sz equals minus h-bar over 2. And those are our two states. The z label indicates that we've passed, presumably, these atoms through a filter in the z-direction, so that we know for certain we're talking about the z-component of angular momentum of this state. It is positive, and the values here again have the label z to remind us that we're talking about states that have been organized using the z-component of angular momentum. You could ask whether this state has some angular momentum-- spin angular momentum-- in the x-direction or in the y-direction, and we will be able to answer that question in an hour from now. So mathematically, we say that this statement, that this state, has Sz equals h-bar over 2 means that there is an operator, Sz hat-- hat for operators. And this operator, we say, acts on this state to give h-bar over 2 times this state. So when we have a measurement in quantum mechanics, we end up talking about operators. So this case is no exception. We think of the operator, Sz, that acts in this state and gives h-bar over 2. And that same operator, Sz, acts on the other state and gives you minus h-bar over 2 times the state. You see, an operator on a state must give a state. So in this equation, we have a state on the right, and the nice thing is that the same state appears on the right. When that happens, you say that the state is an eigenstate of the operator. And, therefore, the states z plus, minus are eigenstates of the operator Sz with eigenvalues-- the number that appears here-- equal to plus, minus h over 2. So the relevant physical assumption here is the following, that these two states, in a sense, suffice. Now, what does that mean? We could do the experiment again with some Stern-Gerlach machine that is along the x-axis, and say, oh, now we've got states x plus and x minus and we should add them there. They are also part of the possible states of the system. Kind of. They are parts of the possible states of the system. They are possible states of the system, but we shouldn't add them to this one. These will be thought as basis states. Just like any vector is the superposition of a number times the x-unit vector plus a number times the y-unit vector and a number times the z-unit vector, we are going to postulate, or try to construct the theory of spin, based on the idea that all possible spin states of an electron are obtained by suitable linear superposition of these two vectors. So, , in fact, what we're going to say is that these two vectors are the basis of a two-dimensional vector space, such that every possible state is a linear superposition. So psi, being any possible spin state, can be written as some constant, C1 times z plus plus C2 times z minus where these constants, C1 and C2 belong to the complex numbers. And by this, we mean that if any possible state is a superposition like that, the set of all possible states are the general vectors in a two-dimensional complex vector space. Complex vector space, because the coefficients are complex, and two-dimensional, because there's two basis vectors. Now this doesn't quite look like a vector. It looks like those things called kets. But kets are really vectors, and we're going to make the correspondence very clear. So this can be called the first basis state and the second basis state. And I want you to realize that the fact that we're talking about the complex vector space really means these coefficients are complex. There's no claim that the vector is complex in any sense, or this one. They're just vectors. This is a vector, and it's not that we say, oh this vector is complex. No. A complex vector space, we think of as a set of vectors, and then we're allowed to multiply them by complex numbers. OK, so we have this, and this way of thinking of the vectors is quite all right. But we want to be more concrete. For that, we're going to use what is called a representation. So I will use the word representation to mean some way of exhibiting a vector or state in a more concrete way. As something that any one of us would call a vector. So as a matter of notation, this being the first basis state is sometimes written as a ket with a 1. Like that. And this being this second basis state is sometimes written this way. But here is the real issue of what we were calling a representation. If this is a two-dimensional vector space, you're accustomed to three-dimensional vector space. What are vectors? They're triplets of numbers. Three numbers. That's a vector. Column vectors, it's perhaps easier to think about them. So column vectors. So here's what we're going to say. We have this state z plus. It's also called 1. It's just a name, but we're going to represent it as a column vector. And as a column vector, I'm going to represent it as the column vector 1, 0. And this is why I put this double arrow. I'm not saying it's the same thing-- although the really it is-- it's just a way of thinking about it as some vector in what we would call canonically a vector space. Yes AUDIENCE: So do the components of the column vector there have any correspondence to the actual. Does it have any basis in the actual physical process going on? Or, what is their connection to the actual physical [INAUDIBLE] represented here? PROFESSOR: Well, we'll see it in a second. It will become a little clearer. But this is like saying, I have a two-dimensional two0 vector space, so I'm going to think of the first state as this vector. But how do I write this vector? Well, it's the vector ex. Well, if I would write them in components, I would say, for a vector, I can put two numbers here, a and b. And this is the a-component and b-component. So here it is, ex would be 1, 0. And ey would be 0, 1. If I have this notation then the point a, b is represented by a and b as a column vector. So at this moment, it's just a way of associating a vector in the two-dimensional canonical vector space. It's just the column here. So the other state, minus-- it's also called 2-- will be represented by 0 1 1. 1 And therefore, this state, psi, which is C1 z plus plus C2 z minus will be represented as C1 times the first vector plus C2 times the second vector. Or multiplying, in C1, C2. So this state can be written as a linear superposition of these two basis vectors in this way-- you can write it this way. You want to save some writing, then you can write them with 1 and 2. But as a vector, it's represented by a column vector with two components. That's our state. Now in doing this, I want to emphasize, we're introducing the physical assumption that this will be enough to describe all possible spin states, which is far from obvious at this stage. Nevertheless, let's use some of the ideas from the experiment, the Stern-Gerlach experiment. We did one example of a box that filtered the plus z states, and then put it against another z machine, and then all the states went through the up. Which is to say that plus states have no amplitude, no probability to be in the minus states. They all went through the plus. So when we're going to introduce now the physical translation of this fact, as saying that these states are orthogonal to each other. So, this will require the whole framework, in detail, of bras and kets to say really, precisely-- but we're going to do that now and explain the minimum necessary for you to understand it. But we'll come back to it later. So this physical statement will be stated as z minus with z plus. The overlap, the bra-ket of this, is 0. The fact that all particles went through and went out through the plus output will state to us, well, these states are well normalized. So z plus, z plus is 1 Similarly, you could have blocked the other input, and you would have concluded that the minus state is orthogonal to the plus. So we also say that these, too, are orthogonal, and the minus states are well normalized. Now here we had to write four equations. And the notation, one and two becomes handy, because we can summarize all these statements by the equation Ij equals delta Ij. Look, if this equation is 2 with 1 equals 0. The bra 2, the ket 1. This is 1 with 1 is equal to 1. Here is is 1 with 2 is equal to 0, and 2 with 2 is equal to 1. So this is exactly what we have here. Now, I didn't define for you these so-called bras. So by completeness, I will define them now. And the way I will define them is as follows. I will say that while for the one vector basis state you associate at 1, 0, you will associate to one bra, the row vector 1, 0. I sometimes tend to write equal, but-- equal is all right-- but it's a little clearer to say that there's arrows here. So we're going to associate to 1, 1, 0-- we did it before-- but now to the bra, we think of the rho vector. Like this. Similarly, I can do the same with 2. 2 was the vector 0, 1. It's a column vector, so 2 was a bra. We will think of it as the row vector 0, 1. We're going to do this now a little more generally. So, suppose you have state, alpha, which is alpha 1, 1 plus alpha 2, 2. Well, to this, you would associate the column vector alpha 1, alpha 2. Suppose you have a beta state, beta 1, 1 plus beta 2, 2. You would associate beta 1, beta 2 as their representations. Now here comes the definition for which this is just a special case. And it's a definition of the general bra. So the general bra here, alpha, is defined to be alpha 1*, bra of the first, plus alpha 2*, bra of the second. So this is alpha 1* times the first bra, which we think of it as 1, 0, plus alpha 2* times the second bra, which is 0, 1. So this whole thing is represented by alpha 1*, alpha 2*. So, here we've had a column vector representation of a state, and the bra is the row vector representation of the state in which this is constructed with complex conjugation. Now these kind of definitions will be discussed in more detail and more axiomatically early very soon, so that you see where you're going. But the intuition that you're going to get from this is quite valuable. So what is the bra-ket? Alpha-beta is the so-called bra-ket. And this is a number. And the reason for complex conjugation is, ultimately, that when these two things are the same, it should give a positive number. It's like the length squared. So that's the reason for complex conjugation, eventually. But, for now, you are supposed to get a number from here. And the a reasonable way to get a number, which is a definition, is that you get a number by a matrix multiplication of the representatives. So you take the representative of alpha, which is alpha 1*, alpha 2*. And do the matrix product with the representative of beta, which is beta 1, beta 2. And that's alpha 1*, beta 1 plus alpha 2*, beta 2. And that's the number called the inner product, or bra-ket product. And this is the true meaning of relations of this kind. If you're given an arbitrary states, you compute the inner product this way. And vectors that satisfy this are called orthonormal because they're orthogonal and normal with respect to each other in the sense of the bra and ket. So this definition, as you can see, is also consistent with what you have up there, and you can check it. If you take I with j, 1, say, with 2-- like this-- you do the inner product, and you get 0. And similarly for all the other states. So let's then complete the issue of representations. We had representations of the states as column vectors-- two by two column vectors or row vectors. Now let's talk about this operator we started with. If this is an operator, acting on states, now I want to think of its representation, which would be the way it acts on these two component vectors. So it must be a two by two matrix, because only a two by two matrix acts naturally on two component vectors. So here is the claim that we have. Claim, that Sz hat is represented-- but we'll just put equal-- by this matrix. You see, it was an operator. We never talked about matrices. But once we start talking about the basis vectors as column vectors, then you can ask if this is correct. So for example, I'm supposed to find that Sz hat acting on this state 1 is supposed to be h-bar over 2 times the state 1. You see? True. Then you say, oh let's put the representation, h-bar over 2, 1 minus 1, 0, 0. State one, what's its representation? 1, 0. OK, let's act on it. So, this gives me h-bar over 2. I do the first product, I get a 1. I do the second product, I get a 0. Oh, that seems right, because this is h over 2 times the representation of the state 1. And if I check this, and as well that Sz on 2 is equal minus h-bar over 2, 2-- which can also be checked-- I need to check no more. Because it suffices that this operator does what it's supposed to do of the basis vectors. And it will do what it's supposed to do on arbitrary vectors. So we're done. This is the operator Sx, and we seem to have put together a lot of the ideas of the experiment into a mathematical framework. But we're not through because we have this question, so what if you align and operate the machine along x? What are the possible spin states along the x-direction? How do you know that all that the spins state that points along x can be described in this vector space? How do I know there exists a number C1, C2 so that this linear combination is a spin state that points along x. Well, at this moment, you really have to invent something. And the process of invention is never a very linear one. You use analogies-- you use whatever you can-- to invent what you need. So, given that that's a possibility, we could follow what Feynman does in his Feynman lectures, of discussing how to begin rotating Stern-Gerlach machines, and doing all kinds of things. It's an interesting argument, and it's a little hard to follow, a little tedious at points. And we're going to follow a different route. I'm going to assume that you remember a little about angular momentum, and I think you do remember this much. I want to say, well, this is spin angular momentum. Well, let's compare it with orbital angular momentum, and see where we are. You see, another way of asking the question would be, well, what are the operators Sx and Sy. Where do I get them? Well, the reason I want to bring in the angular momentum is because there you have Lz, but you also have Lx and Ly. So angular momentum had Lz, just like we had here, but also Lx and Ly. Now these spin things look a lot more mysterious, a lot more basic, because, like Lz, it was xpy minus ypx. So you knew how this operator acts on wave functions. You know, it multiplies by y, takes an x derivative, or it's a dd phi. It has a nice thing, but Sz on the other hand, there's no x, there's no derivatives. It's a different space. It's working in a totally different space, in the space of a two-dimensional complex vector space of column vectors with two numbers. That's where it acts. I'm sorry there's no dd x, nothing familiar about it. But that's what we have been handed. So this thing acts on wave functions, and thus natural things. Well, the other one acts on column vectors. Two-by-two-- two component column vectors, and that's all right. But we also know that Lz is Hermitian. And that was good, because it actually meant that this is good observable. You can measure it. Is Sz Hermitian? Well, yes it is. Hermeticity of a matrix-- as we'll discuss it in a lot of detail, maybe more than you want-- means you can transpose it complex conjugated, and you get the same matrix. Well that matrix is Hermitian. So that's nice. That maybe is important. So what other operators do we have? Lx and Ly. And if we think of Lx as L1, Ly as L2, and Lz as L3, you had a basic computation relation. Li with Lj was equal to i epsilon ijk Lk-hat-- oops i-hbar. And this was called the algebra of angular momentum. These three operators satisfy these identities. i and j are here, k is supposed to be summed over-- repeated in this is our sum from 1, 2, and 3. And epsilon ijk is totally anti-symmetric with epsilon 1, 2, 3 equal to plus 1. You may or may not know this epsilon. You will get some practice on that very soon. Now for all intents and purposes, we might as well write the explicit formulas between Lx, Ly equal i-hbar Lz. Ly Lz equals i-hbar Lx. And Lz Lx-- there are hats all over-- equal i-hbar Ly. So we had this for orbital angular momentum, or for angular momentum in general. So what we're going to do now is we're going to try to figure out what are Sx and Sy by trying to find a complete analogy. We're going to declare that S is going to be angular momentum. So we're going to want that Sx with Sy will be i-hbar Sz. Sy with Sz will be i-hbar Sx. And finally, Sz with Sx is i-hbar Sy. And we're going to try that these things be Hermitian. Sx and Sy. So let me break for a second and ask if there are questions. We're aiming to complete the theory by taking S to be angular momentum, and see what we get. Can we invent operators Sx and Sy that will do the right thing? Yes. AUDIENCE: What's the name for the epsilon ijk? I know there's a special name. Levi-Civita? Levi-Civita. Yeah. PROFESSOR: What's the name? AUDIENCE: Levi-Civita tensor. PROFESSOR: That's right. Levi-Civita tensor. It can be used for cross products. It's very useful for cross products. It's a really useful tensor. Other questions. More questions about what we're going to try to do, or this so far. Yes. AUDIENCE: When you use the term representation, is that like the technical mathematical term of representation, like in algebra? PROFESSOR: Yes. It's representation of operators in vector spaces. So we've used the canonical vector space with column vectors represented by entries one and numbers. And then the operators become matrices, so whenever an operator is viewed as a matrix, we think of it as a representation. Other questions. Yes. AUDIENCE: Will we talk about later why we can make an analogy between L and S? Or is it [INAUDIBLE]? PROFESSOR: Well you see, this is a very strong analogy, but there will be big differences from orbital angular momentum and spin angular momentum. And basically having to do with the fact that the eigenvalues of these operators are plus minus h-bar over 2. And in the orbital case they tend to be plus minus integer values of h-bar. So this is a very deep statement about the algebra of these operators that still allows the physics of them to be quite different. But this is probably the only algebra that makes sense. It's angular momentum. So we're going to try to develop that algebra like that, as well here. You could take it to be an assumption. And as I said, an experiment doesn't tell you the unique way to invent the mathematics. You try to invent the consistent mathematics and see if it coincides with the experiment. And this is a very natural thing to try to invent So what are we facing? We're facing a slightly nontrivial problem of figuring out these operators. And they should be Hermitian. So let's try to think of Hermitian two-by-two matrices. So here is a Hermitian two-by-two matrix. I can put an arbitrary constant here because it should be invariant on their transposition, which doesn't change this diagonal value in complex conjugation. So c should be real. d should be real. For the matrix to be Hermitian, two-by-two matrix, I could put an a here. And then this a would have to appear here as well. I can put minus ib, and then I would have plus ib here. So when I transpose a complex conjugate, I get this one. So this matrix with abc and d real is Hermitian. Hermiticity is some sort of reality condition. Now, for convenience, I would put a 2c and a 2d here. It doesn't change things too much. Now to look at what we're talking about. We're talking about this set of Hermitian matrices. Funnily, you can think of that again as a vector space. Why a vector space? Well, we'll think about it, and in a few seconds, it will become clear. But let me just try to do something here that might help us. We're trying to identify Sx and Sy from here so that this commutation relations hold. Well, if Sx and Sy have anything to do with the identity matrix, they would commute with everything and would do nothing for you. So, I will remove from this matrices then trying to understand something having to do with the identity. So I'll remove a Hermitian matrix, which is c plus d times the identity-- the two-by-two identity matrix. This is a Hermitian matrix, as well. And I can remove it, and then this matrix is still Hermitian, and this piece that I've removed doesn't change commutators as they appear on the left hand side. So if you have an Sx and an xy here, and you're trying to do a computation, it would not contribute, so you might as well just get rid of them. So if we remove this, we are left with-- you're subtracting c plus d from the diagonal. So here you'll have c minus d. Here you'll get b minus c, a minus ib, and a plus ib. And we should keep searching for Sx and Sy among these matrices. But then you say, look, I already got Sz, and that was Hermitian. And Sz was Hermitian, and it had a number, and the opposite number on the other diagonal entry. If Sx and Sy have a little bit of Sz, I don't care. I don't want these to be independent matrices. I don't want to confuse the situation. So if this thing has something along Sz, I want it out. So since precisely this number is opposite to this one, I can add to this matrix some multiple of Sz and kill these things in the diagonal. So add the multiple and Sz multiple, and we finally get this matrix. 0, a minus ib, a plus ib, and 0. So we've made quite some progress. Let's see now what we have. Well, that matrix could be written as a times 0, 1, 1, 0 plus b times 0, minus i, i, 0. Which is to say that it's this Hermitian matrix times a real number, and this Hermitian matrix times a real number. And that makes sense because if you take a Hermitian matrix and multiply by a real number, the matrix is still Hermitian. So this is still Hermitian because these are real. This is still Hermitian because a is real, and if you add Hermitian matrices, it's still Hermitian. So in some sense, the set of Hermitian matrices, two-by-two Hermitian matrices, is a real vector space with four basis vectors. One basis vector is this, another basis vector is this, the third basis vector is the Sz part, and the fourth basis vector is the identity that we subtracted. And I'm listing the other two that we got rid of because physically we're not that interested given that we want Sx and Sz. So, Sx and Sy. But here it is. These four two-by-two matrices are sort of the linearly independent Hermitian matrices. You can think of them as vectors, four basis vectors. You multiply by real numbers, and now you add them, and you got the most general Hermitian matrix. So this is part of the subtlety of this whole idea of vector spaces of matrices, which can be thought of as vectors sometimes, as well. So that's why these matrices are quite famous. But before we just discuss why they are so famous, let's think of this. Where we're looking for Sx and Sy, and we actually seem to have two matrices here that could do the job, as two independent Hermitian two-by-two matrices. But we must add a little extra information. We don't know what the scale is. Should I multiply this by 5 and call that Sx? Or this by 3? We're missing a little more physics. What is the physics? The eigenvalues of Sx should also be plus minus h over 2. And the eigenvalues of Sy should also be plus minus h over 2. Just like for Sz. you could have started the whole Stern-Gerlach things thinking of x, and you would have obtained plus minus h over 2. So that is the physical constraint. I have to figure out those numbers. Maybe Sx is this one, as y is this one. And you can say, oh, you never told us if you're going to get the unique answer here. And yes, I did tell you, and you're not going to get a unique answer. There are some sign notations and some other things, but any answer is perfectly good. So once you get an answer, it's perfectly good. Of course, we're going to get the answer that everybody likes. And the convention is that happily that everybody uses this same convention. Questions. AUDIENCE: So I have a related question, because at the beginning we could have chosen the top right entry to be a plus ib and the bottom left to be a minus ib and that would have yielded a different basis matrix. PROFESSOR: Right, I would have called this plus and minus. Yes. AUDIENCE: Are we going to show that this is the correct form? PROFESSOR: No, it's not the correct form. It is a correct form, and it's equivalent to any other form you could find. That's what we can show. In fact, I will show that there's an obvious ambiguity here. Well, in fact, maybe I can tell it do you, I think. If you let Sx go to minus Sy, and Sy goes to plus Sx, nothing changes in these equations. They become the same equations. You know, Sx would become minus Sy, and this Sx-- this is not changed. But, in fact, if you put minus Sy and Sx as the same commutator then this one will become actually this commutator, and this one will become that. So I could change whatever I get for Sx, change it from minus Sy, for example, and get the same thing. So there are many changes you can do. The only thing we need is one answer that works. And I'm going to write, of course, the one that everybody likes. But don't worry about that. So let's think of eigenvectors and eigenvalues now. I don't know how much you remember that, but we'll just take it at this moment that you do. So 0, 1, 1, 0 has two eigenvalues, and lambda equals 1, with eigenvector 1 over square root of 2, 1, 1. And a lambda equals minus 1 with eigenvector 1 over square root of 2, 1, minus 1. The other one, it's equally easy to do. We'll discuss eigenvectors and eigenvalues later. Minus i, i, 0, 0, plus a lambda equals one eigenvector, with components 1 over square root of 2, 1, and i. I'm pretty sure it's 1 and i. Yes, and a lambda equals minus 1, with components 1 over square root 2, 1, minus i. Now I put the 1 over square root of 2 because I wanted them to be normalized. Remember how you're supposed to normalize these things. You're supposed to take the row vector, complex conjugate, and multiply. Well, you would get 1 for the length of this, 1 for the length of this. You would get one for the length of this, but remember, you have to complex conjugate, otherwise you'll get 0. Also, you will get one for the length of this. So these are our eigenvalues. So actually, with eigenvalues lambda equals 1 and minus 1 for these two, we're in pretty good shape. We could try Sx to be h-bar over 2 times 0, 1, 1, 0. And Sy to be h-bar over 2, 0, minus i, i, 0. These would have the right eigenvalues because if you multiply a matrix by a number, the eigenvalue gets multiplied by this number, so the plus minus 1s become plus minus h-bar over 2. But what are we was supposed to check? If this is to work, we're supposed to check these commutators. So let's do one, at least. Sx commutator with Sy. So what do we get? h-bar over 2, h-bar over 2-- two of them-- then the first matrix, 0, 1, 1, 0 times 0, minus i, i, 0, minus 0, minus i, i, 0 times 0, 1, 1, 0. Which is h-bar over 2 times h-bar over 2, times i, 0, 0, minus i, minus minus i, 0, 0, i. And we're almost there. What do we have? Well, we have h-bar over 2, h-bar over 2. And we've got 2i and minus 2i. So this is h-bar over 2 times h-bar over 2, times 2i times 1, minus 1. And this whole thing is i h-bar, and the other part is h-bar over 2, 1, minus 1, which is i h-bar as z-hat. Good, it works. You know, the only thing that could have gone wrong-- you could have identified 1 with a minus, or something like that It would have been equally good. Once you have these operators, we're fine. So one has to check that the other ones work, and they do. I will leave them for you to check. And therefore, we've got the three matrices. It's a very important result-- the Sx Sy, and Sz. I will not rewrite them, but they should be boxed nicely, the three of them together, with that one there on top of the blackboard. And of course by construction, they're Hermitian. They're famous enough that people have defined the following object. Si is defined to be h-bar over 2, sigma i, the power of the matrix sigmas. And these are Pauli matrices, sigma one is 0, 1, 1, 0. Sigma two is 0, minus i, i, 0. And sigma three is equal to 1, minus 1, 0, 0. OK, so in principle-- yes, question. AUDIENCE: Is it at all significant that the Pauli matrices are all squared [INAUDIBLE]?? PROFESSOR: Yes, it is significant. We'll use it, but at this moment, it's not urgent for us. We'll have no application of that property for a little while, but it will help us do a lot of the algebra of the Pauli matrices. AUDIENCE: [INAUDIBLE] eigenvalues, right? PROFESSOR: Sorry? AUDIENCE: Doesn't that follow from the eigenvalue properties that we've [INAUDIBLE] plus or minus one. Because those were both squared. [INAUDIBLE] PROFESSOR: That's right. I think so. Our eigenvalues-- yes, it's true. That the fact that the eigenvalues are plus minus 1 will imply that these matrices squared themselves. So it's incorporated into our analysis. The thing that I will say is that you don't need it in the expression of the commutators. So in the commentators, it didn't play a role to begin with. Put it as an extra condition. Now what is the next thing we really want to understand? Is that in terms of plain states, we now have the answer for most of the experiments we could do. So in particular, remember that we said that we would have Sx, for example, having states x plus minus, which are h-bar over 2 plus minus, x comma plus minus. The states along the x-direction referred like that would be the eigenstates of the Sx operator. But we've calculated the states of the Sx operator-- they're here. The Sx operator is h-bar over 2 times this matrix. And we have those things. So the plus eigenvalue and the minus eigenvalue will just show up here. So let me write them, and explain, in plain language, what these states are. So the eigenstate with lambda equal 1-- that would correspond to h-bar over two-- so the x plus corresponds to this vector. So what is that state? It's that vector which, if you want more explicitly, it's the z plus, plus z minus. This is the state 1 over square root of 2, 1, 1. The x minus is z plus, minus z minus. As you see on that blackboard, it's 1 minus 1. So here it is. The states that you were looking for, that are aligned along x-- plus x or minus x-- are not new states that have you to add to the state space. They are linear combinations of the states you've got. We can invert this formula and write, for example, that z plus is 1 over square root of 2, x plus, plus 1 over square root of 2, x minus. And z minus is 1 over square root of 2, x plus, minus 1 over square root of-- minus the square root of 2 is already out, I'm sorry-- minus x minus. So actually, this answers the question that you had. For example, you put a z plus state, and you put an x filter-- what amplitude do you have to find a state in the x plus, given that you start with a state on the z plus? Well, you put an x plus from here. You get 1 from this and 0 from this one because the states are always orthogonal. The states are orthogonal-- you should check that. And therefore, this is 1 over square root of 2. If you ask for x minus with respect to z plus, that's also 1 over square root of 2. And these are the amplitudes for this state to be found in this, for this state to be found in them. They're equal. The probabilities are 1/2. And that's good. Our whole theory of angular momentum has given us something that is perfectly consistent with the Stern-Gerlach experiment, and it gives you these probabilities. You can construct in the same way the y states. So the y states are the eigenstates of that second matrix, Sy, that we wrote on the left. So this matrix is Sy, so its eigenstates-- I'm sorry, Sy is there. Sy is there. The eigenstates are those, so immediately you translate that to say that Sy has eigenstates y plus minus, whose eigenvalues are plus minus h-bar over 2y plus minus. And y plus is equal 1 over square root of 2, z plus-- and look at the first eigenvector-- plus iz minus. And, in fact, they can put one formula for both. Here they are. So, it's kind of neat that the x1s were found by linear combinations, and they're orthogonal. Now, if you didn't have complex numbers, you could not form another linear combination of this orthogonal. But thanks to these complex numbers, you can put an i there-- there's no i in the x ones-- and the states are orthogonal, something that you should check. So again, you can invert and find the z states in terms of y, and you would conclude that the amplitudes are really the same up to signs, or maybe complex numbers, but the probabilities are identical. So we've gotten a long way. We basically have a theory that seems to describe the whole result of the Stern-Gerlach experiment, but now your theory can do more for you. Now, in the last few minutes, we're going to calculate the states that are along arbitrary directions. So here I produced a state that is along the x-direction plus, and along the x-direction minus. What I would like to construct, to finish this story, is a state that is along some arbitrary direction. So the state that points along some unit vector n. So here is space, and here's a unit vector n with components nx, , ny, and nz. Or you can write the vector n as nx ex plus ny ey plus nz ez. And I would like to understand how I can construct, in general, a spin state that could be said to be in the n direction. We have the ones along the z, x, and y, but let's try to get something more general, the most general one. So for this, we think of the triplet of operators S, which would be Sx, Sy, and Sz. Now you can, if you wish, write this as Sx-hat ex vector, plus Sy-hat ey vector, plus Sz hat ez vector. But this object, if you write it like that, is really a strange object. Think of it. It's matrices, or operators, multiplied by unit vectors. These vectors have nothing to do with the space in which the matrices act. The matrices act in an abstract, two-dimensional vector space, while these vectors are sort of for accounting purposes. That's why we sometimes don't write them, and say we have a triplet. So this product means almost nothing. They're just sitting together. You could put the e to the left of the x or to the right. It's a vector. You're not supposed to put the vector inside the matrix, either. They don't talk to each. It's an accounting procedure. It is useful sometimes; we will use it to derive identities soon, but it's an accounting procedure. So here's what I want to define. So this is a crazy thing, some sort of vector valued operator, or something like that. But what we really need is what we'll call S-hat n, which will be defined as n dot S. Where we take naively what a dot product is supposed to mean. This component times this component, which happens to be an operator. This times this, this times that. nx Sx plus ny Sy, plus nz Sz. And this thing is something very intuitive. It is just an operator. It doesn't have anymore a vector with it. So it's a single operator. If your vector points in the z-direction, nx and ny z, and you have Sz because it's a unit vector. If the vector points in the x-direction, you get Sx. If the vector points in the y-direction, you get Sy. In general, this we call the spin operator in the direction of the vector n-- spin operator in the direction of n. OK, so what about that spin operator? Well, it had eigenvalues plus minus h-bar over 2 along z, x, and y-- probably does still have those eigenvalues-- but we have to make this a little clearer. So for that we'll take nx and ny and nz to be the polar coordinate things. So this vector is going to have a theta here on the azimuthal angle phi over here. So nz is cosine theta. nx and ny have sine theta. And nx cosine phi, and this one has sine phi. So what is the operator Sn vector hat? Well, it's nx times Sx. So, I'll put a h-bar over 2 in front, so we'll have nx sigma x, or sigma1, plus ny sigma2, , plus nz sigma3. Remember the spin operators are proportional h-bar over 2 times the sigmas-- so sigma1, sigma2, sigma3. And look what we get. h-bar over 2. Sigma1 has an nx here, nx. Sigma2 has minus iny plus iny. And sigma3, we have a nz minus nz. So this is h-bar over 2, nz is cosine theta, nx minus iny-- you'd say, oh it's a pretty awful thing, but it's very simple-- nx minus iny is sine theta times e to the minus i phi. Here it would be sine theta, e to the i phi, and here we'll have minus cosine theta. So this is the whole matrix, Sn-hat, like that. Well, in the last couple of minutes, let's calculate the eigenvectors and eigenvalues. So what do we get? Well, for the eigenvalues, remember what is the computation of an eigenvalue of a matrix. An eigenvalue for matrix a, you write that by solving the determinant of a minus lambda 1 equals 0. So for any matrix a, if we want to find the eigenvalues of this matrix, we would have to write eigenvalues of Sn-hat. We have to ride the determinant of this, minus lambda i, so the determinant of h-bar over 2 cosine theta, minus lambda, minus h-bar over 2 cosine theta, minus lambda. And here, it's sine theta, e to the minus i phi, sine theta e to the i phi, the determinant of this being 0. It's not as bad as it looks. It's actually pretty simple. These are a plus b, a minus b. Here the phases cancel out. The algebra you can read in the notes, but you do get lambda equals plus minus h-bar over 2. Now that is fine, and we now want the eigenvectors. Those are more non-trivial, so they need a little more work. So what are you supposed to do to find an eigenvector? You're supposed to take this a minus lambda i, acting on a vector, and put it equal to zero. And that's the eigenvector. So, for this case, we're going to try to find the eigenvector n plus. So this is the one that has Sn on this state-- well, I'll write it here, plus minus h over 2, n plus minus here. So let's try to find this one that corresponds to the eigenvalue equal to plus h-bar over 2. Now this state is C1 times z plus, plus C2 times z minus. These are our basis states, so it's a little combination. Or it's C1, C2. Think of it as a matrix. So we want the eigenvalues of that-- the eigenvector for that-- so what do we have? Well, we would have Sn-hat minus h-bar over 2 times 1, on this C1, C2 equals 0. The eigenvector equation is that this operator minus the eigenvalue must give you that. So the h-bars over 2, happily, go out, and you don't really need to worry about them anymore. And you get here cosine theta minus 1, sine theta e to the minus i phi, sine theta e to the i phi, and minus cosine theta minus 1, C1, C2 equals 0. All right, so you have two equations, and both relate C1 and C2. Happily, and the reason this works is because with this eigenvalue that we've used that appears here, these two equations are the same. So you can take either one, and they must imply the same relation between C1 and C2. Something you can check. So let me write one of them. C2 is equal to e to the i phi, 1 minus cosine theta over sine theta C1. It's from the first line. So you have to remember, in order to simplify these things, your half angle identities. Sorry. 1 minus cosine theta is 2 sine squared theta over 2, and sine theta is 2 sine theta over 2 cosine theta over 2. So this becomes e to the i phi sine theta over 2, over cosine theta over 2, C1. Now we want these things to be well normalized, so we want C1 squared plus C2 squared equal to 1. So, you know what C2 is, so this gives you C1 squared times 1 plus-- and C2 you use this, when you square the phase goes away-- sine squared theta over 2, cosine squared over 2 must be equal to 1. Well, the numerator is 1, so you learn that C1 squared is equal to cosine squared theta over 2. Now you have to take the square root, and you could put an i or a phase or something. But look, whatever phase you choose, you could choose C1 to be cosine theta over 2, and say, I'm done. I want this one. Somebody would say, no let's put the phase, e so to the i pi over 5. So that doesn't look good, but four or even worse, this phase will show up in C2 because C2 is proportional to C1. So I can get rid of it. I only should put it if I really need it, and I don't think I need it, so I won't put it. And you can always change your mind later-- nobody's going to take your word for this. So, in this case, C2 would be sine theta over 2, e to the i phi. It's nice, but it's [INAUDIBLE]. And therefore, we got this state n plus, which is supposed to be cosine theta over 2, z plus, and plus sine theta over 2, e to the i phi, z minus. This is a great result. It gives the arbitrarily located spin state that point in the n-direction. As a linear superposition of your two basis states, it answers conclusively the question that any spin state in your system can be represented in this two-dimensional vector space. Now moreover, if I take that theta equals 0, I have the z-axis, and it's independent of the angle phi. The phi angle becomes singular at the North Pole, but that's all right. When theta is equal to 0, this term is 0 anyway. And therefore, this goes, and when theta is equal to 0, you recover the plus state. Now you can calculate the minus state. And if you follow exactly the same economical procedure, you will get the following answer. And I think, unless you've done a lot of eigenvalue calculations, this is a calculation you should just redo. So the thing that you get, without thinking much, is that n minus is equal to sine theta over 2 plus, minus cosine theta over 2, e to the i phi minus. At least some way of solving this equation gives you that. You could say, this is natural, and this is fine. But that is not so nice, actually. Take theta equal to pi-- no, I'm sorry. Again, you take theta equal to 0. Theta equal to 0-- this is supposed to be the minus state along the direction. So this is supposed to give you the minus state. Because the vector n is along up, in the z-direction, but you're looking at the minus component. So theta equals 0. Sure, there's no plus, but theta equals 0, and you get the minus state. And this is 1, and phi is ill-defined-- it's not so nice, therefore-- so, at this moment, it's convenient to multiply this state by e to the minus i phi, times minus 1. Just multiply it by that, so that n minus is equal to minus sine theta over 2, e to the minus i phi, plus, plus cosine theta over 2, minus. And that's a nice definition of the state. When theta is equal to 0, you're fine, and it's more naturally equivalent to what you know. Theta equal to 0 gives you the minus state, or z minus. I didn't put the zs here, for laziness. And for theta equal to 0, the way the phase phi doesn't matter. So it's a little nicer. You could work with this one, but you might this well leave it like that. So we have our general states, we've done everything here that required some linear algebra without doing a review of linear algebra, but that's what we'll start to do next time.

MIT_805_Quantum_Physics_II_Fall_2013

21_Angular_Momentum_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. Today we'll be talking a little about angular momentum. Continuing the discussion of those vector operators and their identities that we had last time. So it will allow us to make quite a bit of progress with those operators, and understand them better. Then we'll go through the algebraic analysis of the spectrum. This is something that probably you've seen in some way or another, perhaps in not so much detail. But you're probably somewhat familiar, but it's good to see it again. And finally at the end we'll discuss an application that is related to your last problem in the homework. And it's a rather mysterious thing that I think one should appreciate how unusual the result is, related to the two dimensional harmonic oscillator. So I'll begin by reminding you of a few things. We have L, which is r cross p. And we managed to prove last time that that was equal to p cross r, with a minus sign. And then part of the problem's that you're solving with angular momentum use the concept of a vector and the rotations. So if u is a vector under rotations-- to say that something is a vector under rotations means the following, means that if you compute Li commutator with uj, you can put a hat. All these things are operators, all these vectors. So maybe I won't put a hat here on the blackboard. Then you're supposed to get i, epsilon, ijk, ih bar. Epsilon, ijk, uk. So that's a definition if you wish. Any object that does that is a vector under rotations. And something that in the homework you can verify is that r and p are vectors under rotation. That is, if you put here xj, you get this thing with xk. If you put here pj, you get this thing with pk. If you compute the commutator. So r and p are vectors under rotation. Then comes that little theorem, that is awfully important, that shows that if u and v are vectors under rotations-- u and v vectors under rotations-- then u dot v is a scalar. And u cross v is a vector. And in both cases, under rotations. So this is something you must prove, because if you know how u and v commute with the angular momentum, you know how u times v, in either the dot combination or the cross combination, commute with j, with L. So to say that something is a scalar, the translation is that Li with u dot v will be 0. You don't have to calculate it again. If you've shown that u and v are vectors, that they transform like that, this commutes with this. So r-- so what do you conclude from this? That Li commutes with r squared, commutes-- it's equal to p squared. And it's equal to Li, r dot p. They all are 0. Because r and p are vectors under rotation, so you don't have to compute those ones anymore. Li will commute with r squared, with p squared r cross p. And also, the fact that u cross v is a vector means that Li commutated with u cross v, j-- the j component of u cross v is ih bar, u cross v. I'm sorry-- epsilon, ijk, u cross v, k. Which is to say that u cross v is a vector under rotations. This has a lot of important corollaries. The most important perhaps is the commutation of angular momentum with itself. That is since you've shown that r and p satisfy this, r cross p, which is angular momentum, is also a vector under rotation. So here choosing u equal r, and v equal p, you get that Li, Lj is equal to ih bar, epsilon, ijk, Lk. And it's the end of the story. You got this commutation. The commutation you wanted. In earlier courses, you probably found that this was a fairly complicated calculation. Which you had to put the x's and the p's, the x's and the p's, and start moving them. And it takes quite a while to do it. So, that's important. Another property that follows from all of this, which is sort of interesting, that since L is now also a vector under rotations, Li commutes with L squared. Because l squared is L dot L, therefore it's a scalar. So Li commutes with L squared. And that property is absolutely crucial. It's important that it's worth checking that in fact, it follows just from this algebra. You see, the only thing you need to know to compute the commutator of Li with L squared is how L's commute. Therefore it should be possible to calculate this based on this algebra. So this property is true just because of this algebra, not because of anything we've said before. And that's important to realize it. Because you have algebra like si, sj, ih bar, epsilon, ijk, sk, which was the algebra of spin angular momentum. And we claim that for that same reason that this algebra leads to this result, that si should commute with s squared. And you may remember that in the particular case we examined in this course, s squared-- that would be sx squared plus sy squared plus sz squared-- was in fact h bar over 2 squared. And each matrix was proportional to the identity. So there's a 3 in the identity matrix. And s squared is really in the way we represent that spin, by 2 by 2 matrices, commutes with si. Because it is the identity. So it's no accident that this thing is 0. Because this algebra, whatever l is, implies that this with the thing squared is equal to zero. So whenever we'll be talking about spin angular momentum, orbital angular momentum, total angular momentum, when we add them, there's all kinds of angular momentum. And our another generic name for angular momentum will be j. And we'll say that ji, jj, equal ih bar, epsilon, ijk, jk is the algebra of angular momentum. And by using j, you're sending the signal that you may be talking about l. Or may be talking about s, but it's not obvious which you're talking about. And you're focusing on those properties of angular momentum that hold just because this algebra is supposed to be true. So in this algebra, you will have that ji commutes with j squared. And what is j squared? Of course, j squared is j1 squared plus j2 squared plus j3 squared. Now this is so important, and this derivation is a little bit indirect, that I encourage you all to just do it. Without using any formula, put the jx here, and compute this commutator. And it takes a couple of lines, but just convince yourself that this is true. OK, now we did have a little more discussion. And these are all things that are basically related to what you've been doing in the homework. Another fact is that this algebra is translated into j cross j equal ih bar, j. Another result in transcription of equations is that the statement that u is a vector under rotations corresponds to a vector identity. Just the fact that the algebra here is this, the fact that l with u is this, implies the following algebra. j cross u plus u cross j equal 2i h bar. So this is for a vector under rotations. Under rotations. So this I think is in the notes. It's basically saying that if you want to translate this equation into vector form, which is a nice thing to have, it reads like this. And the way to do that is to just calculate the left hand side. Put and index, i. And just try to get the right hand side. It will work out. OK. Any questions so far with these identities? OK. So we move on to another identity that you've been working on, based on the calculation of what is a cross b dot a cross b. If these things are operators, there's corrections to the classical formula for the answer of of what this product is supposed to be. Actually, the classical formula, so it's not equal to a squared, b squared, minus a dot b squared. But it's actually equal to this, plus dot dot dot. A few more things. Classically it's just that. You put 2 epsilons. Calculate the left hand side. And it's just these 2 terms. Since there are more terms, let's look what they are for a particular case of interest. So our case of interest is L squared, that corresponds to r cross b, times r cross b. And indeed, it's not just r squared, p squared, minus r dot p squared. But there's a little extra. And perhaps you have computed that little extra by now. It's ih bar r dot p. So that's a pretty useful result. And from here, we typically look for what is p squared. So for p squared-- so what we do is pass these other terms to the other side. And therefore we have 1 over r squared, r dot p squared, minus ih bar, r dot p. Yes. Plus 1 over r squared, l squared. And, we've done this with some prudence. The r squared is here in front of the p squared. It may be fairly different from having it to the other side. And therefore, when I apply the inverse 1 over r squared, I apply it from the left. So I write it like that. And that's very different for having the r squared on the other side. Could be completely different. Now, what is this? Well this is a simple computation, when you remember that p vector is h bar over i gradient. And r dot p, therefore is h bar over i, r, dvr. Because r vector is r magnitude times the unit vector in the radial direction. And the radial direction of gradient is dvr. So this can be simplified. I will not do it because it's in the notes. And you get minus h squared, 1 over r, d second, d r squared, r. In a funny notation, the r is on the right. And the 1 over r is on the left. And you would say, this doesn't sound right. You have here all this derivatives and what is an r doing to the right of the derivatives. I see no r. But this is a kind of a trick to rewrite everything in a short way. So if you want, think of this being acting on some function of r. And see what it is. And then you put a function of r here, and calculate it. And you will see, you get the same. So it's a good thing to try that. So p squared is given by this. There's another formula for p squared. p squared is, of course, the Laplacian. So p squared is also equal to minus h squared times the Laplacian operator. And that's equal to minus h squared times-- in fact, the Laplacian operator is 1 over r, d second, dr squared, r, plus 1 over r squared, 1 over sine theta, dd theta, sine theta, dd theta. It's a little bit messy. Plus 1 over sine squared theta, d second, d phi squared, times closing this. So a few things are there to learn. And the first thing is if you compare these 2 expressions, you have a formula for l squared. You have l squared is 1 over r squared on the upper right. And here you have minus h squared times this thing. So l squared, that scalar operator is minus h squared, 1 over sine theta, dd theta, sine theta, dd theta, plus 1 over sine squared theta, d second, d phi squared. So in terms of functions of 3 variables, x, y, and z, L squared, which is a very complicated object, has become just a function of the angular variables. And that this a very important intuitive fact. L squared. L is operator. That's rotation. So it shouldn't really affect the r, shouldn't change r, modify r in any way. So it's a nice thing to confirm here that this operator can be thought as an operator acting on the angular variables. Or you could say, on functions, on the units here for example. It's a good thing. The other thing that you've learned here-- so this is a very nice result. It's not all that easy to get by direct computation. If you had to do Lx squared plus Ly squared plus Lz squared, first all this possible order-- well, there's no ordering problems here. But you would have to write this in terms of x, and py, and pz, and xy, and z, then pass to angular variables. Simplify all that. It's a very bad way to do it. And it's painful. So the fact that we got this like that is very nice. The other thing that we've got is some understanding of the Hamiltonian for a central potential, what we call a central potential problem. v of r. Now, I will write a v of r like this. But then we'll simplify it. In fact, let me just go to a central potential case, which means that the potential just depends on the magnitude of r. So r is the magnitude of the vector r. So at this moment, you have p squared over there. So this whole Hamiltonian is minus h squared over 2m, 1 over r, d second, dr squared, r, plus p squared over 2m. So 1 over 2m, r squared, l squared plus v of r. So our Hamiltonian has also been simplified. So this will be the starting point for writing the Schrodinger equation for central potentials. And you have the operator l squared. And as far as we can, we'll try to avoid computations in theta and phi very explicitly, but try to do things algebraically. So at this moment, the last comment I want to make on this subject is the issue of set of commuting observables. So if you have a Hamiltonian like that, you can try to form a set of commuting observables that are going to help you understand the physics of your particular problem. So the first thing that you would want to put in the list of complete set of observables is the Hamiltonian. We really want to know the energies of this thing. So what other operators do I have? Well I have x1, x2, and x3. And well, can I add them to the Hamiltonian to have a complete set of commuting observables? Well, the x's commute among themselves. So can I add them? Yes or no? No. No you can't add them, because the x's don't commute with the Hamiltonian. There's a p here. p doesn't commute with x's. So that's out of the question. They cannot be added to our list. How about the p's? p1, p2, and p3. Not good either, because they don't commune with the potential term. The potential has x dependents, and will take a miracle for it to commute. In general, it won't commute. So no reason for it to commute, unless the potential is 0. So this is not good. Nor is good to have r squared, or p squared, or r dot p. r squared, p squared, r dot p. No good either. On the other hand, r cross p is interesting. You have the angular momentum, L1, L2, and L3. Well, the angular momentum will commute, I think, with the Hamiltonian. You can see it here. You have p squared, and Li's commute with p squared because p is a vector under rotations. p doesn't communicate with Li, but p squared does. Because that was a scalar. So this term commutes with any angular momentum operator. Moreover, v or r, r is this. So a v of r is a function of r squared. And r squared is the vector r squared. So ultimately, anything that is a function of r is a function of r squared that involves the operator r squared, that also commutes with all the Li's. So h commutes with all the Li's. And that's a great thing. So this is absolutely important. h commutes with all the Li's. That's angular momentum conservation. As we've seen, the rate of change of any operator is equal to expectation value of the commutator of the operator with the Hamiltonian. So if you put any Li, this commutator is 0. And the operator is conserved in the sense of expectation values. Now this conservation law is great. You could add this operators to the commuting set of observables. But this time, you have a different problem. Yes, this commutes with h. This commutes with h. And this commutes with h. But these one's don't commute with each other. So not quite good enough. You cannot add them all. So let's see how many can we add. We can only add 1. Because once you have 2 of them, they don't commute. So you're going to add 1, and everybody has agreed to add L3. So we have H, L3. And happily we have 1 more is L squared. Remember, L squared commutes with all the Li's, so that's another operator. And for a central potential problem, this will be sufficient to label all of our states some. AUDIENCE: So how do we know that we need the L squared? How do we know that we can't get-- how do we know that just H and L3 isn't already a complete set? PROFESSOR: I probably wouldn't know now, but in a little bit, as we calculate the kind of states that we get with angular momentum, I will see that there are many states with the same value of L3 that don't correspond to the same value of the total or length of the angular momentum. So it's almost like saying that there are angular momenta-- here is-- let me draw a plane. Here is z component of angular momentum, Lz. And here you got it. You can have an angular momentum that is like that, and has this Lz. Or you can have an angular momentum that is like this, L prime, that has the same Lz. And then it will be difficult to tell these 2 states apart. And they will correspond to states of this angular momentum, or this angular momentum, have the same Lz. Now drawing these arrows is extraordinarily misleading. Hope you don't get upset that I did it. It's misleading because this vector you cannot measure simultaneously the 3 components. Because they don't commute. So what do I mean by drawing an arrow? Nevertheless, the intuition is sort of there. And it's not wrong, the intuition. It will happen to be the case that states that have same amount of Lz will not be distinguished. But by the time we have this, we will distinguish them. And that's also a peculiarity of a result but we'll use. Even though we're talking about 3 dimensions, the fact that the 1 dimensional Schrodinger equation has non degenerate bound states. You say, what does that have to do with 3 dimensions? What will happen is that the 3 dimensional Schrodinger equation will reduce to a 1 dimensional radial equation. And the fact that that doesn't have degeneracies tells you that for bound state problems, this will be enough to do it. So you will have to wait a little to be sure that this will do it. But this is pretty much the best we can do now. And I don't think you will be able to add anything else to this at this stage. Now there's of course funny things that you could add like-- if there's spin, the particles have spin, well we can add spin and things like that. But let's leave it at that and now begin really our calculation, algebraic calculation, of the angular momentum representations. So at this moment, we really want to make sure we work with this. Only this formula over here. And learn things about the kind of states that can exist in a system in which there are operators like that. So it's a funny thing. You're talking about a vector space. And in fact, you don't know almost anything about this vector space so far. But there is an action of those operators. From that fact alone, and one more important fact-- the j's are Hermitian. From these 2 facts, we're going to derive incredibly powerful results, extremely powerful things. And as we'll see, they have applications even in cases that you would imagine they have nothing to do with angular momentum, which is really surprising. So how do we proceed with this stuff? Well, there's a hermeticity. And you immediately introduce things called J plus minus, which are J1 plus minus i J 2. Or Jx plus minus y Jc. Then you calculate what is J plus J minus. Well J plus J minus will be a J1 squared plus J2 squared. And then you have the cross product that this doesn't cancel. So J plus times J minus would be J1 plus i J2, J1 minus i J2. So the next term would be minus i, J1, J2. And that's i h bar, J3. So this is J1 squared plus J2 squared plus h bar J3. So that's a nice formula for J plus, J minus. J minus, J plus would be J1 squared plus J2 squared minus h bar J3. These 2 formulas are summarized by J plus, J minus-- minus, plus-- is equal to J1 squared plus J2 squared plus minus h bar J3. OK. Things to learn from this. Maybe I'll continue here for a little while to use the blackboards, up to here only. The commutator of J plus and J minus can be obtained from this equation. You just subtract them. And that's 2h bar, J3. And finally, one last thing that we like to know is how to write J squared. So J squared is J1 squared plus J2 squared plus J3 squared, which then show up here. So we might as well add it and subtract it. So I add a J3 squared, and I add it on the left hand side. And pass this term to the other side. So J squared would be J plus, J minus, plus J3 squared, minus h bar, J3. Or J minus, J plus, plus J3 squared, plus h bar, J3. OK. So that's J squared. OK. So we're doing sort of simple things. Basically at this moment, we decided that we like better J plus and J minus. And we tried to figure out everything that we should know about J plus, J minus. If we substitute Lx, and Jx, and Jy for J plus and J minus, you better know what is the commutator of J plus and J minus. And how to write J squared in terms of J plus and J minus. And this is what we've done here. And in particular, we have a whole lot of nice formulas. So one more formula is probably useful. And it's the formula for the commutator of J plus and J minus with Jz. Because after all, the J plus, J minus commutator, you've got it. So if you're systematic about these things you should figure out that at this I would like to know what is the commutator of J plus and J minus with Jz. So I can do Jz, J plus. It's not hard. It's Jz. I'm sorry. I'm calling it 3. So, I think in the notes I call them x, y, and z. But never mind. J1 plus i, J2. The plus is really with a plus i. So J3 with J1 by the cyclic ordering is ih bar, J2. And here you have plus i, and J3 with J2 is minus ih bar, J1. So this is h bar, J1, plus i, J2, which is h bar, J plus. So what you've learned is that J3 with J plus is equal to h bar, J plus. And if you did it with J minus, you'll find a minus, and a plus minus here. So that is the complete result. And that should remind you of the analogous relation in which you have in the harmonic oscillator, N commutator, with a dagger. With a dagger. And N commutator with a was minus a. Because of the fact that I maybe didn't say it here, and I should have, that the dagger of J plus is J minus. Because the operators are Hermitians. So J plus and J minus are daggers of each other, are adjoins of each other. And here you see a very analogous situation. a and a dagger were adjoins of each other. And with respect to N, a counting number operator. One increased it. One decreased it. a dagger increased the number eigenvalue of N. a decreased it, the same way it's going to happen here. J plus is going to increase the C component of angular momentum. And J minus is going to decrease it. OK. So we've done most of the calculations that we need. The rest is pretty easy work. Not that it was difficult so far. But it took a little time. So what happens next is the following. You must make a declaration. There should exist states, basically. We have a vector space. It's very large. It's actually infinite dimensional. Because they will be related to all kinds of functions on the unit sphere. All these angular variables. So it's infinite dimensional. So it's a little scary. But let's not worry about that. Something very nice happens with angular momentum. Something so nice that it didn't happen actually with a and a dagger. With a and a dagger, you build states in the harmonic oscillator. And you build infinitely many ones. The operators x and p, you've learned you cannot represent them by finite dimensional matrices. So this is a lot more complicated, you would say. And you would say, well, this is just much harder. This algebra is so much harder than this algebra. Nevertheless, this algebra is the difficult one. Gives you infinite dimensional representations. You can keep piling the a daggers. Here, this is a very dense algebra. Mathematicians would say this is much simpler than this one. And we'll see the simplicity of this one, in that you will manage to get representations and matrices that are finite dimensional to work these things out. So it's going to be nicer in that sense. So what do we have? We have to think of our commuting observables and the set of Hermitian operators that commute. So we have J squared, and J3-- I call it Jz now, apologies. And we'll declare that there are states. These are Hermitian, and they commute. So they must be diagonalized simultaneously. And there should exist states that represent the diagonalization. In fact, since they commute, and can be diagonalized simultaneously, the vector space must break into a list of vectors. All of them eigenstates of these 2 operators. And all of them orthogonal to each other. Matthew, you had a question? AUDIENCE: I was just wondering when we showed that Jz is Hermitian? PROFESSOR: We didn't show it. We postulated that J's are Hermitian operators. So you know that when J is L, yes it's Hermitian. You know when J is spin, yes it's Hermitian. Whatever you're doing we'll use Hermitian operators. So not only they can diagonalize simultaneously, by our main theorem about Hermitian operators, this should provide an orthonormal basis for the full vector space. So the whole answer is supposed to be here. Let's see. So I'll define states, Jm, that are eigenstates of both of these things. And I have 2 numbers to declare those eigenvalues. You would say J squared. Now, any normal person would put here maybe h squared, for units, time J squared. And then Jm. Don't copy it yet. And Jz for Jm. It has units of angular momentum. So an h, times m, times Jm. But that turns out not to be very convenient to put the J squared there. It ruins the algebra later. So we'll put something different that we hope has the same effect. And I will discuss that. I'll put h squared, J times J plus 1. It's a funny way of declaring how you're going to build the states. But it's a possible thing to do. So here are the states, J and m. And the only thing I know at this moment is that since these are Hermitian operators, their eigenvalues must be real. So J times J plus 1 is real. And m is real. So J and m belong to the reals. And they are orthogonal to-- we can say they're orthonormal states. We will see very soon that these things get quantized. But basically, the overlap of a Jm with a J prime, m prime would be 0 whenever the J's and the m's are different. As you know from our theory, any 2 eigenstates with different eigenvalues are orthonormal. And in fact, you can choose a basis so that in fact, everything is orthonormal. So there's no question like that. So let's explain a little what's happening with this thing. Why do we put this like that? Or why can we get away with this? And the reason is the following. Let's consider Jm, J squared, Jm. If I use this, J squared on this is this number. And Jm with itself will be 1. And therefore I'll put here h-- I'm sorry. This should be an h squared. J has units of angular momentum. h squared, J times J plus 1. And I'm assuming that this will be discretized so I don't have to put the delta function normalization. At any rate, this thing is equal to this. And moreover, it's equal to the following. Jm sum over i, Jm, Ji, Ji, Jm. But since J is Hermitian, this is nothing but the sum over i of the norm squared of Ji with J acting on Jm. The norm squared of this state. Because this times the bra with Ji Hermitian is the norm squared. So this is greater or equal than 0. Perhaps no surprise, this is a vector operator, which is the sum of squares of Hermitian operators. And therefore it should be like that. Now, given that, we have the following-- oops-- the following fact that L times L plus-- no. J times J plus 1 must be greater or equal than 0. J times J plus 1 must be greater or equal than 0. Well, plot it as a function of J. It vanishes at 0. J times J plus 1 vanishes at 0, and vanishes at minus 1. It's a function like this. The function J times J plus 1. And this shows that all you need is this thing to be positive. So to represent all the states that have J times J plus 1 positive, I could label them with J's that are positive. Or J's that are smaller than minus 1. So each way, I can label uniquely those states. So if I get J times J plus 1 equals 3, it may correspond to a J of something and a J of some other thing. I will have just 1 state, so I will choose J positive. So given that J times J plus 1 is positive, I can label states with J positive, or 0. So it allows you to do this. Whatever value of this quantity that is positive corresponds to some J positive that you can put in here. A unique J positive. So this is a fine parametrization of the problem. OK. Now what's next? Next, we have to understand what the J plus operators and J minus operators do to the states. So, first thing is that J plus and J minus commute with J squared. That should not be a surprise. J1 and J2 commute. Every J commutes with J squared. So J plus and J minus commute with J squared. What this means in words is that J plus and J minus do not change the eigenvalue of J squared on a state. That is, if I would have J squared on J plus or minus on Jm-- since I can move the J squared up across the J plus, minus-- it hits here. Then I have J plus minus, J squared, Jm. And that's there for h squared, J times J plus 1, times J plus minus on Jm. So this state is also a state with the same value of J squared. Therefore, it must have the same value of J. In other words, this state J plus minus of Jm must be proportional to a state with J and maybe some different value of m, but the same value of J. J cannot have changed. J must be the same. Then we have to see who changes m, or how does J plus minus changes m. So here comes a little bit of a same calculation. You want to see what is the m value of this thing. So you have J plus minus on Jm. And you act with it with a Jz, to see what it is. And then, you put, well, the commutator first. Jz, J plus minus, plus J plus minus, Jz on the state. The commutator, you've calculated it before, was Jz with J plus minus is there, is plus minus h bar, J plus minus. And this Jz already act. So this is plus h bar m, J plus minus on Jm. So we can get the J plus minus out. And this h bar m plus minus 1, j plus minus, Jm. So look what you got. Jz acting on this state is h bar, m plus minus 1, Jm. So this state has m equal to either m plus 1, or m minus 1. Something that we can write. Clearly-- oops-- in this way, we'll say that J plus minus, Jm-- we know already it's a state with J and m plus minus 1. So it raises m. Just like what we said that the a's and a daggers raise or lower the number. J plus and J minus raise and lower Jz. Therefore, it's this is proportional to this state. But there's a constant of proportionality that we have to figure out. And we'll call it the constant C, Jm. To be calculated. So the way to calculate this constant-- and that will bring us almost pretty close to what we need-- is to take inner products. So we must take the dagger of this equation. So take the dagger, and you get Jm, the adjoin, J minus plus. And hit it with this equation. So you'll have here-- well maybe I'll write it. The dagger of this equation would be C plus minus star of Jm. Jm plus minus 1. And now, sandwich this with that. So you have Jm, J minus plus, J plus minus, Jm equals to norm of C plus minus Jm. And then you have this state times this state, but that's 1. Because it's J, J, m plus 1, m plus 1. So this is an orthonormal basis. So we have just 1. And I don't have to write more. Well the left hand side can be calculated. We have still that formula here. So let's calculate it. The left hand side, I'll write it like this. I will have C plus minus, Jm squared, which is equal to the norm squared of J plus minus, Jm. It's equal to what? Whatever this is, where you substitute that for this formula. So you'll put here Jm. And you'll have-- well, I want actually the formula I just erased. Because I actually would prefer to have J squared. So I would have this is equal to J squared, minus J3 squared, plus minus h, J3. So let's see. I have the sign minus plus, plus minus. So I should change the signs there. So it should be J squared, minus J3 squared, minus plus J3, and Jm, minus plus h bar, J3, Jm. So this is equal to h bar squared, J times J plus 1, minus an m squared, and a minus plus. So minus, plus, minus here. I think I have it here correct. Plus minus 1. And that's it. J squared is h squared this. J3 squared would give that. And the minus plus here is correctly with this one. So m should be here. Plus minus m. So this is h squared, J times J plus 1, minus m, times m plus minus 1. OK. So the C's have been already found. And you can take their square roots. In fact, we can ideally just take the square roots, because these things better be positive numbers because they're norms squared. So whenever we'll be able to do this, these things better be positive, being the square of some states. And therefore the C plus minus is-- C plus minus of Jm can be simply taken to be h bar, square root of J times J plus 1, minus m, times m plus 1. And it's because of this thing, this m times m plus 1, that it was convenient to have J times J plus 1. So that we can compare J's and m's better. Otherwise it would have been pretty disastrous. So, OK, we're almost done now with the calculation of the spectrum. You will say, well, we seem to be getting no where. Learned all these properties, these states, and now you're just manipulating the states. But the main thing is that we need these things to be positive. And that will give us the whole condition. So, for example, we need 1, that the states J plus, Jm, their norm squareds be positive. So for the plus sign-- so you should have J times J plus 1, minus m, times m plus 1 be positive. Or m times m plus 1 be smaller then J times J plus 1. The best way for my mind to solve these kind of things is to just plot them. So here is m. And here is m times m plus 1. So you plot this function. And you want it to be less than some value of J times J plus 1. So here's J times J plus 1, some value. So this is 0 here. This function is 0 at minus 1. So it will be something like this. And there's 2 values at which m becomes equal to this thing. And one is clearly J. When m is equal to J, it's saturates an inequality. And the other one is minus J, minus 1. If m is minus J, minus 1, you will have minus J, minus 1 here, and minus J here, which would be equal to this. So, in order for these states to be good, the value of m must be in between J and minus J, minus 1. Then the other case is that J plus on-- J minus on Jm. If you produce those states, they also must have positive norms. So J times J plus 1, minus m, times m minus 1 this time, must be greater than 0. So m times m minus 1 must be less than or equal then J times J plus 1. And again, we try to do it geometrically. So here it is. Here is m. And what values do you have? Well, if you plot here m times m minus 1. And that should be equal to some value that you get fixed, which is the value J times J plus 1. So you think in terms of m's, how far can they go? So if you take m equals J plus 1 that hits it. So this is 0 here, at 1, at 0. So it's some function like this. And here you have J plus 1. And here you have minus J. Both are the places for m equal J plus 1, and minus J that you get the states. You get the saturation. So you can run m from this range. Now, m can go less than or equal to J plus 1, and greater than or equal to minus J. But these 2 inequalities must hold at the same time. You cannot allow either one to go wrong for any set of states. So if both must hold at the same time for any state, because both things have to happen, you get constrained. This time for the upper range, this is the stronger value. For the lower range, this is the stronger value. So m must go between J and minus J for both to hold. Oops. To hold. Now look what happens. Funny things happen if-- this is reasonable that the strongest value comes from this equation. Because J plus increases m. So at some point you run into trouble if you increase m too much. How much can you increase it? You cannot go beyond J, and that makes sense. In some sense, your intuition should be that J is the length of J squared. And m is mz. So m should not go beyond J. And that's reasonable here. And in fact, when m is equal to J, this whole thing vanishes. So if you reach that state when m is equal to J, only then for m equal to J, or for this state, you get 0. So you cannot raise the state anymore. So actually, you see if you choose some J over here, we need a few things to happen. You choose some J, and some m. Well you're going to be shifting the m's. And if you keep adding J pluses, eventually you will go beyond this point. The only way not to go beyond this point is if m reaches the value J. Because if m reaches the value J, the state is killed. So m should reach the value J over here at some stage. So you fix J, and you try to think what m can be. And m has to reach the value J. So m at some point, whatever m is, you add 1. You add 1. You add 1. And eventually you must reach the value J. Reach with some m prime. m here. You should reach the value J, so that you don't produce another state that is higher. If you reach something before that, that state is not killed. This number is not equal to 0. You produce a state and it's a bad state of bad norm. So you must reach this one. On the other hand, you can lower things. And if you go below minus J, you produce bad states. So you must also, when you decrease m, you must reach this point. Because if you didn't, and you stop half a unit away from it, the next state that you produce is bad. And that can't be. So you must reach this one too. And that's the key logical part of the argument in which this distance 2J plus 1-- no. I'm sorry. This 2J must be equal to some integer. And that's the key thing that must happen, because you must reach this and you must reach here. And m just varies by integers. So the distance between this J and minus J must be twice an integer. And you've discovered something remarkable by getting to that point, because now you see that if this has to be an integer, well it may be 0, 1, 2, 3. And when J-- then J-- this integer is equal to 0, then J is equal to 0. 1/2, 1, 3/2. And you get all these spins with-- consider particles without spin having spin 0. Particles with spin 1/2. Particles of spin 1, or angular momentum 1, orbital angular momentum 1. And both these things have a reason for you. Now if you have 2J being an integer, the values of m go from J to J minus 1, up to minus J. And there are two J plus 1 values. And in fact, that is the main result of the theory of angular momentum. The values of the angular momentum are 0, 1, 1/2, 3/2. So for J equals 0, there's just one state. m is equal to 0. For J equals to 1, there's two states. I'm sorry for 1/2, two states. One with m equals 1/2. And m equals minus 1/2. J equals 1, there's three states. M equals 1, 0, and minus 1. And so on. OK. This is a great result. Let me give you an application in the last 10 minutes. It's a remarkable application. Now actually, you would say, so what do you get-- what vector space were we talking about? And what's sort of the punchline here is that the vector space was infinite dimensional and it breaks down into states with J equals 0. States was J equal 1/2. States with J equal 1. States with J equal 3/2. All these things are possibilities. They can all be present in your vector space. Maybe some are present. Some are not. That is part of figuring out what's going on. When we do central potentials, 0, 1, 2, 4 will be present for the angular momentum theory. When we do spins, we have 1/2. And when we do other things, we can get some funny things as well. So let's do a case where you get something funny. So the 2D, SHO. You have ax's, and ay's, and a daggers, and ay daggers. And this should seem very strange. What are we talking about 2 dimensional oscillators after talking about 3 dimensional angular momentum and all that? Doesn't make any sense. Well, what's going to happen now is something more magical than when a magician takes a bunny out of a hat. Out of this problem, an angular momentum, a 3 dimensional angular momentum, is going to pop out. No reason whatsoever there should be there at first sight. But it's there. And it's an abstract angular momentum, but it's a full angular momentum. Let's see. Let's look at the spectrum. Ground state. First excited state is isotropic. So 2 states degenerate in energy. Next state. ax dagger, ax dagger. ax, ay. ay, ay. 3 states, degenerate. Go up to ax dagger to the n, up to ax-- no ax, or ax dagger to the 0. And ay dagger to the n. And that's n a daggers up to 0 a daggers, so n plus 1 states. 3 states, 2 states, 1 state. And you'll come here and say, that's strange. 1 state, 2 states, 3 states, 4 states. Does that have anything to do with it? Well, the surprise is it has something to do with it. Let's think about it. Well, first thing is to put these aR's and aL oscillators-- these were 1/2, 1 over square root of 2, ax plus iay. And a left was 1 over square root of 2, ax minus iay. I may have-- no, the signs are wrong. Plus and minus. And we had number operators. n right, which were a right dagger, a right. And n left, which was a left dagger, a left. And they don't mix a lefts and a rights. And now, we could build a state the following way. 0. a right dagger on 0. a left dagger on 0. A right dagger squared on 0. a right, a left on 0. and a left dagger, a left dagger on 0. Up to a right dagger to the n on 0. Up to a left dagger to the n on 0. And this is completely analogous to what we had. Now here comes the real thing. You did compute the angular momentum in the z direction. And the angular momentum in the z direction was Lz. And you could compute this. xpy minus ypx. And this was all legal. And the answer was h bar, N right, minus NL. That was the Lz component of angular momentum. So, let's see what Lz's those states have. This one has no n rights, or n lefts, so has Lz equals 0. This state has Nz equal h bar. And this has minus h bar. OK. h bar and minus h bar. That doesn't quite seem to fit here, because the z component of angular momentum is 1/2 of h bar, and minus 1/2 of h bar. That's-- something went wrong. OK. You go here. You say, well, what is Lz? Lz here was h bar, minus h bar. Here is 2h bar, 0, and minus 2h bar. And you look there, and say, no, that's not quite right either. This-- if you would say these 3 states should correspond to angular momentum, they should have m equal plus 1, plus h bar, 0, and minus h bar. So it's not right. OK. Well one other thing maybe we can make sense of this. If we had L plus, should be the kind of thing that you can't annihilate. That you annihilate the top state. Remember L plus, or J plus, kept increasing so it should annihilate the top state. And I could try to devise something that annihilates the top state. And it would be something like aR dagger, a left. Why? Because if aR dagger, a left, goes to the top state, the top state has no a left daggers, so the a left just zooms in, and hits the 0 and kills it. Kills it here. So actually I do have something like an L plus. And I would have the dagger-- would be something like an L minus-- would be aL dagger, a right. And this one should annihilate the bottom one. And it does. Because the bottom state has no aR's, and therefore has no aR daggers. And therefore, the aR comes there, and hits the state, and kills it. So we seem to have more or less everything, but nothing is working. So we have to do a last conceptual step. And say-- you see, this is moving in a plane. There's no 3 dimensional angular momentum. You are fooling yourself with this. But what could exist is an abstract angular momentum. And for that, in order to-- it's time to change the letter from L to J. That means some kind of abstract angular momentum. And I'll put a 1/2 here, now a definition. If this is what I called Jz, oh well, then thing's may look good. Because this one for Jz has now angular momentum 1/2 of h bar, and minus a half of h bar. And that fits with this, these 2 states. And with the 1/2, the other ones, the Jz's, also have something here. So Jz here now becomes h bar, minus h bar, and it looks right. And now you put the 1/2 here, and in fact, if you tried to make these things J-- call it J plus and J minus. Now you put a number here, and a number here. If you would have put a number here, if you try to enforce that the algebra be the algebra of angular momentum, the number would have come out to be 1/2. But now we claim that in this 2 dimensional oscillator, there is-- because there's a number here that works with this 1/2. Something you have to calculate. And with this number, you have some sort of Jx, Jy, Jz, where this is like 1/2 of Lz. And those have come out of thin air. But they form an algebra of angular momentum. And what have we learned today, if you have an algebra of angular momentum, the states must organize themselves into representations of angular momentum. So the whole spectrum of the 2 dimensional harmonic oscillator has in fact all spin representations. J equals 0. J equals 1/2. J equals 1. J equals 2. J equals n, and all of them. So the best example of all the representations of angular momentum are in the states of the 2 dimensional simple harmonic oscillator. It's an abstract angular momentum, but it's very useful. The one step I didn't do here for you is to check. Although you check that all of these Ji commute with the Hamiltonian. Simple calculation to do it. In fact, the Hamiltonian is NL plus N right, and you can check it. Since they commute with them, these operators act in states and don't change the energy. And they're a symmetry of the problem. So that's why they fell into representations. So this is our first example of a hidden symmetry. A problem that there was no reason a priori to expect an angular momentum to exist, but it's there, and helps explain the degeneracies. These degeneracies you could have said they're accidental. But by the time you know they have to fall into angular momentum representations, you have great control over them. You couldn't have found different number of degenerate states at any level here. This was in fact discovered by Julian Schwinger in a very famous paper. And is a classic example of angular momentum. All right. That's it for today. See you on Wednesday if you come I'll be here.

MIT_805_Quantum_Physics_II_Fall_2013

1_Wave_Mechanics.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. So, we'll get started. And as I mentioned, to some degree this is going to be review on the setting of our notation and conventions clear. So, our first topic is the Schrodinger equation. So this Schrodinger equation is an equation that takes the following form. I h bar partial derivative of this object called the wave function that depends on x and t is equal to minus h squared over 2m second derivative with respect to x plus v of x and t Psi of x and t. And that's the full equation. That's the Schrodinger equation. Now actually, this is not the Schrodinger equation in most generality, but it's the Schrodinger equation for the case that you have a potential that depends on x and t. For the case that we are doing non-relativistic physics, because this thing you may remember is p squared over 2m is the kinetic energy operator. So p squared over 2m is non-relativistic. That's a non-relativistic kinetic energy. So this is non-relativistic. Moreover, we have just one x here. That means it's a particle in one dimension. So we've done a few things, but this is generally enough to illustrate our ideas. And the most important thing that should be said at this point is that Psi of x and t-- which is the wave function-- belongs to the complex numbers. It's a complex number. And that's by necessity. If Psi would be real, this quantity-- the right hand side-- would be real. The potential is a real number. On the left hand side, on the other hand, if Psi is real, its derivative would be real, and this would be imaginary. So, it's just impossible to get the solution of this equation if Psi is real. So, Psi complex is really the fundamental thing that can be said about this wave function. Now, you've used complex numbers in physics all the time, and even in electromagnetism, you use complex numbers. But you use them really in an auxiliary way only. You didn't use them in an absolutely necessary way. So, for example. In E&M, you had an electric field, for example, for a circularly polarized wave. And you would write it as this. Let me put the z here. Zero. X hat plus y hat-- those are unit vectors. I is a complex number. It's the square root of minus 1. E to the IKZ minus omega t. You typically wrote things like that, but, in fact, you always meant real part. An electric field is a real quantity. And the Maxwell's equations are real equations. This is a circularly polarized wave. And this whole thing-- by the time you take the real part of this, all these complex numbers play absolutely no role. It's just a neat way of writing a complicated electric field in which the x component and the y component are out of phase, and that you have a wave at the same time propagating in the z direction. So this-- in the here, E is real, and all i's are auxiliary. This is completely different from the case of the Schrodinger equation. This i there is fundamental. The Psi is the dynamical variable, and it has to be complex. So, we make a few remarks about the Schrodinger equation to get started. First remark is that this is first order differential equation in time. This has implications. Those two derivatives are maybe-- for some funny Hamiltonians, you can have even more than two derivatives or more complicated things. But definitely there's just one derivative in time. So, what this means is that if you know the wave function all over space, you can calculate what it's going to be a little time later. Because if you know it all over space, you can calculate this right hand side and know what is the time derivative. And with the time derivative, you can figure it out what it's going to be later. A first order differential equation in time is something that if you know the quantity at one time, the differential equation tells you what it's going to be later. So, that's really sufficient. Psi of x-- of all x's-- at some time t naught determines Psi at all times. Second property, fundamental property. The equation is linear. So, if you have two solutions, you can form a third by superimposing them, and you can superimpose them with complex coefficients. So, if you have two solutions, Psi 1 and Psi 2, then a 1 Psi 1 plus A2 Psi 2 is a solution. And here the a's belong to the complex numbers. So A 1 and A 2 are complex numbers. As far as complex numbers are concerned, the first thing you just need to know is the definition of the length of a complex number. So, if you have z, a typical name people use for a complex number, having two components. A plus ib, where a and b are real. There's the definition of the complex conjugate, which is a minus ib, and there's the definition of the length of the complex number, which is square root of a squared plus b squared, which is the square root of z times z star. So, that's for your complex number. So, the property that this makes this into a physical theory and goes beyond math is what you know is the interpretation of the wave function as a probability. So, what do we construct? We construct p of x and t, or sometimes called the row of x and t as a density. And it's defined as Psi star of x t. Now, here the notation means this Psi star-- we'd put the star here-- it really means Psi of x and t complex conjugate. You complex conjugate the wave function. And you get that. We'd put the star here, and typically don't put the parentheses, unless you have to complex conjugate something that's a little ambiguous. So, Psi star of x and t times Psi of x and t. And this is called the probability density. Probability density. And the interpretation is that if you take p of x and t and multiply by little dx, this is the probability to find the particle in the interval x comma x plus dx at time t. So, this is our probability density. It's a way to make physics out of the wave function. It's a postulate. And so the consequence of this postulate, since we're describing just one particle, is that we must have the particle as somewhere. So, if we add the probabilities that the particle is somewhere all over space, this is the probability that the particle is in this little dx we integrated that must be equal to 1. And this must hold for all times. In terms of things to notice here, maybe one thing you can notice is the units of Psi. The units of Psi must be 1 over square root of length, because when we square it, then we multiply it by length, we get one, which has no units. Key property of the Schrodinger equation. We will revisit the Schrodinger equation later and derive it, sort of the way [? De ?] [? Rack ?] derives it in his textbook. As just a consequence of unitary time evolution, it would be a very neat derivation. It will give you a feeling that you really understand something deep about quantum mechanics. And it will be true, that feeling. But here, we're going to go the other way around. Just simply ask the question-- suppose you have a wave function such that the integral of this quantity at some specific time is equal to one. Will this integral be equal to one for all times, given that it is one at some given time? Now, you say, well, why do you ask that? I ask that because actually this could be a problem. We've said that if you know the wave function all over space at one time, it's determined everywhere. So any time later. Therefore, if I know the wave function at time equal zero is good-- time equal t zero-- is a good wave function, I might warranty that when I saw the Schrodinger equation, the wave function will be normalized, well, later? Yes, you are. And it's a simple or interesting exercise that we'll call it the quick calculation that I'll leave it for you to do. Which is show that d dt of this integral Psi of x and t squared dx is equal to zero. So, basically what this is saying. You got one but, think of this integral-- I'm sorry, I'm missing a dx here-- think of this integral for all times. Now it could be a function of time, because you put an arbitrary time here. The integral might depend on time. So, it's a good question to think of that integral that may be a function of time and take its derivative. If its derivative is zero for all times, and that sometimes equal to one, it will be one forever. So, you must show that this is true. Now, this I think you've done one way or another several ways maybe in 804. But I ask you to do it again. So this is left for you as a way to warm up on this object. And you will see actually that it's a little subtle. It's a little delicate, because how is it going to go? You're going to go in and take the derivative of Psi Psi star. You're going to take the derivative of Psi and you're going to use the Schrodinger equation. You're going to take the derivative of Psi star, and you're going to use the complex conjugate of the Schrodinger equation. It's going to be a little messy. But then you're going to do integration by parts, and you're going to get zero, but only if you throw away the terms at infinity. And what gives you the right to throw them away? You will have to think. And the answer is that you will throw them away if the wave function goes to zero at infinity, which must do it. The wave function must go to zero at infinity, because if it didn't go to zero at infinity, it went to a constant at infinity, it would pick up an un-normalizable thing here. So the wave function definitely has to go to zero at infinity. But that will also not be quite enough if you're careful about what you're doing. You will have to demand that the derivative of the wave function doesn't blow up. It's not asking too much, but it's asking something. A function could go to zero, presumably, and its derivative at the same time blow up, but it would be a very pathological function. This will bring us to something that we said. We're going to try to be precise, but it's not so easy to be precise. When you try to be precise, you can exaggerate and go precise to a point that you're paralyzed with fear with every equation. We don't want to get that far. We want you to notice what happens and just look at it and state what you need. Why can't we be precise? Because at the end of the day, this equation is extraordinarily complicated, and maybe crazy. The potential is crazy enough. So, functions-- mathematicians can invent crazy functions, things like a function that is one for every rational number and zero for every rational number. Put that for a potential here, and who knows what one gets. So, we're going to take mild functions. We're not going to make them a very complicated, and we're going to be stating very soon what we need. So, what you need for this to work is that the function goes to zero and the relative goes to zero. Yes. AUDIENCE: The potential has to be real always? PROFESSOR: The potential is real at this moment. Yes. For the discussion that we're doing here, v is also a real number. AUDIENCE: So it can't be complex? PROFESSOR: Sorry? AUDIENCE: Can it be complex? PROFESSOR: It could be in certain applications for particles in electromagnetic fields. You can have something that looks like a complex Hamiltonian. So we will not discuss that in this couple of lectures, but maybe later. Yes. AUDIENCE: Are there any conditions that the potential has to be time-dependent? PROFESSOR: Well, at this moment, I put it time dependent. Also, it complicated potentials, but they're sometimes necessary. And we will discuss some of them. We will have very simple time dependencies. Otherwise, it's difficult to solve this equation. But very soon-- in about five minutes, I will say-- let's consider time-independent things to review the things that are a little more basic and important and that you should definitely remember well. OK, so that's this part of the Schrodinger equation. I want to remind you of another concept called the current-- probability current. Probability current. What is it? It's a j of x and t-- that you will review in the homework-- is given by h over m, the imaginary part of Psi star d Psi over dx. So, it's a real quantity. And it's called a probability current. And it goes together with this probability density, this probability density that we wrote over here. So it's the current associated to that density. Let's think a second what this means. In electromagnetism, you have currents and charged densities. So in E&M, you have a current. It's a vector and a charged density. Now, this current could also be a vector. If you're working in more than one dimension, it would be a vector. But if you have electromagnetism, the most famous thing associated to electromagnetism currents and charged densities is the so-called conservation law. This differential equations satisfied by the current and the density. Divergence of j plus d Rho dt is equal to zero. That means charge conservation. You may or may not remember that. If you don't, it's a good time to review it in E&M and check on that, discuss it in recitation. Think about it. This means charge conservation as we usually understand, and the way to do it-- I'm saying just in words-- is you think of a volume, you can see how much charge is inside, and you see that the rate of change of the charge is proportional to the current that is escaping the volume. Which is to say, charge is never destroyed or created. It can escape a volume, because the charges are moving, but if it doesn't escape, well, the charge remains the same. So, this is charge conservation. And this is the same thing. So the divergence of j in this case reduces to dj dx plus d Rho dt equals zero. It has a very similar interpretation. So, perhaps in equations, it's easier to think of interpretation. Consider the real line and the point a and b, with a less than b. And define the probability pab of t of finding the particle in this interval between a and b at any time. You should be able to show-- and it's again another thing to review. This you can review. And this review as well. You will use this differential equation, things like that, to show that dpab dt-- the rate at which the probability that you find the particle in this interval changes depends on what the current is doing here and what the current is doing here. So, it's actually given by j of a and t minus j at b at time t. You can show, and please try to show it. So, what does that mean? You can have the particle here at any time. But if you want to know how the probability changing, you must see how it's leaking from a or how it's leaking from b. Now j's are defined, by convention, positive to the right. So, if there's a current-- a bit of current at a, it increases the probability. This particle is sort of moving into the interval. And here at b, there's a positive current decreases the probability. Finally, for wave functions, the last thing we say is that these wave functions are-- you want them normalized, but we can work with them and they're physically equivalent if they differ just by a constant. So Psi 1 and Psi 2 are said to be equivalent if Psi 1 of x and t is equal to some complex constant of Psi 2 of x and t. Now, you would say, well, I don't like that. I like normalized wave functions, and you could have a point there. But even if these are normalized functions, they could differ by a phase. And they would still be physically equivalent. This part of the definition of the theory-- the definition of the theory is that these wave functions are really physically equivalent and indistinguishable. And that puts a constraint on the way we define observables. Any observable should have this property that, whether we used this wave function or the other, they give you the same observables. So, if your wave functions are normalized, this can be complex constant of length one. Then one normalized implies the other is normalized. If they're not normalized, you can say, look, the only reason I'm not normalizing it because I don't gain all that much by normalizing it, in fact. I can do almost everything without normalizing the wave function. So, why should I bother? And we'll explain that also as well very soon. So, this is something that this part of the physical interpretation that we should keep. So, now we've reviewed the Schrodinger equation. Next thing we want to say is the most important solutions of the Schrodinger equations are those energy Eigenstates, stationary states. And let's just go through that subject and explain what it was. So, I'm going to start erasing here. So we're going to look at-- whoops-- stationary solutions. Now, I've used this week wave function with a capital Psi for a purpose, because I want to distinguish it from another Psi that we're going to encounter very soon. So, stationary solutions. And we'll take it-- from now assume v is time-independent. The case is sufficiently important that we may as well do it. So, in this case, the Schrodinger equation is written as I h bar d Psi dt, and we'll write it with something called an h hat acting on Psi. And h hat at this point is nothing else than minus h squared over 2m second derivative with respect to x plus v of x. We say that h hat is an operator acting on the wave function Psi on the right. Operator acting on that-- what does that mean? Basically, when we say an operator acts on some space, we mean that it takes elements of that space and moves them around in the space. So, you've got a wave function, which is a complex number that depends on x and t ultimately, and then you act with this thing, which involves taking derivatives, multiplying by v of x, and you still got some complex function of x and t. So, this is called the Hamiltonian operator, and it's written like that. This Hamiltonian operator is time-independent. So, what is a stationary state? A stationary state-- the way it's defined is as follows. A stationary state of energy e-- which is a real number-- is a Psi of x and t of the following form. It's a simple form. It's a pure exponential in time times a function that just depends on x. So, it's a pretty simple object. So what is it? We say that this is a stationary state. e to the minus i Et over H bar Psi of x. And this Psi is in purpose different from this Psi. It doesn't have the bar at the bottom, and that signals to you that that's the time-independent one. So this also belongs to the complex numbers, but doesn't depend on time. So, it's called stationary because, as it turns out, when we will compute expectation values of any observable on this state, in this stationary state, it will be time-independent. In particular, you know, one thing that observable is the probability density. And when you look at that, you have Psi star and Psi. Since E is real, this phase cancels-- this is really a face, because E is real. Therefore, Psi star Psi, the e cancels, and all the time dependence cancels and goes away. Same thing here for the j. The x derivative over here it doesn't do anything to that phase. Therefore, the phase e to the i Et over H bar cancels from there two. And the current also has no time dependence. So, this will be the case for any operator that is called a time-independent operator. It will have time-independent expectation values. So you can ask anything about some familiar operator-- energy operator, momentum operator, angular momentum operator-- all the famous operators of quantum mechanics, and it will have real expectation values. So, as you, you're supposed to now plug this into this equation. And it's a famous result. Let's just do it. Plug back into the top equation. So, we have I H bar. The DET will only act on the phase, because the Psi has no time-dependence. And on the other hand, on the right hand side, the H has nothing to do with time, and therefore it can slide through the exponential until it hits Psi. So here we have H-- well, I'll put the exponential in front-- H on little Psi. So, we multiply here, and what do we get? Well, the H bars cancel. The i at minus i gives you one. You get that E in front. So you get E times this phase Psi of x. And the phase is supposed to be here, but I cancel it with this phase as well. And I get on the right hand side H Psi. I will put it as a left hand Psi. And this is the time-independent Schrodinger equation. So far this is really a simple matter. We've written a solution that will represent the stationary state, but then this energy should be such that you can solve this equation. And as you've learned before, it's something not so easy to solve that equation. So what do we want to say about this equation? Well, we have a lot to say, and a few things will be pointed out now that are very important. So, we have a differential equation now. This differential equation has second derivatives with respect to x. Now it has no time derivatives. The time has been factored out. Time is not a problem anymore. This equation, in fact, looks quite real in that it seems that Psi could even be real here. And in fact, yes, there's no problem with this Psi being real. The total Psi just can't be real in general. But this one can be a real, and we'll consider those cases as well. So, things that we want to say is that this is a second order differential equations in space. So second order differential equation in space. You could write it here. The H operator has partial derivatives, but this time time, you might as well say that this is minus h squared over 2m. The second Psi vx squared plus v of x tines Psi of x. Because Psi only depends on x, might as well write it as complete derivative. So, second order differential equation. And therefore, the strategy for this equation is a little out there in relation to the Schrodinger equation. We said, in the Schrodinger equation, we know the wave function everywhere, you know it later. Here, if you know it at one point-- the wave function-- and you know the derivative at that one point, you have it everywhere. Why is that? Because that's how you solve a differential equation. If you know the wave function and the derivative at the point, you go to the equation and say, I know the wave function and I know the first derivative, and I know the second derivative. So, a little later I can know what the first derivative is, and if I know what the first derivative is a little later, I can then know what the wave function is a little later, and you just integrate it numerically. So, you just need to know the wave function Psi of x zero and Psi prime at x zero suffice for a solution when v is regular. But this v is not too complicated, or too strange, because you can always find exceptions. You have the square well potential, and you say, oh, I know the wave function is here and its derivative is zero. Does that determine the solution? No, because it's infinite. There's no space here, really, and you should work here. So, basically, unless v is really pathological, Psi and Psi prime are enough to solve for everything. And that actually means something very important, that if Psi is equal to zero at x zero is equal to zero, and Psi prime at x zero is equal to zero, then under these regular conditions, Psi of all x is zero. Because you have a differential equation which the initial value is zero, the Psi prime is zero. And you go through the equation, you see that every solution has to be zero. It's the only possibility here. So what happens now is the following-- that you have a physical understanding that your wave function, when it becomes zero-- it may do it slowly that it's becoming zero, but never quite being zero-- but if it's zero, it does it with Psi prime different from zero, so the wave function is not zero all over. So, this is a pretty important fact that is useful many times when you try to understand the nature of solutions. So what else do we have here? Well, we have energy Eigenstates on the spectrum. So, what is an energy Eigenstate? Well, it's a solution of this equation. So a solution Psi-- a solution for Psi is an energy Eigenstate. Then, this set of values of E is this spectrum. And these two values of E-- if there's a value of E that has more than one solution, we say the spectrum is degenerate. So a degenerate spectrum is more than one Psi for a given E. So, these are just definitions, but they're used all the time. So, our energy Eigenstates are the solutions of this. The funny thing about this equation is that sometimes the requirement that Psi be normalized means that you can't always find the solution for any value of E. So, only specific values of Es are allowed-- you know that for the harmonic oscillator, for example-- and therefore there's something called the spectrum, which is the allowed values. And many times you have degeneracies, and that makes for very interesting physics. Let's say a couple more things about the nature of this wave function. So, what kind of potentials do we allow? We will allow potentials that can fail to be bounded. What do we allow? We allow failure of continuity. Certainly, we must allow that in our potentials that we consider, because you have even the finite square well. The potential is not continuous. You can allow as well failure to be bounded. So, what is a typical example? The harmonic oscillator, the x squared potential. It's not bounded. It goes to infinity. So, we can fail to be continuous, but we can fail at one point, another point, but we shouldn't fail at infinitely many points, presumably. So, it's piecewise continuous. It can fail to be bounded, and it can include delta functions. Which is pretty interesting, because a lot of physics uses delta functions, but a delta function is a complicated thing. We'll include delta functions but not derivatives of them, nor powers. So we won't take anything more strange than delta functions, collections of delta functions. So, this is really how delicate your potentials will be. They will not be more complicated than that. But for that, we will assume, and it will be completely consistent to require the following for the wave functions. So Psi is continuous-- Psi of x-- is continuous and bounded. And its derivative is bounded. Psi prime is bounded. AUDIENCE: What about Psi's behavior at infinity? PROFESSOR: Sorry? AUDIENCE: What kind of extra conditions do we have to impose of Psi's behavior at infinity? PROFESSOR: Well, I will not impose any condition that is further than that, except the condition that they've been normalizable. And even that we will be a little-- how would I say, not too demanding on that. Because there will be wave functions, like momentum Eigenstates that can't be normalized. So, we'll leave it at that. I think probably this is what you should really box, because for a momentum Eigenstate, e to the ipx over h bar. This is a momentum Eigenstate. This is continuous. It's bounded. The derivative is bounded. It is not normalizable, but it's so useful that we must include in the list of things that we allow. So, bound states and non-bound states are things that are not normalizable. So, I don't put normalization. Now, if you put normalization, then the wave function will go to zero at infinity. And that's all you would want to impose. Nothing else. So, really in some sense, this is it. You don't want more than that. AUDIENCE: Is normalization sufficient to ensure the derivative also goes to zero at infinity? PROFESSOR: Sorry? AUDIENCE: Is normalization sufficient to ensure that the-- PROFESSOR: Not that I know. I don't think so. AUDIENCE: Then why is integration by price generically valid? PROFESSOR: It's probably valid for restricted kinds of potentials. So you could not prove it in general. So, you know, there may be things that one can generalize and be a little more general, but I'm trying to be conservative. I know that for any decent potential-- and we definitely need Psi prime bounded. And wave functions that go to zero, the only ones I know that also have Psi prime going to zero. But I don't think it's easy to prove that's generic, unless you make more assumptions. So, all right. So, this we'll have for our wave functions, and now I want to say a couple of things about properties of the Eigenstates. Now, we will calculate many of these Eigenstates, but we need to understand some of the basic properties that they have. And there's really two types of identities that I want you to be very aware that they play some sort of dual role-- a pretty interesting dual role-- that has to do with these wave functions. So, the Eigenstates of-- Eigenstates of H hat-- these are the energy Eigenstates. you can consider them and make a list of them. So, you have an energy E zero less than or equal an E 1, E 2. Just goes like that. And you have a Psi zero, Psi 1. All this wave functions. And then H hat Psi N is equal to E N Psi N. You have a set of solutions. So, this is what will happen if you have a good problem. A reasonable potential, and nothing terribly strange going on. There would be a lot of solutions, and they can be chosen to be orthonormal. Now at first sight, it's a funny term to use-- orthonormal. This is a term that we use for vectors. Two vectors are orthogonal, and we say they're orthonormal if they have unit length, and things like that. But what do we mean the two functions are orthonormal? Well, our function's vectors. Well, that's a little dubious. But the way we will think in quantum mechanics is that, in some sense, functions are vectors in an infinite dimensional space. So, they're just vectors, but not in three dimensions. Why? Think of it. If you have a function, you have to give values-- independent values-- at many points-- Infinitely many. And if you give all those values, you've got the function. If you have a vector, you have to give components, and you've got the vector. So, in a sense, to give a function, I have to give a lot of numbers. And I can say the first vector is the value along the direction-- the first component is the value around zero. The second unit vector is the value of about 0.01, 9.02, going on and on. And then list of them, and you have a vector of infinite dimensions. You say, totally useless. [LAUGHTER] No, it's not totally useless. Actually, if you visualize that-- and we'll do it later more-- you will be able to understand many formulas as natural extensions. So, what does it mean that these two functions are orthonormal? Well, a dot product, or orthonormality, is to say that the dot product is zero. And the way we dot product functions Psi m and Psi n of x is we take their values at the same point with star one, and we integrate. And, if this is equal to delta mn, we say the functions are orthonormal. So, ortho, for orthogonal, which says that if m is different from n, the Kronecker delta, that symbol is equal to one if the two labels are the same, or zero otherwise. If they're different, you get zero. The inner product-- this left hand side is called the inner product-- is zero. On the other hand, if they are the same, if m is equal to n, it says that the Psi squared is one. Kind of like a wave function that is well normalized. So we say normal for orthonormal. So these are orthonormal wave functions, and that's good. This is called orthonormality. But then there is a more subtle property, which is that this set of functions is enough to expand any function in this interval that you're doing your quantum mechanics. So, if you have any reasonable function, it can be written as a superposition of these ones. So, this differential equation furnishes for you a collection of functions that are very useful. So this is orthonormality. This is also completeness, which is to say that any function can be written as a sum of of this function. So I will write it as this. Psi of x-- an arbitrary Psi of x-- can be written as bm Psi n of x n equals zero to infinity, where the bn's are complex. So, this is an assumption, but it's a very solid assumption. When you study differential equations of this type-- Sturm-Liouville problem-- this is one thing that mathematicians prove for you, and it's not all that easy. But the collection of wave functions is good in this sense. It provides you a complete set of things that any function can be written in terms of that. I'm not saying this satisfies any particular equation. You see, this function satisfies very particular equations-- those equations-- but this is an arbitrary function. And it can be written as a sum of this. See, these equations have different energies for different Psi's. This Psi here satisfies no obvious equation. But here is a problem that this is useful for. Suppose you're given a wave function at, at the given time, you know what it looks like. So, here is your wave function. Psi. And you know that Psi at x and time equals to zero happens to be equal to this Psi of x that we wrote above. So, it's equal to bn Psi n of x. Well, if you know that, if you can calculate this coefficient, the wave function of time equals zero is known, say, and it was given by this thing, which is then written in this form. If you can write it in this form, you've solved the problem of time evolution, because then Psi of x at any time is just simply obtained by evolving each component. Which is bn e to the minus iEnt over h bar Psi n of x. So this is the important result. Now, look what has happened. We have replaced each term. We added this exponential. Why? Because then each one of these is a solution of the full Schrodinger equation. And therefore a superposition with complex coefficients is still a solution of the Schrodinger equation. Therefore, this thing I've put by hand is, you would say it's ad hoc. No, it's not. We've put it by hand, yes, but we've produced a solution of the Schrodinger equation, which has another virtue. When t is equal to zero, it becomes what you know the wave function is. So, since this solves the Schrodinger equation-- time equals zero gives you the right answer. And you remember that the Schrodinger equation, if you know that time equals zero, the term is a wave function everywhere-- this is the solution. It's not just a solution. It's the solution. So, you've solved this equation, and it's a very nice thing. It all depends, of course, on having found the coefficients bn. Because typically at time equals zero, you may know what the wave function is, but you may not know how to write it in terms of these coefficients bn. So, what do you do then? If you don't know those coefficients, you can calculate them. How do you calculate them? Well, you use orthonormality. So you actually take this and integrate against another Psi star. So you take a Psi star sub m and integrate-- multiply and integrate. And then the right hand side will get the Kronecker delta that will pick out one term. So, I'm just saying in words a two line calculation that you should do if you don't see this as obvious. Because it's a kind of calculation that you do a few times in life. Then it becomes obvious and you never do it again. It's minus infinity to infinity dx Psi m star of x Psi of x dx. So, bm is given by this quantity, or bn is given by this quantity. You obtain it from here plus orthonormality. So, once you have this bn, you can do something that may-- if you look at these things and say, well, I'm bored, what should I do? I say, well, you have bm. Plug it back. What happens then? You say, why would I plug it back? I don't need to plug it back. And that's true, but it's not a crazy thing to do, because it somehow must lead to some identity. Because you solve an equation and then plug it back and try to see if somehow it makes sense. So either it makes sense, or you learned something new. So, we were supposed to calculate the bn's. And now we have them, so I can plug this back here. So what do I Get Psi of x now is equal to the sum from n equals zero to infinity. bn-- but this bn is the integral of Psi n star of x prime. I put here Psi of x prime. dx prime. I don't want to confuse the x's with x prime, so I should put the x primes all over here. Psi n of x. Well, can I do the integral? No. So, have I gained anything? Well, you've gained something if you write it in a way that Psi is equal to something times Psi. That doesn't look all that simple, but we can at least organize it. Let's assume things are convergent enough that you can change orders of sums and integrals. That's an assumption we always do. I'll write it like this. dx prime. And now I'll put the sum here equals zero to infinity of Psi n star of x prime. And I'll put the other Psi here as well. The Psi n of x over here. I'll put the parentheses, and finally the Psi of x prime here. So, now it's put in a nice way. And it's a nice way because it allows you to learn something new and interesting about this. And what is that? That this must be a very peculiar function, such that integrated against Psi gives you Psi. And what could it be? Well, this is of the form, if you wish-- the x prime-- some function of x and x prime-- times Psi of x prime. So, this k is this thing. Well, you can try to think what this is. If you put the delta function here-- which may be a little bit of a cheat-- you will figure out the right answer. This must be a function that sort of picks out the value of the function at x by integrating. So it only cares about the value at x. So, it must be a delta function. So, in fact, this is a delta function, or should be a delta function. And therefore the claim is that we now have a very curious identity that looks as follows. It looks like n equal zero to infinity, Psi n star of x prime Psi n of x is actually delta of x minus x prime. So, this must be true. If what we said at the beginning is true, that you can expand any function in terms of the Eigenfunctions, then, well, that's not such a trivial assumption. And therefore, it allows you to prove something fairly surprising, that this must be true, that this identity must be true. And I want you to realize and compare and contrast with this identity here. One is completeness. One is orthonormality. There are two kinds of sums going on here. Here is sum over space, and you keep labels arbitrary-- label indices arbitrary. So, sum over space. These functions depend on space and on labels. Sum over space, and keep the labels, and you get sort of a unit matrix in this space, in the space of labels. Here, you keep the positions arbitrary, but sum over labels. And now you get like a unit matrix in the space of positions. Something is one-- but actually infinite, but you couldn't do better-- when x is equal to x prime. So, if you think of it as a matrix, this function in x and x prime is a very strange matrix, with two indices, x and x prime. And when x is different from x prime, it's zero, but when x is equal to x prime, it's one. But it has to be a delta function, because continuous variables. But it's the same idea. So, actually if you think of these two things, x and m as dual variables, this is a matrix variable, and then you're sort of keeping these two indices open and summing over the other index. Multiplying in one way you get a unit matrix. Here, you do the other way around. You have a matrix in m and n. This is a more familiar matrix, but then you sum over the other things. So, they're dual, and two properties that look very different in the way you express them in words. One is that they're orthonormal. The other is that they're complete. And then suddenly then the mathematics tells you there's a nice duality between them. So, the last thing I want to say today is about expectation values, which is another concept we have to review and recall. So let's give those ideas. So, if we have a time-dependent operator-- no, a time independent-- we'll do a time-independent operator, I'm sorry. Time-Independent operator. And this operator will be called A hat. No time dependence on the operator. So, then we have the expectation value of this operator on a normalized state. So what does that mean? The expectation value of this operator on a state-- on a wave function here. Now, this wave function is time-dependent. So this expectation value of this operator is expected to be a function of time. And how is it defined? It's defined by doing the following integral. Again, from minus infinity to infinity, dx Psi star of x and t, and then the operator A acting on Psi of x and t. And Psi is supposed to be a normalized state. So, notice the notation here. We put the Psi here because of the expectation-- whenever somebody asks you the expectation value for an operator, it has to be on a given state. So you put the state. Then you realize that this is a time-dependent wave function typically, so it could depend on time. Now, we said about stationary states that if the state is stationary, there's a single time exponential here. There's just one term, e to the minus iEt over h bar. And if A, of course, is a time-independent operator, you won't care about the exponential. You will cancel this one, and there will not be a time dependence there. But if this state is not stationary-- like most states are not stationary-- remember it's very important. If you have a stationary state, and you superimpose another stationary state, the result is not stationary. Stationary is a single exponential. More than one exponential is not stationary. So when you have this, you could have time dependence. So that's why we wrote it. Whenever you have a state that is not stationary, there is time dependence. Now, you could do the following thing. So here is a simple but important calculation that should be done. And it's the expectation value of H. So what is the expectation value of the Hamiltonian at time t on this wave function Psi that we've computed there? So, we would have to do that whole integral. And in fact, I ask you that you do it. It's not too hard. In fact, I will say it's relatively simple. And you have H on Psi of x and t, and then you must substitute this Psi equal the sum of bn Psi n. And you have two sums. And the H acting on each side n-- you know what it is. And then the two sums-- you can do the integral using orthonormality. It's a relatively standard calculation. You should be able to do it. If you find it hard, you will see it, of course, in the notes. But it's the kind of thing that I want you to review. So, what is the answer here? It's a famous answer. It's bm squared En. So, you get the expected value of the energy. It's a weighted average over all of the stationary states that are involved in this state that you've been building. So your state has a little bit of Psi zero, Psi 1, Psi 2, Psi 3. And for each one, you square its component and multiply by En. And this is time-independent. And you say, well, you told me that only for stationary states, things are time-independent. Yes, only for stationary states, all operators are time-independent, but the Hamiltonian is a very special operator. It's an energy operator, and this is a time independent system. It's not being driven by something, so you would expect the energy to be conserved. And this is pretty much the statement of conservation of energy, the time-independence of this thing. My last remark is technical about normalizations, and it's something you may find useful. If you have a wave function that is Psi, which is not normalized, you may say, OK, let's normalize it. So, what is the normalized wave function? The normalized wave function is Psi divided by the square root of the integral of Psi star Psi dx. You see, this is a number, and you take the square root of it. And this is the Psi of x and t. If the Psi is not normalized, this thing is normalized. So, think of doing this here. Suppose you don't want to work too hard, and you want to normalize your wave function. So, your Psi is not normalized. Well, then this is definitely normalized. You should check that. Square it, an integrate it, and you'll see. You'll get one. But then I can then calculate the expectation value of A on that state, and wherever I see a Psi that should be normalized, I put this whole thing. So what do I end up with? I end up with this integral from infinity to infinity dx Psi star A A hat Psi divided by the integral from minus infinity to infinity of Psi star Psi dx. If you don't want to normalize a wave function, that's OK. You can still calculate its expectation value by working with a not-normalized wave function. So in this definition, Psi is not normalized, but you still get the right value. OK, so that's it for today. Next time we'll do properties of the spectrum in one dimension and begin something new called the variational problem. All right. [APPLAUSE] Thank you, thank you.

MIT_805_Quantum_Physics_II_Fall_2013

23_Angular_Momentum_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So today, let me remind you, for the convenience of also the people that weren't here last time, we don't need too much of what we did last time, except to know, more or less, what's going on. We were solving central potential problems in which you have a potential that just depends on r. And at the end of the day, the wave functions were shown to take the form of a radial part and an angular part with the spherical harmonic here. The radial part was very conveniently presented as a U function divided by r. That's another function, but the differential equation for U is nice. It takes the form of a 1-dimensional Schrodinger equation for a particle under the influence of an effective potential. This potential, effective potential, has the potential that you have in your Hamiltonian plus an extra term, a barrier. It's a potential that grows as r goes to 0, so it's a barrier that explodes at r equals 0. And this being the effective potential that enters into this 1-dimensional Schrodinger equation, we made some observations about this function U. The normalization of this wave function is guaranteed if the integral of U squared over r is equal to 1. So that's a pretty nice thing. U squared meaning absolute value squared of U. And we also noticed that U must go like r to the l plus 1 near r going to 0. So those were the general properties of U. I'm trying to catch up with notes. I hope to put some notes out today. But this material, in fact, you can find parts of it in almost any book. It will just be presented a little differently. But this is not very unusual stuff. Now, the diagram that I wanted to emphasize to you last time was that if you're trying to discuss the spectrum of a central state potential, you do it with a diagram in which you list the energies as a function of l. And it's like a histogram in which for l equals 0, you have to solve some 1-dimensional Schrodinger equation. This 1-dimensional Schrodinger equation will have a bound state spectrum that is non-degenerate. So for l equal 0, there will be one solution, two solutions, three. I don't know how many before the continuous spectrum sets in, or if there is a continuous spectrum. But there are some solutions. For l equal 1, there will be some other solutions. For l equal 2, there might be some other solutions. And that depends on which problem you are solving. In general, there's no rhyme or reason in this diagram, except that the lowest energy state for each level goes up. And that's because the potential goes up and up as you increase l. Notice this is totally positive. So whatever potential you have, it's just going up as you increase l. So the ground state should go up. The ground state energy should go up. So this diagram looks like this. We also emphasized that for every l, there are 2l plus 1 solutions obtained by varying M, because M goes from l to minus l. Therefore, this bar here represents a single multiplate of l equals 1, therefore three states. This is a single multiplate of l equals 1, three more states. Here is five states, five states, but only one l equal 1 multiplate, one l equal 1 multiplate, 1, 1. There are no cases in which you have two multiplates because that would contradict our known statement that the spectrum of the potential of bound states in one dimensions is non-degenerate. So that was one thing we did. And the other thing that we concluded that ties up with what I want to talk now was a discussion of the free particle, free particle. And in the case of a free particle you say, well, so what are you solving? Well, we're solving for solutions that have radial symmetry. So they are functions of r [INAUDIBLE] angular distribution. So what do you find is UEl of r is equal to rJl of kr, as we explained, where these were the spherical Bessel functions. And those are not as bad as the usual Bessel functions, not that complicated. They're finite series constructed with sines and cosines, so these are quite tractable. And that was for a free particle. So we decided that we would solve the case of an infinite spherical well, which is a potential V of r, which is equal to 0 if r is less than a, and infinity if r is greater or equal than a. It's a small-- well, a is whatever size it is. It's a cavity, spherical cavity where you can live. And outside you can't be there. This is the analog of the infinite square well in one dimension. But this is in three dimensions. An infinite spherical well should be imagined as some sort of hole in the material and electrons or particles can move inside and nothing can escape this. So this is a hollow thing. So this is a classic problem. You would say this must be as simple to solve as the infinite square well. And no, it's more complicated. Not conceptually much more complicated, but mathematically more work. You will consider some aspects of the finite spherical well in the homework. The finite square well, you remember, is a bit more complicated. You can't solve it exactly. The finite spherical well, of course, you can't solve exactly either. But you will look at some aspects of it, the most famous result of which is the statement that while any attractive potential in one dimension has a bound state in three dimensions. An attractive potential, so a finite spherical well, may not have a bound state, even a single bound state. So that's a very interesting thing that you will understand in the homework in several ways. You will also understand some things about delta functions, that they're important. So we'll touch base with that. So that's as far as I got last time and just a review. If there are any questions, don't be shy if you weren't here and you have a question. Yes. AUDIENCE: Is there any reason to expect [INAUDIBLE] intuitively should be like [INAUDIBLE]? PROFESSOR: Well, the reason, intuitively the reason is basically the conspiracy between this UEl, as I was saying, UEl as r goes to 0 goes like r to the l plus 1. So first of all, this potential is very repulsive. Is that right? So that tends to ruin things. So you could say, oh, well, this thing is probably not going to get anything because near r equal 0, you're being repelled. But you cay say, no, let's look at that l equal 0. So you don't have that, so just V of r. But we take l equals 0-- I'm sorry, U here, U of El has to go like that. So actually, U will vanish for r equals 0. So the effective potential for the 1-dimensional problem may look like a finite square well, that is like that. But the wave function has to vanish on this side. Even though you would say, it's a finite spherical well, why does it have to vanish Here well, it's the unusual behavior of this U function. So the wave function that you can sort of imagine must vanish here. So in order to get a bound state, it has to have enough time to sort of curve so that it can fall, and it's sometimes difficult to do it. So basically, it's the fact that the wave function has to vanish at the origin, the U wave function has to vanish. Now, the whole wave function doesn't vanish because it's divided by r. But the U does. So it's the reason why you don't have bound states in general. And then there's also funny things like a delta function. You would say, well, a 3-dimensional delta function, how many bound states do you get, or what's going on? With a 1-dimensional delta function, you have one bound state, and that's it. With a 3-dimensional delta function, as you will find, it's [INAUDIBLE] is rather singular, and you tend to get infinitely many bound states. And you cannot even calculate them because they fall off all the way through r and go to minus infinity energy. It's a rather strange situation. All right. Any other questions? So let's do this infinite spherical well. Now, the reason we did the free particle first was that inside here, this is all free, so the solutions will be sort of simple. Nevertheless, we can begin with looking at the differential equation directly for inside. So r less than a, you would have minus d second UEl over d rho squared, actually, plus l times l plus 1 over rho squared UEl equals UEl, where rho is equal to kr. And k-- I'm sorry, I didn't write it there-- is 2mE over h squared as usual. So here I didn't say what k was. That was 2mE over h squared. And this doesn't quite look like the differential equation you have here. Well, V of r is 0 for r less than a, so you just have this term. The h squared's over 2m and the E have been rescaled by changing r to rho. So the differential equation becomes simple and looking like this. So that was a manipulation that was done in detail last time, but you can redo it. Now, this, as I mentioned, is not a simple differential equation. If you didn't have this, it would have a power solution. If you don't have this, it's just a sine or cosines. But if you have both, it's Bessel. So having a differential with two derivatives, 1 over rho squared and 1, brings you into Bessel territory. Anyway, this is the equation that, in fact, is solved by these functions because it's a free Schrodinger equation, and you can take it for l equal 0. This is the only case we can do easily without looking up any Bessel functions or anything like that. You then have d second UE0 d rho squared is equal UE0. And therefore, UE0 goes like A sine of rho plus B cosine rho. Rho is kr. UEl must behave like r to the l plus 1, so UE0 must behave like r. So for this thing to behave, must behave like r. So it must behave like rho as rho goes to 0. Therefore, this term cannot be there. The only solution is UE0 is equal to sine of rho, which is kr. So UE0 of r must be of this form. Then in order to have a solution of the 1-dimensional Schrodinger equation, it's true that the potential becomes infinite for r equal a. So that is familiar. It's not the point r equal 0 that is unusual. r equal a, this must vanish. So we need that UE0 of a will equal to 0. So this requires k equal some kn so that kna is equal to n pi. So for k is equal to kn, where kn,a is equal to n pi, a multiple of pi, then the wave function will vanish at r equals a. So easy enough. We've found the values of k. This is quite analogous to the infinite square well. And now the energies from this formula En will be equal to h squared kn squared over 2m. And it's convenient, of course, to divide by ma squared so that you have kna squared. So the energies are h squared over 2ma squared. Here we have n pi squared. I'll put them like this. En,0 for l equal 0, En,l's energies. Now, if you want to remember something about this, of course, all these constants are kind of irrelevant. But the good thing is that this carries the full units of energy. And you know in a system with length scale a, this is the typical energy. So the energies are essentially that typical energy times n squared pi squared. So it's convenient to define, in general, En,l to be En,l, for any l that you may be solving, divided by h squared over 2ma squared. So that this thing has no units. And it tells you for any level, the calligraphic E, roughly how much bigger it is than the natural energy scale of your problem. So it's a nice definition. And in this way, we've learned that En,0 is equal to n pi squared. And a few values of this are E1,0 about 9,869 [INAUDIBLE], E2,0 equal 39,478, and E3,0 is equal 88,826. Not very dramatic numbers, but they're still kind of interesting. So what else about this problem? Well, we can do the general case. Let me erase a little here so that we can proceed. The general case is based on knowing the zeroes of this spherical Bessel function. So this is something that the first one you can do easily. The zeroes of J1 of rho are points at which tan rho is equal to rho. That is a short calculation if you ever want to do it. That's not that difficult, of course, but you have to do it numerically. So the zeroes of the Bessel functions are known and are tabulated. You can find them on the web, little programs that do it on the web and give you [? directly those ?] zeroes. So how are they defined? Basically, people define Zn,l to be the n-th zero with n equals 1 like that of Jl. So more precisely, Jl of Zn,l is equal to 0. And all the Z and l's are different from 0. There's a trivial zero at 0. And nevertheless, that is not counted. It's just too trivial for it to be interesting. So these numbers, Z and l, are basically it. Why? Because what you need is, if you're looking for the l-th solution, you need UEl of a equal 0. And UEa of that equal 0 means that you need kn,l times a be equal to Zn,l. So kn,l is the value of k. And just like we quantized here, we had kn, well, if you have various l's, put the kn,l. So for every value of l, you have kn,l's that are given by this. And the energy's like this. Let me copy what this would be. En,l would be En,l over this ratio. And En,l h squared, well, let me do it this way. I'm sorry. En,l would be h squared kn,l over 2ma squared, over 2m like that. Then you multiply by a squared again. So you get kn,l a squared over 2ma squared. So what you learn from this is that En,l, you divide this by that, is just kn,l times a, which is Zm,l squared. So that's the simple result. The En,l's are just the squares of the zeroes of the Bessel function. So you divide it again by h squared over 2ma, and that's all that was left. So you need to know the zeroes of the Bessel function. And there's one, you might say, well, what for do I care about this? But it's kind of nice to see them. So Z1,1 is equal to 4.49 Z2,1 is equal to 7.72, and Z3,1 is 10.90, numbers that may have no rhyme or reason. Now, you've done here l equals 1. Of course, it continues down, down, down. You can continue with the first zero, first nontrivial zero, second nontrivial zero, third nontrivial zero, and it goes on. The energies the squares. So the squared goes like 20.19. This goes like 59.7. And this goes like 119 roughly. Then you have the other zeroes. First zero for l equals 2, that is 5.76 roughly. Second zero for l equals to 2 is 9.1 roughly. And if you square those to see those other energies, you would get, by squaring, 33.21 and 82.72. And finally, let me do one more. Z1,3, the first zero of the l equal 3, and the Z2,3, the second zero, are 6.99 and 10.4, which when squared give you 48.83 and 108.5. OK. Why do you want to see those numbers? I think the reason you want to see them is to just look at the diagram of energies, which is kind of interesting. So let's do that. So here I'll plot energies, and here I put l. And now I need a big diagram. Here I'll put the curly energies. And here is 10, 20, 30, 40, 50, 60-- and now I need the next blackboard, let's see, we're 60, let's see, more or less, here is about right-- 70, 80, 90, 100, 110. How far do I need? 120, ooh, OK, 120. There we go. So just for the fun of it. Look at them to see how they look, if you can see any pattern. So the first energy was 986, so that's roughly here. That's l equals 0 is the first state. Second is 39.47, so it's a little below here. Next is 88.82, so we are here, roughly. Then we go l equals 1. What are the values? This one's 20.19. L equals 1, 20.19, so we're around here. Then 59.7 is almost 60. And then 119, so that's why we needed to go that high. So here we are. And then l equals 3, you have 48.83, so that's 50. I'm sorry, l equals 2. 48.83. A little lower than that. No, I'm sorry. It's 33.21. I'm misreading that. 33 over here. And then 82.72, so we are here. And then l equals 3, we have 48.83, so that was the one I wanted, and 108.5. That's it, and there's no pattern whatsoever. The zeroes never match. The only thing that is true is that 0, 1, 2, 3, they were ascending as we predicted. But no level matches with any other level. If you were trying to say, OK, this potential is interesting, is special, it has magic to it, a spherical square well, it doesn't seem to have anything to it, in fact. It's totally random. I cannot prove for you, but it's probably true, and probably not impossible to prove, that these zeroes are never the same. No l and l prime will have the same zero. No degeneracy ever occurs that needs an explanation. For example, this state could have ended up equal to this one or equal to this one, and it doesn't happen. And that's OK, because at this level, we would not be able to predict why it happened. We actually, apart from the fact that this a round, nice box, what symmetries does it have, that box, except rotational symmetry? Nothing all that dramatic. So you would say, OK, let's look for a problem, which we'll deal now, that does have a more surprising structure, and let's try to figure it out. Let's try the three dimensional harmonic oscillator. So 3D SHO. Isotropic. What is the potential? It's 1/2 m omega squared x squared for x plus y squared plus z squared, all the same constant. So it's 1/2 m omega squared r squared. You would say, this potential may or may not be nicer than the spherical well, but actually, it is extraordinarily symmetric in a way that the spherical well is not. So we'll see why is that. Let's look at the states of this. Now, we're going to do it with numerology. Everything will be kind of numerology here because I don't want to calculate things for this problem. So first thing, how you build the spectrum? H is equal to h bar omega N1 plus N2 plus N3 plus 3/2, where these are the three number operators, and 0. Now, just for you to realize, in the language of things that we've been doing, what is the state space? If we call H1 the state space of a one dimensional SHO, what is the state space of the three dimensional SHO? Well, conceptually, how do you build a three dimensional SHO? Well, you have the creation annihilation operators that you had for the x, y, and z. So you have the ax dagger, the ay dagger, and the az dagger, and you could act on the vacuum. So the way you can think of the state space of the one dimensional oscillator is this is one dimensional oscillator and I have all these things. Here is the other one dimensional, here is the last one dimensional. But if I want to build a state of the three dimensional oscillator, I have to say how many ax's, how many ay's, how many az's. So you're really multiplying the states in the sense of tensor products. So the H, for a 3D SHO, is the tensor product of three H1's, the H1 x, the 1y, and the z. You're really multiplying all the states together. Yes? AUDIENCE: So this is generalized to when you have a wave function that's separable into products of different coordinates. Can you express those as tensor products of the different states, basically? BARTON ZWIEBACH: You see, the separable into different coordinates, it's yet another thing because it would be the question of whether the state is separable or is entangled. If you choose, for example, one term like that, a1, x, ax dagger, ay dagger, az dagger, with two of those here, the wave function is the product of an x wave function, a y wave function, and a z wave function. But if you add to this ax dagger squared plus ay plus az, it will also be factorable, but the sum is not factorable. So you get the entanglement kind of thing. So this is the general thing, and the basis vectors of this tensor product are basis vectors of one times basis vectors of the other basis vector. So basically, one basis vector here, you pick some number of ax's, some number of ay's, some number of az's. So this shows, and it's a point I want to emphasize at this moment, it's very important, that even though we started thinking of tensor products of two particles, here, there are no two particles in the three dimensional harmonic oscillator, no three particles. There's just one particle where there's one kind of attribute that that's doing in the x direction, one kind of attribute that it's doing in the y, one kind of attribute that it's doing in the z. And therefore, you need data for each one, and the right data is the tensor product. You're just combining them together. We mentioned that the basis vectors of a tensor product are the products of those basis vectors, of each one, so that's exactly how you build states here. So I think, actually, you probably have this intuition. I just wanted to make it a little more explicit for you. So you don't need to have two particles to get a tensor product. It can happen in simpler cases. So here's the thing that I want to do with you. I would like to find that diagram for the three dimensional SHO. That's our goal that we're going to spend the next 15 minutes probably doing. How does that diagram look? So I'll put it somewhere here maybe, or maybe here. I won't need such a big diagram. So I'll have it here. Here is l, and here are the energies. So ground states. The ground state, you can think of it as a state like that. How should I write it? A state like that. No oscillators acting on it whatsoever, so the N's are N1 equals N2 equals N3 equals 0, and you get E equals h bar omega times 3/2. So 3/2 h bar omega. So actually, we got one state, and it's the lowest energy state. Energy lowest possible. So let me write here, energy equals 3/2 h bar omega. We got one state over here. Now, can it be an l equals 1 state or an l equals 2 state or an l equals 3 state? How much is l for that state? You see, if it's a spherically symmetric problem, it has to give you a table like that. It's guaranteed by angular momentum, so we must find. My question is whether it's l equals 0, 1, 2, 3, or whatever. Anybody would like to say what do they think it is? Kevin? AUDIENCE: It's 0, right? BARTON ZWIEBACH: 0. And why? AUDIENCE: Because we wrote the operator for l in terms of ax, ay, and az, and you need one to be non-zero. You need a difference between them to generate a rotation. BARTON ZWIEBACH: OK, that's true. It's a good answer. It's very explicit. Let me say it some other way, why it couldn't possibly be l equals 1. Yes? AUDIENCE: Because the ground state decreases for l decreasing. BARTON ZWIEBACH: The ground state energy does what? AUDIENCE: It's smaller for smaller l, and so for l equals 0, you have to have a smaller ground state than for l equals 1. BARTON ZWIEBACH: That's true. Absolutely true. The energy increases so it cannot be l equals 1, because then there will be something below which doesn't exist. But there may be a more plain answer. AUDIENCE: The state is non-degenerative. BARTON ZWIEBACH: Yes. There's just one state here. We built one state. If it would be l equals 1, there should be three states because l equals 1 comes with m equals 1, 0, and minus 1. So unless there are three states, you cannot have that. All right. So then we go to the next level. So I can build a state with ax dagger on the vacuum, a state with ay dagger on the vacuum, and a state with az dagger on the vacuum using one oscillator. Here, the N's are 1, different ones, and the energy is h bar omega 1 plus 3/2, so 5/2. And I got three states. What can that be? Well, could it be three states of l equals 0? No. We said there's never a degeneracy here. There's always one thing, so there would be one state here, one state here, one state here maybe. We don't know, but they would not have the same energy, so it cannot be l equals 0. Now, you probably remember that l equals 1 has three states. So without doing any computation, I think I can argue that this must be l equals 1. That cannot be any other thing. It cannot be l equals 2 because you need five states. Cannot be anything with l equals 0. So it must be l equals 1. So here is l equals 0, and here is l equals 1, and there's no state here, but there's one at 5/2 h bar omega. So we obtain one state here. And this corresponds to a degeneracy. This must correspond to l equals 1 because it's three states. And that degeneracy is totally explained by angular momentum's central potential. It has to group in that way. Of course, if my oscillator had not been isotopic, it would not group that way. So we've got that one and we're, I think, reasonably happy. Now, let's list the various l's. l equals 0, l equals 1, l equals 2, l equals 3, l equals 4, l equals 5. How many states? 1, 3, 5, 7, 9, 11. OK, good enough. So we succeeded, so let's proceed to one more level. Let's see how we do. Here, I would have ax dagger squared on the vacuum, ay dagger squared on the vacuum, az dagger squared on the vacuum. Three states, but then I have three more, ax ay, both dagger on the vacuum, ax az, and ay az, for a total of six states. So at N equals 2, the next level, let's call N equals N1 plus N2 plus N3. So this is N equals 2. This is N equals 1. You've got six states. They must organize themselves into representations of angular momentum, so they must be billed by these things. So I cannot have l equals 3. I don't have that many states. I could have two l equals 1 states, three and three. That would give six states, or a five and a one. So what are we looking at? Let's see what we could have. Well, we're trying to figure out the next level, which is 7/2 h bar omega. If I say this is built by two l equals 1's, I would have to put two things here, and that's wrong. There cannot be two multiplates at the same energy. So even though it looks like you could build it with two l equal 1's, you cannot. So it must be an l equals 2 and an l equals 0. So l equals 2 plus l equals 0, this one giving you five states and this giving you one state. So at the next level, this cannot be, but what you get instead, l equals 2. You get one state here and one state there. This is already something a little strange and unexpected. For the first time, you've got things in different columns that are matching together. Why would these ones match with these ones? That requires an explanation. You will see that explanation a little later in the course, and that's something we need to understand. So far, so good. We seem to be making good progress. Let's do one more. In fact, we need to do maybe a couple more to see the whole pattern. Let's do the next one, N total equals 3. And now you have-- I'll be very brief-- ax cubed, ay cubed, az cubed, ax squared times ay or az, ay squared times ax or az, and az squared times ay or ax, and ax ay az, all different. And that builds for three states here, two states here, two states here, two states here, and one state here. So that's 10 states. Yes? AUDIENCE: [INAUDIBLE]? BARTON ZWIEBACH: No. It's just laziness. I just should have put ax squared ay dagger or ax squared az squared. AUDIENCE: [INAUDIBLE]? BARTON ZWIEBACH: No, this is a sum. This is what we used to call the direct sum of vector spaces. This is not the product. That's pretty important. Here, it's a sum. We're saying 6 is 5 plus 1, basically. Six states are five vectors plus one vector. Now, it can seem a little confusing because-- well, it's not confusing. If it would be a product, it would be 1 times 5, which is 5. So here, it's 6. It's a direct sum. It's saying the space of states at this level is six dimensional. This is a five dimensional vector space, this is a one dimensional vector space. This is a direct sum, something we defined a month ago or two months ago, direct sums. So this is funny how this is happening. This tensor product is giving you direct sums of states. Anyway, 10 states here. And now it does look like we finally have an ambiguity. We could have l equals 4, which is nine states, plus l equals 0. You cannot use any one more than once. We've learned that for any energy level, we cannot have some l appear more than once because it would imply degeneracy. So I cannot build this with 10 singlets, or three l equal 1's and one l equals 0. I have to build it with different things, but I can build it as 9 plus 1, or I can build it as l equals 3 plus l equals 1. And the question is, which one is it? AUDIENCE: [INAUDIBLE]. BARTON ZWIEBACH: 3 and 1, is that right? How would you see that? AUDIENCE: Because the lowest energy with l3 has to be lower than the lowest energy with l4. BARTON ZWIEBACH: Yes. Indeed, it would be very strange. It shouldn't happen. The energies are sort of in units, so here is l3 and here is l equals 4. If l4 would be here, where could be l3? It cannot be at a lower energy. We've accounted for all of those. This is terribly unlikely, and it must be this. And therefore, you found here next level, 9/2 h bar omega, you got l equals 3, l equals 1. It's possible to count. You start to get bored counting these things. So if you had to count, for example, the number of states with 4, how would you count them a little easier? Well, you say, I need ax dagger to the nx, ay dagger to the ny, and az dagger to the nz. That's the state. And you must have nx plus ny plus nz equals 4. And you can plot this, make a little diagram like this, in which you put nx, ny, and nz. And you say, well, this can be as far as 4, this can be as high as 4, this can be as high as 4, so you have triangle, but you only have the integer solutions. nx plus ny plus nz equals 4 is that whole hyperplane, but only integers and positive one. So you have here, for example, a solution. This line is when nz plus ny is equal to 4. So here's nz equals 4, nz equals 3, 2, 1, 0. These are solutions. Here, you have just one solution. Then you would have two solutions here, three solutions here, four here, and five there. So the number of states is actually 1 plus 2 plus 3 plus 4 plus 5. The number of states is 1 plus 2 plus 3 plus 4 plus 5, which is 15. And you don't have to write them. So 15 states, what could it be? Well, you go through the numerology and there seem to be several options, but not too many that make sense. You could have something with l equals 5, but by the same argument, it's unlikely. But you could have something with l equals 4 and begin with it. So it must be an l equals 4, which gives me already nine states, and there are left with six states. But you know that with six states, pretty much the only thing you can do is l equals 2 and l equals 0, so that must be it. The next state here, l equals 4, is here. This was 11/2 h bar omega, and then it goes 4, 2, 0. Enough to see the pattern, I think. You could do the next one. Now it's quick because you just need to add 6 here. It adds one more, so it's 21 states, and you can see what can you build. But it does look like you have this, this, and that you jump by two units. So you have 0, then 1, and nothing. Then 2, and you jump the next to 0. And then 3 is the next one, and then you jump 2, and that's it. And here, jump 2 and jump 2. So in jumps of 2, you go to the angular momentum that you need. So how can you understand a little more of what's going on here? Why these things? Well, as you may recall, we used to have this discussion in which you have an a x and ay. You could trade for a right and a left. And with those, the angular momentum in the z-direction was h bar N right minus N left. This is for a two-dimensional oscillator, but the x and y of the three-dimensional oscillator works exactly the same way. So Lz is nicely written in terms of these variables. And it takes a little more work to get the other-- the Lx and Ly, but they can be calculated. And they correspond to the true angular momentum of this particle. It's the real angular momentum. It's not the angular momentum that you found for the two-dimensional harmonic oscillator. It's the real one. So here we go with a little analysis. How would you build now states in this language? You can understand things better in this case because, for example, for N equals 1, you could have a state a right dagger on the vacuum, a z dagger, a left dagger on the vacuum. And then you can say, what is the Lz of this state? Well, a right dagger on the vacuum has Lz equal h bar. This has 0 because Lz doesn't depend on the z-component of the oscillator. And this has minus h bar. So here you see actually, the whole structure of the L equal 1 multiplet. We said that we have at this level L equals 1. And indeed, for L equals 1, you expect the state with Lz equal plus 1, 0, and minus 1. So you see the whole thing. For n equals 2, what do you get? Well, you see a state, for example, of a right dagger a right dagger on the vacuum. And that has Lz equals 2 h bar. And therefore, you must have-- since you cannot have states with higher Lz, you cannot have a state, for example, here with Lz equal 3. So you cannot have an L equal 3. In fact, for any N that you build states, you can only get states with whatever N is is the maximum value that Lz can have, which is something I want to illustrate just generically for a second. So in order to show that, let me go back here and exhibit for you a little of the general structure. So suppose you're building now with N equal n. The total number is N. So you have a state with a right dagger to the n on the vacuum. And this is the state with highest possible Lz because all the oscillators are aR dagger. So Lz is the highest. And highest Lz is, in fact, n h bar. Now, let's try to build a state with a little bit less Lz. You see, if this is a multiplet, this has to be a multiplet with some amount of angular momentum. So it's going to go from Lz equal n, n minus 1, up to minus n. There are going to be 2n plus 1 states of this much angular momentum because this has to be a multiplet. So here you have a state with one unit less of angular momentum, a right dagger to the n minus 1, times an az dagger. I claim that's the only state that you can build with one unit less of angular momentum in the z-direction because I've traded this aR for an az. So this must be the second state in the multiplet. This multiplet with highest value of L, which is equal to n, corresponds to an angular momentum l, little l, equals n. And then, it must have this 2n plus 1 states. And here is the second state. So this is Lz equals nh bar. And here, n minus 1 h bar. And I don't think there's any other state at that level. Let's lower the angular momentum once more. So what do we get? a right dagger n minus 2 az dagger squared. That's another state with one less angular momentum than this. This, in fact, has n minus 2 times h bar. Now, is that the unique state that I can have with two units less of angular momentum? No. What is the other one? AUDIENCE: aR to the n minus 1 a l? PROFESSOR: Correct, that lowers it. third so here you have aR to the n minus 1 a left on the vacuum. That's another state with two units less of angular momentum. So in this situation, a funny thing has happened. And here's why you understand the jump of 2. This state, you actually-- if you're trying to build this multiplet, now you have two states that have the same value of Lz. And you actually don't know whether the next state in the multiplet is this, or that, or some linear combination. It better be some linear combination. But the fact is that at this level, you found another state. So this multiplet will go on and it will be some linear combination. Maybe this diagram doesn't illustrate that. But then you will have another state here. So some other linear combination that builds another multiplet. And this multiplet has two units less of angular momentum. And that explains why this diagram always jumps. It always jumps 2. And you could do that here. If you tried to write the next things here, you will find two states that you can write. But if you go one lower, you will find three states. Which means that at the next level, you built another-- you need another state with two units less of angular momentum each time. So pretty much that's it. That illustrates how this diagram happens. The only thing we haven't answered, and you will see that in an exercise, how could I have understood from the beginning that this would have happened rather than building it this way that there's this thing? And what you will find is that there's some operators that commute with the Hamiltonian that move you from here to here. And that explains it all. Because if you have an operator that commutes with the Hamiltonian, it cannot change the energy. And if it changes the value of L, it explains why it happened. So that's something that you need to discover, what are these operators. I can give you a hint. An operator for the type ax dagger ay destroys a y oscillator, creates an x one. It doesn't change the energy because it adds one thing and loses one. So this kind of thing must commute with the Hamiltonian. And these are the kind of objects-- there are lots of them. So surprising new things that commute with the Hamiltonian. And there's a whole hidden symmetry in here that is generated by operators of this form. So it's something you will see. Now, the last 15 minutes I want to show you what happens with hydrogen. There's a similar phenomenon there that we're going to try to explain in detail. So a couple of things about hydrogen. So hydrogen H is equal to p squared over 2m minus e squared over r. There's a natural length scale that people many times struggle to find it, the Bohr radius. This does it come from here? How do you see immediately what is the Bohr radius? Well, the Bohr radius can be estimated by saying, well, this is an energy but it must be similar to this energy. So p is like h over a distance. So let's call it a0. So that's p squared. m is the reduced mass of the proton electron system roughly equal to the electron mass. And then you set it equal to this one because you're just doing units. You want to find the quantity that has units of length and there you got it. That's the famous Bohr radius. p is h bar over a distance, therefore this thing must be an energy. It must be equal to this. And from this one, you can solve for a0. It's h squared over m e squared. The 1 over e squared is very famous and important. It reflects the fact that if you had the interaction between the electron and the proton go to 0, the radius would be infinite. As it becomes weaker and weaker the interaction, the radius of the hydrogen atom blows up. So this is about 0.529 angstroms, where an angstrom is 10 to the minus 10 meters. And what is the energy scale? Well, e squared over a0 is the energy scale, roughly. And in fact, e squared over 2a 0, if you wish, is a famous number. It's about 13.6 ev. So how about the spectrum? And how do you find that? Well, there's one nice way of doing this, which you will see in the problem, to find at least the ground state. And it's a very elegant way based on factorization. Let we mention it. It is called Hamiltonian. It can be written as a constant gamma plus 1 over 2m sum over k pk plus i beta xk over r times pk minus i beta xk over r. It's a factorized version of the Hamiltonian of a hydrogen atom. Apparently, not a well-known result. Professor [? Jackiw ?] mentioned it to me. I don't think I've seen it in any book. So there's a constant beta and a constant gamma for which this becomes exactly that one. So gamma and beta to be determined. And you have to be a little careful here when you expand that this term and this term don't commute. And this and this don't commute. But after you're done, it comes out. And then, the ground state wave function is-- since this is an operator and here it's dagger, the ground state wave function is-- the lowest possible energy wave function is one in which this would kill the wave function. So pk minus i beta xk over r should kill the ground state wave function. And then the energy, E ground, would be equal to precisely this constant gamma. And you will show, in fact, that yes, this has a solution. And that's the ground state energy of the oscillator of the hydrogen atom, of course. So this looks like three equations, pk with k equals 1 to 3. But it reduces to 1 if the state is spherically symmetric. So it's a nice thing and it gives you the answer. Now, the whole spectrum of the hydrogen atom is as interestingly degenerate as one of the three-dimensional harmonic oscillator. And a reminder of it is that-- should I go here? Yes. You have here energies and here l's. l equals 0 you have one state. l equals 1 you have another state that's here. But actually, l equals 0 will have another state. And then it goes on like that, another state here, state here, state here for l equals 2. And the first state is here. The first state of this one aligns with this one. The first state of that aligns with that. So they go like that, the states just continue to go exactly with this symmetry. So let me use label that is common, to call this the state nu equals 0 for L equals 0. Nu equals 1. Nu equals 2. Nu equals 3. This is the first state with L equals 1 is here. So we'll call it nu equals 0. Nu equals 1. New equals 2. The first state here is nu equals 0, nu equals 1. And then the energies. You define n to be nu plus l plus 1. Therefore, this corresponds to n equals 1. This corresponds to n equals 2. That corresponds to nu can be 1 and l equals 0 or nu can be 0 and l equals 1. This is n equal 3, and things like that. And then the energies of those states, nl is, in fact, minus z squared. Well, forget the z squared. e squared over 2 a0 1 over n squared. So the only thing that happens is that there's a degeneracy, complete degeneracy. Very powerful degeneracy. And then, l can only run up to-- from 0, 1, up to n minus 1 in these variables. So this is the picture of hydrogen. So you've seen several pictures already-- the square well, the three-dimensional harmonic oscillator, and the hydrogen one. Each one has a different picture. Now, in order to understand this one-- this one is not that difficult. But the one of the hydrogen is really more interesting. It all originates with the idea of what is called the Runge-Lenz vector, which I'm going to use the last five minutes to introduce. And think about it a little. So it comes from classical mechanics. So we have an elliptical orbit, orbits, and people figured out there was something very funny about characterizing elliptical orbit. So consider a Hamiltonian, which is p squared over 2m plus v of r, a potential. The force, classically, would be minus the gradient of the potential, which is minus the derivative of the potential with respect to r times the r unit vector. Now classically-- this all begins classically. Except for spin 1/2 systems, classical physics really tells you a lot of what's going on. So classically, dp dt is the force and it's minus v prime over r r vector over r. And dl dt, the angular momentum, it's a central potential. The angular momentum is 0. It's rate of change is 0. There's no torque on the particle, so this should be 0. Now, the interesting thing that happens is that this doesn't exhaust the kind of things that are, in fact, conserved. So there is something more that is conserved. And it's a very surprising quantity. It's so surprising that people have a hard time imagining what it is. I will write it down and show you how it sort of happens. Well, you have to begin with p cross L. Why you would think of p cross l is a little bit of a mystery, but it's an interesting thing. Now, here is a computation that will be in the notes that you can try doing. And it takes a little bit of work, but it's algebra. If you compute this and do a fair amount of work, like five, six lines-- I would suspect it's fairly non-trivial to do it if you don't see how it's being done, but it will be in the notes-- you get the following thing. Just by manipulating the time derivative of p cross L, you get this. Which is equal to m times the potential differentiator times r squared times the time derivative of this. So time derivative, time derivative. You can get the conservation if this is a constant. So when is this a constant? If this is some constant, say, e squared, you would get a conservation. But what is v prime equals e squared over r squared? It would give you that v of r is essentially minus e squared over r. That's the potential of hydrogen. Or the 1 over r potential, 1 over r squared force field. So in 1 over r potentials, this is a number. And then you get an incredible conservation law, d dt of p cross L minus m e squared r hat over r is equal to 0. So something fairly unexpected that something like this could be conserved. So actually, you can try to figure out what this is. And there's two neat-- first, one thing that people do, which is convenient, is to make this into unit-free vector. So define R to be p cross L over m e squared minus r vector over r. This has no units. And it's supposed to be conserved. Now, one thing you will check in the homework is that this is conserved quantum mechanically as well. That is, this is an operator that commutes with a Hamiltonian. Very interesting calculation. This is a Hermitian operator, so you will have to Hermiticize the p cross L to do that. But it will commute with the Hamiltonian. But what I want to finish now is with your intuition as to what this is. And this was a very interesting discovery, this vector. In fact, people didn't appreciate what was the role of this vector for quite some time. So apparently, it was discovered and forgotten, and discovered and forgotten like two or three times. And for us, it's going to be quite crucial because I said to you that this operator commutes with the Hamiltonian. So actually, you will get conservation laws and will help us explain the degeneracy of the hydrogen atom. So it will be very important for us. Now, how does this look? First of all, if you had a circular orbit, how does it work? Have a circular orbit. Let's see, p is here, L is out of the board. p cross L is here over m e squared. And the radial vector is here, the hat vector. So the sum of these two vectors p cross L and the radial vector must be conserved. But how could it be? If they don't cancel, it either points in or points out. And then it would just rotate and it would not be conserved. So actually, for a circular orbit, you can calculate it. See the notes. Actually, it's an easy calculation. And you can verify that this vector is, in fact, precisely opposite this. And it's 0. So you say, great. You discover something that is conserved, but it's 0. No. The thing is that this thing is not 0 for an elliptical orbit. So how can you see that? Well here at this point, p is up here. L is out. And p cross L, just like before, is out and r hat is in. And you say, well, OK. Now the same problem. If they don't cancel, it's going to be a vector and going to rotate. But it has to be conserved. So actually, let's look at it here. Here, the main thing of an ellipse, if you have the focus here, is that this line is not-- the tangent is not horizontal. So the momentum is here. L is out of the blackboard, but p cross L now is like that. And the radial vector is here. And they don't cancel. So the only thing that can happen-- since this is vertical, this is vertical. It's a little bit to the left-- is that the r vector must be a little vector horizontal here. Because the sum of this vector and this vector-- it has to be horizontal. Here we don't know if they can cancel. But if they don't cancel, it's definitely horizontal. We know it's conserved, so it must be horizontal here. So it points in. So the Runge-Lenz vector r points in. And it's, in fact, that. So here you go, this is a vector that is conserved. And its properties that is really about the axis of the ellipse. It tells you where the axis is. In Einstein's theory of gravity, the potential is not 1/r and the ellipsis [? precess ?] and the Runge vector is not conserved. But in 1/r potentials, it is conserved. The final thing-- sorry for taking so long-- is that the magnitude of r is precisely the eccentricity of the orbit. So it's a really nice way of characterizing the orbits and we'll be using it in the next lecture. See you on Wednesday.

MIT_805_Quantum_Physics_II_Fall_2013

7_Linear_Algebra_Vector_Spaces_and_Operators_continued.txt

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare ocw.mit.edu. PROFESSOR: OK, so let's get started. I just wanted to make one announcement before we start the lecture. So Prof. Zwiebach is a way again today, which is why I'm lecturing. And his office hours he's obviously not going to have, but Prof. Harrow has kindly agreed to take them over. So today I'll have office hours four to five, and then Prof. Harrow will have office hours afterwards five to six. So feel free to come and talk to us. So today we're going to try and cover a few things. So we're going to spend a little bit of time talking about eigenvalues and vectors, which we've-- finishing this discussion from last time. Then we'll talk about inner products and inner product spaces. And then we'll talk about-- we'll introduce Dirac's notation, some of which we've already been using. And then, depending on time, we'll also talk a little bit more about linear operators. OK? So let's start with where we were last time. So we were talking about T-invariant subspaces. So we had U is a T-invariant subspace if the following is satisfied. If T of U is equal to-- if this thing, which is all vectors that are generated by T from vectors that live in U. So if T is inside U itself. OK? And we can define this in general for any U. However, one class of these invariant subspaces are very useful. So if we take U to be one dimensional. OK/ and so that really means that U I can write as some whatever field I'm defining my vector space over. Every element of this subspace U is just some scalar multiple of a single vector U. So this is a one dimensional thing. Now if we have a T-invariant subspace of this one-- if this is going to be a T-invariant objective, then we get a very simple equation that you've seen before. So we're taking all vectors in U acting on them with T, and if it stays within U, then it has to be able to be written like this. So we have some operator acting on our vector space producing something in the same vector space, just rescaling it. OK? For sum lambda, which we haven't specified. And you've seen this equation before in terms of matrices and vectors, right? This is an eigenvalue equation. So these are eigenvalues and these are eigenvectors. But now they're just an abstract version of what you've discussed before. And we'll come back to this in a moment. One thing that we just defined at the end is the spectrum of an operator. The spectrum of T is equal to all eigenvalues of that operator. And so later on these will become-- this object will become important. Let's just concentrate on this and ask what does it mean. So if we have lambda being an eigenvalues, what does that tell us? What does this equation tell us? Well, it tells us that on U. So all I'm doing is taking this term over to the other side of the equation and inserting the identity operator I. So this is in itself an operator now, right? And so this tells us also that this operator, because it maps something that's non-zero to the null vector, this is not injective, OK? And you can even write the null space of T-- of T minus I lambda is equal to all eigenvectors with eigenvalue lambda. OK? So every eigenvector with eigenvalue lambda, T acting on it is just going to give me lambda times the eigenvector again, and so this will vanish. So for all eigenvectors with that eigenvalue. And we've previously seen that, if something is not injective, it's also not invertible, right? So this lets us write something quite nice down. So there's a theorem. Let me write it out. So if we let T is in the space of linear operators acting on this vector space v, and we have a set of eigenvalues, lambda 1, lambda 2, lambda n, distinct eigenvalues, eigenvalues of T, and the corresponding eigenvectors, which we will call U. OK, so the sum set U1, U2, up to Un with the correspondence by their label. So then we know that this list is actually a linearly independent set. So we can prove this one very quickly. So let's do that. So let's assume it's false. So the proof is y contradiction, so assume it's false. And what does that mean? Well, that means that there is a non-trivial relation. I could write down some relation C1U1 plus C2U2 plus CkUk equals 0 without all the C's being 0. And what we'll do is we'll actually say OK, let's do let-- so we'll let there be a k, a value of k that's less than or equal to n, such that this holds for Ci not equal to 0. So we're postulating that there is some linear dependence on some of these things. So what we can do is then act on this vector here with T minus lambda k times the identity acting on this. So this is C1U1 plus dot dot dot plus CkUk. OK? And what do we get here? So we're going to get, if act on this piece of it, this is an eigenvector, so T acting on this one would just give us lambda 1, right? And so we're going to get products of lambda 1 minus lambda k for this piece, et cetera. So this will give us C1 lambda 1 minus lambda k U1 plus dot dot dot up to the Ck minus 1 lambda k minus 1 minus lambda k Uk minus 1. And then when we act on this one here, so this one has an eigenvalue-- the eigenvalue corresponding to the eigenvector is lambda k, so that last term gets killed. So we get plus 0 lots of Uk. And we know this is still 0. And now we've established, in fact, these things here are just numbers. All of these things. So we've actually written down a relation that involves less than k. Actually, I should have said this. Let there be a least k less than or equal to, and such that we have linear dependence. But what we've just shown is that, in fact, there's a smaller space that's also linear independent, right? So we've contradicted what we assumed to start with. And you can just repeat this procedure, OK? And so this is a contradiction. And so, in fact, there must be no non-trivial relation even for k equals n between these vectors, OK? Another brief theorem that we won't prove, although we'll sort of see why it works in a moment, is, again, for T in linear operators on v with v being a finite dimensional complex vector space. OK? There is at least one eigenvalue. So I guess for this-- so T has at least one eigenvalue. Now remember, in the last lecture, we looked at a matrix, two by two matrix, that was rotations in the xy-plane and found there were, in fact, no eigenvalues. But that's because we were looking at a real vector space. So we were looking at rotations of the real plane. So this is something that you can prove. We will see why it's true, but we won't prove it. And so one way of saying this is to go to a basis. And so everything we've said so far about eigenvalues and eigenvectors has not been referring to any particular basis. And in fact, eigenvalues are basis independent. But we can use a basis. And then we have matrix representations of operators that we've talked about. And sort of this operator equation, or the operator statement that T minus lambda I-- so as operator statement T minus lambda I U equals 0 is equivalent to saying that-- well, we've said it here. We said that it's equivalent to saying that it's not invertible. This operator is not invertible. But that's also equivalent to saying that the matrix representation of it in any basis is not invertible. And by here we just mean inverses as in the inverses that you've taken the many matrices in your lives. And so what that means then, if-- I'm sure you remember. If a matrix is not invertible, that means it has a vanishing determinant. So it has debt of this. Now you can think of this as a matrix. This determinant has to be 0. And remembering we can write this thing out. And so it has lambdas on the diagonal, and then whatever entries T has wherever it wants. This just gives us a polynomial in lambda, right? So this gives us some f of lambda, which is a polynomial. And if you remember, this is called the characteristic polynomial. Characteristic. Right? And so we can write it, if we want, as just some f of lambda is equal to just, in this case, it's going to be just lambda minus some lambda 1. I have to be able to write it like this. I can just break it up into these terms here, where the lambda I's, the 0's of this polynomial are, in general, complex and can be repeated. Now what can happen is that you have, in the worst case -- I don't know if it's the worst case, but in one case, you could have all of the singularities-- all of the the 0's being at the same place. And you could have a eigenvalue that is in full degenerate here. Right? So if we, say, have lambda 1 occurring twice in this sequence, then we set out to a degenerate eigenvalue. And in principle, you could have just a single eigenvalue that's in full degenerate. But you can always write this. There has to be one lambda there at least, one lambda I there at least. And so you can see why this is true, right? Now if you're in a real vector space, you don't get to say that, because this polynomial may only have complex roots, and they're not part of the space you're talking about. OK? So it can be repeated, and this is called degeneracy. OK, so are there any questions? AUDIENCE: Can you turn it? It should be lambda I minus T, just so that it matches the next line. PROFESSOR: Thank you, OK. Thank you. I could have flipped the sign on the next line as well. So any other questions? No? OK, so let's move on and we can talk about inner products. And so first, what is an inner product? So an inner product is a map, but it's a very specific map. So an inner product on a vector space V is a map from V cross V to the field, F. And that's really what it's going to be. Now who has seen an inner product somewhere? OK, what do we call it? AUDIENCE: Dot product. PROFESSOR: Dot product, right. So we can learn a lot from thinking about this simple case, so the motivation for thinking about this is really the dot product. So we have a vector space Rn. OK? And on that vector space, we might have two vectors, a, which I'm going to write as a1. a2 dot dot dot. a2 dot dot dot. an, and b. So we have two vectors, and these are in vector space V. Then we can define the dot product, which is an example of one of these in a product. So a dot b. We can even put little vectors over these. And so our definitions that we've used for many years is that this is a1 b1 plus a2 b2 plus dot dot dot an bn. And you see that this does what we want it. So it takes two vectors which live in our vector space. And from that, you get a number, right? So this lives in R. So this is a nice example in a product. And we can look at what properties it gives us. So what do we know about this dot product? Well, we know some properties that it has is that a dot b. So it doesn't care which order you give the arguments in, all right? Also, if I take the same vector, I know that this is got to be greater than or equal to 0, right? Because this is going to be our length. And the only case where it's 0 is when the vector is 0. Well we can write this. a dotted in to, say, b to 1 b1 plus b to 1 b2. So this b2's are real numbers, and these b's are vectors, right? So this thing we can just write is equal to b to one a dot b1 plus b to 2 a do b2. And make them vectors everywhere. OK, so we've got three nice properties. And you can write down more if you want, but this will be enough for us. And the other thing that we can do with this is we can define the length of a vector, right? So we can say this is for this defines a length. And more generally, we only call this the norm of the vector. And that, of course, you know is that mod a squared is just equal to a dot a, all right? So this is our definition of the norm. OK so this definition over here is really by no means unique in satisfying these properties. So if I wrote down something where, instead of just say a1 b1 plus a1 b2 et cetera, I wrote down some positive number times a1 b1 times some other positive number a2 b2, et cetera, that would also satisfy all of these properties up here. So it's not unique. And so you could consider another the dot product, which we would write as just c1 a1 b1 plus c2 a2 b2 plus some cn an bn, where the c's are just positive real numbers. That would satisfy all of the things that we know about our standard dot product. But for obvious reasons, we don't choose to do this, because it's not a very natural definition to put these random positive numbers along here. But we could. And I guess one other thing that we have is the Schwarz inequality. And so this is the a dot b. So the absolute value of the dot product of a dot b is less than or equal to the product of the norms of the vectors, right? And so one of the problems in the piece is to consider this in the more abstract sense, but this is very easy to show for real vectors, right? So this is all very nice. So we've talked about Rn. What we really are going to worry about is complex vector spaces. And so there we have a little problem. And the problem comes in defining what we mean by a normal, right? Because if I say now that this vector has complex components and write this thing here, I'm not guaranteed that this is a real number, right? And so I need to be a little bit careful. So let's just talk about complex spaces. And we really want to have a useful definition of a length. So let's let z be in this thing, in interdimensional complex space. So really my z is equal to z1 z2 zn, where the zI line as being in c, right? So how can define a link for this object? Well, we have to do it sort of in two steps. So already know how to define the length for a complex number, right? It's just the absolute value, the distance from the origin in the complex plane. But now we need to do this in terms of a more complicated vector space. And so we can really think of this as equal to the sum of the squares of z1, of the absolute values of these complex numbers. OK? Which if we write it out, looks like z1 star z1 plus. OK? And so we should now, thinking about the inner product, we should be thinking that the appearance of complex conjugation is not entirely unnatural. So if we ask about the length of a vector here, then that's going to arise from an inner product. OK? This object we want to arise from our inner product. So we can now define our general in a product with the following axioms. So firstly, we want to basically maintain the properties that we've written down here, because we don't want to make our dot product not being in an inner product anymore. That'd be kind of silly. So let's define our inner product in the following way. I'm going to write it in a particular way. So the inner product is going to be, again, a map. And it's going to take our vector space, two elements of the vector space to the field. And I'm in a complex vector space. So it's a map that I'm going to right like this that takes v cross v to c. OK? And what I mean here is you put the two elements of your vector space in these positions in this thing, OK? And so really a b is what I mean by this. Where a and b-- so let me write it this way. So this thing is in c for a and b are in the v, right? So these things dots are where I'm going to plug-in my vectors. And so this inner product should satisfy some axioms. And they look very much like what we've written here. So the first one is a slight modification. We want that a b is equal not to b a, but to its complex conjugate, OK? And this is related to what I was discussing here. But from this, we can see that the product of a with itself is always real, because it and its complex conjugate are the same. So we know that a a is real. And we're also going to demand a definition of this inner a product that this is greater than or equal to 0. And it's only 0 if a equals 0. Right? So that's pretty much unchanged. And then we want the same sort of distributivity. We do want to have that a inner producted with B to 1 b plus B to 2 b2 should be equal to B to 1 a b1 plus B to 2 a b2 where the [INAUDIBLE] are just complex numbers, right? And that's what we need to ask of this. And then we can make a sensible definitions of it that will give us a useful norm as well. Now I'll just make one remark. This notation here, this is due to Dirac. And so it's very prevalent in physics. You will see in most purely mathematical literature you will see this written just like this. So let me write it as a b and put these things in explicitly. And sometimes you'll even see a combination of these written like this, OK/ But they all mean the same thing. Compared to what we've written up here, this seems a little asymmetric between the two items, right? Well firstly, these are isometric. And then down here we've shown something about that we demand something about the second argument, but we don't demand the same thing about the first argument. So why not? Can anyone see? I guess what we would demand is exactly the same thing the other way around. So we would demand another thing that would be sort of alpha 1 a plus alpha 2 a2 b is equal to-- well, something like this. Well, we would actually demand this. a1 b. But I don't actually need to demand that, because that follows from number one, right? I take axiom one, apply it to this, and I automatically get this thing here. OK? And notice what's arisen is-- actually let's just go through that, because you really do want to see these complex conjugates appearing here, because they are important. So this follows. So 1 plus 3, imply this. Let's just do this. So let's start with this expression and start with this piece. And we know that this will then be given by axiom one by b alpha 1 a 1 plus alpha 2 a2 complex conjugate, all right? And then by this linearity of the second argument we can now distribute this piece, right? We can write this is alpha 1 b a1 plus alpha 2 b a2, all complex conjugated. Which let's put all the steps in. Is alpha 1 star and this one star. And then, again, by the first argument, by the first axiom, we can flip these and get rid of the complex conjugation. And that gives us this one up here, right? So we only need to define this linearity, distributive property on one side of this thing. We could have chosen to define it here and wouldn't have needed that one, but we didn't. OK, so let's look at a couple of examples. And the first one is a finite dimensional example. And we're going to take v is equal to cn. And our definition is going to be a pretty natural generalization of what we've written down before. So a and b are elements of cn. And this is just going to be a1 star b1 plus a2 star b2 plus an star bn. Another piece of chalk. So the only difference from dot product in real vector space is that we've put this complex conjugates here. And that you can check satisfies all of these axioms. Another example is actually an example of an infinite dimensional vector space. Let's take v is the set of all complex functions, all f of x that are in c with x living in some finite interval. OK? And so a natural norm to define on this space-- and this is something that we can certainly talk about in recitations-- is that if I have f and g in this vector space v, then f g I'm going to define-- this is my definition of what the dot product is-- is the integral from 0 to l f star of x g of x dx. OK? If you think of this as arising from evaluating f at a set of discrete points where you've got a finite dimensional vector space, and then letting the space between those points go to 0, this is kind of the natural thing to arise. It's really an integral as a limit of a sum. And over here, of course, I could write this one as just the sum over i of ai star bi. i equals 1 to n. And so this is the integral is infinite dimensional generalization of the sum, and so we have this. And that might be something to talk about in recitations. OK? So we've gone from having just a vector space to having a vector space where we've added this new operation on it, this inner product operation. And that lets us do things that we couldn't do before. So firstly, it lets us talk about orthogonality. Previously we couldn't ask any question about two objects within our vector space. This let's us ask a question about two objects. So if we have the inner product a b in some vector space V, then if this is 0, we say they're orthogonal. We say that the vectors a and b are orthogonal. And I'm sure you know what orthogonal means in terms of Rn, but this is just the statement of what it means in a abstract vector space. This is the definition of orthogonality. So this is one thing if we have a set of vectors. e1 e2 en, such that ei ej is equal to delta ij, chronic to delta ij, this set is orthonormal. Again, a word you've seen many times. OK, so we can also define the components of vectors now in basis dependent way. We're going to choose ei to be a set of vectors in our vector space V. We previously had things that form a basis, a basis of V. And if we also demand that they're orthonormal, then we can-- well, we can always decompose any vector in V in terms of its basis, right? But if it's also orthonormal, then we can write a, which is a is some vector in V. a is equal to the sum over i equals 1 to n of some ai ei. So we can do that for any basis. But then we can take this vector and form its inner product with the basis vectors. So we can look at what ek a is, right? So we have our basis vectors ek, and we take one of them and we dot product it into this vector here. And this is straightforward to c. This is going to be equal to the sum over i equals 1 to n ai. And then it's going to be the inner product of ek with ei, right? Because of this distributive property here. OK? But we also know that, because this is an orthonormal basis, this thing here is a delta function, delta ik, right? And so I can, in fact, do this sum, and I get and this is equal to ak. And so we've defined what we mean by a component of this vector in this basis ei. They're defined by this inner product. So we can also talk about the norm, which I think, unsurprisingly, we are going to take the norm to be, again, equal to this, just as we did in Rn, but now it's the more general definition of my inner product that defines our norm. And because of our axiom-- so because of number two in particular, this is a sensible norm. It's always going to be greater than or equal to 0. OK? And conveniently we can also change this Schwarz inequality. So instead of the one that's specific to Rn, that becomes a b. All right, so let's cross that one out. This is what it becomes. And in the current p set, you've got to prove this is true, right? We can also write down a triangle inequality, which is really something that norms should satisfy. So the norm of a plus b should be less than or equal to the norm of a plus the norm of b. And the R3 version of this is the longest side of a triangle is shorter than the two shorter sides, right? So this is fine. OK, so you might ask why we're doing all of this seemingly abstract mathematics. Well, so now we're in a place where we can actually talk about the space where all of our quantum states are going to live. And so these inner product space-- these vector spaces that we've given an inner product. We can call them inner product spaces. So we have a vector space with an inner product is actually we call a Hilbert space. And so this needs a little qualifier. So if this is a finite dimensional vector space, then this is straight. It is just a Hilbert space. Let me write it here. So let's write it as a finite dimensional vector space with an inner product is a Hilbert space. But if we have an infinite dimensional vector space, we need to be a little bit careful. For an infinite dimensional vector space, we again need an inner product. We need to make sure that this space is complete. OK? And this is a kind of technical point that I don't want to spend too much time on, but if you think about-- well, let me just write it down. Vector space. Let me write it here. And I haven't defined what this complete vector space means. But if we have an infinite dimensional vector space that is complete or we make it complete, then we have an inner product. We also get a Hilbert space. And all of quantum mechanical states live in a Hilbert space. AUDIENCE: [INAUDIBLE]. PROFESSOR: Yes, that's true. So how's that? So we need to define what we mean by complete though, right? So I don't want to spend much time on this. But we can just do an example. If we take the space of-- let V equal the space of polynomials on an interval 0 to L, say. So this means I've got all pn's. P0 plus p1x plus pn xn. There are things that will live in the completed vector space that are not of this form here. So for example, if I take n larger and larger, I could write down this polynomial. I could write pn of x is the sum over i equals 1 up to n of i-- x to the i over i factorial, right? And all of pn's live in this space of polynomials. But their limit, as n becomes large, there's a sequence of these call it a cushy sequence that, as n goes to infinity, I generate something that's actually not a polynomial. So I generate e of x, which lives in the completion of this, but itself not a polynomial. Don't worry about this too much, but in order to really define a Hilbert space, we have to be a little bit careful for infinite dimensional cases. OK, so a few more things that we can do to talk about. Well how do we make a orthonormal basis? So I presume you've all heard of Gram-Schmidt? The Gram-Schmidt procedure? Yep. OK, so that's how we make a orthonormal basis. And just the way you do it in R3, you do it the same way in your arbitrary vector space. So we have the Gram-Schmidt procedure. So you can define this-- so we have a list v1, v2, vn are just vectors in our vector space that are linearly independent. So we can construct another list. There's also orthonormal, so it's a very useful thing for us to have. And so you could define this recursively. You can just write that ej is equal to vj minus the sum over i less than j of ei. So this thing divided by its length. And so by the sum, you're orthogonalizing ej versus all of the previous ei's that you've already defined. And then you normalize it by dividing by its length, right? So that's something that's very useful. And the last thing I want to say about these inner product spaces is that we can use them-- these inner products at least, is that we can use them to find the orthogonal complement of something, of anything really. So let's let u-- so we have a vector space V, and I can just choose some things in that and make a set. So u is the set of v that are in V. So it doesn't need to be a subspace. It's just a set. For example, if v Rn, I could just choose vectors pointing along two directions, and that would give me my set. But that's not a subspace, because it doesn't contain some multiple of this vector times some multiple of this vector, which would be pointing over here. So this is just a set so far. We can define u perpendicular, which we'll call the orthogonal complement of u. And this is defined as u perpendicular is equal to the set of v's in V such that v u is equal to 0 for all u in U. All of the things that live in this space are orthogonal to everything that lives in U. OK And in fact, this one is a subspace automatically. So it is a vector space. So if I took my example of choosing the x direction and y direction for my set here, then everything perpendicular to the x direction and y direction is actually everything perpendicular to the xy-plane, and so that is actually a subspace of R3. And so there's a nice theorem that you can think about, but it's actually kind of obvious. So if u is a subspace, then I can actually write that V is equal to the direct sum of U plus its orthogonal complement, OK? So that one's kind of fairly straightforward to prove, but we won't do it now. OK, so in the last little bit, I want to talk more about this notation that I've introduced, that Dirac introduced. What can we say? If I can find a [INAUDIBLE] here. Are there any questions about this? Yep. AUDIENCE: So when we find space and the idea of basis balance, why is that [INAUDIBLE] decompose things into plane waves when we're not actually [INAUDIBLE]? PROFESSOR: So it's because it's-- basically it works. Mathematically, we're doing things that are not quite legitimate. And so we can generalize the Hilbert space a little bit, such that these non normalizable things can live in this generalized space. But really the answer is that it works, but no physical system is going to correspond to something like that. So if I take plane waves, that's not a physically realizable thing. It gives us an easy way to, instead of talking about some wave packet that some superposition of plane waves, we can talk about the plane waves by themselves and then form the wave packet afterwards, for example. Does that answer the question a little bit at least? Yep. AUDIENCE: If p could be written as a sum of U [INAUDIBLE], why is U not [INAUDIBLE]? PROFESSOR: Well, just think about the case that I was talking about. So if we're looking at R3 and we take U to be the set the unit vector in the x direction, the unit vector in the y direction, that's not a subspace, as I said, because I can take the unit vector in the x direction plus the unit vector in the y direction. It goes in the 45 degree direction. And it's not in the things I've written down originally. So then if I talk about the subspace, the things spanned by x hat and y hat, then I have a subspace. It's the whole xy-plane. And the things are orthogonal to it in R3 are just the things proportion to z hat. And so then I've got the things in this x hat and y hat, and the thing that's in here is z hat. And so that really is the basis for my R3 that I started with. That contains everything. And more generally, the reason I need to make this a subspace is just because-- so I define U by some set of vectors that I'm putting into it. The things that are orthogonal to that are automatically already everything that's orthogonal to it, so there's no combination of the things in the orthogonal complement that's not already in that complement. Because I'm saying that this is everything in V that's orthogonal to these things in this subspace. So I could write down some arbitrary vector v, and I could aways write it as a projection onto things that live in here and things that don't live in this one, right? And what I'm doing by defining this complement is I'm getting rid of the bits that are proportional to things in this, OK? All right any-- yep? AUDIENCE: So an orthogonal complement is automatically a subspace? PROFESSOR: Yes. AUDIENCE: But that doesn't necessarily mean that any random collection of vectors is a subspace. PROFESSOR: No. All right, so let's move on and talk about the Dirac's notation. And let's do it here. So three or four lectures ago, we started talking about these objects, and we were calling them kets, right? And they were things that live in our vector space V. So these are just a way of writing down our vectors, OK? So when I write down the inner product, which we have on the wall above, one of the bits of it looks lot like this. So we can really think of a b, the b being a ket. We know that b is a vector, and here we're writing in a particular way of writing things in terms of a ket. And what we can do is actually think about breaking this object, this inner product up into two pieces. So remember the dot product is taking two vectors, a and b. One of them, we already have written it like a vector, because a ket is a vector. What Dirac did in breaking this up is he said, OK, well this thing is a bracket, and so he's going to call this one a ket, and this is a bra. So this object with something it. The things inside these you should think of as just labeling these things. OK? Now we already know this thing here. So these kets are things that live in-- I should say this is direct notation. OK, so we already know these kets are things that live in the vector space. But what are the bras? Well, they're not vectors in V. So b is a vector, so maybe I should've called this one b to be a little less confusing. So b is a ket, and this is something that lives in our vector space V. This inner product we're writing in terms of bra and a ket. The bra, what does it actually do? So I'm going to use it to make this inner product. And so what it's doing is it's taking a vector and returning a complex number. The inner product takes v cross v goes to c. But if I think of it as the action of this bra on this ket, then the action is that this bra eats a vector and spits back a complex number, OK? So a is actually a map. OK? So these bras live in a very different place than the kets do. Although they are going to be very closely related. So firstly, it's not in V. You should be careful if you ever say that, because it's not right. We actually say that it belongs to a dual space, which we label as V star, because it is very dependent on V, right? It's mapped from V to c. And I shouldn't even say this is a linear map. Now what is V star? Well, at the moment it's just the space of all linear maps from V to c. Me But it itself is a vector space. So we can define addition of these maps. We can define addition on V star and also a scalar modification of these maps. And so what that means is that I can define some bra w That's equal to alpha lots of another one plus B to b. And all of these live in this V star space. Let me write that explicitly. So a b and w live in V star, OK? And the way we define this is actually through the inner product. We define it such that-- so I take all vectors v in the vector space big V, and the definition of w is that this holds. And then basically from the properties of the inner product, you inherit the vector structure, the vector space structure. So this tells us V star is a vector space. Let's go over here. And there's actually a correspondence between the objects in the original vector space V and those that live in V star. So we can say for any v in V, there's a unique-- I should write it like this. Any ket v in the vector space, there is a unique bra, which I'm also going to label by v, and this lives in V star. And so we can show uniqueness by assuming it doesn't work. So let's assume that there exists a v and a v prime in here such that v-- so we'll assume that this one is not unique, but there are two things, v and v prime. And then we can construct-- from this, I can take this over to this side here, and I just get that v w minus v prime w is equal to 0, which I can then use the skew symmetry of these objects to write as w v minus w v prime star. So I've just changed the order of both of them. And then I can use the property of kets. I can combine them linearly. So I know this is equal to w v minus v prime star. And essentially, that's it, because I know this has to be true for every w in the vector space V. So this thing is equal to 0. And so the only thing that can annihilate every other vector is going to be 0 for my definition, in fact, of the inner product. So this implies that v minus v prime equals 0, the null vector, which implies that v equals v prime. So our assumption was wrong, and so this is unique. OK, let's see. And so we actually have really a one to one correspondence between things in the vector space and things in the joule space, OK? And so we can actually label the bras by the same thing that's labeling the kets. So I can really do what I've done in the top line up there and have something-- everything is labeled by the same little v. Both the thing in the big vector space, big V, and the thing in V star are label by the same thing. And more generally, I could say that v-- so there's a correspondence between this thing and this thing. And notice the stars appearing here. They came out of how we define the inner product. OK, so really, in fact, any linear map you write down, any linear map like this defines one of these bras, because every linear map that takes to V to c lives in V star. So there has to be an element that corresponds to it. And just if you want to think about kind of a concrete way of talking about these, I can think of-- if I think of this as a column vector, v1 to vn, the way I should think about the bras is that they are really what you want to write as row vectors. And they may have to have the conjugates of the thing. The components are conjugated. OK, and now you can ask what the dot product looks like. Alpha v is then just this matrix multiplication. But it's matrix multiplication of an n by 1 thing by 1 by n thing. Alpha 1 star alpha 2 star alpha n star times this one here. So v1 vn. And this is now just matrix multiplication. I guess I can write it like this. So they're really, really quite concrete. They're as concrete as the kets are. So you can construct them as vectors like strings of numbers in this way. So I guess we should finish. So I didn't get to talk about linear operators, but we will resume there next week. Are there any questions about this last stuff or anything? No? OK. So you next week, or see you tomorrow, some of you. Thanks. [APPLAUSE]

MIT_1401_Principles_of_Microeconomics_Fall_2018

5_Production_Theory.txt

JONATHAN GRUBER: All right, so we finished the first unit of the course, or consumer theory, and we've sort of gotten to the demand curve. Now we move onto the second year of the course, which is producer theory, and talk about where the supply curve comes from. Now the good news is that a lot of the tools and skills we developed in the first few lectures will translate quite nicely to thinking about the supply curve and production. The bad news is, supply is a lot harder. It's a lot harder, fundamentally, the big picture, because we did consumer theory. We sort of told you what your income was. We told you income and prices, and then we said, OK, here's how to optimize. With firms, they sort of get to decide what their income is. They get to decide how much they produce, and that means we're going to need an extra whole constraint we're going need to model, an extra other part of the process. So it's going to be an extra step with producer theory, but a lot of the tools will be the same. So let's dive in. We're going to start by talking about, just as consumers have a utility function, producers have a production function. So the first parallel is that producers have production function. Now here, their goal is not to maximize production. Their goal is to maximize profits. So as consumers want to maximize utility, producers want to maximize profits, which equals revenues minus costs. So the goal of a producer is to maximize profits, which equals revenues minus costs. And what that's going to mean, it's going to mean producing goods as efficiently as possible, maximizing your profits. We are going to focus for the first few lectures on the cost part. In particular, we're going to focus on maximizing profits through minimizing costs. And we minimize costs by producing as efficiently as possible, OK. And that's what we'll focus on in the next few lectures. Now, what firms can produce comes from their production function. A production function is of the general form q-- that's units of goods produced-- is a function of the amount of labor input and capital input used by the firm, so q, little q-- let me just highlight right here. I will hopefully get this right. I never in the semester have gotten it totally right. Little q refers to a firm. Big Q refers to a market, OK. We're going to try to keep this straight. So little q means a firm's production function. Big Q means a market production function, OK. So if I get that wrong, I'm sure you guys will tell me. OK, so basically what a production function does is it converts inputs, which are labor and capital, into output through some function, just like utility function converts goods into happiness through some function. It's the same idea here, but here, it's more tangible. Unlike utils, output can actually be measured. So literally, it's not some preference mapping. It's literally a technological function. OK, you get your hands around it more than a utility function. It's literally a technologial function by which inputs get converted to an output. Now, we call these inputs the factors of production, OK. Labor and capital, we call the factors of production. They're the inputs they get used to produce things. Now, what are labor and capital? Labor is pretty easy. Labor is workers, OK, either number of workers or hours of work. we'll use those interchangeably, but the bottom line is, labor is workers. It's you all. OK, that's sort of the easy part. Capital is harder. Capital is machines land, building, all the stuff that workers use to make things, OK. So capital's a vaguer concept. But for now, think of it as like machines and buildings, OK, the stuff that workers use to produce goods. And outputs are the goods and services that got produced. Now, when we talk about inputs or factors of production, we're going to talk about them being variable or fixed. Variable means changeable, fixed means not changeable, OK. So variable inputs are inputs that can be easily changed, like hours of work. You can easily work. You guys work different amounts of hours every day. You can easily change the hours that you work. You can pull an all nighter if something's due the next day. You can work less if there's something good on TV, OK. Fixed inputs are those which are harder to change quickly, like the size of a plant. Let's say you're siding a bigger plant. You can't just instantly do that. It takes a lot of production process. You can see by the giant production going on as you pass every day as you walk, if you come from east campus, you walk to this building, it's going to take years to build out those new MIT facilities, OK. So it's not simple. So fixed inputs are inputs that are hard to change quickly. And the key distinction we draw-- and we think about variable fixed-- is the short run versus the long run. And the way we define these is that basically, in the short run, some inputs are fixed and some are variable. In particular, we're going to stay in the short run, labor is variable and capital is fixed. So in the short run, you have labor and then some fixed level of capital, k bar. So in the short run, you've got some building. You can't change it, but you can always change how hard people work in that building. In long run everything's variable. Labor and capital are both variable, OK, so there's no k bar. Capital's variable in the long run. So the question then is, what is the short and the long run. Well, there's no good answer to that. Intuitively, think of the short run as a matter of days or weeks or months and the long run as a matter of years or decades, for your own intuition. But technically, the definition is, the long one is the period of time over which all inputs are variable. That's the technical definition. The long run is a period of time over which all inputs are variable. That's our technical definition. So think about how long it takes to build a plant or make new machines, OK. That's the long run. So I'm never going to ask you, is the short run 8.3 days or 9.7 days, OK. There's no right answer. The right answer is, the technical answer is, the short run is a period of time over which some inputs are fixed and some are variable. The long run is the period of time in which all inputs are variable, OK. And it's not a clean distinction. Obviously, in reality, there's a whole range of inputs ranging from workers to the gas you pipe in to use for your thing, to the raw materials you have to buy to the machines, the buildings. Obviously, in reality, there's a whole range of variability. But once again, to make life easier, we're going to shrink this down to two dimensions, labor and capital. Labor is going to always be variable. Capital's going to be fixed in the short run, variable in the long run, OK. So that's how we'll boil this down to make life easy. Yeah. AUDIENCE: Can you give an example of what capital is again? JONATHAN GRUBER: Capital is the buildings, machines, the stuff that workers use to make things. Yeah, OK. Other questions? With these definitions in mind, let's talk about short run production. Let's start by talking about production in the short run. Someone needs to invent me some more indestructible chalk. OK, so let's start in the short run where labor is variable and capital's fixed. So in the short run production function, q equals f of L and k bar, OK. That's our short run production function. OK, now that means in the short run, the firm's only decision, the firm is given a stock of capital. So think of the short run as, you are hired to manage a plant. And the plant, you don't get to decide if the plant's big or small or what machines. It's there. Your only decision is how many workers to hire, how many hours of labor to employ. And once again, I'll go back and forth through a number of workers and hours of labor. The bottom line is the amount of labor being provided, OK. The way you're going to decide that is, you are going to look at when we're going to call the marginal product of labor, the marginal product of labor, which is simply the change in output for the next unit of labor input. This is very much like a marginal utility of a good. Once again, going back to our powerless consumer theory, the marginal utility of pizza was the delta in utility for the next unit of pizza. The marginal product of labor is the delta in the amount produced for the next unit of labor. And we are going to assume, much as we assume diminishing marginal utility, we are going to assume diminishing marginal product. Now, that's a little less intuitive than diminishing marginal utility was. At least, I hope you found diminishing marginal utility kind of intuitive, that the second slice of pizza would be worth less to you than the first slice of pizza. Hope you found that intuitive, OK. Now, this is less intuitive. Once again, like with utility, it's not saying the next worker doesn't help. The next worker does help. It's just that the next worker helps less than the previous worker, OK. Now, this isn't true everywhere. Obviously, there are tasks where having two workers together makes both better, and we'll talk about that later on. But we're going to focus on the range of production where this is true. Think about production functions as being non-monotonic, they can go up and down. But we're going to focus on the range of production where this is true, where the next worker is not as productive, OK, because eventually that's going to be true for every firm. And why is that going to be true? Because we're holding capital fixed. The reason eventually workers will get less productive is because there are only so many machines and buildings they can work in. The classic example we think is the example of digging a hole with one shovel. You've got one worker digging a hole. She can do a certain amount of effort. Then a second worker comes along. She can add value. The first can rest. They can trade off, and maybe she's almost as productive, or probably not quite. Then the third worker, and the fourth worker. By time you have six workers, they're mostly just standing around because there's only one shovel. Now, each one's more productive because they can help you optimize the shift and get rest and stuff, but the truth is, clearly the sixth worker's less productive the fifth worker given there's only one shovel. So the key to understanding the intuition of diminishing Marshall practice to remember that there's a limited amount of capital. There is a fixed amount of capital. So if there's a given building and you've got 1,000 workers and you try to shove the 1,001st in there, he's not going to do a whole lot of good, OK. So marginal product of labor comes from the notion that there's a fixed amount of capital, so each additional worker does less and less, adds less and less to the production, OK. That's the intuition for diminishing marginal product of labor. And that's pretty much it for short run production. That's sort of what you've got to know. The more interesting action comes when we go to long run production. That gets more interesting because now, you have an optimization decision over labor versus capital. In the short run, you just decide how many workers to hire. In the long run, now you're back to the kind of utility framework we used. We had to trade off pizza and cookies. Now, you get to trade off workers and machines. Now you own the firm. You're going to own it forever, OK, and you get to trade off. You get to think about workers versus machines. So now, you're going to have to make a decision on that. And that decision is going to be, just as your decision of how to trade off cookies and pizza is driven by utility function, your decision about whether to employ workers or machines it's being driven by your production function. So make life easy. Let's start a production function that looks just like the utility function we're using, q equals L times k. Familiar form. Before, we said how happy pizza and cookies made you. It was the square root of pizza times cookies. Now we're going to say how many goods you can produce is the square root of capital times labor. OK, and figure 5-1 shows you what that delivers in terms of graphically. Just as to graphically represent utility, we graft indifference curves, to graphically represent production, we are going to graph isoquants. Isoquants are like firm indifference curves, but once again, they're sort of more tangible. An indifference curve is this weird, intangible idea of points along which are indifferent. And isoquant's a tangible thing. It's the combinations of capital and labor that produce the same amount of output, OK. So for any given production function, there's different combination of capital and labor that produce the same amount of output. So for example, in our example, two units of capital and two units of labor produces two units of output. Four units of capital and one unit labor also produced two units of output, so they would be on the same isoquant. They're this combinations of inputs that deliver the same level of output, OK. And isoquants, all the stuff we learned about indifference curves apply here. More is better, so further out is better. They can't cross, OK. And they slope downwards. All the set of things we learned about indifference curves, that same set of intuitions applies here. The difference with production is it's more plausible to have extreme cases, OK. So let's consider two extreme cases. Let's first consider the case of inputs that are perfectly substitutable, so like a Harvard graduate and a Beanie Baby, OK, perfectly substitutable inputs. Those are goods where the production function would be of the form, q equals L plus k. That's perfectly substitute because you're indifferent between the unit value and the unit value. They're perfectly substitutable. So you can see that in figure 5-2a, I do x and y instead of L and k, but it's the same idea. If there's two inputs, x and y, then with that production function, a perfectly substitutable, that would lead to linear isoquants. Perfectly substitutable inputs will lead to linear isoqaunts with a slope of minus 1, you're perfectly indfiferent between one or the other at all levels. At any point in time, you're indifferent between 1 more unit of x and 1 more unit of y, 1 more unit of labor and 1 unit of capital, OK. At the other extreme would be perfectly non-substitutable inputs, inputs where you can't produce one more unit without one of each input, OK. That would look like figure 5-2b. We call this a Leontieff production function. A Leontieff production function is one where there's non-substitutable inputs, where the production function is the min of x and y, that one more unit of y does you no good unless you also get one more unit of x. So what's an example? What's a real world example of a Leontieff production function? What's a good which would have non-substitutable inputs? We'd need at least one of each. Yeah. AUDIENCE: If you have like programmers and computers, [INAUDIBLE]. JONATHAN GRUBER: Programmers and computers, that's a good one. AUDIENCE: It's like, you need like a right shoe. JONATHAN GRUBER: You need a right shoe and a left shoe. That's a classic example I would use. Cereal and cereal boxes stuff like that, you know, stuff where you basically need both. Shoes are the sort of classic example, OK. And that will give you sort of a Leontieff production function, OK. So basically, those extremes help you think about what isoquants are and what they mean. Now, continuing our parallel to consumer theory, what is the slope of the isoqaunt? What is the slope of the isoquant? The slope of the isoqaunt, just as we call the slope of the indifference curve the marginal rate of substitution, since we're not very creative in economics, we call the slope of the isoqaunt the marginal rate of technical substitution, because it's the same idea, but now it's technical. It come from a technical production function, not from your preferences, OK. So marginal rate of technical substitution is the slope of the isoquant, or delta k over delta L. And as with indifference curves, that slope varies along the isoquant. So we can see that in figure 5-3, OK. Figure 5-3 is once again drawn for our production function, q equals square root of k times L. So let's say for example, we start with one worker and four machines at point A, OK, and now we consider adding a second worker. Well, at point A, that second worker is so productive because of diminishing marginal products, OK. You already got four machines, only one guy to run them. Like, he's not doing a lot of good. So adding a second worker and two machines helps a lot. It's not perfectly Leontieff, but you can get the intuition that two workers and two machines are the same as one worker and four machines. You're not better off. You're the same off. You have the same isoquant. So the marginal rate of technical substitution is minus 2. That is, one worker substitutes for two machines, OK, one worker substitutes for two machines. But now, starting from point B and moving to point C, it takes two more workers to substitute for one machine because, if you go down one machine, you need a lot more workers to make up for it. So then the MRTS falls to minus 1/2. So, going from A to B, one worker makes up for two machines. Going from B to C, it takes two workers to make up for one machine. And that's because of diminishing marginal products, OK? That's because of diminishing marginal products. OK, indeed, there's a convenient way mathematically to relate the marginal rate of technical substitution to marginal products. Think of what the MRTS is asking. Think of what we're asking along the isoquant. We're saying what combinations of capital and labor yield the same output. That's what we're asking, OK? So another way to think about it is what change in capital plus an equivalent change in labor leads to the same level of output. So you can ask, well, the change in labor times the marginal product of a unit of labor, times the marginal product of a unit of labor, plus the change in capital times the marginal product of a unit of capital, which is the same. I didn't define marginal product of capital. It's the same idea as marginal product of labor. It's dq dk is the margin product of capital. That equals 0 along an isoquant. Think about it. Along an isoquant, the next unit of labor times how productive that labor is plus the next unit of capital times how productive that capital is equals 0 because you're staying along an isoquant. So, if you're taking away one unit of labor, if this is minus 1, and this is plus 1, then, based along the isoquant, you're choosing the point where the MPL equals the MPK. Or, more generally, if you reorganize this, you get that delta k over delta L, which is the slope, equals minus MPL over MPK. And that is the MRTS, OK? The marginal rate of technical substitution is the negative of the ratio of the marginal product of labor and the marginal product of capital. Once again, should look familiar. It's just like the marginal rate of substitution. It's the negative of the marginal utility of the good on the x-axis for the marginal utility of the good on the y-axis, same idea. I derived it in a slightly different way here, but it's the same idea. And it comes from the notion that you wanted to stay-- that you're staying constant production, as you change labor and capital along this curve. Yeah? AUDIENCE: [INAUDIBLE] MPK over MPL [INAUDIBLE] give you the same ratio or no? JONATHAN GRUBER: Well, MPK over MPL would give you-- I mean, basically, we're defining the marginal rate of technical substitution the way we define-- the way we define the marginal rate of substitution. You basically want-- because you want it to be downward sloping. If you define that, the inverse would be upward sloping. So what we're defining is the downward-sloping concept, which is the marginal product of the good on the x-axis and marginal product of the good on the y-axis. So it's not invertible. It's not freely invertible. Yeah? AUDIENCE: What's the marginal rate of technical substitution for a Leontief production function? JONATHAN GRUBER: Ah, great question, great question. So, the marginal rate of technical substitution, so let's go back to Leontief. OK, the marginal rate of technical substitution actually sort of depends on-- it sort of depends on where you are. It's sort of a nonlinear marginal rate of technical substitution, right? So, basically, it's going to very much depend-- depend on where you are. So, basically, it can be negative infinity or positive infinity or 0, depending on where you are on the curve. So we'll actually-- I don't want to give you more on that because this problem may-- I'm not giving anything away-- could obviously be a problem set problem. So I don't want to give more answers than that away, but, certainly, it's going to be-- it's not going to be constant, OK? Other questions? Yeah? AUDIENCE: It's just like a line, right? JONATHAN GRUBER: The marginal rate-- if the curve is just a line, the marginal rate of substitution would be constant. For perfectly substitutable inputs, it would be constant. That's right, just like the marginal rate of substation would be constant if your indifference curves were linear, OK? Good questions. OK, so that's production, OK? Other questions about production? OK, that's the basics, and we went fast because, basically, a lot of it's just parallel to what we did with consumer theory, OK? Now but I want to talk about two other aspects of production that we need to keep in mind as we move forward. The first and the fourth topic for today is returns to scale, returns to scale, OK? This is what returns to scale are asking is what happens to production when you increase all inputs proportionally. So, if you double all inputs or triple all inputs or whatever, cut all inputs by 73%, what happens to production? So it's not about K versus L. It's about a scale, a scaling up or down of the operation, OK? Now we know, obviously, if you double inputs, production will go up. More is better. The question is by how much. So our baseline we can think about as what we call a constant returns to scale production function. That would be one where f of 2L, 2K equals 2 times f of L, K. So a constant returns to scale function means, if you double inputs, you double output. If you double inputs, you double output. That's a constant returns to scale production function. But you could also define increasing returns to scale, where doubling inputs leads to more than double the output, or decreasing returns to scale where doubling the inputs leads to less than double the output. So constant returns to scale means doubling the inputs leads to double the output. Increasing returns to scale means more than doubling the inputs more than doubles the output-- I'm sorry, means double the inputs more than doubles the output. Decreasing returns the scale means doubling the inputs less than doubles the output, OK? And that gives you-- that's your definition of returns to scale. Now where could these come from? So increasing returns to scale, for example, where could increasing-- that's the world's worst S. Where could increasing returns to scale come from, OK? So, for example, one reason for increasing returns to scale is that, basically, as a firm gets bigger, it might learn to specialize. So maybe a firm with two workers and two computers, and you double, and you get four workers and four computers, and then you could specialize the tasks more. And each worker is more efficient in their specialized task. That could lead to increasing returns to scale. That's an example of something that could lead to increasing returns to scale. Decreasing returns to scale could come through something like difficulty of coordination. Maybe, when I've got two workers and two computers, I can keep an eye on them and make sure they don't slack off. But, with four workers and four computers, there's more slacking off because I can't keep an eye on them all the time and more so with 8 and 16, et cetera. Yeah? AUDIENCE: So, when-- I have to ask, why is doubling the inputs greater than two times the outputs? JONATHAN GRUBER: Yeah, so, basically, doubling the input-- so, when I move to two workers and two computers to four workers and four computers, I more than double my output. And that's because maybe they specialize and get more productive. AUDIENCE: Oh, so f of L, K equals the original output. JONATHAN GRUBER: Yeah, so, well, it's one function. f of L, it's literally one function. It's literally-- so I'll write it out. It's literally saying doubling my inputs leads to more than twice what I get with just-- without doubling the outputs, OK? Is that another way to think about it? Yeah? AUDIENCE: So, when we're talking about the returns of our-- our return to scale, is that how much product is being produced or how much profit is being made? JONATHAN GRUBER: How much product. We're only-- we haven't gotten to profit yet. We're only talking about quantity. We're only talking about quantity. f is-- remember, f is the function that translates inputs to q. Yeah? AUDIENCE: Are things intrinsically like increasing return to scale functions or decreasing return to scale functions? JONATHAN GRUBER: Well, great question. So what do you think? What's the right answer? What do we think in reality? AUDIENCE: I mean, like, perhaps, maybe there could be ways to shift it or not. Like, going back to the whole example you gave about decreasing-- like, if there's more computers, people can start slacking off-- if he set like-- if you set some parameters or like you deactivated social media on those computers that they couldn't go on Facebook when like you weren't watching them, you could make them be more productive per se. JONATHAN GRUBER: Well, it's a great question. Let's start by looking at figure 5-4 and show some examples of what we think about like decreasing, increasing returns to scale. So figure 5-4 has some examples of the kind of industries people think are potentially decreasing, increasing returns to scale, OK? So, for example, we think the production of tobacco is a decreasing returns to scale activity. That is you're kind of farming tobacco. You're growing it. You're producing it. Then, if you kind of double it up, there's still a certain amount of land you're working on. You can't-- there's still sort of a certain amount of crop. You're not going to produce twice as much by having twice as many threshers and workers, whereas, maybe something like producing primary metal, OK, you could basically maybe work a lot more efficiently by having more machines and more workers together producing that metal. So what is the right answer? The right answer is we don't know, but the one thing we do know is there can never be forever increasing returns to scale. And why is that? Well, at least, we used to think this maybe 15 years ago. Why is that? Why can there-- what would happen in an economy if a firm had forever increasing returns to scale? Yeah? AUDIENCE: You'd get a monopoly. JONATHAN GRUBER: It would own the economy because, the bigger I got, the more productive I'd get. So I would just eventually grow and own the whole economy. Now, actually, that may be happening. So maybe it's not as weird as we thought it was 15-- maybe Google and the big five have increasing returns to scale. But, eventually, we think returns to scale must decrease. We think your scale of production must get so unwieldy that doubling it means you just can't manage it as effectively. Eventually, we think returns to scale must decrease. That's sort of the one sort of principle we have that we don't know-- we think, generally, probably, in the life cycle of firms, returns to scale are probably increasing and then decreasing. But we don't know where it happens. And, certainly, companies like Google and Amazon are showing us that point of decreasing may happen a lot later than we thought, OK? And that's because I think what we didn't account for in our traditional producer theory is networks, the fact that networks get ever more productive. We always thought about buildings and workers, and there's a limit to how productive they can get. But networks, by bringing in more and more people, can get ever more productive. But, at some point, we think these things have to decrease. At least, we traditionally thought so, but maybe, in 10 years when Google owns everything, I'll change my tune, OK? But that's sort of the one sort of rule of thumb we have in thinking about this, OK? Other questions about that? OK, let's talk about the last topic then, which is productivity, how this stuff all matters in the real world. So we're going to come back next lecture and come back to maximizing profits and all that stuff, but I want to sort of step aside now and ask why does this all matter. And, to do so, let's step way back to the original dismal scientist, Thomas Malthus. Thomas Malthus in 1798 wrote a book, which said-- which was really pretty depressing. He said, look, let's think about how basic economics works. Now he didn't do the math. This is pre-math. So let's get the basic intuition. Think about the production of food. OK, the production of food has two inputs, labor and land. There's workers, and then-- you know, there's machines, but they're pretty simple machines. OK, you sort of till the land, and there's land. Well, in the long run-- in the short run, labor is variable. In the long run, labor is variable. But land is never variable. Land is a forever fixed input. There's no long run. Unless we discover a new planet, there's no long run over which land is variable. What that means is that there will be ever diminishing marginal product to farming. He didn't say it this way, but this was sort of his intuition that more and more workers will try to cram on a given acre of land. Each additional worker can only do so much. And, eventually, the marginal product will be diminishing, OK? The result is that productivity will fall, the marginal product of labor, when each additional worker will be less and less. And, as a result, we'll starve because, basically, we have all these people looking for work. There's nothing to do because only a certain amount of land. They won't have anything to do, and, eventually, they'll starve. So Malthus actually predicted we would see cycles of mass starvation through history, fun guy to have at a party. OK, he'd basically say we're going to get overpopulated. These guys will have nothing to do because there's only so much land they can work on. They'll die off. We'll eventually grow overpopulated again. We'll get these cycles of mass starvation. That was his prediction. Now, since he wrote that book, world population has increased about 1,000%. And, yet, we're fatter than ever. I'm not saying food deprivation isn't a problem around the world, but, certainly, the world is much better fed that it was in 1798, OK? What did Malthus miss? What did Malthus miss? What did we get wrong? Yeah? AUDIENCE: The classic example against it is like he didn't account for innovations. JONATHAN GRUBER: He didn't account for innovation or what we call productivity-- productivity, or you can also call it innovation-- and neither have we so far. We have written production functions of the form q equals f of L and K, but, in reality, the production function is actually q equals A times f of L and K, maybe A of t, A sub t. And that's a production factor that, basically, for a given amount of labor and capital, as you get more productive, you can produce more things. The production function itself changes. You get more productive over time, OK? In agriculture, how did we do this? Well, we did it in lots of ways. We invented cool new ways to harvest the crop, tractors. We invented fertilizer, chemical fertilizer. We invented seed-resistant crops. We invented lots of things that Malthus didn't see coming. So, as a result, even though the land is just as fixed as it was in Malthus' time-- we still haven't discovered a new planet we can farm on-- we produce a lot more food because of the factor A. The production function itself has changed. We've become more productive. So productivity is the factor that allows us-- or innovation is the factor that allows us to produce more and more with a given amount of inputs. So, actually, food consumption per capita is rising. Since 1950, food consumption per person in the world is up 40%, OK? So, while we have starvation, and it's terrible, it's up. One side note, some of you may have heard of a very famous economist named Amartya Sen. He's a Nobel Prize winning economist. His biggest-- one of his main contributions was he studied famines, and he said famines are not a technological problem. He said there's never been in history a famine in a democracy. No democratic nation has ever had a famine. Famines are not about technology. Famines are about politics and corruption and the things that get in the way of proper food distribution. So really we have enough food, OK? The food is there. Malthus was wrong. This is not just true in agriculture. It's true all over the world. Let's look at car production, one of most famous examples, OK? Cars have been around since the late 1800s, OK? And they're basically-- when cars were first invented, they were essentially craftsmanship. Someone would sit down and make a car if you can believe it. They'd literally make all the parts. They'd make a car or a couple people together. In the early 1900s, Henry Ford introduced the idea of mass production-- it seems sensible now, but it's not the way they used to do it-- a series of workers who each did a discrete task along the way, constructing a car. So no worker did a whole bunch of the car. Each worker did a little piece, which massively led to increasing returns to scale by specialization. He did that, and, basically, this was radical at the time. It seems obvious to us now, but he cut the price of building a car more than in half almost overnight and basically wiped out all his rivals through the introduction of mass production. Now you might think that's pretty cool, but, you know, that's and old-time story. But it's not over. Innovation in car production continues. The Indian company Tata, you may have heard of them. They do a lot of MIT-- they finance a lot of stuff at MIT. They have a car called the Nano that they produce for $2,500, OK? It's a tiny car. It's lighter. They use extra light materials. It's smaller because they do things like putting wheels on the extreme outside of the car, rather than sort of underneath the car, OK? And they minimize the parts that are used to make it easily fixed and interchangeable with other cars. So innovation is going on all the time. Look at hybrid. Look at the innovation in the fuel space with hybrid cars and electric cars and Tesla. OK, innovation is happening all the time. Now what's key about this, besides the fact, technically, what this means is that, when you write-- when we think about production-- now we're not going to talk about this a lot. We'll assume that there's constant production functions. But what that means technically is, when we think about over time production, innovation is a key factor. But what this means, in terms of all of us sitting in this room, is that productivity innovation is fundamentally what determines the standard of living in a country. Our standard of living is determined by productivity, OK? So, basically, if you think about us as workers, if we're going to get richer, we're going to have to make more stuff, OK? We're going to have to make more q or more valuable q, OK? Now, given our amount of labor, that's either going to happen through more K, through more capital, or through a faster A, through faster innovation. So, ultimately, what determines our standard of living-- that is what determines how much shit we have for a given amount of work-- is going to be how much we save and how innovative we are. I'm sorry, back, how much capital we have and how innovative we are. Capital it turns out is going to come from savings. I sort of cheated there. We're going to talk about that in about-- about maybe 12 lectures from now. We'll talk about where capital comes from. The hint is capital comes from how much we save, and I'll explain why that is. But our standard of living is determined of how much capital we have, which is a function of how much we save, but it's mostly determined by how innovative we are, how productive we are, how much more we can produce for a given level of inputs, OK? And, if you look at-- if you ask how does production go up given an amount of capital and labor, we call that total factor productivity. That is, conditioning on all the factors, how much does productivity go up? Now it turns out we have seen a massive shift in productivity in the US. From 1947, after World War II, to 1973, productivity growth in the US was very rapid, about 2 and 1/2% a year. What that meant-- let's think about what that meant. That meant, not doing anything else, working just as hard as we were working, we could get 2 and 1/2% more stuff every year, OK? That's what I mean by our standard of living. Literally, it's saying, working the same 40-hour week, every year, we got 2 and 1/2% more stuff, OK? However, from 1973 until the early 1990s, productivity growth slowed down massively down to about 1% a year. It dropped massively, OK? Now what happened? Well, one thing that happened is we started saving less. K went down. K is driven by savings, and we started saving less. We save a lot less than other nations. But, in fact, that's not much of it because, even though we don't save much, productivity jumped again in the-- about from 1995 to 2005, productivity jumped again and went up again to about 2 and 1/2% a year. And, essentially, we felt, ah, this is the IT boom. OK, computers were around since the 1970s. And, throughout the late 1980s and early 1990s, people kept saying where's the productivity gain from computers. And it appeared to show up in the mid 1990s. Suddenly, things got more productive in the mid 1990s to the mid 2000s, OK? Productivity rose to about 2.3% a year. But, much to our chagrin, productivity has stopped growing rapidly, and it's back to about 1 and 1/2% a year. So we're not as slow as we were. So we were-- so, from 1947, '47 to '73, we grew at about 2 and 1/2% a year. '73 to '95, it was about 1% a year. So that meant, with the same amount of work, we only got 1% more stuff, OK? '95 to '05, we went up to about 2.3%. We jumped back up. But, since '05, we're down at about 1.5%, so better than we were at our minimum, but not nearly as high as we were at our peak, OK? So, basically, this raises three key questions. Yeah? AUDIENCE: How is that productivity measured? JONATHAN GRUBER: Oh, great question. So, basically, we look at, essentially, a way-- roughly speaking, we look at how much stuff gets produced given how many hours of labor there are submitted to the economy. Roughly speaking, we say how much do people work. How much stuff gets made? Boom, that's productivity, nothing super fancy. Yeah? AUDIENCE: I might have missed it, but what does TFP stand for? JONATHAN GRUBER: Total factor productivity. That's productivity controlling for capital. But the productivity numbers here are not total factor productivity. They're just labor productivity, allowing capital to change. Yeah? AUDIENCE: You said that K would decrease when people save less. But, if you save less, isn't your spending someone else's likely [INAUDIBLE]? JONATHAN GRUBER: You know what? I don't want to go there. We're going to spend a whole lecture on that. So I don't want to go there. K depends on savings. Just take that as a given for now, and we'll come back to that. We'll spend two lectures on it actually, OK? Now I want to raise three questions, before we go, about these facts, OK? The first question is why didn't the IT revolution and the computer revolution lead to longer lasting productivity gains? Why did productivity slow back down after 2005, OK? We don't really know. Folks thought that computers would be the next Industrial Revolution. This was going to be a-- this was going to transform our lives, OK? It looks like what it mostly did is transform how we watch porn, OK? And, basically, it looks like, in terms of productivity, it did not actually change things that much. And we don't quite know why, but it is still a bit worrisome that, in terms of the long run, that, in some sense, there wasn't longer lasting gains from innovation. If it's a question about porn, I'm not answering it. [LAUGHTER] AUDIENCE: Maybe with like how, if you pull more people into like a team, working on team projects, the rate at which the project is worked on tends to decrease. JONATHAN GRUBER: You know, there's lots of theories. We could hypothesize all day about why it is. I'm just going to state the facts and say that it's disappointing. And we need to figure out what to do about it. The second question this raises is how do we spend increases in productivity. What do I mean by that? What I mean by that is, if there's an increase in productivity of 2 and 1/2%, that means we have 2 and 1/2% more stuff for the same amount of work. But why do we have the same amount of work? Another way to say that is we can work 2 and 1/2% less and have the same amount of stuff, roughly speaking, OK? So I have assumed we work the same, and we get 2 and 1/2% more stuff, but why is that the right answer, OK? And, in fact, the US and Europe, since World War II, have taken very different paths in this dimension. In the US, we've taken all our productivity and put it into cooler stuff, and we work harder than ever. In Europe, they work less hard. I mean, starting jobs in Europe have six weeks vacation, OK? Nothing gets done in August in Europe, OK? They've said-- and, you know, if you go to Europe, it's a little bit more rundown, OK? It's not quite as gleaming and cutting edge as the US in many places, OK? Basically, Europe has decided to take some of that productivity increase and put it into more leisure time. We've decided take all that productivity increase and put it into better phones and gadgets, OK? So the question is who's right. We don't know. But the important point is that's an open question. Just because we're more productive doesn't mean we should just consume more stuff. There's an open question of how you spend your productivity gains. And then there's the final question and maybe the most important, which is who actually gains from productivity increases. So, from 1947 to 1973, productivity went up 2.5%, and virtually every group in society saw their incomes go up 2 and 1/2% a year. Since 1973, on average, productivity growth has been about 1.5%, 1.6% on average. You average these three series, about 1.6%. And average incomes have only gone up 0.4%. So productivity has gone up 1.6%, but average income has only gone up 0.4%. The difference is the gains have all gone to the top of the income distribution. So, basically, virtually all of the gains from 1973 until a couple of years ago-- it's started to get better-- essentially, the bottom 80% of people saw no improvement of their standard of living over a 45-year period, whereas the top 20% saw a massive improvement. And, even within that, the top 1% saw a really massive improvement. And, even within that, the top 0.1% and 0.01%, et cetera, saw massive improvements. So, as a result, in 1995, the richest 10%-- or the richest 10% of the population earned 15% of the income. Today, it's close to 25% of the income. It's getting even worse. Since 2009, if you look from 2009 to 2016-- I don't have it updated-- and you look at all the money that was made in society, on net, all of it went to the top 1%. What do I mean by that? The top 99% were, in 2016, in the same place they were in 2009, even though the economy had grown. And all the growth went to the top 1%. So we're actually in an interesting world here where productivity gains by itself may not be enough if we care about what it does to the average standard of living. And that leads to a very interesting set of issues around equity and fairness that we'll spend time on later in the semester. But I want to raise that issue, both that productivity gains can be spent in different ways on goods and leisure, and they can be distributed in different ways. And those are the sorts of things we need to be thinking about as we think about economic policy, OK? So let me stop there. We'll come back-- I guess no section on Friday, right? It's an MIT holiday. And we'll come back on Monday and talk more on producer theory.

MIT_1401_Principles_of_Microeconomics_Fall_2018

13_Oligopoly.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: All right, why don't we get started? Today we're going to move on to, finally, the most realistic market structure. We talked about perfectly competitive markets. Now, that was a very useful, extreme example to help us think about economic efficiency. We then flipped over to talk about the somewhat more real estate case of monopoly, but still, very few markets have only one participant. A true monopoly is rare in the private market. What most markets are marked by are probably more features of oligopoly, which is a market with a small group of firms competing with each other, but with barriers to entry that keep out an unlimited number of firms. Think about these as markets where there are some barriers to entry, so firms just can't consciously enter and exit like they could in our IBM/Dell example, but where there's small enough barriers to entry that a few firms have gotten in, not just one. So it's not a natural monopoly. It's not like only one firm can be in there. Multiple firms are in there, but they only know they have to compete with each other, not with the big, wide world. So for example, the classic example of an oligopoly industry would be the auto industry. Auto manufacturers clearly compete. Clearly, if you watch any sporting event and watch how much advertising that goes, they're clearly competing with each other. They're comparing to each other all the time. But most of the cars in the world are produced by fewer than 10 auto manufacturers. The notion that we have a perfectly competitive market of thousands of sellers selling identical goods is clearly not right when it comes to buying a car. So that's the model we're going to want to focus on for the next few lectures. Now, within an oligopoly market, whenever we think about this market, we want to start by noting that within this market, these limited sets of competitors can behave in two ways. They can behave cooperatively or non-cooperatively. Cooperatively means that they can form what's called a cartel. So when there's an oligopoly market and the firms cooperatively get together and make decisions, that's called the cartel, the most famous example of which is OPEC, the Organization of Petroleum Exporting Countries, which are the set of countries that control about 2/3 of the world's oil, led by Saudi Arabia, is the major player in OPEC. It's a cartel of about a dozen nations. And what they do is they control the vast majority of the world's oil reserves. And by behaving cooperatively, they essentially turn themselves into a monopoly. OPEC acts as if they've got the monopoly in oil. Certainly they used to. Now it's getting harder. Other non-OPEC countries are starting to produce more oil and it's breaking down. But for a long time, they were essentially the cooperative producer of oil, and act essential like a monopoly, and they made lots of profits like a monopoly. That kept prices high, they kept production inefficiently low, and they made lots of money. However, that's a great outcome for producers, but as we'll talk about next time, it's actually a hard outcome to enforce. Turns out to be hard to keep cartels together. And so typically, oligopoly markets behave in a non-cooperative way, with the participants competing with each other, not cooperating with each other. In this case, you can actually get them driving their profits down far below the monopoly level, and indeed, perhaps even all the way to the competitive level. So you can think about markets as a competitive as one extreme and the monopoly as the other extreme, oligopoly in between. A cooperative oligopoly market, like cartel, will end up close to the monopoly outcome. A non-cooperative market will end up somewhere in between, and we're going to model today's where in between do they end up. Now, to think about this, we're going to have to turn to a tool, which has really become a dominant tool in economics over the last 40 years, which is the tool of game theory. Game theory. So basically, we're going to think of oligopoly firms as engaging in a game. And as with any game, you need to know two things. One is you need to what's the strategy, and the second is you need to know when is the game over. What's the equilibrium? And that's, essentially, what you do with any game. And so basically, the key with game theory is that we are going to find the equilibrium, and that's going to yield for us the strategy that players are going to use. However, equilibrium in a game is not well-defined. It's not like a set of rules that are printed out, like monopoly. In a non-cooperative oligopoly market, the equilibrium, you have to actually come up with different concepts of what equilibrium is. There's not a hard and fast scientific rule. And the typical one that's used is called the Nash equilibrium, the Nash equilibrium, named for John Nash, the famous mathematician, who economists have claimed as their own, even though he was really a mathematician. But we gave him the Nobel Prize anyway. And if you think of economists, probably one of the most famous, you all know about the movie and book Beautiful Mind. He is based on the father of game theory. So basically, what is the Nash equilibrium? The Nash equilibrium is defined as the point at which no player wants to change their strategy, given what the other players are doing. So the point at which no player wants to change its strategy, given what the other players are doing. So in other words, every player is happy with where they are. Given what every other player's decided, I'm happy to do what I've decided. So I've got a strategy, and given the strategy other players are using, if I'm happy with my strategy, then that's in equilibrium. So this is a super abstract concept, so let's illustrate it with an example. And the classic example of game theory is the prisoner's dilemma, which many of you, I'm sure, know about, maybe the most of you, but let's just go through it. This is the thing from the old cop movies you see, where they arrest two guys and they put them in separate rooms and they basically interrogate them separately. They're put in separate rooms, and let's say that these guys get told the following. They each get told separately the following thing. They get told that right now, if nothing else happens, there's enough evidence to send them each away for one year. However, they're told, if they turned on their friend and say their friend's guilty, then they go free and their friend gets five years. If their friend turns on them, then the friend goes free and they get five years. But if they both turn on each other, they both get two years. Set up as if they both stay silent, they both get one year. If one turns, then that person gets to leave and the person gets five years. But if they both turn, then they each get two years. So how do we think about decision-making in that context? The way we do that is we write down, we call, a payoff matrix. We write down in matrix form this decision. So let's think about what a playoff matrix looks like. Up here is prisoner B, and here you have prisoner A. Prisoner A. And prisoner A can remain silent or they can talk, and prisoner B can remain silent or they can talk. And then we just write down, what are the outcomes, or the payoffs, from these different strategies? So prisoner A says nothing and prisoner B says nothing, then A gets one year and B gets one year. If prisoner A says nothing and prisoner B says, oh yeah, prisoner A is definitely guilty, then prisoner A gets five years and prisoner gets zero years. If the opposite happens, if prison A says, yeah, B's guilty, and B doesn't say anything about A, then A gets zero years and B gets five years. But if they both say the other one's guilty, then they each get two years. OK, that's the payoff matrix. And now we want to ask, given this payoff matrix, what is the right strategy for each prisoner to pursue? And the way we do this in the Nash equilibrium concept is we look for a dominant strategy. Is there a strategy that I would pursue regardless of what the other person does? And if there is, I'll pursue that. Because remember, the Nash equilibrium concept is, what do I want to do, given what the other person is doing? If I have a strategy I want to do no matter what the other person is doing, then I'll do it. So when asked, is there a dominant strategy? Is there a strategy that is the best thing to do, no matter what the other guy does? Well, clearly, if they're cooperating, if these were stupid police and they sat them in the same room, told them and then left, the two guys could cooperate. Well, clearly, the dominant cooperative strategy is for both of us to remain silent. That's the dominant cooperate strategy. And as a team, we only get two total years in jail, where everything gets many more years in jail. So if they're buddies and they trust each other and they cooperate, then that's clearly the right strategy. But let's say the police are smart and put them in separate rooms. Well, what's the dominant non-cooperative strategy? What is the strategy that A or B, say A, should produce? Yeah? Why? AUDIENCE: Either way, you're going to get less years. Like if you're the only person silent and you talk, you get zero, and if they talk and you talk, you only get two versus five. JONATHAN GRUBER: Exactly. For prisoner A in this first row-- compare the first column. We'll say prisoner B is silent. Then clearly, you're better off talking than not talking, zero rather than one. Let's say prisoner B talks. Then you're still better off talking than not talking. So no matter what B does, you should talk. Likewise, prisoner B, no matter what A does, B should talk. So the non-cooperative equilibrium is actually this outcome. They both end up talking. You get sort of a race to the bottom. The non-cooperative outcome is much worse than if they could have cooperated. So basically, what you get is that the non-cooperative equilibrium is always worse for the players than the cooperative equilibrium. And this was like an unbelievable insight of Nash. Before Nash, we always thought competition was always and everywhere good. We always thought more competition is better, for the reasons we talked in the first 10 or 12 lectures of this class. Nash was the first one to say, no, actually, competition can be bad. Cooperation can be better. I don't know if you remember the scene in a Beautiful Mind where they're picking up girls in the bar. And he described basically a Nash strategy, how competition will lead to the worst outcome. And basically, that's what you see here, that competition can actually lead to a worse outcome than cooperation, and that was really Nash's brilliant insight. Now, this is a cute example with prisoners, but actually-- well first, two points. First of all, this generally shows you how you do gain favor with Nash equilibrium. Basically, you look at the payoff matrix, you find the dominant strategy, and then you find where those dominant strategies intersect. And here, the dominant strategies intersect at this cell, therefore that's the equilibrium. So that's basically how you do game theory in a game theory kindergarten level. You look at the matrix. You find each player's dominant strategy. And you find the point at which those dominant strategies intersect, and at that point, that's the equilibrium. Now, that's all well and good for a simple example like this, but let's actually apply to an economics example. Let's think about advertising. So think about Coke and Pepsi. Right now, let's think about their decision to advertise. Now, obviously it's a simple problem. Obviously Pepsi should just be illegal because Coke is way better. But sadly, it's not, and sometimes I have to drink Pepsi and I'm very sad. But nonetheless, in the real world, we have Coke and Pepsi and they have to decide how much to advertise. Now, the dominant cooperative strategy would be to say, look, advertising costs us a ton of money. Let's just split the market. Let's have a monopoly market and just split it. We're close to splitting it anyway. Coke's got some more of it. We're close to splitting it. Let's just split it. Yeah? AUDIENCE: Can you actually do that? JONATHAN GRUBER: What? AUDIENCE: Can you actually do that? Because I remember, there were places that you get where you aren't allowed to sell in the same place. JONATHAN GRUBER: OK, but that's different than the cooperative. That's imposed not by Coke and Pepsi jointly. That's imposed by Pepsi saying to a university campus, for example, we will cut you a better deal if you'll agree not to sell Coke. That's not cooperation. That's competition. So there's cooperative strategy. What if they don't cooperate? Well, let's imagine we have the following payoff matrix. You've got Pepsi up here, and they can advertise or not advertise. And you've got Coke here, and they can advertise or not advertise. And let's say the payoff matrix is the following. Let's say the total amount of profit to be made is 16 whatever, billion, whatever units you want to make it, $16 billion. And let's say if there's no advertising, Coke gets 8 and Pepsi gets 8. But let's say advertising costs money. It costs 5, $5 billion. So let's say if they both advertise, then they still end up splitting the market, but they only make 3. C equals 3, P equals 3. I'm sorry, advertise. yes, you're right. C equals 3, P equals 3. And here C equals 8, P equals 8. So basically, you have a situation where they both end of splitting the market either way, but they just split a smaller net profit if they advertise. So clearly, they'd rather be here than here. But what happens in the off diagonal elements? Well, let's say also that if Coke advertises but Pepsi does not, then let's say Coke ends up making $13 billion and Pepsi ends up making minus-- I'm sorry, if Coke advertises and Pepsi does not, they split money. And Pepsi makes negative 2. They actually lose money because they have fixed costs and they don't sell anything. Nobody buys Pepsi. It'll lose money. And let's say if Pepsi advertises and Coke doesn't, then Coke makes negative 2 and Pepsi makes 13. So actually, if you don't advertise, you're really screwed, and the other guy is really screwed. Yeah? AUDIENCE: Does this include the cost of advertising? JONATHAN GRUBER: This does include the cost of advertising. But it's just Coke gets a huge market, expands its market. So now let's play the game. Well, now let's say you're Coke. You say, well, if I advertise and Pepsi advertises, I make 3. But if I don't advertise and Pepsi advertises, I make negative 2. So I should advertise. If I advertise and Pepsi doesn't advertise, I make 13. If I don't advertise and Pepsi doesn't advertise, I make 8. So either way, my dominant strategy is to advertise. And likewise, Pepsi does the same thing. I screwed up writing this compared to my notes, but it's good because it shows you-- I flipped the matrix, but the logic is the same. It helps you not just memorize cells of the matrix but learn the logic. The point is either way, the dominant strategy is to advertise, so they both advertise. So real world example of how you can end up. Now, so much of Pepsi and Coke do this. Actually, there was an industry that did this. So when I was a kid, you never, ever saw ads for liquor on TV. There were beer ads and wine ads, but no hard alcohol ad. No bourbon, no whiskey, no nothing, gin. All these Captain Morgan's ads we see now, they didn't exist when I was a kid. But it wasn't because the law. It was because the hard liquor industry cooperatively agreed none of them would advertise. So they actually imposed the cooperative equilibrium, and then that broke down. I don't know the story of how it broke down. But it broke down. Now they all advertise, and they're probably all worse off than they were when they didn't advertise. We'll talk next time about why it probably broke down. I don't know the stories. I have a rough sense, and we'll talk about that next time. But this is the point of how a non-cooperative equilibrium can drive you to a bad outcome. Now, basically, this doesn't just apply to prisoners or businesses. It applies to people, too. So let's say poor Hector back there has had a fight with his girlfriend. And they've had a big fight. They've going a little while. They've had a big fight. And Hector has got to decide, do I apologize or do I wait for her to apologize? Well, the last thing Hector wants is to go up there and apologize and have her say, forget it. I'm breaking up with you. That'd be the worst. If he knows she's going to be like, oh, I'm sorry, too, then he'd be happy to do it. But what if he goes, no I'm breaking up with you, and she's thinking the same thing. So what happens, they break up. We've been through this many times in our lives. This is the non-cooperative strategy. Basically, if you know what the other person is going to do, your dominant strategies to be an asshole, and basically that happens a lot in the context of the real world. So now we have this sad-sounding outcome, that basically game theory leads to bad outcomes for producers, at least. But this is what's exciting about game theory. So when I went to grad school, back when dinosaurs roamed the earth, game theory was taught barely in the sequence. It was like an extra course, taught a little bit. Now it dominates the teaching of microeconomics, in economics. And it doesn't dominate, but it's a whole like component of our core microeconomics education, because it's given such a cool set of tools to think about these decisions. Now, I can't give you even 1% of the flavor of game theory. If you want to learn more, I highly suggest you take 1412, which our game theory class, and you can learn a ton. But let me show you one interesting wrinkle of the things game theory can do, to go beyond this. And that's to imagine that Coke and Pepsi are not playing a one shot game, but a repeated game. Repeated game. So now imagine that Coke says to Pepsi the following, I promise to not advertise as long as you don't advertise. But if you ever advertise, I will advertise forever. Coke says to Pepsi, I promise not to advertise as long as you don't advertise, but if I ever catch you advertising, I'm going to advertise forever. So think about Pepsi choice in period 1. Pepsi's choice in period 1. In period 1, they could say, ha, stupid Coke. I'm going to jump on and advertise. They promised not to advertise. So if Pepsi advertises, they're going to make 13 in period 1 because Coke's taken themselves off to the side. But after period 1, they're going to make 8 forever. No, I'm sorry. They're going to make 3 forever. Because Coke's going to advertise. They're going to advertise. They break down to the non-cooperative equilibrium, if Pepsi advertises. Now, what if Pepsi doesn't advertise? As long as it doesn't advertise, then it gets to deal with Coke, so it makes 8 forever. We'll talk later in the course about how you combine numbers that happen at different times, but trust me, 8 forever is a way better deal than 13, than 3 forever. So actually, by having this be a repeated game, Coke has solved the prisoner's dilemma. It's essentially imposed a cooperative equilibrium on the problem. So that's how repeated game can fix this. But-- this is where the game gets really exciting-- that only works if this game never ends, because once Coke or Pepsi thinks there's an end to the game, the entire thing breaks down. So imagine, for example, that Coke makes the offer to Pepsi, but Pepsi is worried that in 10 years, the government is going to outlaw soda. The government said, look, we're heading that direction. Soda is going to be illegal in 10 years, so I don't want to do this. I'm sorry, I have that in my mind. So Coke offers the deal now, what do I think? Well, let's think about Pepsi's decision in the ninth year. They've made 8, 8, 8, 8, 8, and they get to year 9. Now in year 9, they know that next year there's no more game. So what should they do? Advertise. Grab the 13 in the last period, because Coke can't punish them because the game's over. But Coke knows this. So what's Coke going to do in the ninth year? Advertise. It's going to advertise, so they're both going to make 3. Well, if Pepsi knows Coke's going to advertise in the ninth year no matter what, what should Pepsi in the eighth year? Advertise. And if Coke knows Pepsi is going to advertise in the eighth year, what should Coke do? And so on, and it ends up that they both advertise all the way through. So the game breaks down if it's an end. This is really kind of neat, and this is what game theory is all about, is how do you think through these more complicated scenarios that are much more complicated than the prisoner's dilemma, and actually think about how firms and individuals might actually behave? Yeah? AUDIENCE: Wouldn't it also be advantageous if they just advertised the first year instead of these contracts, kind of what we were talking about earlier? JONATHAN GRUBER: Sure. And once again, that's what you cover in a field course like game theory. What about alternative forms of contracting, with exclusionary contracting, what we call tying in contracting? That's great, and they would. But that's why you got to take 1412, OK? Yeah? AUDIENCE: Would it be a better outcome if they cooperated and switched periods of advertising? Like for the first period, they get 13, they get minus 2. JONATHAN GRUBER: Yeah, the way I've set this problem up, if they could commit to that, that would be right. But you'd have to commit to it. Because then the period that Pepsi promised to take off, if they actually advertised that period, then Coke's screwed. So that would work as a repeated game solution, but it wouldn't work as a non-repeating game. It would work as an infinite repeated game but not a non-infinite repeated game. Good question. OK, other questions? All right, so that's the basis of game theory. That's just a taste for the excitement that you can learn with game theory. But in fact, in economics, we like to write those as fun examples, but we really prefer to do math. So let's actually think about the math of how we take game theory concepts and put them in practice. And the way we do that is through the concept of the Cournot model. The Cournot model of non-cooperative oligopolies. So the Cournot model of non-cooperative oligopoly is the standard workhorse model. It takes this intuition and puts it into the optimizing math we've been doing so far in this class. Now let's imagine non-cooperative case, but now let's imagine not just two choices, but realistically, there's a whole set of choices. Then how would you behave in that case? So let's imagine that there's two airlines, United and American. So we have an oligopolistic two-firm airline industry. Obviously, the math can extend to more firms, but just to start, and I'll talk about that next lecture. But for now, imagine a two-firm industry, United and American. And because the hub and spoke system we discussed last time, let's imagine that they're the only two folks that go from Boston to Chicago. Because it's hub and spoke system. The only folks that go from Boston to Chicago are United and American, and they do, in fact, dominate that line. So let's imagine they're the only folks, and say no other firms can compete on this route because they can't get slots at the airport. So the question is, how do these firms decide how many flights to run? It's not just advertise, don't. It's literally a continuous decision of how many flights to run every day and how much to charge. They've got to make that decision. And the Nash equilibrium here, the subset of Nash, for this example, is called the Cournot equilibrium. And the Cournot equilibrium exists when a firm chooses a quantity such that, given the quantity chosen by the other firm, they don't want to change. So a firm chooses, essentially, a profit-maximizing quantity, given the quantity chosen by the other firm. And that profit-maximizing quantity, then you're in Cournot equilibrium, if you have chosen a quantity that is profit-maximizing, given what the other firm is doing. So basically, how do we actually carry this out? Let's talk about the steps. So the first step, I'm going to talk intuitively about the math, what we're going to do, and then I'm going to talk mathematically and graph what we actually do. There's essentially three steps in solving for the Cournot equilibrium. The first is ask how your demand changes when some of it's absorbed by other firms. So the first is solve for your residual demand function. What does your demand curve look like, given what the other firm does? That's step 1. Step 2 is then you develop a marginal revenue, which is a function of the other firm's quantity. Little q. It's multiple firms. The other firm's-- that's really bad, hard to read-- the other firm's quality. So your marginal revenue is a function. Typically, it's a function-- I'm sorry-- of both your quantity and the other firm's quantity. A function of both your quantity and the other firm's quantity. We develop marginal revenue as a function of your own quantity. We know how to do that. Now we develop a margin as a function of your quantity and the other firm's quantity. Then you simply set this marginal revenue equal to marginal cost, and that delivers you a conditional answer. That delivers you your optimal quantity as a function of the other firm's quantity. Well, that doesn't do us a whole lot of good, except there's two firms. So the fourth step is we do the same thing for the other firm and get the same kind of equation. Then what do we have? Two equations and two unknowns, so we solve. So what we do here is essentially the same thing we did before, but now your marginal revenue is not just a function of your own quantity, it's a function of the other guy's quantity. Same with the other guy. That gives you two equations, two unknowns. We solve. And the point at which both firms are happy is the Cournot equilibrium. That's confusing, so let's actually look at that. We'll do this both graphically and mathematically. Let's start with figure 13.1. To make things easy, let's start by imagining that American Airlines is a monopoly. Let's start with the world with an American Airlines monopoly. And let's say that the demand function is P equals 339 minus Q. That's the demand for flights from Boston to Chicago. And let's say that the marginal cost, to make life easier-- it could have a cost function and make your life difficult, and maybe someday I'll do that. But for now, to make your life easy, let's just say it's a flat marginal cost of $147. I'm not going to make life difficult with solving for marginal cost functions. For now, it's just a flat marginal cost of $147. No matter how many flights they do, it's $147 per passenger. So if you're a monopolist, how do you solve this problem? Well, first you derive your marginal revenue function. Well, what's marginal revenue function? Well, revenues is P times Q, which is 339 minus q squared. So your marginal revenue function is 339 minus 2Q. That's your marginal revenue function, if you're the monopolist. What's your marginal cost? Well, I just said it's $147. And then you just solve. And when you solve that, you get that Q, the optimal quantity, is 96 flights. And then how do you get the price? How do you get the price of monopoly problem? How do we know what the price is? Yeah? AUDIENCE: Where the quantity intersects the demand curve. JONATHAN GRUBER: You've got to plug it back into the demand curve. Take that quantity, plug it back into the demand curve. So the price is 339 minus 96, or 243. So I just solved the monopoly problem quickly, but that's what we've done already in this class. And you could see that in the graph here. In figure 13.1, you've got demand curve, which is P equals 339 minus Q. You've got a supply curve, which is the flat marginal cost of $147. You develop a marginal revenue function, which is 339 minus 2Q. As in our previous example, that's just basically an inward shift of the demand function. That intersects marginal costs at 96 flights. We have 1,000 passengers per quarter. It doesn't really matter. It's just all standardization. And then to get the price, you read it off the demand curve. You say 96 flights means the price of $243 per flight. OK, that's what we do if American was a monopolist. Now, however, American is not a monopolist. American deals with United, and American doesn't know what United is going to do. So what does American do? Well, let's say American has to deal with the fact-- it now has to recognize that it's got its own demand function, qa, which is the total quantity in the market minus qu. So it has a residual demand function, which is the total demand in the market minus what United sells. So suppose, for example, American thinks-- American's got a spy inside United-- and American says, ha, I think United is going to fly 64 flights. So imagine American thinks United's going to fly 64 flights. Well, in that case, if they're going to fly 64 flights, then my demand function is p sub a equals 339 minus q sub a minus 64, because the big quantity is little qa plus little qu. So my demand function is 339 minus q sub a minus 64. Or in other words, my residual demand function is that p sub a equals 339 equals 275 minus qa. So if I think United's going to fly 64 flights, then my effective demand function is 275 minus q. And then I'm done. Then I just solve for, what would I do as a monopolist, given the other guy's flying 64 flights? So you can see that in figure 13.2. So I have a demand function. I say, well, if United's going to fly 64 flights, that demand function gets shifted in by 64. And then I'm going to do the same thing I did before. I solve for marginal revenue. I'm going to solve for marginal revenue and I intersect that with marginal cost. That's going to happen at 64 flights and a price of $211. So basically, it's the same exercise. It's not that hard. You just first take out what United is going to do. The problem is American doesn't have a spy. They don't really know what United's going to do. They have to essentially develop a strategy, given the possibilities of what United might do. They have to say, look, I don't know what q sub u is, so I have to devise my optimal strategy given q sub u. In other words, I have to simultaneously solve for what I would do at every possible quantity United would sell. I have to solve what I would do for every possible quantity United would sell. And we call this developing your reaction, or best response curve. Your reaction curve or your best response curve, which is, what is the best thing to do, given what the other guy's doing? What is the best thing to do, given what the other guys doing? You could see that in figure 13.3, we show how that works. That shows best response curves. So for example, look at the intersection on the y-axis, where the red line hits the y-axis. That was our monopoly equilibrium. I'm sorry, where the blue line hits the x-axis, my bad. We're doing American. Look at where the blue line hits the x-axis. That is assuming zero United flights. Where the blue line is the x-axis is where there's zero United flights. Well, we know what American would do there. They would fly 96 flights. We already solved that. Now look at the point where United is flying 64 flights. Well, we also know what American would do then. We know that we solved, in the previous figure, they would then do 64 flights. And in general, what that blue line is is for every quantity that United flies, what does American want to fly? So meanwhile, United is doing the same mathematics. Imagine, to make life easier-- we'll almost always do this to make life easy-- imagine United has the same marginal costs as American, and obviously faces the same market demand curve. Well then, literally, their math is totally symmetric. If American wasn't in the market, you'd have where the red line intersects the vertical axis. If American was flying zero, United would flight 96 flights, because their problem is identical to American's monopoly problem. So the red line is United's best response curve. So we've graphed, for every possible amount of flights that United does, what's American's optimal amount of flights. We've solved for every amount of flights that American does with United's also amount of flights. Where those lines intersect is the Cournot equilibrium. Why is that the Cournot equilibrium? Because at that point, both firms are doing the best they can, given what the other firm's chosen. Or in other words, to say this is given what the other firm's doing, neither firm wants to deviate. The profit-maximizing choice is to be where they are, given the other firm's behavior. So basically, the Cournot equilibrium is the only equilibrium that's possible in this market. And why is that? So for example, imagine that American came in and said, look, I like doing 96 flights. I love being a monopolist. I'm just going to do 96 flights. I'm going to do 96 flights, I'm going to charge $243. Well, in that case, American-- United, I'm sorry-- would happily come in at $242 and undercut them and sell lots of flights, because that's still well above marginal cost. So that's not an equilibrium because United and American are choosing different outcomes. It's only equilibrium if they're both to the point where the same outcome makes them both happy. So that's the graphics. Let's do the math here. Let's do the Cournot math. In general, the residual demand for American is that p equals 339 minus qa minus qu. Remember, big Q is qa plus qu. Since the demand function is 339 minus big Q, I simply broke big Q into qa and qu. Stop me if this is all unclear. Simply broke the big Q into those two components. So that means that American's revenue function-- it's called revenue A, revenue for American-- is 339 times qa minus qa squared minus qaqu. This is a new term. This was the old revenue function we had when they were monopolists. Now we've got this new term that didn't exist before. So that means the marginal revenue for American is now 339 minus 2qa minus qu. So now their marginal revenue is actually a function now of their own behavior, but their competitor's behavior. That's the new margin revenue function. But the profit maximization rule is the same. We just set that equal to marginal cost. We set it equal to 147, and you solve. And what you end up getting is that q sub a star-- the outcome of q sub a is 96 minus 1/2qu star, or qu. qa star is 96 minus 1/2qu. If you solve this equation, that's what you get. That's 1/2, 1/2 qu. So now we have the optimal quantity, but it's a function of what the other guy does. That's a problem, except that we use the same math for United. Now, if the problem's symmetric, you don't have to do the math again. You know the best response function will be symmetric, but that won't always be the case. So I'm going to shortcut here of saying the best response function for qu is q star u equals 96 minus 1/2qa. So I've just written down the best response function. This corresponds to the graph. So q star a, that's the blue line. It's 96 minus 1/2u. q star u, that's the red line, 96 minus 1/2qa. That's their best response function. Now once again, to remind you, I could simply skip to this step, but normally you'd have to solve through for both firms. They might not have identical best response functions, or symmetrical best response function. Well now we're golden. We have two equations and two unknowns. We know how to deal with that. And you solve them and you get the qa star equals qu star equals 64. You solve those two equations and two unknowns. So 64 is the solution of that system. What's the price? Someone raise their hand and tell me. What's the price? Without looking at the graph. Yeah? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: And how did you get that? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: You got to plug in 64 twice. A lot of people get this wrong. They'll say, oh, 339 minus 64. But no, it's 339 minus 128, because they're each flying 64 and the price comes from the total demand in the market. So the price is $211. That's an important mistake to avoid. A lot of people get here. They'll be super excited. They're tired. They throw the 64 back at the demand equation. But remember, demand's a function of the total market. If symmetrically they're each doing 64, then the price is going to be $211. And that is the Nash or Cournot equilibrium. Both firms are happy to fly 64 flights at a price of $211. Neither firm wants to deviate. And you know that because you've maximized their profits. When United is flying 64, the profits of American are maximized at flying 64. When American's flying 64, the profits for United are maximized at flying 64. Therefore, that is the Nash or Cournot equilibrium. Now, when we get to reality, things might not always work out so neatly. Things might not be symmetric. You might also not have an equilibrium. How could you not have an equilibrium here? How could that happen graphically? What would that mean? Yeah? AUDIENCE: The curves don't intersect. JONATHAN GRUBER: Yeah. The best response curves might not intersect. You might not get an equilibrium. We don't know what the hell to do then. All chaos breaks loose. But you might not get an equilibrium in this market because the best response curves might not intersect. In reality, in life, you could have funky best response curves that are non-linear or you could have multiple intersections. We call it multiple equilibria. And then it becomes an indeterminate problem and you have to figure out which equilibrium they settle at, and that involves higher order mathematics that you talk about in more advanced classes. So this is the simplest, easiest cases. Symmetric case where linear best response functions intersect is your easiest case. But in general, the general way to solve this is the same, which is use the principle of game theory. Look, go back to the prisoner's dilemma. All we're doing here was creating best response functions. It's just there wasn't a line. It was just a point. The best response function was what we laid out here. All we did with these United and American examples was go to a continuum and develop best response functions around the best response point. Yeah? AUDIENCE: If the Nash equilibrium is always worse than when they're cooperating, why is it so hard to maintain a [INAUDIBLE]?? JONATHAN GRUBER: We'll talk about that next time. Other questions? OK, let's stop there. We'll come back. Next time we'll talk about, why don't we all just get along with Mr. Rogers once?

MIT_1401_Principles_of_Microeconomics_Fall_2018

17_Making_Choices_Over_Time.txt

[SQUEAKING][RUSTLING][CLICKING] JONATHAN GRUBER: Today, what we're going to do is continue our discussion of factor markets by essentially talking about how capital markets impact real world decisions. So last time, we talked about the capital market. We talked about, essentially, the way that firms finance their capital is by going to a pool of savings that individuals decided how much to make. Individuals make it into temporal choice on how much to save. Actually, technically, they can make a choice about how much to consume each period. That then yields an amount of savings. And then based on that pool of savings, firms borrow at some interest rate, i, and they decide how much to invest. Now, today we're going to talk about a number of interesting applications that arise from capital markets that are important in the real world. And I'm going to start by talking about the concept of present value. Present value. Now, the key insight when we think about capital markets is that $1 tomorrow is worth less than $1 today. $1 tomorrow is worth less than $1 today. That's because if you had the dollar today, you could productively invest it and have more than $1 tomorrow. So $1 today is worth more, because you could do something productive with that $1 if you had it today. What that means is that dollars in different periods are worth different things. You can't just add them up. OK, the analogy I like is thinking about-- imagine you had a pound of apples, a pound of steak, and a pound of gold. You wouldn't just add them up and say, I have three pounds. That'd be useless. You'd want to know what each is worth, and you'd want add up the dollar value of them. It's a similar thing with money received over time. You can't just say, I'm getting $1 today, $1 next year, and $1 five years from now. They aren't the same thing. Money received at different points of time are worth different amounts, because money received in the future forgoes the possibility of investing it today. So what do we do to deal with this? Well, in economics, we deal with this by the concept of present value. Present value is the value of every period's payments in terms of today. So the way we deal with the fact that money in different periods is worth different amounts is by what we call discounting it back to today. We essentially take future dollars and discount them back to today to get present value. We discount them, because future dollars are worth less. So let's think of it this way. Suppose that the interest rate is 10%. Let's do an example. Imagine the interest rate, i, is 10%. And let's say that you want to have $100 next year. Next year, you want to have $100. Well, how much do you have to put in the bank today? Well, we can solve the equation, which is that you want the amount you put in the bank today-- let's call that y-- times one plus the interest rate, because that's what you make by having it in the bank, equal to 100. That's the equation we want to solve. The money amount you put in today, y, times one plus the interest rate. You want that to be equal to 100. So what that says is that y equals 90.9. If you put $90.90 in the bank today at a 10% interest rate, you will have $100 tomorrow. More generally, we say that the present value of any future payment is the future value, the amount you get in the future, over one plus i to the power little t, where t is the periods in the future. So money received t periods in the future today is worth the amount you get in the future over one plus i to the t. So that is our general formula for how we think about present value. Take all the future payments, and you discount them back to today by dividing by one plus i to the t. Now, that works well if there's one future payment coming. But what if, as in many cases, there's a whole stream of future payments coming? Well, the logic then is the same. You just want to take each future payment and discount it by how far it is in the future. So suppose that you say that you want me to loan you $30 and that you'll pay me back $10 each of the next three years. And suppose the interest rate's 10%. Well, I will say no to that. Because I will say, if you're going to pay me back $10 each of the next three years, then what is the present value of that? The present value is next year's $10 over 1.1, because the interest rate is 10%, plus the $10 the year after that over 1.1 squared, plus the $10 the year after that over 1.1 cubed, which if I add it up, is 24.87. So I am losing money. If I give you $30 today and if you give me $10 back each of the next three years, I'm losing money. Why? Because if I had simply taken that money and invested it in the bank, I would've had a lot more than the $30 I'm going to get from you after three years. OK, so the money that comes in the future must be discounted back to today. That's the key insight of present value. You can't just add it up. You've got to put it in today's terms by discounting it by the interest rate and how far into the future. Yeah. AUDIENCE: So aside from just the change in value that's caused by interest, how do you take into consideration the fact that the value of the currency itself hasn't changed over time? JONATHAN GRUBER: We're assuming now inflation is zero. So we're assuming right now, we're not dealing with inflation. We're assuming a world where prices don't change. We'll come back to inflation in a few minutes. But for now, just assume that prices don't change. OK, so basically, essentially, the formula then for the present value, your general formula for present value you need to know is that present value equals-- if you get a flat stream of payment f for a number of periods, it's f times one over one plus i to the one plus one over one plus i squared plus one over one plus i cubed plus dot dot dot dot dot dot for how many periods you get it. So if you're getting a payment f over a certain number of periods, you've got to discount it by how many periods you're going to get it. Or more generally, a common formula we'll ask you to use in this class is to think about the formula for perpetuity. A perpetuity is a flat payment you get forever. So if we take the infinite sum of this equation, we can summarize it as present value approximately equals f over the interest rate. So if I promise you to pay you forever a certain amount f, that is worth f over the interest rate. So in other words, if the interest rate is 10%, a promise to pay you $10 forever is worth about $100. That's just the Infinite sum. If you've read A Beautiful Mind, John Nash can do that in his head. But I can't, but that's just the formula. So basically, that is the general formula for perpetuity. And that makes life easy. We'll give you a lot of examples. Because if I say I want you to pay me for eight years in the future, I've got to write out eight terms here. It's a pain in the ass. So what we'll say is money forever, and then you'll just use the shortcut, which is the present value for perpetuity. It's just f over i. Questions about that? Yeah? AUDIENCE: Does this also assume the rate of change over time? JONATHAN GRUBER: That's a great point. Yes, I'm assuming constant interest right over time. Now in reality, you'd want to have each period's interest rate here, and then you can't use this formula, because the interest rate itself is changing. Very good point. OK, now the other way to think about this is instead of thinking just-- this will be easy for most of you, but just it's useful to think it through. Let's flip it on its head and not think about present value. Let's think about future value. This is a useful tool as well. Let's flip it on its head. OK, so let's think about the value of a future stream of payments. Well, by the same logic we wrote before, future value-- by the same logic we wrote before, if you're going to have money that's tiers in the future, the future value of that money that you're going to get-- if you're going to basically take your money and invest it, you're going to have money that you are going to invest today and put it in the bank at some interest rate, then the future value is the amount you put in times one plus the interest rate to the t. So I've simply reversed that formula, present value, future value. So basically, future value, if you put money in the bank at some interest rate, you're going to get to have over time that money times one plus i to the t. Now, the reason we write this formula out and flip it for you is because I want to highlight the key feature here that I want to drill into your heads. One of the, I don't know, I guess 10 things I most care you leave this course thinking about if you're not going to major in economics or remembering-- is the beauty of compounding, which is that with a formula like this, you earn interest on your interest. If you leave money in the bank, you don't just earn interest on the initial amount you put in. You earn interest on the interest you earn over time. OK, and this can be quite large. So let's do a simple example just to show you how big this can be and to get you all thinking about what you should do with your money when you get a job. OK, imagine you plan to work full time from age 22 to age 70. A little daunting to think about now. Probably, most people will retire after 70 by the time you guys retire, but let's just think about 22 to 70. And let's say that you can save at a constant 7% interest rate. Inflation is zero. The interest rate is constant. Make life easy. 7% interest rate. And let's consider two different plans you have for savings. Plan one. Plan one is that you're going to save $3,000 a year for the first 15 years that you work and then stop saving. 3,000 a year for 15 years, then you're going to leave that in the bank, leave it alone, never save anymore. Just let that money sit in the bank. Well in that case, what will you have? Well, after the first 15 years of putting $3,000 in the bank every year, if you work out the math, you will have $75,000. $75,387 after 15 years. OK, now that's not just 15 years times-- that's bigger than 15 times 3,000, because you're earning interest along the way. OK, that's those 15 years. Then, you're just going to let that sit there. And that's going to sit there. Remember, after 15 years-- you started working at 22. You're only 37 years old. That's going to then sit there for the next 33 years. You're not going to touch it. You're not going to save anymore. What that means is after 33 years, this turns into $75,387 times 1.07 to the 33rd. OK, after 33 years, or $703,000. 703,010. OK, so you save $3,000. It's not a lot of money. You guys can make a lot of money. $3,000 for 15 years, and then you never have to save again. Contrast that with a different approach. Let's say you say, look. That's stupid. I'm young. I'm going to party. I'll worry about retirement when retirement is closer. I'm going to save nothing the first 15 years, and then I'm going to save $3,000 every year. The first 15 years I'm going to save nothing, then I'll save $3,000 every year. OK, well if you do that and do the math, you end up with when you retire, $356,800. Think about this for one second. In this case, you saved for more than twice as many years. You saved for 33 years, and you ended up with half as much. That's the miracle of compounding. The earlier you save, the more money you can make along the way. And that's why you guys need to start saving right away. Yeah? AUDIENCE: What was plan two? JONATHAN GRUBER: Plan two was I do nothing the first 15 years, then I save $3,000 a year for the remaining 30 years of my career. So literally, in plan one, I save for 15 years, and then I stop. Here, I save for 33 years. But by starting earlier and using the miracle of compounding, I end up with twice as much money. OK, now this is actually pretty-- this is one of the few things in this class my kids could understand when they were little. Because have any of you been to the Boston Science Museum? Have any of you guys been? OK, there's a little kid area, where they've got this ramp. And you can essentially drop balls down a ramp. And one ramp is flat and then steep, and one ramp is steep and then flat. And of course, the one that's steep and then flat wins. It's faster. And that's just because of compounding. That's because acceleration is compounding. OK, so basically, the point is that the earlier you start saving, the more money you'll have. And that's why you guys should pay attention when you're offered a job and offered a 401k and not think, retirement, I'll never retire. You say, no, I want to save now, because the more I save now, the more that will compound by the time I retire. OK, now you can actually see this. It's not just MIT students who have to think about this but professional athletes. So probably, none of you have heard of Bobby Bonilla Any of you guys heard of Bobby Bonilla? OK, well if you were 25 years ago in this class, if you were into sports, you would have heard of Bobby Bonilla. He was a pretty good player in his time. And he retired. But by the end of his career, he was kind of a slacker. He wasn't really worth much. He was playing for the Mets. And the Mets said, look. In 1999, they said, look. We basically want to pay you off to leave the team. You have a contract. We're just going to basically pay the remaining 5.9 million on your contract in 1999, give you the money. Most athletes would've said great, I'm off to Vegas. Bobby Bonilla didn't. He said, well, look. I've got enough money right now, but I might need the money later. So instead, why don't you defer the money at an 8% interest rate and pay me starting in 2011 when I'm getting close to retirement and need the money? They were like, great. That's really great. We don't have to pay you now. That's great. Well Bobby Bonilla, by the time his payments started in 2011, they'd grown to $30 million. And every year on a certain day-- it just passed recently. They call it Bobby Bonilla. Bobby Bonilla, who's now like 70 years old, gets a million dollar check from the Mets every year. Because he was patient enough to put this off and get the virtue of compounding. So this is sort of-- if Bobby Bonilla, a stupid baseball player, can do it, you guys can do it. So make sure that you guys are saving when you start your jobs. OK, questions about that? OK, now let's get a little bit realistic, and let's recognize that in life, prices aren't constant, but rather we have inflation. And how does that affect our thinking here? How does inflation affect the way we think about this problem? Well, it turns out, it actually adds one step, but it's actually a pretty easy step to put in. It actually turns out that you can add inflation without doing a whole lot of work. So let's talk about first what inflation is. Inflation is the rise in the price level year over year. Technically, the inflation rate is a percentage concept. It's the percent rise in the price level year after year. You might say, the price level of what? Of a banana? Of a computer? Of what? Well, what the government does is they form something called the consumer price index, the CPI. Did I talk about the CPI yet? OK, I'm sorry. I just hate repeating myself. OK, so they form something called the consumer price index. What's the consumer price index? Literally, the government every quarter, I believe-- it may be every month-- goes out and gets the prices of a basket of hundreds of goods. They literally say, what does a banana cost this month compared to last month? What does a laptop cost this month compared to last month, et cetera? So they go out, and they price this bundle of goods. And then literally, they just ask, how has the price-- they then take a weighted average of that bundle, where the weights are consumer spending. So consumers spend a lot more of their income on housing than bananas. So the price of housing gets a lot more weight in the CPI than does the price of bananas. Essentially, a weighted average of prices in society, and then they create an index. 1982 is normalized to one, and they just say how much in percentage terms did that weighted average bundle go up in price? OK, and you can see that in Figure 17.1. Here's historical CPI. So basically, what you see is, this is the level of the CPI, which is sort of meaningless. What we care about is inflation, which is the year to year percentage change in the CPI. And what you can see is, basically it's going up. Prices are going up. It went up very steeply if you look from 1970 to 1980. The slope there was much higher than the slope before or after. We had very rapid inflation in the 1970s. It then has then since flattened, and inflation has been much slower. And inflation averages about 3% a year. OK, so basically, that's how we measure inflation. Now, the question is, how does that affect our thinking about present value if there's actually inflation? And the bottom line is, we don't care about dollars. We care about how many goods we can buy. Therefore, it doesn't matter what's happening to how much money we have. It matters what's happening to how many goods we can buy. Therefore, what we care about is not what we call the nominal interest rate. We care about what we call the real interest rate, r, which we define as the nominal interest rate, which is what we've been talking about, we see advertised on a bank, minus the rate of inflation, which for some reason we use pi, even though that's also profits. So sorry about that. OK, so we define the real interest rate as the nominal interest rate minus the inflation rate. The real interest rate is the nominal rate minus the inflation rate. And what the real interest rate measures is how much more I have in terms of goods I can consume, not how much more I have in terms of dollars, which actually in the end doesn't matter. OK, so suppose that I'm going to save $100 at a 10% interest rate. Let's go a simple example. I save $100 at a 10% interest rate. OK, then next year, I have $110. But that's irrelevant. What I want to know is, how many goods can I buy next year? So for example, let's say you spend all your money on Skittles. That's all you buy. OK, and let's say Skittles cost $1 this year. And there's no inflation, so they cost $1 next year. Then, what that means with a 10% interest rate is you can buy 10 more bags of Skittles next year. This year, your $100 could buy you 100 bags of Skittles. Next year, your $110 can buy 110 bags of Skittles, so you are 10% richer. But now, let's say the price of Skittles goes up 10%. Well, what that means is you can only buy the same amount of Skittles next year as you could buy this year. You could buy 100 bags this year and 100 bags next year. So it doesn't matter that you have $110 next year. Who cares? You only get the same amount of Skittles. What you care about is the goods you can buy. We wrote down utility. We didn't put dollars of utility function. We put consumption. So the interest rate you care about is the real interest rate. If the nominal interest rate is 10% but inflation is 10%, then the real interest rate is zero. You can't buy any more goods next year. All the money you made by putting it in the bank got eaten up by how much more expensive things are. So what that means is, all the math we've done and everything we'll talk about all goes through. You just need to be thinking about using the real interest rate, not the nominal interest rate. But otherwise, everything we've done goes through. You just need to essentially think about this in terms of how many goods you can buy, not how much money you have. Now, this isn't quite-- let me just take two minutes and do a little macroeconomics-- this isn't quite as simple as it sounds, because of course, you see in the bank, i, in the bank window, i. You don't see pi. You don't see inflation. That's what we revealed ex post. So really, technically-- you don't know this. It's just for those who care. You don't have to know this for the test. Technically, what you really want is expected inflation. When you think of putting in the money in the bank, you know you'll learn 3%. The real interest rate is that minus what you think inflation is going to be. So it actually becomes complicated. It's not as simple. Ex post, it's easy to find the real interest rate. Ex ante, it's not so easy, because it depends on what you think inflation is going to be. So there's some tricks there. There's also a bunch of tricks in measuring the inflation rate. So for example, like I said, the Bureau of Labor Statistics goes out and has a bundle of 600 goods and gets their prices. But what is a good? I mean, a banana is a banana. But a laptop, what the hell is a laptop? How much ram does it have? What's the graphics card? How fancy is the display? Well, the Bureau of Labor Statistics doesn't go out and price literally hundreds of laptops. It prices one or two. And the problem with that is the following. Let's say you find that today, a laptop is 1,000, and tomorrow it's 1500. But it can do-- let's say today, a laptop is 1,000, and tomorrow, it's 1500. But it can do a ton more stuff. Well, we would say inflation's 50%, but that's not really right. Because the good you're consuming is not the laptop, it's the computing ability of the laptop. And that's gone up. So to say inflation's 50% is wrong. Inflation's 50% minus the quality improvement of the better laptop you got. Think of it another way. Imagine laptop prices didn't go up, but ram doubled. Would you say you're no better off buying a laptop with twice as much ram at the same price? No, you're better off. But our inflation concept would say, no, you're the same off. So the trick here is, it's simple in practice, and we'll pretend it's simple just to say r is i minus pi. But two tricks-- a, it's expected pi, which is hard to guess. And b, inflation is really hard to measure, because there's things like quality bias and other things. There's a whole field of macroeconomics worrying about inflation measurement, so we won't spend a lot of time on it. But it's just sort of interesting just to talk about how at high level we go through things here. Like everything else in this class, largely, we get it right. Largely, expected inflation is not too badly modeled by last year's inflation. And quality bias, we can model and stuff. So this isn't a bad model, but it just points to some of the subtleties you have to deal with in reality when you try to implement these basic sort of formulations. Now, with that in mind, I'm going to now say, let's assume inflation is zero again. So we'll go back and use i, and we'll use i interchangeably with r for the rest of this course unless asked differently. Unless told definitely, assume inflation is zero, so i and r are interchangeable. With that in mind, let's go to the next topic, which is taking these tools, how do we model choices over time? How do people model? How do people make decisions over time? And there's a simple answer. So this is tricky, because if I said to you, hey, I'm going to give-- do you want $30 or $50? You would say, I want $50. But actually, you shouldn't say that. You should say, over what period of time am I getting the 30, and over what period of time am I getting the 50? If it's today, I want 50. But if the 50 is 20 years in the future and the 30's today, I might want the 30. What that means is, you have to evaluate choices in present value terms. You can't just add up the money, you have to evaluate those choices in present value terms. And then, you need to pick the option with the highest present value. So once again, let's come back to athletes, because athletic contracts deal with this all the time. Let's imagine an athlete considering two different contracts. Contract one, contract a, pays $1 million today. Contract b pays 500,000 today and 1.5 million in 10 years. OK, now when you read it in the newspaper, you'll see this guy got offered a million, this guy got offered two million. That's what the newspaper will say. But that's wrong, because these are paid at different periods of times, so they're different amounts. Indeed, the present value of the first contract is what? $1 million. It's today. What's the present value of the second contract? Someone tell me how I'd write that down. I'd write down the present value of the second contract. Yeah? AUDIENCE: 500,000 plus 1.5 million over one plus whatever the interest is based on the time. JONATHAN GRUBER: Exactly. Which is one plus the interest rate to the 10, because you're getting it in 10 years. So essentially, that means that whether it's a better deal or not depends on the interest rate. Indeed, if the interest rate is 7%, and the interest rate is 7%, this has a present value of 1.3 million. So it's a good deal. If the interest rate was 14%, then this deal would have a present value of 0.9 million. So it's a worse deal. So whether or not this a better or worse deal depends on the interest rate. Why is that? Why is this a worse deal the higher the interest rate? AUDIENCE: Because if he'd gotten the money earlier, he wouldn't have benefited from being able to collect that interest earlier. JONATHAN GRUBER: Exactly, he could have invested it earlier, gotten the compounding, and had way more money in 10 years. So the higher the interest rate, the more you want to get your money upfront and save it, the less valuable is money in the future. Now, essentially, what this says is that you have to always use present value to bring things into current dollars. Now, this is not an abstract concept. Let's take Max Scherzer, who's a pitcher with the Washington Nationals. Max Scherzer a couple of years ago signed a seven year $210 million contract, which he was able to brag was the second highest ever signed by a pitcher and the 10th highest contract ever signed by any baseball player. Seven years, 210. But in fact, that contract was not-- we're going to pay you $30 million a year for seven years. It was, we're going to pay you $15 million a year for 14 years. You're going to play for only seven. We're going to pay you over 14 years. So in fact, in present value terms, that was worth somewhat less. If you use the current interest rate when he signed it which was about 4.7%, it was only worth actually 166 million. Not too shabby, but suddenly it drops to the 20th most valuable baseball contract and about fourth among pitchers. So Max Scherzer was able to feel better about himself that he signed this valuable contract, but in fact, it was worth less than he thought. I mean, shed no tears for Max Scherzer. He's doing fine. But it was worth less than he thought. Or maybe an example that's more in our mind with the Mega Millions-- think about a lottery winner. So if a Mega Millions winner gets $290 million, which sounds like a great deal, that's paid out over 20 years. So it's not $290 million, it's 14 and and a half million for each of 20 years. So in present value terms, if we think about a 7% interest rate, then that's not 290 million, it's 164 million. Once again, not too shabby. OK, but a lot less than the advertised amount. So we have to take these dollars and put them in present value. So now, here's an interesting question for that. If you look at the recent lottery, you had a choice of one person won 1.6 billion, one person one that, a person in South Carolina won that. And they were given a choice of the 1.6 billion, which is paid actually over 30 years, or a lump sum they could get right away. How should they decide which of those options to take? How should they decide whether to take the 1.6 billion paid over 30 years, so 1.6 over 30, paid in equal installments over time versus just getting a lump sum today. Not of 1.6 million, but a lower amount. How should they think about that? AUDIENCE: They should evaluate the present value. JONATHAN GRUBER: They should add the present value. They should say, well, what do I think the interest rate is going to be? If I think it's going to be really low-- let's say the interest rate is going to be zero. Then which deal should I take? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: I should spread it out over time, because the money in the future is worth the same as today, so I might as well take the 1.6 billion. But if I think the interest rate is going to be higher, than I should take the money upfront and invest that money and earn my own interest. So essentially, it becomes a debate of what you think the interest rate is versus what the state thinks the interest rate is. They're setting those two to be equal under some assumption of the interest rate. I don't know what number they chose. Whatever number they chose. It was some number they chose, because this is our assumption of the interest rate. You've got to decide, do you think the interest rate's going to be higher or lower? If you think it's going to be higher than the state thinks it is, then you want the money upfront, and you'll invest it. If you think it's going to be lower, than you should take the money over time, because the state is giving you basically a better deal. Do you understand that? OK, so now armed with this, let's go and think about how do firms make investment decisions? How do firms make investment decisions? Remember, investment is about the delay of current consumption for future assumptions. It's about putting some aside today by spending money on a machine which will deliver you benefits in the future. Now, this adds one wrinkle to what we've done so far, which is that we've only talked about money that's always positive, some amount you get get in the future. When you're making an investment decision, it's a little bit more complicated, because you're actually spending money today to make money tomorrow. So in that case, we talk about we call net present value, which is the same thing, it just allows for negative values. Net present value, which is essentially saying in every period, you want to account for the cost of that period and the benefits of that period, and you want to invest only if the net present value is greater than zero. So for example, think about a project that has a stream of payments in every period of r sub i, every period it's got a stream of payments of r sub i, and a set of costs that's c sub i. The costs are the upfront investments, maybe the maintenance of the machine. Whatever it costs to run it. Then, the net present value of that investment is r0 minus c0, comma r1 minus c1 over one plus i-- over one plus i plus r2 minus c2 over one plus i squared plus dot dot dot for as many periods as the investment lasts. That's the net present value of an investment. So basically, what you want to ask is, take each period's costs of benefits into account. On net, is it greater than zero? OK, so the key point is that basically, sometimes investments which have upfront costs can be valuable as long as the long run benefits are large enough. So if you think about an investment that's got 100-- so think about a simple, trivial example of the first period you buy a machine, and it costs $100. So c0 is 100. And let's say for every period thereafter, that machine will deliver you revenues, revenue i-- revenues for i greater than zero-- of 200. But it will have maintenance costs, costs i greater than zero, of 50. So what's the net present value? Well, the net present value is simply minus 100. And let's say the machine is going to last forever. Minus 100 plus 150 over i. Think about that for a second. Basically, what we're saying is, you're throwing 100 at it today, so that's negative. But every period in the future, you're going to net $150, because you're going to make $200 and you have $50 maintenance costs. And we have the formula, so we just apply the formula for perpetuity. We have a set of future payments of $150, so your net present value is minus 100 plus 150 over i. So let's think about, look at this formula for a second. What does that say? What is the relationship between whether the firm's going to want to invest and the interest rate? What does this imply the relationship is between a firm's desire for investment and the interest rate? If the interest rate goes up, will firms want to do more investment, or less investment, and why? Will firms want to be more eager to buy machines or less eager to buy machines, and why, as the interest rate goes up? Yeah. AUDIENCE: They'll be less eager. JONATHAN GRUBER: Less eager. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: They'll be less eager. Say it again, because why? AUDIENCE: Because if they're borrowing. JONATHAN GRUBER: Yeah, or I think an easier way to think of borrowing, just think about, they've got a bunch of money. Apple's got a bunch of money today they're sitting on. If they buy the machine, they're going to get this return. If they don't buy the machine, they could put it in the bank or invest it in Apple stock and get some interest rate, i. The higher the interest rate, the less they want invest. Or think of it this way. The interest rate is the opportunity cost of investment. The more the firm invests in their machines, the less they can earn saving through some other mechanism. And the price that they pay by foregoing that other savings is i. Yeah. AUDIENCE: So does that mean if you're looking at a machine, it's better to try to then push the cost further into the future, because then they can't also divide by the [INAUDIBLE] sign? JONATHAN GRUBER: Basically, you always want to try to make as much money early as you can and make the cost as late as you can. For any given amount, you'd absolutely like to do that. That's why we'll talk about this in my public finance class. We talk about why, when you're paying taxes, you always want to try to find shenanigans that allow you to pay your taxes later. So for any given amount of taxes, the later you pay it, the less it costs you, because you get to earn the interest along the way and then pay it later on. This is a key macroeconomic concept. You'll often hear in the news, high interest rates are bad for business. And you might have thought, why is that? This is why. High interest rates are bad for the economy, you'll hear. Why are high interest rates bad for the economy? You might say, wait a second. That makes no sense. A high interest rate means I earn more on my savings. Why is that bad for the economy? It's bad for the economy, because it lowers the demand for capital. Because the higher the interest rate, the less firms actually want to invest. The more they just want to stock their money away in the bank. So that is the key question. Now, then let me ask another question. If you're a firm, what's the right i to use? If you're a firm thinking about this investment decision, we know the higher i is, the less you want invest. But what is i? Once again, forget inflation. Inflation is zero. What's the right what we call firm discount rate? The discount rate is the amount by which firms are going to discount future dollars to bring them back to today. How does the firm think about what i to use? What i should the firm use? Your Apple. Yeah, go ahead. AUDIENCE: [INAUDIBLE] i that makes that positive, right? JONATHAN GRUBER: Well, they don't want to invest unless it's positive. That's a good point. But the way that is, they write down the math, and they plug in an i. What i do they plug in? Yeah. AUDIENCE: Is it published by the government? JONATHAN GRUBER: Well, i might be. But there's not one answer. What's the general answer the firm wants to use? What's the general answer? That the firm-- if you're a firm thinking about making an investment, you want to discount that investment. What do you want to discount it by? Yeah. AUDIENCE: The opportunity cost of the next best return. JONATHAN GRUBER: The next best thing you could do. So if I'm thinking about buying this machine, when I discount it, I want to think about what's the next best thing I could do with that money? That's the discount rate I want to use. So in a world where firms either can buy a machine or put it in the bank, it's easy. It's the bank interest rate. But life's not that easy. Firms have dozens of investments. So for every investment, you want to discount it by the next best thing you can do with the money. It's the concept of opportunity cost. That's why it's the very first thing we taught in this class. Opportunity cost is always what drives things. Questions about that? Now, this isn't just for firms. This same math applies to consumers as well. Let's think about me. A number of years ago, I had to decide whether to insulate my ancient house. Let's write down the numbers to think how this worked. I had heating bills at that time, back when gas was cheaper, of about $2,000 a year was my heating bills for the house. The best estimate I could get was that if I insulated my house, I would lower my heating costs by 25%. So my heating costs would fall by $500 per year if I escalated my house. But to insulate my house, I had to pay the guy to insulate it. And the insulation cost $4,000. How do I think about whether I should insulate or not? How do I think about that decision? What equation should I write down? Yeah. AUDIENCE: Minus 4,000 plus 500 over i. JONATHAN GRUBER: Exactly. I should say, I'll assume I'm going to own the house forever, or at least long enough that I can treat it as forever. And I write down that formula. And what that formula says is that if I think the interest rate is less than 12.5%, I should insulate. If I think the interest rate is more than 12.5%, I should just invest the money and use the returns that I invested to pay my higher heating bills. So it all depends on what the interest rate is. So that's why-- I did it. I insulated. So the same logic we can think of is basically, essentially the same idea as firms. You want to think about the upfront costs and the long run returns. And here's a fun economics question. What if I don't intend to hold the house forever? I would argue I should still use this formula. Why? Yeah? AUDIENCE: Because whenever you decide to sell the house, you increase the value. JONATHAN GRUBER: Exactly, because I'm increasing the value of an asset that I'll then sell. So presumably, by insulating, I've raised the price of my house. How much have I raised it by? Exactly 500 over i, so I'm going to insulate and sell next year. I should still insulate, because I should get 500 over i more dollars for my house. So in fact, it doesn't matter. If you can sell an asset, then actually your horizon is always infinite. It's not just the short horizon, which is kind of an interesting insight. So the last thing I want to talk about is the fact that these decisions are not just relevant-- you guys are like retirement, business machines, insulation-- god, you're old, John. I don't care about any of this stuff. Well, let's talk about something you care about, which is going to college. Let's talk about your decision. You've already made it, but you've got a little sibling, and they're deciding whether to go to college. And they're not going to go to MIT. They're going to go to a more typical school. And they've got to decide whether to go to college. Well in fact, their decision is an investment decision, just like any other investment decision. What they're investing in is what we call their human capital. When you get education, you're investing in yourself, just like you invest in a machine. Because you are, you hope, raising the value of what you can do, of what you can earn, by investing in learning stuff. Well, that human capital investment has the same features of any other investment. There's an opportunity cost, which is what? What's the opportunity cost of investing your time in going to college-- what's the opportunity cost of going to college? AUDIENCE: You could get a job. JONATHAN GRUBER: You could-- there's two. One is, you can get a job. AUDIENCE: You could also invest your tuition. JONATHAN GRUBER: You could not pay tuition. So if you think about going to college, you're sacrificing two things. All that money you're paying, you could basically invest instead of giving it to some college. And you could be out earning money instead of sitting here listening to me. OK, so if you think about that, it actually becomes a harder decision than you might think. So let's think about a simple example. Let's imagine that if you don't go to college, you work from age 18 to 70. And if you do go to college, you work from age 22 to 70. So we're going to ignore grad school. Four years of college, you either start working at 18, or you start working at 22. And let's say college costs $35,000 a year. Obviously, not MIT. OK, let's say college costs $35,000 a year. And let's say that if you worked starting in high school, you could have earned $20,000. You could have started at $20,000 if you'd gone to work at age 18. Well, we can actually graph what this looks like in figure 17.2 17.2. If you think about age 18, from age 18 to age 22, that's the green area. If you go to college, you give up the $35,000 in tuition and the 20,000 you could have earned. The bottom line is basically-- the red line is your lifetime earnings if you go to high school, if you don't go to college. The blue line is your lifetime earnings if you do go to college. Empirical estimates suggest that at age 22, the typical college graduate earns $45,000. Yes, you're not the typical college graduate. The typical college graduate earns $45,000. And the typical high school non-college education person earns $28,000. So at age 22, you come out of college earning 45, and if you'd not gone to college, you earn 28. Not you, but a normal person. But moreover, knowledge you earn more when you leave college, your earnings grows faster. So if you're college educated, it not only means you earn 17,000 more at 22, it also means your earnings grows faster. So that by age 51, the average college educated person earns $80,000, while the average high school person earns $45,000. So what you see here is, the blue line starts above the red line and the gap widens over time. And maybe, I would have learned to clip this on if I hadn't gone to college. OK, so the gap widens over time. So how do we think about this decision? Well, the cost is the green area. The cost is over four years, you could have earned money and you wouldn't have had to pay tuition. The benefit is the yellow area. Over that entire time after graduation, you're making more money. Now obviously, in terms of size, the yellow area is much, much bigger than the green area. But the yellow area comes later. That's the key thing. So if I look and say, look, it's obvious-- before this lecture, you might say, well, it's obvious you should go to college. Look, the yellow is way bigger than the green. But that's not necessarily true, because the green comes now, and the yellow comes later. Indeed, if you look at the table, this actually shows the net present value of going to college and high school. And what this shows is at low interest rates, you're much better off going to college. So if there's interest rate, then college is a much, much better deal. Your net present value of earnings if if you go to college is 2.6 million, while it's only 1.6 million if you don't go to college. But once the interest rate gets above 8%, it suddenly becomes a worse deal to go to college. That is at only a 9% interest rate, which existed not that long ago in our history. It was actually a worse deal for the average person to go to college. Yeah. AUDIENCE: Isn't that not accounting for financial aid, though? Because if you didn't have the 35,000 to spend, you wouldn't have been able to invest it? JONATHAN GRUBER: Well, actually, it's interesting. It depends on the form of financial aid. Why? Someone tell me why it's dependent on the form of financial aid? Financial aid comes in different forms. So why does it depend on how you get the financial aid? Yeah. AUDIENCE: Because if you have to pay it back. JONATHAN GRUBER: If it's a grant, then yeah, you should just take that out of the cost. But if it's a loan, it depends what interest rate you get the loan at. If the loan is at the market interest rate, then it's no different. But that's a great point, which is why college financial aid comes in two forms. Grants for very low income people, and low interest loans for other people. Why do we give low interest loans for college? Because of this graph. Because we're saying, we think people need to invest in their education. We're afraid that if they faced a regular interest rate, they won't be willing to do it, because the green would be bigger than the yellow. So we're actually going to subsidize their interest rate. So you might have thought to yourself, sort of a weird way to get people to go to college is to have a lower student loan interest rate. But in fact, it makes perfect sense. By having a student loan interest rate that's lower than the market rate, you encourage people to go to college, because essentially you lower this discount rate, at least on the part that's tuition payments. So that's actually very exciting way to think about and a very important part of public policy is how we set the interest rate on student loans. For any of you-- how many of you guys have a student loan? Do you know? Any of you guys have a student loan? You might not know. Anyway, I set the interest rate on that student loan. So thank you. Actually, when I was in the government, I was in the government 14 months, the Clinton Administration. It was super fun. But looking back, I only got one thing done, which I got to set the interest rate for student loans. So that was kind of fun. But otherwise, it was just a lot of fun being there. So anyway, let's stop there, and we will continue. What's today, Wednesday? So no class on Monday. That's Veterans Day, so we'll meet in a week, next Wednesday.

MIT_1401_Principles_of_Microeconomics_Fall_2018

3_Budget_Constraints_and_Constrained_Choice.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: Today, we're going to continue our discussion of consumer choice. And we're going to talk now about what happens when we take that unconstrained choice we talked about on Monday and impose budget constraints. We'll talk about what budget constraints are. We'll then come to talking about how consumers make constrained choices. And then we'll end with an example of food stamps. So let's start by talking about budget constraints. And we'll start by talking about their construction, the construction of budget constraints. So, basically, last time, we talked about the fundamental axiom of consumer choice that more is better. So what stops people from just bingeing on everything? It's their budget constraint. It's their limited resources. Now, for most of this course, we're going to make a simplifying assumption that your budget-- that is what you spend-- equals your income. That is what you earn, OK? That is there won't be any savings or borrowing, OK? Now that is a simplifying assumption. And, indeed, we'll spend a couple lectures at the end of the semester talking about what happens when people can save or borrow. That said, this is not a terrible description of most Americans. The median American household has $400 in the bank. So this is not kind of a terrible description of the way most people live their lives in America, which is what they earn each week is what they spend each week. So that's what we'll do. It also might not be sort of a terrible description of your life. I presume, in college, you're not doing a lot of savings. You maybe do a little borrowing, but not a lot of savings or borrowing. So what we're going to do is we're going to assume that's true for you as well. We're going to assume your parents have given you some amount of money to spend. We'll call it Y. Your income Y is the amount of money your parents have given you to spend for say the semester or the month. And, once again, let's say all you spend your money on is pizza and cookies, OK? That's all you want to spend your money on. We write the budget constraint as saying that your resources, your income Y, can be spent on either pizza or cookies. And the constraint is that you could spend it-- that budget has to be divided between pizza, where there's the price per slice of pizza times the number of slice of pizza, or cookies. We have the price per cookie times the number of cookies. So p sub p is the price per slice of pizza. p sub c is the price per cookie. P is the number of pizzas, and C is the number of cookies. That's your budget constraint. You can essentially devote your income to some combination of pizza and cookies, but you have to consider how much they actually cost in doing that. I find this easier to see graphically. So let's turn to figure 3-1. Figure 3-1 shows a budget constraint. So how does the budget constraint look? Well, the x-axis is your income divided by the price of cookies. That is, if you decide to devote all your income to cookies, then how many cookies can you have? Y over pc. If your income is $100, and cookies are $10-- that means you're going to Insomnia Cookies-- then you can only have 10 cookies, et cetera. Likewise, the y-intercept is the income divided by the price of pizza. That's how many pizzas you can have. The budget constraint represents-- the budget constraint, the slope of the budget constraint, is the price ratio, the negative of the price ratio because it's a downward-sloping line, pc over pp. That is every extra cookie that you buy, holding your income constant, lowers the amount of pizza you can have by p sub p, OK? So let's consider an example. Suppose that Y is $96, that the price of pizza-- it's an expensive pizza place-- is $12, and the price of a cookie is $6, OK? $12 for pizza, this is like downtown San Francisco or New York. $96 income, $12 for a slice of pizza, $6 for a cookie, OK? I'm sorry. Y is-- I wanted to make Y 72, my bad. So Y is 72. Your income is $72, OK? And you can spend it on pizza and cookies, and those are the prices. Now what that means is, if you wanted just pizza, you could get six pizzas. If you wanted just cookies, you can get 12 cookies. And, generally, the rate at which you can trade off pizza for cookies is minus 1/2, OK? That is every additional cookie would require giving up half a slice of pizza, OK? Every additional cookie requires giving half a slice of pizza. That's why the slope would be negative 1/2, OK? So, basically, we're going to call the slope of the budget constraint-- the slope, we are going to call the Marginal Rate of Transformation, the MRT. Last time, we did the MRS, the Marginal Rate of Substitution. Now we're going to have MRT, the marginal rate of transformation, which is equal to minus pc over pp. Or the slope of the budget constraint, OK? That is the marginal rate of transformation. Now this class is not alchemy. We are not literally transforming pizza into cookies. That would be kind of cool, but we're not doing that. That's somewhere else at MIT, OK? But it's effectively doing the same thing. What we're doing is, given that we have a fixed amount of money and given that we're going to spend it all, the more you spend on pizza, the less you spend on cookies. So you're effectively transforming pizza into cookies and vice versa because you're going to spend all your money. You've got to spend it on something. So, the more you spend on one, the less you get of another. So, through the budget constraint, we are effectively transforming one good to the other. By having more of one, we're getting less of the other. So that's the sense in which we call it the marginal rate of transportation-- of transformation. So, basically, this comes back to the key concept we talked about in the very first lecture, opportunity cost. The opportunity cost of a slice of pizza is two cookies. Remember, opportunity cost is the value of the next best alternative, OK? The opportunity cost is the next best alternative. Well, here you only have two alternatives, pizza and cookies. So the opportunity cost of a slice of pizza is two cookies. And that's the sense in which you're transforming pizza into cookies or cookies into pizza, OK? Now this seems kind of abstract, but let's actually think of an organization which has taken this principle to heart to develop the best method of weight loss in America, which is Weight Watchers, OK, Weight Watchers. Now it turns out that dieting is super hard and basically doesn't work, OK? There's a large literature, which says that people go on diets all the time. Then they stop them, and they gain the weight back. OK, dieting is incredibly hard and basically doesn't work, OK? But a much more successful approach has been established by Weight Watchers. It's not the only approach, but it's been proven much more successful, OK? And, essentially, what does Weight Watchers do? They set up a budget constraint and ask you to follow it. So, for example, they essentially assign point values to every good you might consume. You go on the website, and everything in the world you might want to eat has a point value. They then ask, well, what weight are you today? What's your age and gender? That stuff matters for weight loss. And what weight do you want achieve? And they say, if you want to achieve a weight loss of x over y days, then you've got to limit yourself to z points. So, essentially, your goal is to lose weight. So we're going to give you the budget constraint. We're not going to tell you what to eat. That's why it's better than dieting because, once again, Adam Smith was right. People like to have choices. They like to let choice drive things. But we are going to tell you a total budget. So, for example, vegetables are like zero points. Snickers bars are like six points, et cetera. They have various point systems, OK? So, for example, suppose your budget is 30 points, which would be pretty typical, OK? Suppose you go to McDonald's for lunch, and you get a number one. The number one at McDonald's is a Big Mac, which has 14 points, fries, which have 10 points, and a Coke, which has six points. That's 30 points, and it's only lunch, OK? You've blown your whole budget for the day on lunch. Now you could just get depressed and say screw it. I'll just be fat. But, clearly, looking around the room, you guys have not made that choice. Or you could look at the budget constraint and say, well, what else can I get. Well, it turns out you can get a 10-piece nugget, which is 12 points, apple slices, which is one point, and a Diet Coke, which is zero points, for a total of only 13 points. Now you have 13 points and plenty of room for dinner. Now, to be honest, anyone who tells you that second lunch is as good as that first lunch is a liar, OK? I'd much rather a Big Mac and fries and a Coke than nuggets and apple slice and Diet Coke. Give me a break. But I'd also much rather have dinner, OK? So, basically, this lets you make the trade-off by imposing a budget constraint, by setting relative prices across goods. The points are like utils. They're not meaningful. They're only meaningful relatively. It lets you set relative prices across goods and then it lets you, essentially, optimize across those various-- across those various goods. So budget constraints, essentially, by setting up this marginal rate of transformation, can help with a lot of kind of decisions in life. OK, questions about that? OK, now what happens if we shock the budget constraint? So we talked about constructing them. What about shocking the budget constraint? We're going to do a lot in this class of what we call comparative statics, which is, essentially, making changes in one thing or another and seeing what it does to the system. So let's talk about shocking the budget constraint. Let's start first with a change in prices. Suppose the price of pizza goes from $12 up to $18. This is a really good slice of pizza, OK? Well, what happens to the budget constraint? Let's look at figure 3-2. Figure 3-2 shows what happens. You have your original budget constraint BC1. The equation of that line is 12P plus 6C equals 72, OK? The price of pizza and the number of slices of pizza plus the price of cookies times the number of cookies equals 72. Now the price of pizza has gone up. What that's done is that has pivoted inward your budget constraint to BC2. It has flattened the budget constraint because the slope, remember, is the ratio of the price of cookies to the price of pizza, right? That's a ratio. Well, that ratio has just fallen. It used to be a 1/2. Now it's a 1/3. Negative 1/2-- well, it used to be a half. Now it's a 1/3. So the slope has fallen from negative 1/2 to negative 1/3. So what's happened is you can still have as many cookies as you had before. The y-intercept has not changed, but you can have fewer slices of pizza. That's why it's a pivot because one price has not changed, only the other price. So it's a pivot inward. The other thing here, you'll notice we have all these funny dots and stuff, OK? That represents what has happened to what we call your opportunity set, your opportunity set, which is an important concept, OK? Your opportunity set is the set of choices available to you given your income and market prices, the set of choices available to you given your income and market prices. So your opportunity set initially was the black dots plus the red dots. Now your opportunity set has shrunk. Your opportunity set is now just the black dots. Given your income, you can now get less stuff, same amount of cookies, but less pizza. And you are worse off. Your opportunity set has shrunk. Your opportunity set-- even though your parents are still sending you the same check, you are worse off because you can now buy less pizza with it, OK? So that's what happens to the opportunity set when a price changes. And, likewise, you should show to yourself the same thing will happen when the price of cookies change. In that case, you'll get an increase in the steepness of the budget constraint, OK? But your opportunity set will still-- your opportunity set will still shrink, OK? Now what about-- yeah? AUDIENCE: Don't we not care about all the dots below the line, though, because we're assuming we're spending all the money? JONATHAN GRUBER: Well, that's a good point, and we're going to come back to that. We haven't-- we assume they're spending all their money, but it's just a way of representing. You could think of the line being lower as the same thing. We care about-- we just care about the area because it represents the set, but you're right. You could just focus on the line and say the line is everywhere lower. So they're worse off. That's another way to look at it. But we like to think about as a set. It comes in handy later for various reasons, OK? But that's a good question. Now let's ask about a second thing. What if your income goes up? What if prices are back to 12 and 6, but your parents decide to send you more money? Suppose your parents-- or send you less money. It turns out you haven't been paying enough attention in 14.01. You're parents are mad. They're monitoring you. That's why we have the camera here. This goes directly to all your parents, OK? I'm sort of joking. And so let's say parents cut your allowance to $60, OK? Well, what does that do? That's in figure 3-3. OK, in figure 3-3, the old budget constraint was that you get pizzas and cookies at a price of $6 and $12, and you could get them until you spend $72. Now you can only get them until you spend $60. Now what we see is not a pivot in the budget constraint, but an inward shift in the budget constraint, because the relative price of pizza and cookies has not changed. Therefore, the slope hasn't changed. OK, the slope is dictated solely-- you don't do anything to control the slope. The market controls the slope, OK? But you and your family control the level, and the level has shrunk. So you're pivoting inwards, OK? And, once again, now, instead of being able to buy say 12 cookies and six pizzas, now you can only buy say 10 cookies and five pizzas. That's the most you can get, OK? So, once again, your opportunity set has been restricted, but in a different kind of way through this pivot inward, OK? So that's how we sort of manipulate these budget constraints. And we're going to come back to that next lecture. That'll be important. Yeah? AUDIENCE: So, in looking at the differences, can like an increase in the price of pizza or like a decrease in your budget-- is it more showing that like the change in slopes doesn't really affect you if you're like say buying more cookies than pizza? But like, in terms of if your budget as a whole decreases, then it affects you overall. JONATHAN GRUBER: That's a great question, and we're going to actually answer that question next lecture very explicitly. So hold on to that question, and we'll talk about we're going to compare explicitly why income changes differ from price changes and what are the underlying mechanisms. Yeah? AUDIENCE: How do you determine your marginal rate of transformation? How do determine your-- like say it wasn't just pizza and cookies. Like say it was more products. How would you determine that value? JONATHAN GRUBER: Great, great question. So, as I said, we always are going to start with simplifying assumptions to make life easy. There's no reason that this couldn't be written in three dimensions. And you'd have relative marginal rates of transformation, rates at which you're willing to trade off various things. So you could just extend the math in all dimensions. It wouldn't add any richness, and it'd just make your head spin. But the basic-- so all the basic ideas can come across with two goods, but it'd be the same mechanics with more goods, OK? You essentially, when we get to the constrained optimization, you'll essentially have more first-order conditions in your constrained optimization. That's the way to think about it. OK, so let's-- actually, that's a great segue. Let's turn to the second part, which is how we use budget constraints and the utility function we learned about last time to actually describe how consumers make choices. So we're going to take utility. Remember, I said last time consumers are going to maximize their utility subject to a budget constraint. Well, now we've taught you about utility. We've taught you about budget constraints. Let's put them together, OK? How to consume-- how do consumers put them together? Well, graphically, the representation of preferences was our indifference curves. That represented people's indifference with further out indifference curves made people happy, right? That was last time. So, essentially, what we're going to ask graphically is what is the highest indifference curve you can achieve given your budget, right? We know you want to be that highest indifference curve possible by more is better. So we're simply going to ask what is the highest indifference curve you can reach given your budget, OK? So let's consider the same utility from last time. Utility is square root of P times C, OK? And let's consider the same budget we wrote down up here-- $72 income, $12 price of pizza, $6 price of cookies. And now let's ask where can you go with that. So let's turn to figure 3-4 and do it graphically. We'll do it mathematically in a minute, OK? So, in figure 3-4, you have our budget constraint, which runs from 6 pizzas to 12 cookies. That's the original budget constraint. And you have a series of indifference curves. And these indifference curves, I1, I2, I3, I4, they all come directly from this utility function. So, simply, I've solved this utility function. I'll talk about the math in a little bit, and you'll do more math in section on Friday, OK? But, essentially, you can solve-- we'll show you-- you'll drive on Friday how you take this utility function and literally can draw the indifference curves from it, OK? But, for now, take my word that these indifference curves represent this utility function. And what we see is that point D is the furthest out indifference curve you can achieve while still meeting your budget, while still meeting your budget constraint. And, therefore, we say that the optimum, graphically, is the tangency between your indifference curve and your budget constraint is the optimal constrained bundle. You see how we brought-- last time, we talked about further out indifference curves make you happier. Today, we talked about the fact that you're limited by your budget. So we have the furthest indifference curve you can get to is going to be, definitionally, at the tangent of the indifference curve and the budget constraint. And, once again, that gives you-- we realize we don't want to measure utils, but, just for mathematical, for mathematical purpose, that gives utility at the tangency of square root of 18, OK? At that point, you are choosing six cookies and three pizzas. That is the choice you are making. That is the best off you can get given your budget. And, to see this, let's talk about some other points and why they're not better, OK? Let's talk about point A. Why isn't point A better? Why isn't it better to have two-- maybe you just-- maybe you like cookies a lot and don't like-- or like pizza a lot and don't like cookies that much. How can we say that point D is better than point A? Yeah? AUDIENCE: Because point D is on a higher indifference curve. JONATHAN GRUBER: It's on a higher indifference curve. So point D dominates point A because it's a higher indifference curve. Well, fine. Same person, by that logic, why not choose point E? AUDIENCE: It's above the budget. JONATHAN GRUBER: Yeah, you can't afford it. So the bottom line is you can see graphically why the tangency is the best you're going-- is the best you're going to do. OK, likewise, point C you wouldn't choose. Point C has the same slope. It has the same slope as point D. In other words, the slope is minus 1/2 at point C. You've drew a line tangent to point C. The slope will be minus 1/2, just like it is at point D, but you wouldn't be spending all your money. So you wouldn't choose that point either. Yeah? AUDIENCE: What if you have just three indifference curves so there is none that hit the tangent? Do you just go for one that's like the most tangent I guess? JONATHAN GRUBER: We're going to come to-- we're going to-- well, first of all, we're not going to have discrete indifference. We could have lines, and the lines could end up-- you could end up lying along. You could end up lying along a budget constraint for example. Or you could have-- you could even have utility functions, which just touch a budget constraint at one extreme or another. And we'll talk about those cases. Yeah? AUDIENCE: So [INAUDIBLE] utility function go through lines and the budget constraint, right? JONATHAN GRUBER: Yeah. AUDIENCE: Isn't this just Lagrange [INAUDIBLE]?? JONATHAN GRUBER: Well, let's come to the math then. OK, let's come to the mathematical derivation. So that's the graphic. So let's come to the math, OK? Now, always a bit of a tightrope act when I'm doing math up here on the board, so bear with me, OK? But the key thing is the math of constraint optimization is all about the marginal decision. Remember, it's hard to say how many cookies you want. It's easier to say should I have the next cookie, OK? It's about constraint optimization. And what we want to ask is we essentially want to compare how do you feel about trading off pizzas versus cookies versus what will the market let you do in sort of trading off pizzas versus cookies. That is the optimum is going to occur when we set your marginal rate of substitution, which, remember, we defined as minus MUc over MUp, equal-- I'm going to get rid of this-- equal to your marginal rate of transformation, which we defined as minus pc over pp. And this is the fundamental equation of consumer choice. If you understand this equation, you can solve virtually every consumer choice problem I'll give you, OK? That basically, at the optimum, the ratio of marginal utilities equals the ratio prices. That is the rate at which you want to trade off pizza for cookies is the rate at which the market will allow you to trade off pizza for cookies, OK? Basically, it's saying the ratio of the benefits. Think of this as the benefits and this as the costs. Think of the MRS as the benefits. It's what you want. MRT is the costs. It's where you're constrained. You want to set the ratio of the benefits equal to the ratio of the costs, OK? Now I find it actually easier to think of it this way. If you just rearrange terms, you can write it as MUc over pc equals MUp over p sub p. I like this way of writing it because I call this the bang for the buck equation. What this is saying, your marginal happiness per dollar should be equal. This is sort of the happiness per dollar spent on cookies. This is the happiness per dollar spent on pizza. And you want those to be equal. You want the bang for the-- you want to put your next dollar where it's going to make you happiest, OK? And so, basically, think of that as your bang for your buck. So, for example, suppose you were in a position where the MRS was greater than the MRT. You're in a position where the marginal utility of cookies-- and I'm getting rid the negatives. There's negative on both sides. So I'm just going to get rid of the negatives, OK? The marginal utility of cookies over the marginal utility of pizza was greater than the price of cookies over the price of pizza, OK? That is the slope of the indifference curve was greater than the slope of the budget constraint. This is the slope of the indifference curve. OK, this is slope of the indifference curve. This is the slope of the budget constraint. In absolute value, the slope of the indifference curve is greater in absolute value than the slope of the budget constraint, OK? That would be true at points like point A, point A where you intersect-- where you basically intersect from above the budget constraint by the indifference curve. So a point like point A has a steeper slope of the indifference curve than does the budget constraint. What that says is intuitively-- and, once again, I want you to understand the intuition-- the rate at which you are willing to give up, the rate at which you are willing to give up cookies for pizzas-- I'm sorry. Let me say it-- let me say it a better way. The marginal benefit to you of another cookie relative to another pizza is higher than what the market will charge you to turn pizza into cookies. Let me say it again. The marginal benefit to you of another cookie, which is this-- this is how much more you want the next cookie relative to how much more you want the next pizza-- is greater than what the market is going to charge you to trade in your pizza for cookies. Therefore, you should trade in your pizza for cookies, OK? So let's say this mathematically. At a point like A, point A, OK, you have your marginal utility for pizza is the derivative of the utility function with respect to the number of slices of pizza. It's the marginal utility. It's derivative of the utility function. So it's dU dp, which is equal to 0.5 times C over square root of P times C, OK? And, at point A, at point A, we had two cookies and five pizzas. At point A, P was five. C was two. OK, that's true of point A. So we can evaluate the marginal utility dU dp, which equals 0.5 times C over square root of P times C. So that's 1 over the square root of 10. That's the marginal utility of the next slice of pizza. The next slice of pizza makes you 1 over square root of 10 happy. Once again, that number is meaningless. So we only care about it in ratios. So we need the ratio. So let's do the marginal utility of cookies. That's dU dC, which is 0.5 times P over square root of P times C, which is 2.5 over the square root of 10, OK? So the marginal utility of pizza is 1 over square root of 10. Marginal utility of cookies is 2.5 over the square root of 10. Therefore, your marginal rate of substitution is minus 2.5. Remember, marginal rate of substitution is MUc over MUp. So your marginal rate of substitution is minus 2.5. What does that mean? Can anyone tell me what that means? Your marginal rate of substitution is 2.5. What does that mean? That is a meaningful concept. Utils are not, but that is. Yeah, say it loudly so we can hear. AUDIENCE: You're willing to trade-- you're willing to trade two pizzas for one cookie. JONATHAN GRUBER: You're willing to trade. Exactly, you're willing to give up 2.5 slices of pizza for one cookie. That's what that number means. And that is a meaningful number. That's not an ordinal. That's cardinal. We can use that. You are willing to give up 2.5 slices of pizza to get one cookie. What is the market asking you to give up? How much pizza do you have to give up to get one cookie? Half a slice. You are happy to give up 2 and 1/2 slices of pizza to get a cookie, but the market is saying we'll let you have a cookie for half a slice of pizza. So what should you do? AUDIENCE: Trade. JONATHAN GRUBER: Eat less pizza. Eat more cookies. That will unambiguously make you happier. And that's why you should move from point A towards point D. OK, that's the intuition, OK? You basically want to trade pizza for cookies until these things are equal. Indeed, I'd like you to go home and do the same math starting at point B. If you do the same math starting at point B, you'll find the MRS is much below 1/2. That is, at that point, you are happy to give up tons of cookies to get pizza because, jeez, you've got 10 cookies and one slice of pizza. You'd give up tons of cookies to get pizza. But the market says you only have to give up two cookies to get pizza. So you'll happily do it, and you move back towards point D. And that's sort of in a bundle sort of the intuition and math and graphics of how we do constrained optimization. OK, that is hard and very important. Questions about that? Don't hesitate to ask. OK, that is hard and very important. If you understand this, you're sort of done with consumer theory, OK? This is sort of the core of what consumer theory is all about. It's all about this balancing act. The whole course is fundamentally all about one equation, which is marginal benefits equals marginal costs, OK? Everything we do is going to be about weighing the marginal benefit of an activity against its marginal costs. If we take the next step, what's the benefit? And what's the cost? Well, here the marginal benefit is the MRS. The marginal cost is the MRT. We want to set them equal. And this sort of example I hope explained why, OK? So that is how we think about constrained choice. Now I want apply it. I want to apply it by looking at the example of food stamps, OK? Now food stamps are not actually called food stamps anymore. When I was a kid, they were called food stamps. It's basically a program the government has that provides money for individuals to buy food if they're low income. Essentially, we have in the US what's called the poverty line. And I'll talk a lot more about this at the end of the class, but the poverty line is essentially a measure of what's a minimum level of resources you need to live in America. The poverty line for an individual is about $14,000. OK, for a family of four, it's about $28,000. How you feel about that number obviously is going depend on where you're from. If you're from Boston, you'll say that's insane. If you're from some rural part of the country, you think, yeah, that's poor, but manageable. OK, we'll talk later about the poverty line, what's good and bad about it. But, in any case, if you're below the poverty line in America, roughly speaking, you get help with buying food. And that comes through a program we now call SNAP. It used to be called food stamps. I've got to update my notes. Supplemental Nutrition-- I don't know. I know the N is for nutrition. OK, so, basically, what the SNAP program does is it gives you a debit card. If you qualify on income grounds, you get a debit card, and that debit card can be used to buy food and food only, OK? So you essentially get a debit card from the government that you can use to buy food if you're poor enough. And they give you sort of a fixed amount every month, and that amount can be used to purchase food. So here's the question. Why go through this rigmarole? Why not just give people cash? This fancy thing, if we want to give poor people money, why don't you just give them money? And we're going to-- I don't want the answer yet, OK? What I want to do is show you graphically how we think about the trade-off, and then we'll come to the answer. So hold your thoughts. So let's actually graph how we think about food stamps. Let's go to figure 3-5A. And let's start with a cash transfer. So here's the setup. Imagine people start with an income of $5,000. That's super poor, OK? $5,000 is their whole family income for the year, OK? And let's say all they can spend it on is food or shelter. Remember, as this gentleman pointed out, in life, there's more than two goods, but it makes it a lot easier to have two goods. So imagine this case. Your two goods are food and shelter. And, actually, quite frankly, if you're that poor, that probably is the only two goods you have to-- you can worry about at that level of income. OK, it's food and shelter. So you $5,000 to devote to food and shelter. So you have some original budget line, which is labeled there original budget line, that runs from 5,000 in food to 5,000 in shelter. And then you can have some of in between, some along the way, OK? Now let's say we give someone $500 in cash. Obviously, this graph is not to scale, OK? It looks like you're doubling his income, but it's only $500. This just sort of makes it easier, a not to scale graph. Let's say we give someone-- we say to them, look, you're poor. We're going to give you $500 in cash. Well, now all we've done is shift out your budget constraint from 5,000 to 5,500. OK, we've shifted out your budget constraint from 5,000 to 5,500. What does that do to your choices? Well, consider two different types of people. Person y, OK, they used to be on indifference curve I0. They used to spend almost all their income on food and not a lot on shelter. They were probably homeless, OK? So they spent all their money on food and were basically homeless. Now what do they do? Well, they spend a little more on food and a lot more on shelter. Maybe now they get-- you know, $400 still doesn't buy you much shelter. They spend a little more, OK? Maybe, a night a week, they can get shelter, OK? So, basically, that's what they do. That's their constrained optimization. We're not saying it's right or wrong. This is not normative economics. It's positive. The positive thing is, given their utility function, they move from point y1 to y2. Now imagine someone like individual x. They're different. Their tastes are such that they don't need to eat. They just want to have shelter. So they're up at point x1 initially. And you give them that $500, and they spend just a little bit more of it on food and even more of it on shelter. They just love their shelter, OK? And they're just super-- they're super Weight Watchers. They don't eat, OK? So, basically, they move from x1 to x2. Once again, not normative right or wrong, it's just these are feasible choices people could make given the opportunity set with which they're faced. And that's what happens when you give them the $500 in cash. Questions about what I did here on this graph alone? Yeah? AUDIENCE: Like, even if like you gave them money specifically for food, couldn't they then just reallocate their other money? JONATHAN GRUBER: OK, that's a good point. We'll come back to that. That's time out if you're not a sports fan. OK, so we will come back to that. And, in fact-- OK, but do people understand what the cash transfer is, how it works? OK, now let's go to SNAP. And let's say, with SNAP, instead of giving them $500, we'll give them the debit card. Instead of handing them a $500 check, we give them a debit card with $500 on it that can only be used on food. How does this affect their budget constraint? Now we see where budget constraints start to get interesting and fun and the kind of challenges you're going to face in this course in drawing budget constraints. The original budget constraint continues to be the original budget line running from 5,000 to 5,000. The new budget constraint is this kinked line that runs from 5,000 on the y-axis to the point x2 at 5,000 on the y-axis. So it starts at 5,000 on the y-axis, 0 on the x-axis. There's a flat line that goes to 5,000 on the y-axis, 500 on the x-axis. And then it slopes down parallel to the original budget constraint to 5,500. Can someone explain to me why that's the new budget constraint? Yeah? AUDIENCE: You can't spend a negative amount. So you can't spend like negative amounts of your non-food-stamp money on food. JONATHAN GRUBER: Exactly, you have-- we are forcing you to spend at least $500. Compared to cash, where you can do whatever the hell you want, we are forcing you to spend $500 of your money on food. Coming to the question back there, it doesn't have to be a specifically labeled 500. It can be any 500. But we're forcing you to spend at least $500 on food. Well, what does that do to your choices? Well, for person y, it makes no difference whether they get cash or whether they get food stamps. Now the person, light blue shirt, turquoise shirt, asked that question. Why does it make no difference? Yeah? Why does it-- whatever, greenish, I don't know, yeah, you. Why does it make no difference for person y if I give him food stamps or cash? AUDIENCE: He's already spending a lot of his money on food. So any money he gets he can just reallocate differently so he can spend some of the money he would have used on food on shelter. JONATHAN GRUBER: Exactly, he can just reallocate his money, OK? That's exactly right. So, for person y, there's no difference. Look, they're already spending, what, $4,900 on food. You give him a thing labeled $500 for food. It's not going to affect their life. They'll just take 500. They'll just spend-- they'll just treat it as $500 more in cash. They're indifferent. So nothing affects them. But what about person x? Well, person x, remember, the dashed portion of this budget constraint is from the old cash example. And the dotted indifference curve is what they would have chosen with cash. Remember, person x with cash would have chosen to still spend less than $500 on food. Even when you gave them $500, they still only spent $300 on food. So we are forcing them to not be on their preferred budget constraint. Rather, we're forcing them down to point x2, which is they'll spend the minimum they can on food, but the minimum is $500, OK? We are forcing them down to point x2. Now why do I say forcing them? Why do I know for sure they are being forced, that they're less happy at x2 than they would have been when they gave them the cash? How do I know that for sure? Yeah? AUDIENCE: They're at a lower indifference curve. JONATHAN GRUBER: Exactly. Think of it this way. The fundamental-- one of the important things is people always get to the point that makes them happiest, OK? We call it the robustness of economic equilibria. People get to the point that makes them happiest. They want-- they always before had the choice of spending $500 on food, and they chose not to. Therefore, if you force them to spend $500 on food, they must be less happy, OK? Think of it that way. They always could have spent $500 on food. They didn't. Therefore, in forcing them, you're making them less happy, OK? So they are worse off, OK? They are forced to spend. They'd rather spend some of that money and find a nicer place to live, but we're not letting them. We're making them buy food, OK? Do people-- I don't want-- I just want to know if people understand the graphics here and the conclusions I drew. OK, now why? Why are we doing this? Why would you-- they're better off with cash. Why would we force them to have food? Yeah? AUDIENCE: Say because what makes-- what puts people on the highest indifference is just what makes them happiest, but not necessarily what makes them like live the longest or like have the best health So, perhaps, like if you never spend money on food, and then you die, that would be really bad. JONATHAN GRUBER: OK, but, basically, what you're saying is you know better than the guy. Let me-- I'm not accusing you. I'm just saying, look, if people knew best, maybe they'd like to just like have a nice house and die, OK? If people knew best, then there'd be no reason to do this. The reason to do this is because we think they don't know best. So, for example, let's change the label on the y-axis, just a small change. Let's cross out shelter and write cocaine. [LAUGHTER] OK? Well, in that case, maybe we don't feel so bad about forcing the guy to buy food instead of cocaine, OK? In other words, this a program which might make sense if we are paternalistic. Now we're getting into normative economics, paternalistic. If we think that people won't necessarily make the right decisions for themselves, then it may be worth actually making them worse off because they're not worse off. Their perceived benefits are worse, but they don't know what they're doing, OK? Now you can see why-- I hope you can sort of immediately see why this concept makes economists a little nervous because why do we know what they want better than they do, OK? So it makes people a little bit nervous, economists a little bit nervous, and a lot of people a little bit nervous to say, gee, maybe they're just happier doing cocaine. And how do we know that that's the wrong way for them to spend their resources? Yeah? AUDIENCE: Well, like can't you look at it from the perspective of like this is taxpayer money, right? So then aren't you also just factoring in how the taxpayer wants to spend their money and then their indifference curve and all their information? JONATHAN GRUBER: That's a very good point. Now but there's sort of two points there. First of all, if the taxpayers' goal is to help poor people, then why shouldn't you make them as happy as possible, right? If tax-- why am I giving money to this poor guy? Because I'm sad his poor. But, what you're saying, I'm not actually that sad he's poor. I'm sad he's not eating. If you're really just sad he's poor, then you should give him money. If what you're sad about is, gee, I don't like how he's living-- I don't like his-- I'm sad he can't have better food to eat, sad at the place he lives. Then you're starting to impose your preferences, but let's be important. That's imposing your preferences. Yeah? AUDIENCE: I feel like the indifference curve only goes for happiness or like contentedness, but, really, the point of SNAP isn't really with contentedness or happiness, but rather like what would be to a more sustainable life. JONATHAN GRUBER: Well, that's a related point of the taxpayer. If the taxpayer cares about, look, we want a healthy populace that's going to live a long time and be productive and pay taxes, then that would be a reason to do this. But, once again, I want to emphasize, OK, this is paternalism. If you really just care what makes people happiest, you should give them cash, OK? So that raises two questions, OK? First of all, first question-- yeah? AUDIENCE: So how about like negative [INAUDIBLE].. Because, for example, if we pump a lot of money-- if we allow people to spend a lot on shelter, that's not really going to help people. It would just make the real estate developers rich. And say the amount of shelter is kind of fixed, but like the amount of food that eaten [INAUDIBLE].. So, if we let people spend more money on food-- JONATHAN GRUBER: Yeah, yeah, so, basically, that's a great question. And, in general, we're going to-- I'm going to answer a lot of those questions with the same cheat this semester, which is we're going to assume the markets are perfectly functioning. So there's no-- you're imposing sort of a market failure. If there's no market-- once there's market failures, all bets are off. But, with no market failure and no paternalism, you'd want to give them cash. So this raises an important question. Do food stamps actually increase food purchases? First of all, there's two reasons why they might not. Reason one is everybody could be like y. x is sort of a silly case, right? You're going to die if you eat that little. And food stamps aren't that much. They're maybe like $3,000 a year. Everybody is going spend $3,000 on food. So the first issue is the first reason why food stamps may not matter is that, in fact, everybody is spending at least that amount. Everybody is like y, and nobody is like x. What's another reason why it might not matter? What's a way people could get around food stamps? Yeah? AUDIENCE: Buy food with food stamps and sell it. JONATHAN GRUBER: Yeah, they could set up a black market where they, essentially, say, look, I only want $2,000 of food. The government is making it worth $3,000. I'll buy my extra $1,000 of food, and I'll sell it to people who do want it. And I'll end up still eating $2,000 worth of food. So we actually want to know do food stamps actually increase food consumption in practice. Are they making a difference? Well, actually, we've run an experiment on this, OK? We're going to talk in this class a lot about empirical results in economics. This class is mostly going to be a theoretical class. That is we'll talk about models and ideas. But we're also-- since, basically, I'm an empirical economist, we're going to talk about empirical economics, which is results and testing the theories we develop. Empirical economics, here's a great example of empirical economics is we set up a theoretical model. You always want to start with the theory, but the theory sometimes has predictions, which are uncertain. Here we have an uncertain prediction from theory about whether food stamps will affect food purchases or not. So let's test it. And the way we test it is we actually have run food stamps cash out experiments where we literally take guys on food stamps and give them cash instead and watch what happens to their consumption before and after. It's a real randomized trial. We literally flip a coin. Heads, you keep your food stamps. Tails, we replace those food stamps with an equal amount of cash. Then we watch what happens. What happens is that people spend about 15% less on food when you give them cash instead of food stamps. That is food stamps is forcing people to spend about 15% more on food than they would like to unconstrained by the cash. Yeah? AUDIENCE: Yeah, this gets you into the behavior of [INAUDIBLE]. I remember reading an experiment like, if you have the price of gas go down, the actual like amount of money spent on gas is constant. And this might translate to food stamps because like food stamps are like explicitly on food. JONATHAN GRUBER: Yeah, you know, that's a great question. And that's you're asking about richer theory, richer theory. And I'm telling you that I'm going to give you the empirical evidence. So, whatever the theory is, the empirical evidence tells you what happens. And there's different explanations for why. So the empirical evidence is that, basically, the price of our paternalism is 15%, OK? We are making people, effectively, 15% worse off. We're making them spend 15% more food than they want to. So is it worth it? Well, actually, the evidence is starting to pour in that it might not be worth it because there's starting to be a lot of experiments where we're giving people just cash, especially in developing countries. In developing countries, the answer seems to be just giving people cash makes them better off, that actually, especially in developing countries, people use the cash in productive ways. So, for example, they have a series of evaluation programs where they've given people cash, mostly in developing countries, in Africa in particular, some in the US. And they find that people spend relatively little of that on drugs and alcohol, but they actually tend to spend it productively. And, in fact, they found, in developing countries, this often provides valuable resources for individuals to start businesses. So they ran experiment Uganda where a nonprofit company randomly offered a group of women $150, which is huge relative to their income. That's 50% to 100% of annual income in Uganda, $150. And what they found was, after two months-- after 18 months, these women had used that money to start businesses. And that actually raised their earnings. That actually effectively doubled their earnings. From that one injection of cash, it led them to actually double their annual earnings, OK? So that leads one to think that maybe we should stop being paternalistic and just give cash. Unfortunately, if you're a reader of policy websites like I am, the best one of which is vox.com-- it's a great website-- they had an article just the other day pointing out how they actually followed these women up nine years later. And, nine years later, the effect had totally gone away. So the story isn't quite necessarily positive, but it's not negative. They're not worse off, but it looks like, at least what in the short run made them better off, well, that effect fades over time. But the bottom line is, at this point, I think the evidence is sort of probably in favor of being less paternalistic and just giving people cash, but that runs into a lot of difficulties in terms of our concerns about how people will spend it. So let me stop there. We will come back on Monday, and we'll talk about how we actually go from this stuff to the demand curves we started the class with.

MIT_1401_Principles_of_Microeconomics_Fall_2018

24_Market_Failures_II_Informational_Asymmetry.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: So today, we're going to talk about social insurance. So why do we have this thing called "social insurance?" Let's first talk about what social insurance is, and then ask why we have it. So basically, social insurance is government-provided insurance programs. This is the largest single category of government expenditure in the US today, is government-provided insurance programs. Now, why do we have these? You might say, well, we know why we have these. We learned about uncertainty. We already talked about how people dislike uncertainty, and about how as a result, insurance is big business in America. Private insurance for health, for auto, for life, for property and casualty adds up to about $1.5 trillion every year. So we already have big business of private insurance. So why does the government need to get involved? I mean, after all, people want insurance if they're risk averse. We talked about insurance markets can work. And as always in economics, the question is, what's wrong with the private market? What is the market failure that might generate interest in government being involved? And the market failure in the context of insurance is a different kind of market failure than we've talked about. We've talked about market failures like imperfect competition as a market failure. We talked about externalities as a market failure. The new kind of market failure we want to talk about today is what we call "information asymmetry." Information asymmetry, basically which is the difference in the information available to sellers and to buyers in a given market. So far in this course, we sort of assumed full information. We've assumed everybody knows everything. We weaken that assumption a little bit with uncertainty, and say that people don't know whether they're going to be sick or healthy, but they still knew the probabilities. Now we weaken it further by saying not only is the information imperfect, but different parties in a transaction might have different levels of information. And that's going to turn out to cause market failure. That information asymmetry will cause market failure. Now, the math here is quite hard, harder than we do in this class. So we'll just sort of do this by an example or two. And the best example to start with is the so-called "lemons problem" that was laid out by the Nobel Prize-winning economist George Akerlof in 1970, Nobel Prize-winning economist and husband of former Fed Chairman Janet Yellen-- quite a power couple. And here's what Akerlof laid out-- he said, let's look at the market for used cars. This is the market for used cars as of 1970. There's no CarFax. There was none of this information. In 1970, when you went to buy a used car, you sort of went, and kicked the tires, and decided if you're going to buy it. So this is a classic case of an information asymmetry, in that someone selling a car knows what's wrong with it, whereas the person buying the car doesn't. I'll go sort of kick the tires and hope for the best. So basically, in particular, sellers of cars might be selling them because they're not good. After all, why sell a car? Maybe because it's what's called a "lemon." A lemon is something which is a poorly performing product, in this case a car that's got something wrong with it. So when you go to buy a car, you're worried. You want to buy a car in the used car market, but you're worried. Why is someone selling this car? You should be. Why is someone selling this car? If it's really in good shape, why would they be selling it? Therefore, as a result, Akerlof argued, there might be a collapse of the entire used car market. There might not be transactions that happen that can make both parties better off. Remember, a market failure is whenever the private market fails to maximize welfare. What that means is a market failure arises whatever the private market does not deliver all transactions that make buyer and seller better off. And so let's look at an example of this. Suppose that I have a 10-year-old car, and I keep it in pristine shape. Hint-- that's not true for me. I'm terrible at cars, but imagine I was someone who wasn't. I kept my 10-year-old car in pristine shape. And let's say that a 10-year-old car in pristine shape is-- and let's say I'm trying to sell this car. And let's say that I would happily take $5,000 for this 10-year-old car that's in pristine shape. So I would be willing to sell at $5 K. And let's say that you-- let's say Patricia-- needs that used car, and she is willing to buy a car that's in good shape for $6 K. So my willingness to provide, willingness to supply, is $5 K. Her willingness to pay is $6 K. So that is a transaction that should happen. Given the quality of my car, given that's in good shape, she is willing to pay $1,000 more than I'm willing to sell it for. So that transaction should happen, and it would be welfare maximizing. But let's say that most 10-year-old cars are not in good shape. Most 10-year-old cars, in fact, are in kind of crappy shape. And in fact, for the typical 10-year-old car, to get it up and running well, you'd have to throw $2,000 in once you bought it. And Patricia knows this. She knows that for the typical 10-year-old car, she would have to put $2,000 in. So her willingness to pay is not $6 K, it's $4 K for an average 10-year-old car. Now I say to Patricia, well, that's an average 10-year-old car, but I have a perfect 10-year-old car. You don't need to put $2 K into it. It's good to go. So why don't we split the difference and pay $5.5 K. She says, no way. You're a damn liar. I have no way of knowing your car is better than average. All I know is the average 10-year-old car needs $2,000 of work. So I'm not going to pay more than $4,000 for your car. As a result, Patricia doesn't buy my car. And a transaction that would have made both parties better off does not happen. A transaction where there was full information, like we have much more of today-- we can get the entire record of the car, all the crashes its been in, how much care it's taken care of-- that problem would go away, should go away, because now, Patricia could look at my CarFax and note this, in fact, is a pristine car. And she's more willing to pay the $6,000 for it. But the bottom line is, in this world of 1970, this was a market failure, because a transaction that made both parties better off did not happen because of imperfect information. The buyer was perfectly happy to buy. Patricia is perfectly happy to buy my car, but because I had information that she didn't and she was just suspicious that I was lying, as a result, that transaction didn't happen. Now, questions about that? People understand it's a market failure. Now we come to insurance. The story is flipped. Now it's not the seller that has the information. It's the buyer that has the information. In particular, when you buy insurance, you know how healthy you are. You know your genetic history. You know whether you're a risk taker. You know whether you're around a lot of snotty kids who might get you sick. You know a lot of stuff about yourself that the insurer doesn't know. As a result, the information asymmetry is flipped. With insurance, the insurer is worried that when you come looking for insurance, they're worried you're looking for insurance because you're sick. You might be looking for insurance because you're risk averse and that's great for insurers. We talked about insurance, how there's essentially a game. I'm willing to pay a risk premium so insurers can make money by selling to me. But what if I'm not coming to you because I'm risk averse? What if I'm coming to you because I'm a huge skydiving fan? And you don't know that. You might be afraid to sell me insurance, because you might lose money on me. So let's work out another example to show this. Imagine you graduate, and you decide a great business model is to offer health insurance to recent MIT grads. You say, look, we're a bunch of kind of careful nerds. We're likely not going to go skydiving. We're just going to sit at our desks and work. Maybe there'll be carpal tunnel risk, but other than that, we're a pretty safe bunch. So I'm going to offer health insurance to recent MIT grads, because they're a healthy group. And let's say suppose that of every 100 MIT grads, 90 are healthy, and 10 are sickly. Let's just suppose you know those facts. You know those facts. You've collected the data to know that on average, of every 100 MIT grads, 90 are healthy, 10 are sickly. You don't know which are which, but you know the proportions. And let's say that with a healthy person over the next year, there's a 10% chance that they will need-- that they will incur a $10,000 charge, and a 90% chance that they'll have zero costs. So there's a 10% chance of a $10,000 cost, 90% chance they'll have a zero cost. So your expected cost for insuring this person is $1,000. You expect someone like that will cost you $1,000. Now suppose for the sickly guy, there's a 50% chance that they'll cost you $10 K, and a 50% chance that they'll cost you 0. So your expected costs for them, the expected costs for this person, is $5,000. Do people understand the setup? There's two types. I know these facts, but I don't know who's who. I just know these facts, because I'm good at math. I've done all the actuarial calculations. Now, if everyone buys in, now I'm going to set my price. What am I doing to do to the price? I'm going to say, look, if everyone buys health insurance, then I've got my expected cost is 0.9 times 1,000 plus 0.1 times 5,000. My expected cost is 1,400, so I expect to have to spend $1,400 a year. In fact, on average, I'll spend $1,400 a year. With large enough samples, I can predict that with certainty-- that if everyone buys insurance, I'll spend $1,400 a year. So let's say you're risk neutral, because you're rich, and so forth. This is sort of risk neutral for you. So you say, look, I'll just charge-- I'll set a premium of $1,500, and I'll make $100 per person. If 100 will buy, that's $10,000 profit. That's pretty good. There are 1,000 kids in the graduating class. If 1,000 kids buy, and I make $100 profit, then that's $100,000. That's pretty good money for a year. Now, what is wrong with this calculation? What, in fact, will happen if you sell insurance for $1,500? Yeah. AUDIENCE: We'll buy insurance. JONATHAN GRUBER: Well let's go step by step. What about sick people? If you sell $1,500, what will they do? AUDIENCE: They'll buy. JONATHAN GRUBER: Yes, so if you set up for $1,500, you are certainly going to sell to all the sick. So if you sell for $1,500, you'll certainly sell to all the sick. What about the healthy? What will determine whether or not they buy? Yeah. AUDIENCE: When the price is smaller than the expected amount they'd have to pay on their own [INAUDIBLE]. JONATHAN GRUBER: Not quite, not quite. There's another piece, too. Don't forget. What else? What else? It's not just the expected cost. What else? AUDIENCE: The risk aversion. JONATHAN GRUBER: The risk aversion-- remember, there's a risk premium that they'll pay. So whether that healthy person will buy or not, if it's just expected cost, then they wouldn't buy-- the $1,000 expected cost, buy the $1,500. But some might be risk averse and buy. So let's just say half the guys are risk averse, and they're willing to pay $1,500 for $1,000 expected costs in half are. So let's say you end up selling to all the sick and half the healthy. So how much money do you make? Well, you sell the 60 people at $1,500 each, so your revenues are $75,000. You sell to 50 healthy and 10 sick, $75,000 revenues. What are your costs? Your costs are-- you have 10 people that are cost you $5,000, so it's $50,000, plus 50 people who are going to cost you $1,000, plus another $50,000 equals $100,000. And you've lost money. You priced at above the expected total, and lost money. Why did you lose money? You lost money because of the problem of adverse selection-- the problem of we call "adverse selection." Adverse selection is the problem that, due to information asymmetries, only the worse risks will participate in the market. And that will cause people selling in the market to lose money, or likewise here, the concern is that only the worst cars will participate in the market. And so if people buy cars, we'll be worse off. Yeah. AUDIENCE: Wouldn't you make $90,000? JONATHAN GRUBER: No. You sell to-- you sell $1,500 each, and you sell to-- yes, you're right. You make $90,000, my bad. Yes, $1,500 each times 60 people is $90,000. You still lose money, but not as much. Right. Now, you might say, look, you're not losing that much. Your solution-- just raise the price. What if you said, fine, let's just raise the price, and let's charge $2,000 a person. Then that would cover, because $2,000 a person, 60 will make $120,000, the costs are $100,000. You'd be golden, right? Yeah. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Who would you lose? AUDIENCE: The healthy. JONATHAN GRUBER: Not the sick. The sick are delighted by $10,000. But once you raise the price, more healthy people drop out, because it's higher than their risk premium. So what happens is by raising the price, you're not necessarily going to make money. It depends on how many healthy people drop out. So for example, imagine that you raise it to $2,000, but now the number of healthy people that buys drops to 20 from 50. Then you lose money again. So the point is, you can't actually solve this problem just by raising the price, because there's this what we call "death spiral." This is a term called "death spiral," which is as you raise the price, you chase out the healthier people, which means you have to raise the price more, which shakes out even more healthy people. And you end up in this death spiral. So that is the problem of adverse selection. And that leads you to say, you know what, I'm not going to offer this product. I can't make money on it, because if I set the price, whatever price I set, I'm going to lose money. So I'm not going to offer the product. Therefore, the market has failed. A market that might have existed-- on average, this was a market that made people better off, but the market that might have existed doesn't does exist. Yeah. AUDIENCE: With the death spiral, wouldn't it converge like something equivalent [INAUDIBLE] market forecasting is [INAUDIBLE] still [INAUDIBLE].. JONATHAN GRUBER: Right, so if you leave this alone-- it's an excellent point-- what should the new equilibrium be? The price now should potentially have chased all the healthy people out. And then you price, but you'd have to price it, then, at what? AUDIENCE: $5,000 [INAUDIBLE] JONATHAN GRUBER: $5,000 plus something. So as long as sick people are risk averse, you could still make money. You could still make money if you sold, say, $5,500 with even modest risk aversion. Why is that still a market failure? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Because there's all these healthy people who now can't get health insurance. So yes, it doesn't mean the market collapse-- market failures doesn't necessarily mean market collapse. It means a reduction in welfare, because transactions that might make some people better off aren't happening. Here, you might be able to offer insurance for healthy people that makes them better off, but you're not. You're only offering insurance for the sick. So it's a market failure, because healthy people who might want the insurance end up being kept out of the market by adverse selection. Questions about that? And that is the fundamental market failure we face in insurance markets. That's why we think private insurance markets will not function well. Because private insurers-- in some sense the fundamental problem is that you're setting one price for multiple products. A great case of adverse selection is going to buy fruit at the beginning versus the end of the day. What's the difference? Buy fruit at the beginning or the end of the day? You guys probably don't buy a lot of fruit, but try to think about it. Yeah. AUDIENCE: Normally, better is at the beginning of the day. Then you have-- JONATHAN GRUBER: And at the end of the day, in particular, what's left? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: All the shitty fruit is left, because you set one price. You didn't say good apples $1.80, shitty apples $1.40. You said apples, $1.70. So people come, and they buy apples. They go and they feel it. They feel around. They find the good ones. The ones that left are crap. And that is the adverse selection problem. Now, with apples, the market still exists. Why? Because they charge so much they can live with a few bad apples being at the bottom, so to speak-- a few bad apples being at the bottom. With health insurance, if I get one bad risk-- someone, say, who's really, really sick and costs $1, million-- I go out of business. So adverse selection may not destroy markets. It doesn't destroy the apple market, but it can destroy or significantly impede insurance markets. Questions about that? Yeah. Manny. AUDIENCE: Is there some way insurance companies that hassle hospitals, like lower the prices or they give them better discounts so they can increase the price of people-- JONATHAN GRUBER: Well, that's a separate issue. We'll talk about that next lecture when we talk about health care. So that's separate, about the cost of health care. This is the reason why insurance companies make you fill out a lot of forms before you go in. So it's for this reason-- insurance companies are not powerless against this problem. They could try to collect as much information about you as they can. As I get more and more [INAUDIBLE] can learn more and more who's healthy, who's sick, then I can solve this problem. AUDIENCE: So your familiar with those home kits that [INAUDIBLE] and 23andme will send you. JONATHAN GRUBER: Yeah, 23andme. Yeah. AUDIENCE: So is it possible that at some point in the foreseeable future, those are going to become part of [INAUDIBLE]? Those are going to become part of how insurance is determined, like if it's in your DNA, get some condition when you get old that we can say you have a preexisting condition now that hasn't manifested yet? JONATHAN GRUBER: So this is a great point. I was going to talk about it next time. I'll talk about it now, which is in some sense, we are eventually moving to a point where there'll be no adverse selection. Now you might say, on the one hand, that's because ultimately, we'll know everything about you from the moment you're born. We know your genes. We won't know if you're a skydiver, but we'll know-- we'll probably know genes that determine risk taking. And we'll charge more for people who like taking risks. So the good news is that then, I can make the market work. The bad news is in that world, how would I set my insurance? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: I would charge-- because what I would do is I'd say, your genes say you're healthy, so I want $1,100 from you. Your genes say you're sick, so I want $6,000 from you. So essentially, insurance wouldn't exist anymore. There'd be no insurance. What is insurance? Insurance is pooling people with different probabilities of adverse events, and letting us all benefit from the fact that if it happens to us, at least we're protected. Well, if you charge me my expected cost, I'm no longer protected. So here's the example that makes it perfectly clear, one of the most famous examples. Ken Arrow was one of the great economists of the 20th century, died recently. He had his famous islands example. Ken Arrow's island example is the following-- imagine there's two islands somewhere in the South Pacific that are very small, that's got one farmer on each. And the farmers know a hurricane is coming and it's going to wipe out one of their islands, but they don't know which. They just know one's getting wiped out. What will they naturally do? They'll naturally get together and say, look, islands get wiped out, but let's insure each other. If I get wiped out, you give me a bunch of your crop. If you get wiped out, I'll give you a bunch of my crop. That will improve both our welfare, because getting wiped out is going to zero. You die. That's a terrible outcome. So that will improve our welfare if we insure each other. Now let's say a weather service comes along, and provides information, and tells you that farmer A's island is getting wiped out and farmer B's island is going to be fine. What has happened to welfare? It's gotten worse. Why? Because farmer A goes to farmer B, says it turns out I'm going to be wiped out. Farmer B says, well, see you. I'm just going to keep consuming my high level. Farmer B is somewhat better off. Farmer A is dead. Total social welfare has fallen because the concavity, because diminishing market utility. More information has made us worse. We say more is better in economics. Once you get into topics like, this you realize more is not always better. More is worse. More information has destroyed the insurance market that might function. So in fact, this issue I'm talking about is becoming paramount as we move more and more towards perfect information environment. So the kind of government policies I'm talking about next become critical as you move towards that environment. But first, I want to make sure we all understand why the private markets failed, why it's a failure. Now, what can the government do about this? What are some potential government solutions? And we've tried all of these in the US and around the world. Let's talk about three categories of government solutions. The first is subsidization. The government could subsidize the purchase of health insurance. So for example, what if the US government said to all the MIT grads, I'm going to give you a $400 tax credit that you could have-- or $500 tax credit if you buy health insurance. Well, if there's a $500 tax credit if I buy health insurance, and I charge $1,500, then what's the effective price now to the healthy guy? AUDIENCE: [INAUDIBLE]. JONATHAN GRUBER: $1,000, so he buys. Even if he's risk neutral, he buys, as long as he's a tiny bit risk averse. So I do sell to everyone. I make my money. So one way to solve this problem is to basically pay the healthy people to get into the market. They can't just give money to the healthy people. You've got to give it to everyone, because you can't tell who's healthy. But if we give everyone a tax credit, then we could bring everyone to the market and solve this problem. Well, in fact, we do this in America. It's actually perhaps the largest hidden government expenditure in our country, which is the tax subsidy to employer-sponsored insurance, employer health insurance. The tax subsidy on employer health insurance-- what do I mean by that? What I mean is the following-- when MIT pays me in wages, I am taxed on that, like the taxation we talked about a couple lectures ago. When MIT pays me in health insurance, I am not taxed on that. So what does that mean? If MIT comes to me, and they say, would you like $1,000 raise or $1,000 orthodontic benefits for your daughter? I say, well, $1,000 raise in today's tax rates, I'm going to take home about $550. If you add up all the tax I'll pay, then I'll take home about $550. $1,000 of orthodontic benefits for my daughter, I get the whole $1,000. So why not? So I got these cool braces. They spin and change color. And every two weeks, she's in for a different kind of braces. It's great, because it's free. So we do subsidize health insurance in America. And this amounts to-- this program that I just talked about amounts to almost $300 billion per year. We spend almost $300 billion per year giving a tax break to people to buy health insurance. So that's one tactic we take to try to solve this problem to get healthy people into the market. That's approach one. A second approach one can use to try to get people into the market is a mandate. Suppose I just pass a law that says everyone has to buy health insurance. Then I've solved the problem. I know what my expected costs are if everyone has to buy. I know my expected costs are $1,400, so I know I can make money at $1,500. That's easier at one level. I don't have to spend-- $30 billion is a lot of money. This cost me $0. It's harder on another level. Why? What's the problem with that solution? Yeah. AUDIENCE: Not having the money for insurance. [INTERPOSING VOICES] JONATHAN GRUBER: Well, it may not have it. That's right. What else? Yeah. AUDIENCE: He may not want it. JONATHAN GRUBER: The healthy people are going to be pissed. They're like, look, if I had chosen not-- you're going to basically-- the mandate only has an effect if it changes people's behavior. But changing people's behavior means you're making them do something they didn't want to do beforehand. So the problem with that the problem with this is you spend a lot of money. The problem this is you piss off healthy people. The third approach we could do-- there's lots of examples of a mandate. Obviously, we know about the health insurance mandate that was originally part of Obamacare. But that's not the biggest example. The biggest example in the US is what's called "Workers' Comp Insurance," which is insurance that you have for on-the-job injuries. If you get hurt at work, your employer pays money so that you get reimbursed when you're-- it pays your medical bills when you get hurt at work and gives you partial replacement of your wages. That is mandated insurance on all employers in America, except in Texas. Texas, they can choose. Every other state, it's mandated. Mandated insurance for every employer in America. They have to buy Workers' Comp. So we've examples of that. And that's an $80 billion a year program. That's a big deal. Finally, we can just provide the insurance. That's actually the most common thing we do in America. Social Security is our program that provides insurance for the elderly, for the costs for survival after retirement. Medicare is insurance for the elderly we provide. Unemployment insurance is insurance we provide against losing your job. Disability insurance is insurance we provide against having a career-ending disability. So this is actually the most common thing. Indeed, provision of social insurance in America costs almost-- costs more than private insurance. So we spend about a trillion and a half on private insurance in America. Social insurance is probably about $1.7 to $2 trillion, depending how you measure it. So actually, this is the biggest thing we do is we just provide insurance, and that is a very large solution. Now once again, what's the problem with this? You don't make the healthy people unhappy, because everyone, you just give it to them. The problem is you have to spend money on this. This is $1.7 trillion in taxes we've got to raise every year. That's non-trivial. So basically, each of these solutions has potential problems. So the adverse selection problem will cause the private market to fail. There are potential government solutions, but they each have limitations. This one's pretty expensive, this one's super expensive, this one pisses off healthy people. Now, you'll note the middle one, the pissed-off healthy people is kind of subtle. You don't see a lot of healthy people railing against mandated [INAUDIBLE] Workers' Comp like they did against the health insurance mandate, because people don't know. So in some sense, this one's a little bit subtle, because people have to know. Basically, it's sort of crazy that I'm paying tax that I'm never getting hurt at work. Am I going to have-- what am I going to do, like slip at my desk or something? Never going to get hurt at work, but I pay taxes all the time just in case someone else at MIT gets hurt. Some of the janitorial staff has a risk of being hurt. I'm paying taxes in case the janitor gets hurt. I should be upset about that, but I'm not. And in some sense, it's about what people know, what they don't. So that is the basic argument for social insurance. But when we provide social insurance despite all these problems, we enter into a fundamental trade-off, which is, let's decide we've determined some optimal government policy. Let's decide that the markets failed, so we're going to do one of these things or are some combination of these things and solve the problem. The problem is that when you insure people for risks you create a new problem called "moral hazard." Moral hazard is basically the adverse behavior that is encouraged by insurance. When you insure people, you encourage adverse behavior. So the classic example of this is-- if I have health insurance, I ride my bike less carefully, because if I get in a crash-- I'm not crazy. I certainly don't want to get in a crash, but I'm a little bit less careful because I know I'm insured in case I get in a crash. If I have fire insurance, I don't buy a fire extinguisher for my house, because if it burns down, I'm just going to get the money back anyway. Or if workers have insurance against losing a job that pays them when out of work, they might search less hard for a new job. Basically, if I lose my job and I got nothing, I'm going to work my ass off to get a new job. If I lose my job and the government says, well, for 26 weeks, we'll give you half your salary while you look for a job, I'll be a little bit less rushed. And there's lots of evidence that moral hazard is a problem. I comes with two types of evidence. The first type evidence is fun anecdotes. So the great effect that-- workers' compensation, let's take that. Workers' compensation, it's a program [INAUDIBLE] needed. Lots of people get hurt at work. I don't, but lots of people do get hurt at work. And so it's a sensible social insurance program. The problem is it has a huge moral hazard component. And there's fun examples of this, like the prison guard in Massachusetts who claimed he got hurt on the job, collected $82,000 in benefits, while the whole time running a karate school and teaching students karate. And finally someone noticed online this guy who couldn't work was running a karate school and doing karate kicks and stuff online. So there's all sorts of fun examples about that. But more convincing for economists is statistical evidence. And the statistical evidence is clear that moral hazard is a big problem. For example, if you raise the benefits people get under workers' comp, suddenly they become injured more often and stay out of work longer. There's no reason-- injuries should be because you got hurt. So how can it be that suddenly, when a state raises its benefits, suddenly, there's more injuries? The answer is moral hazard. When states raise their unemployment insurance benefits, more people leave their jobs and they stay unemployed longer-- moral hazard. So the moral hazard problem is real. It's an inherent trade-off, actually not just with public insurance. Private insurance, too-- anytime you insure people and something bad happens, you're providing less of an incentive for them to try themselves to avoid that bad thing happening. So moral hazard is a real problem and it's essentially the trade-off. On the one hand, we talked about why people like insurance. We talked about why people like private insurers because of risk aversion. We talked about why government intervention insurance in markets is necessary. But that comes with the trade-off, which is the more insurance you provide, the less people take care of themselves. And that's the trade-off. Now, why do we care? Let's just sit back and say, that's an interesting economics concept, but why do I care? Why do I care if someone stays out of work longer, fakes an injury, or whatever? Why do I care about this? Why is this a problem? It's a problem for two reasons. There's two costs to moral hazard. The first cost to moral hazard is lower efficiency. And the best way to see this is just to think about the economics of the consumption/leisure trade-off. Think about how I make my decision of how hard to work. Basically, if there's no insurance, no social insurance, no workers' comp, no unemployment insurance, how do I choose how hard to work? How do I choose how hard to work? How do I do that? What's the trade-off I consider in deciding how hard to work? Yeah. AUDIENCE: Consumption versus leisure. JONATHAN GRUBER: Consumption versus leisure, in particular, I will trade them off until the marginal value of the next hour of leisure-- marginal value of leisure-- equals the wage. Because the marginal value of leisure is above the wage, I should work less hard. That means I'd rather be at home. If the marginal value of leisure is below the wage, that means I'm just wasting my time at home. I should work harder. So I'll continue to trade off work and leisure till the next hour of leisure makes me just as happy as the next hour of working. And that is the efficient outcome. That is the socially efficient outcome, because leisure is not a social bad. There's nothing wrong with leisure. People value leisure. They should get to trade off the leisure versus what they get from working till they choose the right amount. That's what makes society best off. Now, what happens if I say, if you sit at home, you're also going to get a check from the government? Now, what's my new equation? Now, if I work, I still get the wage. But what happens if I sit at home? I get the marginal value of leisure plus the government check. So now I sit at home until this equation is true, which means that I sit at home until the marginal value of leisure equals the wage minus the government transfer. That means the marginal value of leisure will be lower than it would be without the government transfer, which means I work what? More hard or less hard? If the marginal value of leisure is forced down, that means I'm doing what? Someone raised their hand. Yeah. AUDIENCE: More leisure. JONATHAN GRUBER: More leisure, because remember, there's diminishing marginal value of everything, so I'm taking more leisure, less work. So the government is causing me to work less by essentially saying, look, I'm going to reward you more for staying at home. What does that do? That means that people work less than is socially optimal. This is the social optimum. This means people are taking more leisure and working less than is socially optimal. When people work less, that shifts in the supply curve and creates a deadweight loss. Social welfare has fallen. Let me remind you, it's not falling because people take some time off. Many people on the conservative side of the spectrum will act as if work is a virtue. Work is not a virtue. The optimal solution is to work until your value of working equals your value of leisure. If you're someone who has a job that you hate and doesn't pay well, and you love watching TV, you should work less. That's what's optimal for society. But you shouldn't work even less because the government's paying you to stay home. That reduces efficiency. So that's a problem of moral hazard is it lowers efficiency. There is a second problem, of course, of moral hazard, which is if you work less, then we have to tax people who do work more to pay for these programs. So it raises taxation, raises the required tax revenues, raises the tax revenues required. Because if you're sitting at home more, I've got to make more money to pay for you to sit at home. And we know taxation also causes deadweight loss. So it's a double win. I cause you to stay at home and I cause other people to have to pay more taxes to pay for you to sit at home, which causes them to work less, too. There's a second round effect. As a result, moral hazard causes inefficiency in society. And that is the trade-off. Once again, I told you, this course is annoying. We don't give you right answers. We just tell you trade-offs. The trade-off here is we need programs like unemployment insurance, because otherwise-- let's take the case of unemployment insurance. We'll go through it one more time. Imagine there's no government unemployment insurance, and you said, that's great. I'll offer private unemployment insurance. Well, that's not going to work. Why? Because people know way more than you do about whether they're going to lose their job. If you tried to offer private insurance, you'd lose your shirt because of adverse selection. So absent government-provided unemployment insurance, there would be no unemployment insurance. And that would be bad. That would mean people would be subject to a risk that would drive their consumption to zero. Remember, most Americans have no savings. That mean Americans would be subject to risk where if they lost their job, they would starve. That's a very bad outcome. So it is socially valuable to insure against unemployment risk. The private market can't do it because of adverse selection. Therefore, there is a compelling case for government unemployment insurance. But with government-provided unemployment insurance, that causes people to sit at home extra and not work as hard. And that's the sort of chain of logic which teaches you the trade-off. What this says is optimal social insurance-- that in these markets, we're going to want some social insurance, but not too much. We're going to want enough to protect people against starving, but not so much that it causes people to sit at home. So for example, if I told you I'm going to set up an unemployment insurance program, and the way it's going to work is if you lose your job, I'm going to pay your entire wage for as long as you need until you find a new job, that would not be a good idea. That would cause a huge amount of moral hazard. And remember, compare that program to one where I'll pay you 50% of your wage till you find a new job. Well, 50%, going from 0% to 50%, 0% of your wage to 50% of your wage is a huge consumption smoothing benefit. You go from starving to being able to eat decently. 50% to 100% is an increase, but not as much. But 50% to 100% has a huge moral hazard effect. So you want something more towards the middle, where you're getting people away from starving, but not so much that they don't work. So that's the trade-off. So let's talk about that trade in practice. Let's talk about the US Social Security program. The Social Security program in the US-- Social Security is our biggest single social insurance program in the US. Currently, the Social Security program is about $800 billion per year. That's real money. That's even more than Jeff Bezos has, $800 billion a year. That's more than he's worth, every year. What does this program do? What this program does, in a nutshell, is it insures you against the income loss you're going to face when you retire. When people retire, they suddenly go from having a lot of income to having no income. And basically, the idea of Social Security is to make sure you don't starve when you're old. So the way it works is you pay a tax. And if you ever see a line on your pay stub that says FICA, that's what this is for. You pay a FICA tax. It is 12.4% of payroll, half on you, half on your employer. But it doesn't matter that half is on your employer, because we learned two lectures ago it doesn't matter who pays the tax. It's a 12.4% tax. That's what matters on you. That money then provides that when you retire, starting at age 62, you get a check from the government. And you get a check from the government that lasts until you die. The check from the government you get is what's called an "annuity." An annuity is a payment. Annuities are the opposite of life insurance. Life insurance is money that your family gets when you die. Annuity is a regular payment you get until you die. The way it works, you pay 12 and 1/2 percent of your income all the way through your working life till we turn 62, or you can collect it later. You then get a payment for the rest of your life. That payment is typically about half of what you made when you were working, but it's very progressive, in the sense that for someone who's very poor, it would probably more than half of what they made. For someone who's rich, it would be much less than half what they made. It's a progressive payment. Everyone gets it. Everyone gets Social Security. But how much you get from it depends on your income. The poorer you are, the more generous it is relatively when you retire. Now the-- yeah. AUDIENCE: Is it possible as life expectancies get larger, it's going to be harder to have Social Security because if you're going to-- JONATHAN GRUBER: That's a huge problem. It sounds like you should definitely be enrolled in 1441. That's a whole half a lecture we spend on that. I don't have time to talk about it here, but clearly this is a huge, huge-- so just to give you a couple numbers just to keep you up at night. We all know, we're all talking about the deficit is $500 billion. It's a big deal. If you ask how much has America promised to pay to our senior citizens over the foreseeable future minus how much we'll collected taxes, we are currently, as a nation, $75 trillion in debt. And it's because of the aging society and things like that. We've got big problems coming down the road. We can talk about that another time. But let's focus on the program itself at a point in time right now. So basically, we see here the moral hazard trade-off. On the one hand, we don't want people to starve when they're old. On the other hand, if I pay you once you're retired, that could cause you to retire. If I say, once you're retired you're going to get a check, 50% of your wage, you might say, 50% of the wage isn't that much, but I really don't like working. I'd rather just hit the links at 50% of my wage. So that's the trade-off. Now, how do we think about evaluating that trade-off? Evaluating that trade-off, different countries think about it differently. In the US, we think about it in what I would say is a fairly rational way, which is let's consider your decision to retire at 62 versus 63. The way it works in the United States is we say, look, if you work one year more, since it's an annuity, you will get one less year off payment. If you start at one year later, you're going to die the same time, you get one year less. So what we do is we pay you more every month. Indeed, for every year you delay, you get 6.7% more every month, reflecting the trade-off that you're going to get it for a shorter period of time. And that turns out to be roughly fair. Given the expected life of Americans, that's a roughly fair trade-off. Every year you delay gets 6.7% more. In Europe, they don't have this. So every year delay, you just get less money before you die. So let's take the example of the Netherlands. In the Netherlands, you can retire at 55 with a benefit that is 90% of what you made. So if you earned $30,000, you can retire at 55 with a benefit of $27,000. And if you decide to work instead, you just forego that $27,000. There's no bump up of your benefits. That's just one less year of $27,000 you get. So what that means, if you're in the Netherlands, your choice is work and get 30,000 or stay home and get 27,000. In other words, it's sort of like a 90% tax. Think about it-- by working relative to staying home, I'm only keeping 3,000 of the 30,000 I made. It's basically like a 90% tax. But that's not all. How do they pay for this program? They tax people. You can't tax people who are at home. You've got to tax workers. So if you work, you also have to pay a 45% tax to pay for this program and lots of other things. 45% tax is a high tax rate for everything. What that means is if you stay at home, you get 27,000. If you go to work, you get 30,000 times 1.55, or about 18,000. That's 1.6, It's 16, 5. So your choice is stay at home and get 27, work and get 16, 5. Guess what? No one works. No one over 55 in the Netherlands works, like zero. And they might work on the black market and ways they don't report to the government. But basically, they just sort of sit around coffee shops and spend their retirement money. So here's a case where they've made a very different decision about how to make this trade-off, which is it's a pretty sweet life for the elderly in the Netherlands, but no one's working over age 55. And that's a different way to resolve this trade-off. So basically, this illustrates different design features of the program. What makes the Netherlands program have much more moral hazard than the US is the benefits are higher and they don't increase your benefits if you work more. So essentially, these are little kind of tweaky details that turn out to matter enormously for how we think about the program. Now, I hope you find that interesting. That was a lot to put in one lecture. Like I said, if you find this interesting, there's the whole third of a semester in my class 14.41. So take that. We can learn a lot more about it.

MIT_1401_Principles_of_Microeconomics_Fall_2018

8_Competition_II.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: OK, why don't we get started? Since I had some problems with the end of last lecture, I'm going to pick up right where things got a little dicey in the last lecture, and we're going to start over. So we're looking back at figure 7-3, which, if you remember, was the cost curves for our cost function 10 plus 5q squared. And you remember where it came from. This cost function we derived ourselves from the production function and wages and rental rates. We derived this cost function. We're now graphing the cost curves that come out of this cost function, and we're talking about profit maximization. And we're talking about measuring profit. So we're talking about perfect competition. And remember, we said that profits are revenues minus costs. That means the profits per unit are revenues per unit minus costs per unit. Revenues per unit are price, and cost per unit is average cost. So profits per unit is price minus the average cost. So we just go to the diagram. You see the height of the profit rectangle is price minus the average cost. So what do you do? You start by finding the point. So what are your steps here? Step one is you find the point where price equals marginal cost. That gives you your production level. That gives your optimal q star-- we derived that last time-- of 3. We derived that last time that you want to optimize the price equals marginal cost. That gives our q star of 3. Then at that level, we compute the average cost. Average cost at q of 3 is simply going to be this cost function divided by 3-- so 10 over 3 plus 5q squared over 3, or just 5q, plus 15, OK? So the average cost at three is going to be $18.33. That's going to be the average cost at production of three. That means that the profits we're making for that third unit, the profits on our third unit, is the price-- profits per unit are the price, which is a fixed level of 30, minus the average cost, which is $18.33. So profits per unit equals $11.67. So that's the profits. That's the height of the rectangle. The profits per unit is $11.67. The price minus the average cost is $11.67. We're selling three units, so that means our total profit rectangle is 35, OK? So our total profits are 35, which is three units at a profit of $11.67 each. And that's how we get that rectangle. Questions about that? OK. So now, let's return to what we did last time. Let's imagine there's a tax of $10 per unit. That would shift the cost curve to C equals 10 plus 5q squared plus 10q. Remember, it's a tax per unit, plus 10q. That would mean-- so that is illustrated in figure 7-4. That means that the marginal cost curve and the average cost curve both shift up. Marginal cost is now equal to 10q plus 10. We want to set that equal to the price to get the optimal q, and you solve this and you get a new q star equals 2. So now your optimal production level is two. You set your new marginal cost equal to the price. Marginal cost equals price. Marginal cost equals price at a new optimal quantity of two. What's the profits there? Well, once again, profits per unit are just price, which is $30, minus average cost. Well, what's the average cost at two? The average cost at two is going to be 10 over q-- so 5. 10 over 2 is 5, OK? Plus 5q because we're dividing this by q, 5q, which is 10, OK? Plus 10, OK? 30 minus 25, which equals 5. So your profits per unit, the height of that rectangle, has fallen from $11.67 to $5. So you're now making less profit per unit and you're selling fewer units. The height of the rectangle has shrunk. The width of the rectangle has shrunk. So the total profits have fallen from $30 to $10. You used to make profits of $11.67 on each of three units. Now you make profits of $5 on each of two units. So that tax has lowered your profits from $30 to $10, OK? Question about that? Yeah. AUDIENCE: Is that by changing the cost [INAUDIBLE]?? JONATHAN GRUBER: Yes, exactly. Because you have a higher cost-- now, it's not just that your costs change. Your costs changing fed through to your production as well. So because your costs change, you produce at a different level and you made different profit per unit, OK? All right. Now, let's go to the other point I tried to cover last time that I want to get straight on, which is the shutdown decision. Now remember, we are, in the short run, shut down. We're in the short run. In the short run, there's no entry and exit. The firm can't literally leave, but it can just produce zero. That we call the short-run shutdown decision. A short-run shutdown decision, you're still in the market. You still paid your fixed costs. You just produce zero. That's what we mean by shutdown, as opposed to exit, which is literally you take your toys and leave. This is you're still in the market. You've paid your fixed costs. You can't go anywhere-- remember, those costs are fixed in the short run-- but you just produce zero. So now let's ask, for example, what happens if the price suddenly dropped to $10. Let's say the price dropped from $30 to $10. Well, the fundamental profit-maximization rule never changes. We're not having the tax anymore. We're back to the original cost function. So original cost function is C equals 10 plus 5q squared, OK? So the marginal cost equals 10q, and our profit-maximization rule does not change. Our profit-maximization rule is that price equals marginal cost. So we set 10q to 10, marginal cost equal to price. $10 is the price. And we get that the optimal quantity is now one. We now want to produce one unit. That's the optimal quantity. Well, what is our profits if we produce one unit? Well, profits equals revenues, $10, minus costs. Well, what's our cost if we produce one unit? 10 plus 5, 15. So profits are negative 5. I'm going fast, so jump in if I get the math wrong here, OK? Profits are negative 5. So you might think that's terrible, using money. You should get out of there, shut down. But the answer is, as we discussed last time, you should not shut down because shutting down still means paying your fixed costs. No matter what you do, you have to pay those fixed costs. So shutting down means production of zero. What are your profits at a production of zero? Your profits at a production of zero are zero revenues minus the fixed cost of $10. If you produce zero, what's your cost? Plug 0 in here. Your costs are still 10. So if you produce zero, your profits are minus 10. So you should continue to sell even though you're losing money. Even though you're losing money, you continue to sell that one unit. You don't go to zero. Because at zero, you're even worse off. This is the key thing about short-run shutdown decision. Yeah? AUDIENCE: What happens if you're in a position where, like, you are afraid you're going to keep selling one unit the next couple of months? JONATHAN GRUBER: Ah, but that's the key thing. Once you move to the long run, you can reoptimize your capital and you can exit. But in the short run, you've already paid that. You've already laid down your blanket, which cost you $10. You might as well sell at a loss. If you just sell zero, then you've laid down your blanket for $10 and gotten nothing out of it, OK? That's the transition to the long run. And so basically, that's how you think about the short run versus the long run. So what do we think about how this works? So basically, the shutdown rule, the way you think of the math of the shutdown rule is that basically you only want to shut down if your revenues are less than your variable costs because you got to pay your fixed cost no matter what. That's done. You're in the short run. That's gone. Forget it. You laid your blanket down in the bazaar. That's done. So you only want to shut down if your revenues are less than your variable costs. You don't care about your fixed costs. That's done. You only shut down if your revenues are less than your variable costs. Well, that's the same as saying you only want to shut down if your price-- variable costs are pq-- I'm sorry, your variable cost. You only shut down if your price are less than your average variable cost-- so divide by q. You only shut down if your price is less than your average variable costs. That's the shutdown rule. Shut down if the price you get is less than, on average, what you'll get unit for the units you sell. So in our example, you never shut down. And why is that? Let's look at the math of our example. Well, in our example, what are variable costs? The variable cost is 5q squared. I'm going to put these down here so I don't have to-- the variable costs in our example, variable costs are 5q squared. So what are average variable costs? 5q. Those are average variable costs. Now, remember that you want to shut down if variable costs are greater than the price. We can express the price in terms of q. Because remember, marginal cost equals price. So 10q equals p. So at the optimum, it is always true that p equals q over 10. That's a profit-maximizing condition, that p equals q over 10. We can just substitute that in, and we have that average variable costs equals 5 times q. So average variable costs equals 0.5 times p at the optimum. Because we know that the optimum marginal cost equals price, we can plug that in and say, at the optimum, average cost is 0.5 times p. Yeah? AUDIENCE: Do you mean q equals p over 10? JONATHAN GRUBER: q-- let me see if I got that wrong. Yes, I'm sorry. q equals p over 10. You're right. q equals p over 10, my bad. Sorry. So the average variable cost is 0.5 times p. What's the shutdown rule? The shutdown rule is that price is less than average variable cost or price is less than 0.5 price, which can never be true. So you'd never shut down with this cost function. To say it again, you only shut down if price is less than average variable cost. We've computed average variable cost in terms of price, and it's 0.5 times price. Therefore, you never shut down. And you could see this when we actually go to-- so you can actually see this in figure 7-5 when we look at the firm's supply decision. So what figure 7-5 does is show you, at every price, what the firm wants to produce in the short run. At a price of $10, it wants to produce one. We showed you that. At a price of $20, it wants to produce two. At a price of $30, it wants to produce three, and so on. That dash-- that line that runs from 0, 0 all the way up to 4, 40, that's the marginal cost line. At each point, price equals marginal cost, the optimal production decision. And you never shut down. As long as price is positive, you produce. Yeah? AUDIENCE: So in this case, we don't set price to 10 minus p? Because if it's 10, then [INAUDIBLE] will be higher than 10. JONATHAN GRUBER: I'm sorry. I don't understand. AUDIENCE: So-- JONATHAN GRUBER: Price is always above average variable cost, so you'd never shut down, right? That's the shutdown rule. This can never be true with this function. Price is forever above average variable cost. You can see that in this graph, right? The price equals the marginal cost, and that is always above the average variable cost, OK? So the bottom line is you'd never shut down. But here's the other cool thing. After all this confusing math, guess what this line is, this marginal cost line. What else do we call that? That's the supply curve, people. We just derived the supply curve. What's the supply curve? The supply curve is the relationship between the price in the market and the amount the producer desires to produce. Well, that's what this marginal cost curve is. So we've just shown you that the firm's supply curve is simply its marginal cost curve. So all you need to know to derive what a firm's supply curve looks like is what its marginal cost curve looks like, and you're done. So literally, you take production plus input prices, and that gives you the supply curve. Why? Because production plus input prices gives you the cost function. Take the derivative of the cost function with respect to quantity. That gives you the marginal cost function. That's the supply curve. So just like I could give you a utility function and a budget constraint and you got the demand curve, here, I can give you a production function and input prices and it gets the supply curve-- however, only under perfect competition, OK? This is only for the case of perfect competition. That's different from consumer theory. That was an everywhere rule. Because I've given you an extra constraint, which is a constraint on the market, I've allowed you to derive the supply curve just like we easily derived the demand curve. So we have now derived a firm's supply curve. Questions about that? So let's go back one more time. Just like if I give you on, say, a problem set or an exam a utility function and a budget constraint, you should be able to draw a demand curve. If I give you a production function and input prices and I tell you you're in perfect competition, then you should be able to draw a supply curve because it's just the marginal cost curve, OK? Now, that's the firm's supply curve. What we care about, actually, in the end, is the market supply curve, right? I've been doing little q. We care about big Q. That's what I drew in the lecture, was a big Q diagram. So how do we get there? Well, all you do to get the market supply is to horizontally sum each firm's supply. So we can see that in figure 7-7, OK? What figure 7-7 does is take multiple identical firms and put them in the market. So for example, if there's one firm in the market, the supply curve is what we just drew. At a price of $10, you get 10. At a price of $10, you get one. At a price of $30, you get three. Now let's add a second identical firm. Well, that firm behaves the same way. At a price of $10, it produces one. And at a price of $30, it produces three. So the market now suddenly has twice as much. At a price of $10, it has two. At a price of $30, it has six. Now let's add a third firm. It behaves the same way. At a price of $10, it produces one. At a price of $3, it produces three. So each additional firm you add just literally shifts the supply curve down. So in other words, the point is that a market supply curve, as you add more and more firms, the market supply curve is always more elastic than the firm's supply curve because any given increase in price calls for, from each firm, an increase in quantity. As long as there's multiple firms, that means a flatter market supply curve than a firm's supply curve. So that's how we get the market supply curve. We solve for the firm's supply curve. We then just horizontally sum it over the number of firms in the market. Get the firm supply curve, horizontally sum, OK? Questions about that? Now, now we have everything we need to go back to the first lecture, back to the future, back to the first lecture. And we can actually get to market equilibrium. In the first lecture, we started with a supply curve and a demand curve and got equilibrium. Well, we derived the demand curve a few lectures ago. We've just derived the supply curve. So let's do short-run equilibrium. Let's do short-run equilibrium, OK? Now, the key thing is that-- so how do we do short-run equilibrium? Let's go through the steps. Step one, each firm picks a fixed amount of capital it's going to have in the short run. So each firm has a K bar. And based on that, it has a production function. Each firm has a production function, q equals f of k, K bar, L. And we have some input prices. We have some w and some r. Taken together, we can use those to create a cost function, which, in our example, was C equals 10 plus 5q squared, step one. Step two. Step two, based on this, we can get optimal production levels from the fundamental profit-maximization rule, MC equals p. So this yields-- so here, we say 10q equals p. And this yields a supply function which is q equals p over 10, and that's our supply curve. That's what I drew here. q equals p over 10 is the firm's supply curve. That's step two. Step three is to create a market supply curve. Well, let's say there's six firms in the market. I'm just pulling this out of thin air. In a minute, we'll get to how many firms there are. But for now, in the short run, there's no entry and exit. So whatever number of firms I tell you, that's what's there. It's just given. So let's say n equals 6. Let's say there's six firms. And I'll come in a minute to where six comes from. But for now, let's just assume it. That means that the total market supply, big Q, is just equal to 6 little q, 6 little q, which equals 3/5 p or 6/10 p. That is our market supply curve. Yeah? AUDIENCE: Would firms that change [INAUDIBLE] who are out there? Would that change the overall-- JONATHAN GRUBER: The number of firms that are out there? Yeah, but once again, once they're in, they're done. They know there's six firms. So they're done. I'm going to come to this. You're talking about the long run, OK? Other questions? Yeah-- OK. So that gives you your market supply. Finally, we go back to lecture one and the first recitation where you solved mathematically for equilibrium. We have a demand curve. I'm just going to make this up. Let's say the demand curve is Q equals 40 minus p. I just made that up to make the math easy. Where does this come from? Well, you know where that comes from. You solved where that comes from. That comes from consumer maximization, 48 minus p. So to get equilibrium, we just set 48 minus p equal to 3/5 p, demand equal to supply. And we end up with p equals 30, conveniently familiar number, which means that the total demand, big Q, is 48 minus 30 equals 18, also convenient number. And I'll show you why. The fifth step, each firm says, given this price, what do I want to produce? Well, we know that given a price of 30, each firm's little q star is 3, right? That's what we solved. Each firm's little q star is 3 given a price of 30. So each firm produces three units. How many firms are there? AUDIENCE: Six. AUDIENCE: Six. JONATHAN GRUBER: What's 6 times 3? Supply equals demand. Six firms produce three each at a price of 30. That's 18. At a price of 30, people want 18, equilibrium. That's how it all works. So leaving aside where the six came from, everything else you see here is just taking what we did and working out the math. And we get six firms in the market. They each want to produce three. That's the 18 that people want. So to get equilibrium, you just need-- to get equilibrium, you need a demand curve, you need a cost function, and you need the number of firms in the market. You need a demand curve. You need the cost function and the number of firms in the market. Given that, you can solve for the equilibrium. Questions about that? Yeah. AUDIENCE: So thinking about this in terms of the intersection of the graphs, would intersecting the demand curve with the whole market's supply curve-- JONATHAN GRUBER: Yeah, because it's the whole market demand curve. AUDIENCE: OK. JONATHAN GRUBER: Yeah? AUDIENCE: Q equals 48 minus p is the demand curve. JONATHAN GRUBER: Q equals 48 minus p is the demand curve. Other questions? Good questions. Those are good, clarifying questions. OK, so now let's ask where the hell six comes from. And where six comes from is the fact-- where six comes from is from the long run. So now we get to long-run competition, which is, in the short run, there's a certain number of firms in the market. But where those firms come from? Well, they come from the fact that each short run is a repeated exercise that makes up the long run. So in the long run, perfect competition in the long run looks just like perfect competition in the short run-- full information, no transaction cost, lots of firms, with one difference. We're now going to allow entry and exit. Nothing else is going to change in the short run except we're now going to allow entry and exit. So now, what-- and one other thing. I'm sorry. One other thing changes. Since all costs are variable, there's no shutdown decision. There's no issue of this shutdown rule. There's no more fixed costs. So if you lose money, in the long run, you shut down. So all you have to worry about is profits. You don't have to worry about this extra shut-- so in the short run, we had an extra shutdown condition. You don't worry about that in the long run. You just worry about if you're making money or losing money. In the long run, if you're making money, you're in. If you're losing money, you're out. So now let's ask how that pins down the number of firms in the market. This is complicated, so the best way to do it, I think, is an example. And think about our-- the fundamental rule I want you to keep in mind-- and the example gives you this rule. The simple rule is if there's profits to be made, you enter. If there's losses, you exit. And these two things together imply the fundamental rule of competitive equilibrium-- that in the long run, profits are zero. Because if there's ever profit in the market, firms will come in. And if there's ever loss in the market, firms will exit. And that process will continue until profits are zero. So our fundamental conclusion is the long-run competitive equilibrium features zero profits. Now let's see why. Let's go to figure 8-1. I want to talk about the market for personal computers. Let's see. You guys were born what year? So you're now, like, 18. So you're born, like, 1990. This is about when you guys are born, OK? 1990 was a very interesting time for PC market. You all grew up in a PC world, a personal computer world. But basically, in 1990, we're in a very different world. We're in a world that was dominated by giant mainframe computers. My graduation speaker from MIT in 1987 was Ken Olsen, the chairman of DEC, which was one of the premier manufacturer of giant computers. He gave the worst fucking graduation speech you've ever seen. If you ever become famous and give a graduation speech, the number one rule is don't spend the whole speech talking about yourself. He spent the whole speech talking about how wonderful he was, how DEC was everything. And five years later, he was bankrupt. So take that, Ken Olsen. That's why you don't give a speech talking about how wonderful you are. What happened? Well, let's talk about what happened. Let's talk about the PC market, OK? I hope Ken Olsen doesn't watch this. [LAUGHTER] Let's talk about the PC market. Let's talk about Dell, who was an early PC manufacturer. And what we're going to do in figure 8-1, there's going to be two side-by-side graphs. I'm going to go back and forth between them. If you're ever not clear about which graph I'm talking about, stop me. But I'll try to be very clear because you have to think about both these graphs in tandem. On the right-hand side is the market for PCs, the market graph. On the left-hand side is Dell's cost curves. So the left-hand side graph is firm-specific Dell information. The right-hand side graph is the market. In the short run, we were at a position where people wanted PCs, and there weren't many people making them. You guys, PCs were super cool. I mean, I guess, you're on laptops and everything now. But a desktop PC was an amazing thing at this point. Everyone wanted them, and there weren't that many people making them. So at that point, Dell could-- the short-run market cost curve was SR1, and the demand was D. That was the short-- so we're starting on the right. We're going to go from the right to the left. Start in the right. The initial market in 1990 was the short-run cost curve SR1 and the demand curve of D. At that intersection, the price was p1. Meanwhile, Dell, in the short run, had a marginal cost curve such-- now shift to the left-hand diagram. At a price of p1, Dell wants to produce little q1 PCs-- marginal cost equals price, right? So find the intersection of that price with its marginal cost curve. Once again, this is very important. So stop me if I'm going too fast. Find where the price intersects the marginal cost curve on the left. Dell wants to produce little q1. Well, what's its profits? Profits are price minus average cost. Well, its average cost was way below that. So Dell made this huge profit of the lightly shaded rectangle. That was its profit because it produced little q1. At little q1, average cost was way below price. So it made this lighted dot rectangle on profits. Now what happens? Other companies see this and say, hey, we want in. What happens as more companies enter a market? The supply curve flattens. You're horizontally summing firm supply curve. Supply curve flattens. As it flattens, you move from SR1 to SR2. So now we're back on the right. In the market, as more firms enter-- and let's assume there are more firms identical to Dell. That's our perfect competition assumption, that each firm that enters is sort of identical. So more firms just like Dell enter. Gateway and all these guys start entering. And you shift to SR2, a flatter curve. That intersects the demand curve at the new price p2. Now shift to the left-hand diagram. The price facing Dell falls to p2, but their cost function hasn't changed. They're still Dell. They're still the same underlying technology, paying the same wages and rental rates. So now, their marginal cost curve is the same. So at this new lower price, their production drops to little q2. Their production drops to little q2 and their profits fall to the darkly shaded rectangle, pi 2. They make less money as firms enter. And this process will continue until profits go to zero. Even at pi 2, another firm-- I forget what the third PC firm was. Some other firm will come in and say, wait a second. There's still money to be made. I want to come in. And it'll keep going till profits go to zero. These are repeated short runs. So SR1 was a short run. Dell made a ton of money. Then we get to the next short run. Gateway enters. SR2's the new short run. Gateway and Dell still make money. So the third period, another firm's going to enter. And it's going to go till profits equal zero. Yeah? AUDIENCE: Does this assume that the manufacturers or all the different companies produce the same amount of quantity and that they're the same quality, right? JONATHAN GRUBER: Yes, absolutely. I'm going to come to those-- there's a large set of assumptions under this. But once again, think of Dell and Gateway selling their computers on rugs in front of the Eiffel Tower. It's all the same. You just go. You can compare equally. They're all the same thing. So it's that kind of market, OK? Now let's think about poor, old IBM. Flip the page. IBM dominated-- despite Ken Olsen's claims, IBM dominated the big computer market. And so they were initially, on the right-hand side, in equilibrium with supply curve SR1. That intersected demand-- now we're on the mainframe market. So in figure 8-1, we're in the PC market. Figure 8-1 is the PC market. Figure 8-2 is the mainframe market. The mainframe market, they're initially at supply curve one, which is very flat because there's lots of firms making mainframes. It intersects demand at a price p1, OK? Now we go to the left. Well, remember, IBM has to produce where its marginal cost equals price. That occurs at production level q1. At production level of q1, it is losing money. IBM is losing money. It is losing that entire rectangle, the entire large rectangle. Now, does IBM exit? It can't in the short run, but in the long run, the next period, some firms will exit. Some firms will say, yeah, we lost money in the short run. We're stuck. But in the long run, we just don't think this is a winning game. In particular, firms with high fixed costs will exit. They don't want to pay those fixed costs again in the second period. They know they have to build a new factory. They're not going to do it. They exit. Yeah? AUDIENCE: What constitutes as a period? I know we said the long run is when-- JONATHAN GRUBER: I told you I can't tell you that. It's just some period of time more than a month, less than 10 years, OK? It's the period over which capital is variable. Think of it as a period of time over which you can build a new plant to make PCs. Think of it that way. So it's years, OK? So what happens, then, somebody exits, says, I'm out of this. I'm shutting down the plant and moving somewhere else. That's steepens the market supply curve because now fewer firms are in the market. As that steepens the supply curve, the new intersection of supply and demand is at the higher price p2. IBM stayed in the market. They just built a new plant. At p2, they now produce an amount q2. And at that combination, they are literally zero profit. That's the point where marginal cost equals average cost, which is a zero-profit point. Why is that zero profits? It's zero profits because, remember, profits are price minus average cost. What's price? Marginal cost. So when marginal cost equals average cost, profits are zero. You should be able to see that from the math we did before. So what that means is that basically, when firms are losing money, they will leave and drive profits from below zero toward zero. So when they're making money, they enter and drive profits from above zero towards zero. When they're losing money, they leave and drive profits from below zero towards zero. What does that mean? That means that our long-run perfectly competitive supply curve is in figure 8-3. It's flat. Long-run perfectly competitive market supply is flat at the price level. The market long-run supply is the point where marginal cost equals average cost or where supply equals average cost, OK? And why is this? This is true because at any price above that point, if any firm tries to charge more than $10, they'll be driven out of business. If any firm tries to charge below $10, they won't make any money. Remember before, I said earlier-- you said, well, gee why don't firms just-- somebody asked, why don't firms just come in and take the whole market? This is why. Because cost is upward sloping. You don't want to come and take the whole market. You'll lose money. That's why firms don't want the whole market. If marginal cost was flat, then you would then be undefined. You would want the whole market. But you don't. Marginal cost is rising. So you never want to produce at a price above $10 because no one will buy from you. You never want to set a price below $10 because you'll lose money. Therefore, supply is perfectly elastic at a price of $10. Yeah? AUDIENCE: Thinking about this long-run logic then, wouldn't [INAUDIBLE]? JONATHAN GRUBER: No, no, no. Once again, there's a lot of assumptions under this. But under these assumptions, you never make money in the long run because they shut down. There are short-run periods. So let's say what happened was-- let's go back to figure 8-2. Let's say that another firm shut down in the next period and suddenly IBM started making money. What would happen? Someone would enter and drive profits back to zero. So under the assumptions we've laid out here, profits are zero. Because in the long run, we've achieved cost minimization. Firms minimize their costs where marginal cost equals average cost. Look at the average cost curve. Firms are producing at the minimum. The minimum of average cost is where average cost equals marginal cost. That is, competition has forced cost minimization. I'm just doing all sorts of mind-blowing stuff here for you guys. It can take hours to recover from this. Competition forces cost minimization. Why? Because competition forces each firm to produce where price equals marginal cost, and it forces entry and exit until marginal cost equals average cost. Therefore, it forces each firm to produce at the most efficient point, which is where marginal cost equals average cost, because that is the point at where average cost is minimized. So under competition, in the long run, every firm is cost-minimizing. They're producing the minimum of their average costs. And therefore, the supply curve is elastic, and it's defined purely by the minimum of average costs. That is, if you know-- here's a cheat. If you know you're in a perfectly competitive market and I give you a cost function, then you know-- I guess you still need the demand function too. If I give a cost function and a demand function, you know you don't need to know how many firms are going to be in the market. You don't need to know little n. You know, in the long run, profits are going to be zero. So long run, firms are just going to produce at the minimum of average cost. You find that, use the demand to find the price, and you're done, OK? So competition leads to cost minimization. Questions about that? OK. Now, you're all thinking, well, wait a second. We've got rich parents, many of us who make money in these businesses. I don't see zero profits. There's a stock market that's been booming. Where's the zero profits, buddy? I don't see zero profits. Well, the answer is twofold. First of all, remember, firms can make money in the short run in this model. But that doesn't explain the stock market. The stock market's supposed to be forward-looking-- not months, but years and decades. So firms-- really, long run profit's zero. Why would stocks be expensive? Why would people want to invest in these companies? And the answer is because these assumptions are unrealistic-- that this is an extreme version of the model that delivers some nice intuition, but doesn't apply to the real world. So I want to take the last few minutes to talk about the assumptions that we've made that don't really work in the real world to make this model work. It doesn't mean the model's invalid. We learn a huge amount. And the key lesson from this model is competition pushes you towards cost minimization. Always think about these models as not delivering a level truth, but a directional truth. The directional lesson is this is why competition forces firms towards cost minimization, but firms won't actually necessarily get to zero profits. And there's at least three complications. The first one is limited entry. I assumed that firms could costlessly enter and exit in this market. But in fact, that might be hard because, in reality, we have sunk costs, which I talked about last time. We have sunk costs-- or two times ago-- costs which, once paid, can never be recovered. And therefore, firms might say, look, I don't want to get into this market because it's not like I can get my fixed cost back out next period. So if my fixed costs are building a building, next period, I can just sell that building to someone else. But my fixed costs are going to med school. I can't sell that med school degree. Therefore, if doctors aren't going to be profitable in the long run-- there'll be zero profit in the long run-- I'm going to go be a lawyer instead. There are sunk costs, and those sunk costs create what we call barriers to entry. There are barriers to entry that come from costs that are sunk in the long run-- med schools. But there are other sorts of barriers to entry. Take our vendor market. Even in vendor markets, they're not perfectly elastic. One barrier to entry could be they could come in the middle of the night and steal your stuff and beat you up if you tried to enter the market. There's lots of reasons why entry and exit is not costless and easy. There's lots of barriers to entry. Once there's a barrier to entry, this graph goes away. Because go back to figure 8-1. In the second short-run equilibrium, we've got these small profits, this small, dark gray rectangle, right? I then told you another firm would enter and squeeze those away, but another firm will enter only if what? Only under what-- somebody raise their hand and tell me. Under what condition would that third firm enter? Yeah? AUDIENCE: Profits are greater than zero. JONATHAN GRUBER: No, not profits are greater than zero. In the model, that was true. But in reality, what has to be true? Profits have to be greater than what? AUDIENCE: Sunk cost. JONATHAN GRUBER: Sunk costs. Raise your hand, people. I'm going to give you credit because you raised your hand regardless of who yelled out the answer. I'll assume you were right. Sunk costs. Profits have to be greater than the barriers to entry. So in the short run, you enter if profits are greater than zero. In the long run, you only enter if profits are greater than the barriers to entry, which might not be true. There's always some cost to starting a firm. So profits will never really go to zero. They'll only go down to the barriers to entry. So that's problem one. Problem two with this model, problem two is firms may differ. I've assumed-- and this was raised in one question. I've assumed identical firms here. But in fact, firms differ. And in particular, different firms have different cost functions. And with different cost functions, you can get some firms making long-run profits. So for example, let's consider, in figure 8-4, the long-run market supply for cotton. This is from a textbook example. And this is from estimates that people have made of the minimum average cost of producing cotton by country. So in other words, this is old now, but it doesn't matter. The country names don't matter. The example's what matters. In the period of time this study was done, the cheapest place to produce cotton was Pakistan. You produce cotton in Pakistan for $0.71 per-- I don't know-- kilogram, $0.71 per kilogram, OK? However, that was only for a certain amount. At some point, Pakistan ran out of cheap cotton and they had to start producing cotton using more expensive methods. Remember, marginal cost, at some point, has to slope up. So the point is marginal cost was flat for a while in Pakistan, then has to slope up. And suddenly, it becomes cheaper to produce cotton in Argentina at $1.08. And then it becomes cheaper to produce cotton in Australia at $1.15. And eventually, if the price gets to $1.56, it's finally cheap enough to produce cotton in the United States. The point is these flat segments represent the minimum average cost in each country. It's just, instead of making it a point, there's an amount of production they can do with that minimum average cost. Now, let's say world demand for cotton was 1 billion kilograms per year. Then what would happen would be what I taught. You would have Pakistan-- competitors in Pakistan would compete, driving profits down to zero. And price would be $0.71 per kilogram. Now let's say, however, demand for cotton is 5 billion kilograms per year. Well, now, that intersects this supply curve. This is the world supply curve. There's the supply curve at $1.71. So now, US producers are making zero profits because their marginal cost's $1.71. The price is $1.71. But what about producers in Pakistan? They still make cotton. They make the first almost 2 billion kilos. But they're selling at $1.71, and it's costing them $0.71. So they just made profits. The point is that if there are firms which have rising costs and demand is high enough that the high-cost firms are actually producing, that higher price means profits for the low-cost firm. Yeah? AUDIENCE: Would this increase the price variable in Pakistan? JONATHAN GRUBER: You could imagine those profits could then have a feedback effect in asset markets. And you could imagine the long run, in the very, very, very long run, as people buy land, that could dissipate the profits. That's a good point. We'll come back to that. But no, that's very long run, OK? But for now, people have their land in Pakistan, and they're making their money on it. And so that becomes long-run profits. So long-run profits can come from heterogeneous costs. If some firms are particularly efficient in a multimarket firm, those firms can make money. That's a second feature. Now, the third feature is, in some sense, the most interesting, which is input prices may not be fixed. And in fact, input prices, input prices-- ah. Input may have an upward-sloping supply. There could be an upward-sloping supply for inputs. We've assumed input prices are, everywhere, fixed, but that's not true. And in a few lectures, we'll come and teach you about that. But for now, let's recognize that inputs may have an upward-sloping supply. So let's go through that for a couple minutes before we stop. Let's take a market in figure 8-5. This is the market for labor. Now, we've only been doing markets for goods so far in this course, and I'm sort of shortcutting by jumping to here. We'll spend a lot more time on this graph. But the bottom line is this is a graph of the amount of labor supply to the market. In this graph, people are now the suppliers because they're supplying labor. Firms are the demanders. They're demanding labor. So what happens is you have supply curve of labor, and let's assume it's upward sloping. What I mean by that is as the wage goes up, people want to work harder. You want to work harder as the wage goes up. So it's an upward-sloping supply. It makes sense, right? Upward-sloping supply of labor. Now let's imagine a firm suddenly wants to produce more. They used to produce little q1. Now they want to produce little q2. To do so, they need more workers. That represents a shift out in the demand for workers. With upward-sloping supply, what does that do? Raises the wage. If you want more workers, you've got to pay them more. We didn't do that before. We assumed W was a constant. But imagine if, to produce more, you have to pay more. What does that do? Well, you see that in figure 8-6. Now, I used to produce-- the market equilibrium used to be at point E1 with n1 firms producing little q1. So that point big E1-- biggie, big comma E1-- is little n1. There was n firms producing little q1 units per firm. And they were making these profits. Their profits were where price equaled marginal cost 1, now shifting to the left. p equals marginal cost one at little e1. So they were producing where marginal cost equaled average cost. So little q1 at price p1 meant that firms producing at little e1, which was the zero-profit point, OK? Now what happens? Now the firm wants to produce more. Demand goes up. Firm wants to produce more. It wants to produce q2. You have a new long-run equilibrium with n2 firms producing q2. Well, in that case, now if you want to produce q2, you're going to have to pay a higher wage. A higher wage means a higher marginal average cost. Higher marginal average cost means that you're now producing at a higher price and, therefore, the supply curve slopes upward. Now, this is very different. One thing to think about-- I'll let you go. One last thing to think, think about the difference between this third case and the other two cases. In the other two cases, firms made profits. In this case, firms still don't make profits. So notice that profit's still zero, but the supply curve's upward sloping. So you don't need positive profits have an upward-sloping supply curve. So let's stop there. I've given you a lot to think about. And we will come back and talk more about this stuff on Wednesday.

MIT_1401_Principles_of_Microeconomics_Fall_2018

20_Uncertainty.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: All right, let's get started. We have three weeks left in the class. And what we'll be doing for the next three weeks is really a series of applications of what we've learned so far to sort of help you understand how we add some richness to what we've learned and sort of take it to some more real world applications. So we're going to start that today by talking about something we've really ignored in the course so far, which is uncertainty and how uncertainty affects your decision making. So what we've done so far in this class is we've sort of said, look, we've assumed whenever you make decisions you make them with full knowledge and full certainty. But many, many decisions in life are made under conditions of uncertainty. So consider your decision to study for the final in this class. In our models so far, you could optimize your studying across different units of the class. So our model so far, you'd know on the final which unit would be represented with what proportion. You'd optimize your studying appropriately. But of course you don't know that. You face uncertainty about what will be covered in the final or upper proportion. You know the whole course-- you're responsible for the whole class. But obviously you allocate your time. You're uncertain about how to allocate your time across the different subjects on the test. So how do you make that decision? We need to bring our tools to bear on thinking about these kinds of decision making under uncertainty situations. And this isn't just about the test. I decide whether to bring my umbrella today. If I bring my umbrella, there's a chance I'm going to lose it. But I don't want to get wet. I have to think about whether it's going to rain. It all depends on how certain I am it's going to rain, et cetera. There's decisions about whether to bet on a sporting event. That's a decision of uncertainty. And that's all the fun stuff in your life. When you get to the anxiety ridden adult life, you've got things like whether to buy a seven year mortgage or a 30 year mortgage, whether to buy health insurance, what school to put your kids in. All these things involve a huge amount of uncertainty. And we have not yet developed the tools to deal with this. What's really cool is that economics has a very useful tool to think about exactly these kinds of situations. That much like the other tools we've dealt with this semester, it's pretty easy once you understand it. But it's a huge amount of power for explaining the world. And that's the tool of expected utility theory. And that's what we'll focus on today. That's what we'll learn today, the tool of expected utility theory. Now, I'm going to ask you a question. As always, I don't want you try to outsmart me. I just want a quick, gut reaction. I'm going to offer you a bet. Not really, but imagine I was. I'm going to flip a coin. Heads you get $125. Tails you give me $100. So you win $125 versus lose $100. How many of you take that bet? So yeah. You're a slightly more aggressive class than usual. About 40% of you taking the bet. Usually I get more about 20%, 25%. OK. Now-- yeah. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: No. It's a one time bet. So basically-- now how do we think about this, whether it's a good idea or not? Now those who raised your hand probably quickly did the math and did in your head what we call an expected value calculation. What's the expected value of any gamble? The expected value of any gamble is the probability that you win times what you win plus the probability that you lose times what you lose. So you did that calculation in your head. You said, as long as he's not cheating, using a fair coin, there's a 0.5 times 125 plus 0.5 times minus 100. And that is an expected value of $12.50. So those of you who raised your hand probably did that calculation quickly in your head and said, yeah, this is a positive expected value. This is what we call in economics a more than fair bet. More than fair. A fair bet is one with an expected value of 0. So a fair bet is one with an expected value of 0. So a bet with a positive expected value is more than fair. And you in your head said, I did that through. It's more than fair. And that's why some of you raised your hands. But many of you didn't. And that doesn't mean you were wrong. It just means this is not the right way to think about it. The right way to think about it is that what you care about is not dollars. What you care about is utility. Dollars are meaningless. What you care about as a consumer is your utility. And so we don't want to think about expected value. We want to think about expected utility. Now, what is expected utility? It's the same kind of formula as this but with one change. Expected utility is the probability that you win times your utility if you win plus the probability that you lose times your utility if you lose. And that is somewhat different than expected value. And the reason is because utility is not a linear weighting of dollars. Utility is a concave weighting of dollars. As such, because you utility is a concave weighting of dollars, it exhibits diminishing marginal utility. Diminishing marginal rate of substitution. Diminishing marginal rate of substitution, as we talked about ad nauseam in consumer theory. That means that the next dollar is worth less to you than the previous dollar. The next dollars worth diminishing utility. The next slice of pizza is worth less to you than the previous slice of pizza. Likewise, the next dollar's worth less to you than the previous dollar. As a result, losing $1 makes you sadder than winning $1 makes you happy. There's a nonlinearity that comes from diminishing marginal utility and diminishing marginal rate of substitution. So for example, let's think about our typical utility function that we use previously in consumer theory. Utility equals the square root of consumption. And let's just say you consume all your income, as we always did. We've talked about savings the last couple of lectures. Let's put savings aside again and just assume people consume all their income. And so you utility is the square root of consumption. And let's say you start your initial consumption, c0 is 100. You start with $100 of consumption. Your c0 is 100. So your initial utility u0 is 10. Now, I give you-- I offer you this bet. What's the expected utility of this bet? Well, the expected utility of this bet is the probability you win, 0.5, times utility if you win, where utility if you win is the square root of your consumption if you win. What is your consumption if you win? If you win the bet, what is your consumption? Somebody raise your hand and tell me. Starting with 100 and I made this bet with. Yeah. AUDIENCE: 225. JONATHAN GRUBER: 225. You started with 100 and you won 125. If you lose the bet, what do you have? What do you have if you lose the bet? 0. OK? So your expected utility is 0.5 times this utility of what you get if you win plus 0.5 times utility what if you lose. Now if you do that math, you will find that that is 7.5. Your expected utility of this bet is 7.5, which is less than your initial utility. So you should not take the bet, which is through the mechanism of psychology why many of you didn't. You should not take the bet. And the reason is because you are what we call risk-- and humans are what we call risk averse. That risk is inherently negative value to us. A certain dollar is worth much more to us than an uncertain dollar. Just like $1 today is worth more than $1 tomorrow, a certain dollar is worth much more than an uncertain dollar. To see why, the best way-- this intuition here is graphical. So let's go to figure 20-1 and just sort of slowly walk through this. It's a little confusing graph. On the x-axis is your wealth, your total consumption or money you have. You consume everything you have. So it's wealth, or alternatively consumption. On the y-axis of figure 20-1 is your utility. It's the graph of how much you consume against utility. And as you can see, this is a concave graph. It's exhibiting diminishing marginal utility. Each dollar of wealth adds utility but less and less over time. Just like a slice of pizza makes you happier but less and less over time. You get that. So the shape of this curve is true for any utility function that features diminishing marginal utility. That pretty much any utility function we haven't used this semester. It's not as if we're trying to trick you. Has diminishing marginal utility. So as a result, it has this shape. Now, with the utility function of this shape, let's evaluate the gamble I just gave you. You start with wealth of 100. So you see on the x-axis the 100 point, that corresponds to utility of 10. So you can trace up that 100 to utility curve, and then you go over to the y-axis. So that corresponds to a utility of 10. Now, think about what the gamble does. What the gamble does is say look. There's two possible outcomes with 50% chance each. One is wealth of 225. That is the point all the way to the right. That leaves utility of 15. The other's wealth of 0. That's the point all the way to the left. That was utility of 0. What is the average of those two? Well, it's just a linear combination. So 50% chance of each. So the average of those two is a wealth of 112.5 but an expected utility of 7.5. So you draw a cord between those two points, between the 0 point and point B. You find the midpoint of that cord. That's wealth of 112.5. But then you trace that over to utility function. And you see the expected utility is only 7.5. And that's because we're not using a linear combination. We're using a nonlinear combination-- a nonlinear concave combination-- which means that moving up in terms of wealth makes you less happy than moving down in terms of wealth makes you sad. And you're really sad at 0. So going for 100 down to 0 makes you way sadder than going from 100 up to 225 makes you happier. And that's all because diminishing marginal utility of income. So it's natural that this gamble would make you worse off, even though it's more than fair, because, yeah, you're somewhat happier if you win. If you win, utility goes up from 10 to 15. That's great. But if you lose, utility goes down from 10 to 0. That's really bad. So you don't want to take this risk, which is why, although you may not realize it, many of you wouldn't want to take that gamble. So using this graphic, let's ask-- are there questions about that? Yeah. AUDIENCE: So would we take the bet if our gain utility outweighs our loss utility, or we would not take the bet if our utility [INAUDIBLE].. JONATHAN GRUBER: Well, you've answered-- it's the same-- those two questions are the same. It just depends on the probabilities. If the probabilities are 0.5 each, then those two answers-- two questions are exactly the same because 0.5 each would be gain outweighing losses the same as on net positive. But if the probabilities aren't 0.5, you want to use this equation. So you basically want to say is the weighted-- it's about the weighted average change in utility, essentially, where the weights are these probabilities. Yeah. AUDIENCE: Is there any utility calculation in actually gambling. Like can someone dislike gambling-- JONATHAN GRUBER: Hold on. I'm going to come to that. We're going come back to gambling. We talk all about the lottery at the end. OK. But do people understand the basics of this graph? So using this graph, tell me the following. How much-- answer the following question. How much-- let's say-- it's a hard question. See if you can get this. Let's say that I said, you know what, class, I'm going to force you to take this gamble. I'm going to come in here, and I'm going to tell you I'm locking the door. You're not leaving without taking this gamble unless you pay me not to force you. So I'm going to offer you a more than fair bet. Would you be willing to pay me to get out of that bet? And how much would you be willing to pay me? Yeah. AUDIENCE: I'd be willing to pay you up to 2 and 1/2 dollars. JONATHAN GRUBER: Up to 2 and 1/2 dollars. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: 2 and 1/2 dollars. I don't understand. AUDIENCE: Sorry, [INAUDIBLE]. JONATHAN GRUBER: Well, OK, that's-- AUDIENCE: $100 start, right? JONATHAN GRUBER: Yeah. $100 start. AUDIENCE: $25. JONATHAN GRUBER: OK. You're thinking about that sort of right, but you're still thinking in a linear world, not nonlinear world. Look at the graph. OK. Yeah. AUDIENCE: Would it be $43.75? JONATHAN GRUBER: It would be exactly $43.75. Why? AUDIENCE: Because right now the bet essentially gives me the utility of $56.25. JONATHAN GRUBER: So your answer in the front was exactly right in a linear world. But in a nonlinear world, you've got to account for the curvature. So the way to think about it is right now your utility from that bet, if I force you to take it, it leaves you a utility of $56.25. You can see that. Just go backwards from the 7.5 utility down to what level of wealth that's equivalent. What that means is you would rather pay me $43.74 than take that bet. Think about how crazy that is for one second. I'm offering you a bet that is more than fair, and you will pay me almost half of your entire wealth to avoid taking that bet. And that's only with our standard utility function we always use. That's risk aversion. And the reason is because 0 sucks so badly. The reason is because you're so sad going to 0 that you really don't want to be in that situation. And so you will actually pay me $43.75 to avoid a more than fair bet. That's what's crazy. So another way to see this-- let me ask you another question. How large-- let's offer you the same bet. Flip a coin. Tails, you lose 100. Heads, you win x. How large would x have to be for you to take the bet? Tails, you lose 100. Heads, you win x. Yeah. AUDIENCE: $300. JONATHAN GRUBER: $300. Why? AUDIENCE: Because then your expected utility is 10. JONATHAN GRUBER: Right. Exactly. If it's $300, then I'm doing the square root of 400, which is 20, and the square root of 0. You average those, and you're going to get expected utility of 10. So for you to take that bet, I would have to say, tails, you lose 100, heads, you win 300. I need to give you a monstrously more than fair bet. And that is due to the principle of risk aversion. That basically, because of diminishing marginal utility-- risk aversion isn't something made up. It's not some crazy concept. It just falls naturally out of diminishing margin utility because making you sad, because losing, because moving down makes you sadder than moving up makes you happier. Any questions about that? Yeah. AUDIENCE: I guess the question is, doesn't this kind of more depende on utility? Because I know that-- JONATHAN GRUBER: OK. Stop there. Let's go to the next section. You guys are way ahead of me as always. Let's talk about a couple extensions. Let's talk about a couple extensions. First extension, change utility function. Suppose your utility function was of the form u equals 0.1 times c. Now I've chose this particular form because the initial conditions are the same. With c0 of 100, u0 is still 10. So I'm starting from the same point as I was before. But now, would you take the bet, and why? If that's utility function, would you take the bet and why? Just do the math. Do the math. What's your expected utility of that bet? Your expected utility is 0.5 times your utility of 225. So it's times 0.1 times 225 plus 0.5 times your utility of 0. So it's 0.1 times 0. And if you write that out and solve it, you get 11.25, which is greater than 10. So you would take the bet. Any questions about the math? I just did the expected utility evaluation. So you would take the bet. What's the difference? Why do you take the bet here? Yeah. Yeah. Let me get-- go ahead. AUDIENCE: This is a linear utility function. So-- JONATHAN GRUBER: If you're a linear utility function, what you care about is expected value. So you can see that-- do I have that, yeah-- in figure 22. This is the case we call risk neutrality. With the linear utility function, you're risk neutral because your linear utility function does not have what? Does not have diminishing margin utility. As a result, you just care about expected value. There's no difference in expected value and expected utility. Now, the numbers are a little bit different because the functional form, but it gives the same outcome. You always take a more than fair bet and you reject a less than fair bet. So as people move from risk averse to risk neutral, or as utility features less and less diminishing margin utility, they'll be more and more willing to take gambles. But it doesn't have to stop there. We can go further. Imagine that I wrote utility of the form-- utility was of the form c squared over 1,000. A weird utility function but once again created so that if c0 equals 100, u0 equals 10. Now let's do the expected value calculation. Well, if you take the gamble, there's a 0.5 probability that you win. So then you would have-- utility B would be 225 squared over 1,000 and a 0.5 probability that you lose. So you just get 0. And the expected utility in that case is 25.3. The expected utility equals 25.3, which is way bigger than 10. So you take this gamble. Why? Because now you're risk loving. Because what does this say about the diminishing marginal utility of income? This actually says you have increasing margin utility of consumption. A utility function like this says the next slice of pizza makes you even happier than the previous slice of pizza. So go figure 22-3. This is the risk loving case, where you actually have increasing marginal utility of consumption. We don't really talk about this case because it doesn't make sense. But just to understand how this works, same calculations before you start at a point like A. If you win, you go to B. If you lose, you go to 0. Well, this shape utility function, going to B makes you way happier than going to 0 makes you sad. So you love the gamble. In fact, with this utility function, you would take an unfair bet. So for example, imagine I change the gamble to one where it's win 100, lose-- I'm sorry, win 75, lose 100. So I changed the gamble now. Win 75, lose 100. I made an unfair bet. The expected value was negative. Well, this person will still take that bet. If you do the math, if they win, 75, they get 175. Just replace this with 175. And you do the math. You're going to find that the expected utility in that case is 15.3, which is greater than 10. So even an unfair bet-- lose 100, win 75-- will still leave these risk loving people better off than if they hadn't taken a bet. Yeah. AUDIENCE: So when we're doing the losses, when we set that to 0, we're assuming that they're starting off with the amount that they could possibly lose, correct? JONATHAN GRUBER: Oh, that's-- OK. Great. I'm going to come to that next. That's in these examples. But you've called me on an important assumption I'm making. These examples I have. Let's make sure we understand risk neutral and risk loving. People understand that? So now let's-- now you actually raised another issue. Let me ask a new question. Once again, forget everything you've learned. Gut instincts. Here's the bet I'm offering you. Flip a coin. Heads, you win 1,250. Tails, you lose $10. Who takes that bet? Raise your hands if you take that bet. OK. That's backwards. More of you should take that bet, not less of you. And why is that? It's because of exactly what you just pointed out. It's because basically if you think about that-- think about that bet. Think about-- let's go back to our old utility function, u equals square root of c. Let's think about that bet. So what's the expected utility? It's 0.5 times you win 1,250. So it's square root of 112.5 plus 0.5 times you lose 10 square root of 90. And that expected utility is 10.5, which is greater than 10. So you should take that bet. What changed? You're still risk averse. The guy who would've paid me $44 to avoid the other bet is now happy to take this bet. Same person. What changed? What changed? Yeah. AUDIENCE: Smaller portion of [INAUDIBLE].. JONATHAN GRUBER: Right. And why does a smaller portion of the income change things here? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Exactly. Because ultimately, infinitesimally, it's a linear curve. So if you go back to Figure 21, for any given epsilon change from point A it's linear. So essentially as gambles get smaller and smaller relative to your starting point, you become more and more risk neutral. Because, yeah, you're a bit sadder than happier but just a bit. And remember, it's linear. You just take-- so if I did 12 and 1/2 cents versus $0.10, then unless you were crazy risk averse, you should take that bet because essentially you only care-- at that point it's so tiny relative to your wealth you might as well use expected value. So as gambles become smaller relative to your income, the utility function becomes locally flatter. Utility function goes locally flatter. And as a result, you become more and more risk neutral. Question about that? Last point. Why did so many of you still not take that bet? Well, the answer is that even the model we've talked so far misses an important psychological phenomena. So now I'm stepping out of standard microeconomics into the realm of behavioral economics. Unfortunately, due to time this semester, I'm not going to get to my lecture on behavioral economics. But you guys want to learn-- I'll talk a little bit about it in the next few lectures. I'll sprinkle it in. But course 1113 is a fascinating course we offer here about how you build psychology when you think about economics. And here's one example. Why is it that even with this gamble, even if I'd done the 12 and 1/2, $0.10 gamble, a bunch of you still wouldn't have taken it. Why is that? That's because humans not only feature risk aversion, humans also feature loss aversion, which is we have an irrational behavioral bias that losing by itself makes us sad relative to winning. Taking away something we have makes us sadder than getting something new. So here's a standard experiment that's run. They get a bunch of $5 mugs. Mugs worth about $5. They take people. And randomly half of the people they ask, how much would you pay for this mug? And half the people they give them the mug and say, how much would you sell your mug for? Half people they say, here's a mug, how much would you pay for it? Half the people say, here's a mug. It's yours. But I want to buy it back. How much would you sell it to me for? The average person will pay $3, but the average person wants $7 to sell it back. That makes no sense. Either way, it's trivial. It's just a mug. It's trivial relative to your wealth. It's just a mug. It doesn't really matter. But once people have it, they feel like, no, that's already mine. I don't want to sell it. That's loss aversion. People are biased by their starting points. Your very starting point dictates your willingness to take a gamble, which is not true in a standard economic model. But it's true in all laboratory experiments in psychology. So the reason people don't like gambles are not only because they're risk averse. But even more, they're loss averse. So for example, there's great economic studies which show that there's a massive bias against selling your house for anything under the price you paid for it. That people sort of-- they're very sort of linear, and they're willing to sell the house above the price you paid for it. But there's this huge notch at the price you paid for it. They simply don't want to sell less they paid for it. And that's just loss aversion idea. This comes up in many other contexts. So there's two reasons why people are risk averse, that people won't take gambles-- a standard reason and the sort of extra psychological bias. Now, this raises-- this leads us naturally to the next section I want to go to, which is to talk about applications of this theory and why it's important. And the first application I want to talk about is insurance. Insurance is big business in America. Individuals in America pay individuals-- forgetting the government, just people-- spend $1.5 trillion a year on insurance products. Almost 10% of GDP. 10% of our entire economy is people buying insurance. Health insurance is the biggest, life insurance, casualty and property insurance, auto insurance, et cetera. Added all up, it's almost a tenth of our entire economy. Why? Because they're risk averse, and also loss averse. I'm going to put loss aversion aside and just use the standard framework. That just strengthens the argument. But it's because they're risk averse. So let's do the math. Imagine you're a single 25-year-old male. I'm only being gender biased here because there's no risk of pregnancy. So you're a single 25-year-old male in perfect health. So the only risk you face health wise is getting hit by a car. That's basically your risk. Otherwise, you're basically not going to go to the doctor. So essentially, let's say your income is $40,000. That's your income. And let's say that there's a 1%-- since it's Cambridge, there's a non-trivial chance you get hit by a car. Let's say there's a 1% chance, probability 0.01, you'll get hit by a car every year. And if you do, you're going to face $30,000 in medical bills. So you have a $40,000 income. There's a 1% chance you get hit by a car. And if you do, you'll face $30,000 in medical bills. And let's assume for the minute you'll still get to work. Your income will always be there. You're just going to have to face a bunch of medical bills. There's a separate issue about whether you might have to miss your job. That makes this even worse. But let's ignore that. You get to go to your job. You're just going to have to take a week to get patched up. And $30,000 is nothing, by the way, for a hospital bill. A typical, for example, heart attack hospital bill is well over $100,000. So $30,000 is pretty modest. Yeah. AUDIENCE: Wouldn't the person that hit the guy have to get their insurance cover the medical bill? JONATHAN GRUBER: Well, let's ignore that for a second. Right now we're talking about why you want insurance overall. We'll later get into who-- we can discuss later who should own that insurance, who should bear the risk. Right now it's just simply why you'd want insurance. So the expected cost of getting hit by a car-- the expected value or expected cost because it's negative-- is minus 300. So every year you have a $300 expected cost of getting hit by a car. And let's say your utility function is u equal square root of c. And let's assume there's no savings. You consume all your income. How much will this person be willing to pay for insurance? Well, we can solve that by asking at what insurance price would they be better off being insured versus uninsured. So let's just do the math. If they're uninsured what's going to happen to them? Well, what's their expected utility? Well, there's a 0.1 chance that they are going to get hit. And so their net income will drop from 40,000 to 10,000 because they'll have to pay $30,000 in medical bills. So 0.1% chance their net income be $10,000. And there's a 99% chance that their net income will be $40,000. That's utility without insurance. Add that up and you get 199. So their expected utility without insurance is 199. That's their expected utility with no insurance. Now let's do their expected utility with insurance but with insurance with an uncertain price. Let's call the price x. What's the utility with insurance? Well, with insurance there's a 0.1% chance that they get hit. 0.01, I'm sorry. It should be an .01. My bad. Wow. I can't believe you guys missed that. You guys are a little tired today. 0.01. 0.01% chance that I get hit. Now, if I get hit with insurance, I don't have to pay my medical bill. But I do have to pay my insurance premium. So what I have is I have $40,000 minus x, which is my insurance premium. I always have to pay my insurance premium every year, no matter what. If I don't get hit, then I get my 40,000, but I also have to pay my insurance premium. So basically these things are the same. So my expected utility is square root of $40,000 minus x. That's my expected utility. So how do I solve for the optimal x? Well, I ask at what x am I better off than being uninsured? So I set this equal to 199. I ask at what x would I be better off than being uninsured? And if you solve that, you get that the x star, the point at which you would rather be insured than uninsured, is a premium of $399. So you will pay-- you would rather have insurance at a cost of 399 than you would go uninsured. Clear on that? You'd rather pay a premium of 399 than you would go uninsured. Now think about what that means. Remember, the expected cost of this accident was only $300. That means that you have a $99 risk premium. You are willing to pay $99 to avoid bearing this risk. That is what you're willing to pay-- before you were willing to pay me $43.75 to get out of that gamble. That was your risk premium. It's how much you'll pay to get out of taking a gamble. Now before I was offering a gamble. But here's the thing. Being uninsured is a gamble. Being uninsured is like taking the gamble. So the example I had before, I locked you in the room. And you have to pay me if you want avoid the bet. That's insurance. You are locked in this life. You're locked into being in Cambridge. You're dealing with a 1% risk getting hit by a car always. So the question is, how much will you pay to avoid at least the financial cost-- forget the trauma, the financial costs-- of being hit by that car? And the answer is, you'll pay $99 above the expected damage it will do. And that is why insurance is big business because people will pay to avoid being put in risky situations. So insurance is a very profitable exploit. Now, in fact, of course, the insurance industry is like any other industry. The supply side and competition, and whether that will lead to profits and stuff like that. Depends on the whole supply side. This is just the demand side. But the point is that there's going to be huge demand for insurance, and people will be willing to pay much more than it costs the insurance companies. The consumers expect to pay the $300 a year. And they're getting almost $400 a year in premium. So the insurance company makes that money. And basically, that's why insurance is big business. Now, here's a couple of things I'd like you to show yourself in your copious spare time. First of all, this risk premium should be bigger as the loss gets bigger. Why? Because you're moving away from that linear part towards the nonlinear part of the utility function. Similarly, this risk premium should fall as your income is higher. Why? Because once again, that makes you more towards the linear part. The bottom line is the bigger the risk is relative to your income, the more risk averse you become. The more you move from that linear part of the curve onto the nonlinear part of the curve. So that's the key thing. What matters is risk relative to your income. That's going to determine your value, your willingness to pay for insurance. So for example, let's think of your decision to go buy consumer electronics and the warranty. And they always offer you a warranty. Now, those warranties are expected value negative. If you take the odds of your machine breaking-- if you go to buy a new stereo. As if people buy stereos anymore. You're going to buy a car stereo. People still buy those. You're going to buy a car stereo. You take the odds of that breaking times the cost it would take to fix it. Those multiplied are less than what they'll charge you for the insurance premium. It's a bad bet but it's insurance. And so the question is, should you take that? Well, that depends on how wealthy you are. I should never take that because my car radio cost is tiny relative to my income. Someone who has low income might decide that's a large gamble relative my income. I don't want to take that. So I want to buy the insurance offered by the manufacturer. So once again, it's all about the size of the risk relative to your income. Yeah. AUDIENCE: What if I have increased chance of breaking my phone? JONATHAN GRUBER: Well, that's a separate issue. We'll talk about that later. That's called moral hazard. You might-- well, it's not moral hazard. What you're saying is there's heterogeneity. And you know you're actually-- that for you it is a fair bet because you're clumsy. Well, then you should definitely take it. This is saying risk aversion works in favor of you taking it. Clumsiness further works in favor of you taking it. And we'll talk about that in a lecture or two when we talk about government provision of insurance. So that's sort of the first application, which is thinking about insurance and why it's such big business in America. And it is huge business in America. The second application is thinking about our friend the lottery. Big news lately. We talked. There was a couple of huge lottery payouts. So let's talk about-- actually, let's do this. Let's talk about the lottery. Now, the lottery is a total rip off. The expected value of a lottery purchase is 50%. So every dollar you spend on the lottery, over all lottery options-- for every dollar you spend, the expected payout is $0.50, is a much, much less than fair bet. And yet lotteries are wildly popular. Actually, the beginnings of this country, the US were financed by a lottery. Much of the money they government raised initially to set up America came from a lottery. And state lotteries are a huge source of public financing right now across America. Now, why do people play lotteries. Well, there's basically four theories for why people play lotteries. The first is that people are risk loving. In fact, that were wrong, people are risk loving. That's why they play lotteries. How do we know that theory is wrong? How do we know that theory is its face wrong, that people are risk loving? Yeah. AUDIENCE: The demonstration in class. Lots of didn't raise their hands. JONATHAN GRUBER: Well, that's one way we might know. But how do we know more globally? You guys could just be a weird bunch. How do you we know more globally? Yeah. AUDIENCE: Because of [INAUDIBLE].. JONATHAN GRUBER: Yeah. But that's absolutely right, theoretically. But that's theoretical. In the real world, what piece of evidence can you immediately point to that I recently pointed out that could show you this is wrong. Yeah. AUDIENCE: People buy insurance. JONATHAN GRUBER: People buy insurance. If they're risk loving, why are they buying insurance? So clearly people aren't risk loving. We wouldn't spend 10% of GDP on insurance. So that theory is clearly wrong. OK. That's theory one. Now, the second theory is a somewhat subtler version of this theory, which is quite interesting, which is that people are both risk averse and risk loving. So that risk aversion varies. Risk tolerance, let's call it, varies. And then in particular, people are risk averse over small gambles, but risk loving over big gambles. So let's look at figure 20-4. This is an example of what we call Friedman-Savage preferences. You don't need to know that. But they had this idea that maybe people are locally risk averse but globally risk loving. Let's see what this-- let me explain this. This is sort of complicated. So imagine that I'm going to offer you a 50-50 gamble between w1 and w3. So a gamble leaves you at w1 and a gamble leaves you at w3 at 50% chance. Well, if we look at between w1 and w3, we're on the concave part of the utility function. And as a result, I will not take that gamble. That gamble leaves me at point B, which is below B star. So basically, I will not take that gamble. That leaves me worse off than just having w1 plus w3 over 2 with certainty. This is just if you hold your hand-- if you hold your hands between w1 and w3, that's just the graph we saw before. You won't take that bet. But now let's say once you get above w3, once you're rich enough, you're risk loving. Let's say people are risk averse at first but then get risk loving. So if you started it-- if you said you were starting at w3 plus w5 over 2, I'd offer you a gamble between w3 and w5. Then you're risk loving in that range. And you take it. So once you get rich enough, you start to get risk loving. So I talked about before getting rich or getting more and more risk neutral as you get richer. What if it goes the other way? What if you actually get risk loving when you get richer? Well, then, if you think about the whole Mega Millions thing, you could see that over the whole distribution from w1 to w5, people might want to take that risk. That you could actually be risk loving over these gambles, over these giant gambles. And that could explain why people play Mega Millions. That over the giant gambles, they're risk loving, even if over more moderate gambles they're risk averse. You don't insure yourself for a billion dollars. You insure yourself for a few thousand dollars. Yeah. AUDIENCE: How does the size of the gamble we're talking about determine-- we're talking about small bets that are more linear, is it based on what the player puts in or what they get in return? JONATHAN GRUBER: Both. Both. It's basically about-- it's about expected utility calculations. Yes, it's true, for Mega Millions, you put in $2 for the chance of winning $1.6 billion. But your probability is way, way lower than 2 in 1.6 billion. So it's still unfair. So the Friedman-Savage hypothesis-- you don't need to know the name, once again. This hypothesis is that what's happening is that people are first risk averse, but then risk loving. Well, how could we test and actually disprove this theory? Yeah. AUDIENCE: Like scratch offs. JONATHAN GRUBER: Yeah. If this is true, people would love Mega Millions but hate $10 scratch offs. In fact, the vast majority of lottery playing is not Mega Millions. It's $10 scratch off. Most money spent on lotteries are $10 and $20 gambles where you bet $1 to win $10 or $20. You should be risk averse over that, or at best risk neutral. You shouldn't be risk loving over those tiny gambles. And yet that's most of what lottery players do. AUDIENCE: Wouldn't not really a scratch off be counted internally as not getting money instead of losing money? JONATHAN GRUBER: Well, no, but you've lost the dollar you spent. And that dollar-- I still offered you a gamble. Spend $1, win $10, with a 0.05 probability. AUDIENCE: I mean, in the loss averse sense. JONATHAN GRUBER: Well, we're not doing loss aversion. Loss aversion is sort of hard. You don't have to think about gambling. Loss aversion is more about losing. But in the regular-- you can see you shouldn't do that unless you're risk loving. But even this theory would say you're not locally risk loving. So this theory is out, which leaves us with two more theories. The first theory is that this is entertainment. That in people's utility function is not just consumption but the thrill of finding out if they won. My wife, against my better judgment, went out and bought a Mega Millions ticket. And she got utility out of waiting for that number to come up. And it was pretty cheap utility. Cost her $2. So in that sense, maybe you play the lottery a lot for entertainment. That's one theory, unfortunately, the other theory is ignorance. That basically, the saying is the lottery is a tax on the stupid. Basically just don't understand what a bad deal this is. And the problem is we don't know which of these theories is right. And they have very different implications for government policy. If this theory is right, if this theory is right, then the government should support lotteries, where essentially the government is essentially getting paid for providing entertainment. It's what the lotteries often call the voluntary tax. That I am basically giving the government money that can run our schools in return for the government giving me the entertainment value of seeing if my scratch off won. That's great. That's welfare improving. Under this theory, we should be discouraging lotteries. That all we're doing is taking a bunch of ignorant people and getting them to waste their money. Yeah. AUDIENCE: Is there almost a possibility that there is this concept of having nothing to lose. If you're already too poor to be able to afford your basic needs, then you might feel like, I may as well try and win the lottery and then I would be all set. That is-- [INTERPOSING VOICES] AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: That's exactly this. That wouldn't explain why I'd pay the $20 scratch off. That's an absolute reason why I'd go ahead, even if I was starving, and play the Mega Millions. And that's the Friedman-Savage hypothesis, absolutely. But that can't explain why in fact, in some low income communities, among some low income groups, they'll spend as much as 20% of their income every year on the lottery, a net. They're losing huge amounts of money on scratch off tickets. So the question is, is that a rational decision because they find it entertaining or an irrational decision because they just don't understand what's going on? And unfortunately, we don't know the answer. But we do know it's very important. It's important because there's big bucks and in many low income communities it's a huge source of expenditure. So I can't give you the answer to that. I can just tell you it's an important question. I hope someday someone to figure out how to think about this because it's got very important implications. So let me stop there. That's all I want to say about uncertainty. And we'll come back and do another topic on Wednesday.

MIT_1401_Principles_of_Microeconomics_Fall_2018

2_Preferences_and_Utility_Functions.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: Today we're going to start talking about what's underneath the demand curve. So basically, what we did last time, and what you did in section on Friday is talk about sort of the workhorse model of economics, which is supply and demand model. And we always start the class with that, because that's the model in the course. But I think as any good sort of scientists and inquisitive minds, you're probably immediately asking, well, where do these supply and demand curves come from? They don't just come out of thin air. How do we think about them? Where do they come from? And that's what we'll spend basically the first 1/2 of the course going through. And so we're going to start today with the demand curve, and the demand curve is going to come from how consumers make choices, OK? And that will help us drive the demand curve. Then we'll turn next to supply curve, which will come from how firms make production decisions. But let's start with the demand curve, and we're going to start by talking about people's preferences, and then the utility functions, OK? So our model of consumer decision making is going to be a model of utility maximization. That's going to be our fundamental-- remember, this course is all about constrain maximization. Our model today is going to be a model of utility maximization. And this model's going to have two components. There's going to be consumer preferences, which is what people want, and there's going to be a budget constraint, which is what they can afford. And we're going to put these two things together. We're going to maximize people's happiness, or their choice-- or their happiness given their preferences, subject to the budget constraint they face. And that's going to be the constraint maximization exercise that actually, through the magic of economics, is going to yield the demand curve, and yield a very sensible demand curve that you'll understand intuitively. Now, so what we're going to do is do this in three steps. Step one-- over the next two lectures. Step one is we'll talk about preferences, how do we model people's tastes. We'll do that today. Step two is we'll talk about how we translate this to utility function, how we mathematically represent people's preferences in utility function. We'll do that today as well. And then next time, we'll talk about the budget constraints that people face. So today, we're going to talk about the max demand. Next time we'll talk about the budget constraint. That means today's lecture is quite fun. Today's lecture is about unconstrained choice. We're not going to worry at all about what you can afford, what anything costs. We're not going to worry about what things cost. We're not going to worry about what you can afford, OK? Today's the lecture where you won the lottery, OK? You won the lottery. Money is no object. How do you think about what you want, OK? Next time, we'll say, well, you didn't win the lottery. In fact, as we learn later in the semester, no one wins the lottery. It's an incredibly bad deal. But next time, we'll impose the budget constraints. But for today, we're just going to ignore that and talk about what do you want, OK? And to start this, we're going to start with a series of preference assumptions. A series-- remember, as I talked about last time, models rely on simplifying assumptions. Otherwise, we could never write down a model. It'll go on forever, OK? And the key question is, are those simplifying assumptions sensible? Do they do violence to reality in a way which makes you not believe the model, or are they roughly consistent with reality in a way that allows you to go on with the model? OK? And we're going to pose three preference assumptions, which I hope will not violate your sense of reasonableness. The first is completeness. What I mean by that is you have preferences over any set of goods you might choose from. You might be indifferent. You might say, "I like A as much as B," but you can't say, "I don't care," or, "I don't know." You can say, "I don't care." That's indifference. You can't say, "I don't know." You can't literally say, "I don't know how I feel about this." You might say you're indifferent to two things, but you won't say, "I don't know how I feel about something." That's completeness, OK? The second is the assumption we've all become familiar with since kindergarten math, which is transitivity. If you prefer A to B and B to C, you prefer A to C, OK? That's kind of-- I'm sure that's pretty clear. You've done this a lot in other classes. So these two are sort of standard assumptions you might make in any math class. The third assumption is the one where the economics comes in, which is the assumption of nonsatiation or the assumption of more is better. In this class, we will assume more is always better than less, OK? We'll assume more is better than less. Now, to be clear, we're not going to say that the next unit makes you equally happy as the last unit. In fact, I'll talk about that in a few minutes. Well, in fact, the next unit makes you less happy. But we will say you always want more, that faced with the chance of more or less, you'll always be happier with more, OK? And that's the nonsatiation assumption, OK? And I'll talk about that some during the lecture, but that's sort of what's going to give our models their power. That's a sort of new economics assumption. That's going to give-- beyond your typical math assumptions-- this is going to give our models their power, OK? So that's our assumptions. So armed with those, I want to start with a graphical representation of preferences. I want to graphically represent people's preferences, and I'll do so through something we call indifference curves. Indifference curves, OK? These are-- indifference curves are basically preference maps. Essentially, indifference curves are graphical maps of preferences, OK? So for example, suppose your parents gave you some money to begin the semester, and you spent that money on two things. Your parents are rich. They gave you tons of money. You spent your money on two things, buying pizza or eating cookies, OK? So consider preferences between pizza and cookies. That's your two things you can do. Once again, this is a constrained model. Obviously, in life, you can do a million things with your money. But it turns out, if we consider the contrast between doing two different things with your money, you get a rich set of intuition that you can apply to a much more multi-dimensional decision case. So let's start with a two dimensional decision case. You've got your money. Either you can have pizza or you can have cookies, OK? Now, consider three choices, OK? Choice A is two pizzas and one cookie. Choice B is one pizza and two cookies, and choice C is two pizzas, two cookies. OK, that's the three packages I want to compare. And I am going to assume-- and I'll mathematically rationalize in a few minutes-- but for now, I'm going to assume you are indifferent between these two packages. I'm going to assume you're equally happy with two slices of pizza and one cookie or two cookies and one slice of pizza, OK? I'm going to assume that. But I'm also going to assume you prefer option C to both of these. In fact, I'm going to assume that, because that is what more is better gives you, OK? So you're indifferent between this. This indifference doesn't come from any property I wrote up. That's an assumption. That's just-- for this case, I'm assuming that. This comes to the third property I wrote up there. You prefer package C because more is always better than less, OK? So now, let's graph your preferences, and we do so in figure 2-1, OK, in the handout. OK, so here's your indifference curve. So we've graphed on the x-axis your number of cookies, on the y-axis slices of pizza, OK? Now, you have-- we've graphed the three choices I laid here, choice A, which is two slices of pizza and one cookie, choice B, which is two cookies and one slice of pizza, and choice C, which is two of both. And I've drawn on this graph your indifference curves. The way your indifference curves looks is there's one indifference curve between A and B, because those are the points among which you're indifferent. So what an indifference curve represents is all combinations of consumption among which you are indifferent. That's why we call it indifference curve. So an indifference curve, which will be sort of one of the big workhorses of this course, an indifference curve represents all combinations along which you are in different. You're indifferent between A and B. Therefore, they lie on the same curve, OK? So that's sort of our preference map, our indifference curves. And these indifference curves are going to have four properties, four properties that you have to-- that follow naturally from this-- it's really three and 1/2. The third and fourth are really pretty much the same, but I like to write them out as four. Four properties that follow from these underlying assumptions-- Property one is, consumers prefer higher indifference curves. Consumers prefer higher indifference curves, OK? And that's just all from more is better. That is, an indifference curve that's higher goes through package that has at least as much of one thing and more of the other thing. Therefore, you prefer it, OK? So as indifference curve shifts out, people are happier, OK? So on that higher indifference curve, point C, you are happier than points A and B, because more is better, OK? The second is that indifference curves never cross. Indifference curves never cross, OK? Actually, that's third, actually. I want to come to that in order. Second-- third is the indifference curves never-- Second is indifference curves are downward sloping. Second is indifference curves are downward sloping. Indifference curves are downward sloping. Let's talk about that first, OK? That simply comes from the principle of nonsatiation. So look at figure 2-2. Here's an upward sloping indifference curve, OK? Why does that violate the principle of nonsatiation? Why does that violate that? Yeah. AUDIENCE: Either, if you're-- either you're less happy with you have more cookies, or you're less happy if you have more pizza. And like there's-- and that violates nonsatiation. JONATHAN GRUBER: Exactly. So basically, you're indifferent-- on this curve, you're indifferent with one of each and two of each. You can't be indifferent. Two of each has got to be better than one of each. So an upward sloping indifference curve would violate nonsatiation. So that's the second property of indifference curve. The third property of indifference curve is the indifference curves never cross, OK? We could see that in figure 2-3, OK? Someone else tell me why this violates the properties I wrote up there, indifference curves crossing. Yeah. AUDIENCE: Because B and C [is strictly better. JONATHAN GRUBER: What's that? AUDIENCE: Because B and C, B is strictly better. JONATHAN GRUBER: Because the B and C, B is strictly better. That's right. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: But they're also both on the same curve as A. So you're saying they're both-- you're indifferent with A for both B and C, but you can't be, because B is strictly better than C. So it violates transitivity, OK? So the problem with crossing indifference curves is they violate transitivity. And then finally, the fourth is sort of a cute extra assumption, but I think it's important to clarify, which is that there is only one indifference curve through every possible consumption bundle, only one IC through every bundle. OK, you can't have two indifference curves going through the same bundle, OK? And that's because of completeness. If you have two indifference curves going through the same bundle, you wouldn't know how you felt, OK? So there can only be one going through every bundle, because you know how you feel. You may feel indifferent, but you know how you feel. You can't say I don't know, OK? So that's sort of a extra assumption that sort of completes the link to the properties, OK? So that's basically how indifference curves work. Now, I find-- when I took this course, before you were-- god, maybe before your parents were born, I don't know, certainly before you guys were born-- when I took this course, I found this course full of a lot of light bulb moments, that is, stuff was just sort of confusing, and then boom, an example really made it work for me. And the example that made indifference curves work to me was actually doing my first UROP. When my UROP was with a grad student, and that grad student had to decide whether he was going to accept a job. He had a series of job offers, so he had to decide. And basically, he said, "Here's the way I'm thinking about it. I am indifferent-- I have an indifference map and I care about two things. I care about school location and I care about economics department quality. I care about the quality of my colleagues, and the research it's done there, and the location." And basically, he had two offers. One was from Princeton, which he put up here. No offense to New Jerseyans, but Princeton as a young single person sucks. OK, fine when you're married and have kids, but deadly as a young single person. And the other-- so that's Princeton. Down here was Santa Cruz. OK, awesome. [INAUDIBLE] is the most beautiful university in America, OK? But not as good an economics department. And he decided he was roughly indifferent between the two. But he had a third offer from the IMF, which is a research institution in DC, which has-- he had a lot of good colleagues, and DC is way better than Princeton, New Jersey, even though it's not as good as Santa Cruz. So he decided he would take the offer at the IMF, OK? Even though the IMF had worse colleagues than Princeton and worse location than Santa Cruz, it was still better in combination of the two of them, given his preferences. And that's how he used indifference curves to make his decision, OK? So that's sort of an example of applying it. Once again, no offense to the New Jerseyans in the room, of which I am one, but believe me, you'd rather be in Santa Cruz. OK, so now, let's go from preferences to utility functions. OK, so now, we're going to move from preferences, which we've represented graphically, to utility functions, which we're going to represent mathematically. Remember, I want you understand, everything this course at three levels, graphically, mathematically, and most importantly of all, intuitively, OK? So graphic is indifference curves. Now we come to the mathematical representation, which is utility function, OK? And the idea is that every individual, all of you in this room, have a stable, well behaved, underlying mathematical representation of your preferences, which we call utility function. Now, once again, that's going to be very complicated, your preference over lots of different things. We're going to make things simple by writing out a two dimensional representation for now of your indifference curve. We're going to say, how do we act mathematically represent your feelings about pizza versus cookies? OK? Imagine that's all you care about in the world, is pizza and cookies. How do we mathematically represent that? So for example, we could write down that your utility function is equal to the square root of the number of slices of pizza times the number of cookies. We could write that down. I'm not saying that's right. I'm not saying it works for anyone in this room or even everyone this room, but that is a possible way to represent utility, OK? What this would say-- this is convenient. We will use-- we'll end up using square root form a lot for utility functions and a lot of convenient mathematical properties. And it happens to jive with our example, right? Because in this example, you're indifferent between two pizza and one cookie or one pizza and two cookie. They're both square root of 2. And you prefer two pizza and two cookies. That's two, OK? So this gives you a high utility for two pizza and two cookies, OK, than one pizza and two cookie, or two pizza and one cookie. So now, the question is, what does this mean? What is utility? Well, utility doesn't actually mean anything. There's not really a thing out there called utiles OK? In other words, utility is not a cardinal concept. It is only an ordinal concept. You cannot say your utility, you are-- you cannot literally say, "My utility is x% higher than your utility," but you can rank them. So we're going to assume that utility can be ranked to allow you to rank choices. Even if generally, we might slip some and sort of pretend utility is cardinal for some cute examples, but by and large, we're going to think of utility as purely ordinal. It's just a way to rank your choices. It's just when you have a set of choices out there over many dimensions-- like if your choice in life was always over one dimension and more was better, it would always be easy to rank it, right? You'd never have a problem. Once your choice is over more than one dimension, now if you want to rank them, you need some way to combine them. That's what this function does. It allows you essentially to weight the different elements of your consumption bundle, so you can rank them when it comes time to choose, OK? Now, this is obviously incredibly simple, but it turns out to be amazingly powerful in explaining real world behavior, OK? And so what I want to do today is work with the underlying mathematics of utility, and then we'll come back. We'll see in the next few lectures how it could actually be used to explain decisions. So a key concept we're going to talk about in this class is marginal utility. Marginal utility is just a derivative of the utility function with respect to one of the elements. So the marginal utility for cookies, of cookies, is the utility of the next cookie, given how many cookies you've had. This class is going to be very focused on marginal decision making. In economics, it's all about how you think about the next unit. Turns out, that makes life a ton easier. Turns out, it's way easier to say, "Do you want the next cookie," than to say, "How many cookies do you want?" Because if you want the next cookie, that's sort of a very isolated decision. You say, OK, I had this many cookies. Do I want the next cookie? Whereas before you start eating, if you say, how many cookies do you want, that's sort of a harder, more global decision. So we're going to focus on this stepwise decision making process of do you want the next unit, the next cookie, or the next slice of pizza, OK? And the key feature of utility functions we'll work with throughout the semester is that they will feature diminishing marginal utility. Marginal utility will fall as you have more of a good. The more of a good you've had, the less happiness you'll derive from the next unit, OK? Now, we can see that graphically in figure 2-4. Figure 2-4 graphs on the x-axis the number of cookies holding constant pizza. So let's say you're having two pizza slices, and you want to say, what's my benefit from the next cookie? And on the left axis, violating what I just said like 15 seconds ago, we graph utility. Now, once again, the utile numbers don't mean anything. It's just to give you an ordinal sense. What you see here is that if you have 1 cookie, your utility is 1.4, square root of 2 times 1. If you have 2 cookies, your utility goes up to square root of 4, which is 2. You are happier with 2 cookies, but you are less happy from the second cookie than the first cookie, OK? And you could see that in figure-- if you flip back and forth between 2-4 and 2-5, you can see that, OK? The first cookie, going from 0 to 1 cookie, gave you one-- so in this case, we're now graphing the marginal utility. So figure 2-4 is the level of utility, which is not really something you can measure, in fact. Figure 2-5 is something you can measure, which is marginal utility, what's your happiness-- and we'll talk about measuring this-- from the next cookie. You see, the first cookie gives you a utility increment of 1.4, OK? You go from utility of 0 to utility of 1.4. The next cookie gives you utility increment of 0.59. OK, you go from utility of 1.41 to utility of 2. The next cookie gives utility increment of 0.45, the square root of 3. So now we flip back to the previous page. We're going from the square root of 4, we're going from the square root of 4-- I'm sorry-- to the square root of 6. Square root of 6 is only 0.45 more than the square root of 4, and so on. So each additional cookie makes you less and less happy. It makes you happier, it has to, because more is better, but it makes you less and less happy, OK? And this makes sense. Just think about any decision life starting with nothing of something and having the first one, slice of pizza, a cookie, deciding on which movie to go to. The first movie, the one you want to see the most, is going to make you happier than the one you want to see not quite as much. The first cookie when you're hungry will make you happier than the second cookie. The first slice of pizza make you happier-- Now, you may be close to indifferent. Maybe the second slice of pizza makes you almost as happy as the first. But the first will make you happier, OK? If you think about-- that's really sort of that first step. You were hungry, and that first one makes you feel happier. Now, but you got to remember, you always want more cookies. Now, you might say, "Wait a second. This is stupid. Once I've had 10 cookies, I'm going to barf. The 11th cookie can actually make me worse off, because I don't like barfing." But in economics, we have to remember, you don't have to eat the 11th cookie. You can give it away. So if like say, you don't want the 11th cookie, you can save it for later. You can give it to a friend. So you always want it. In the worst case, you throw it out. It can't make you worse off, it can only make you better off. And that's what our sort of more is better assumption comes from. Obviously, the limit-- you know, if you get a million cookies, your garbage can gets full. You have no friends to give them to. I understand at the limit, these things fall apart, OK? But that's the basic idea of more is better and the basic idea of diminishing marginal utility. OK, any questions about that? Yeah. AUDIENCE: Can the utility function ever be negative? JONATHAN GRUBER: Utility function can never be negative because we have-- well, utility-- once again, utility is not an ordinal concept. You can set up utility functions such that the number is negative. You can set that up. OK, the marginal utility is always positive. You always get some benefit from the next unit. Utility, once again, the measurement's relevant. So it could be negative. You could set it up-- I could write my utility function like this, you know, something like that. So it could be negative. That's just a sort of scaling factor. But marginal utility is always positive. You're always happier, or it's non-negative. You're always happier or at least indifferent to getting the next unit. Yeah. AUDIENCE: So when you're looking at 2-5, if you get like a fraction of a cookie, is the marginal utility still going to go up? JONATHAN GRUBER: I'm sorry, you look-- figure 2-5-- no, the marginal is going to go down. Each fraction of a cookie, the marginal utility-- marginal utility is always diminishing. AUDIENCE: So if you start with zero, and you get 1/2 a cookie based on this graph-- JONATHAN GRUBER: Well, it's really hard to do it from zero. That's really tricky. It's sort of much easier to start from one. So corner solutions, we'll talk about corner solutions in this class, they get ugly. Think of it starting from one. Starting with that first cookie, every fraction of a cookie makes you happier, but less and less happy with each fraction. Good question. All right, good questions. All right, so now, let's talk about-- let's flip back from the math to the graphics, and talk about where indifference curves come from. I just drew them out. But in fact, indifference curves are the graphical representation of what comes out of utility function, OK? And indeed, the slope of the indifference curve, we're going to call the marginal rate of substitution, the rate essentially at which you're willing to substitute one good for the other. The rate at which you're willing to substitute cookies for pizza is your marginal rate of substitution. And we'll define that as the slope of the indifference curve, delta p over delta c. That is your marginal rate of substitution. Literally, the indifference curve tells you the rate at which you're willing to substitute. You just follow along and say, "Look, I'm willing to give up--" So in other words, if you look at figure 2-6, you say, "Look, I'm indifferent between point A to point B. One slice of pizza-- I'm sorry-- one cookie and four slices of pizza is the same to me as two cookies and two slices of pizza." Why is it the same? Because they both give me utility square root of four, right? So given this mathematical-- I'm not saying you are. I'm saying, given this mathematical representation, OK, you are indifferent between point A and point B. So what that says-- and what's the slope with the indifference curve? What's the arc slope between point A and point B? The slope is negative 2. So your marginal rate of substitution is negative 2. You are indifferent, OK? You are indifferent between 1, 4 and 2, 2. Therefore, you're willing to substitute or give away two slices of pizza to get one cookie. Delta p delta c is negative 2, OK? Now, it turns out you can define the marginal rate of substitution over any segment of indifference curve, and what's interesting is it changes. It diminishes. Look what happens when we move from two pizzas and two cookies, from point B to point C. Now the marginal rate of substitution is only negative of 1/2. Now I'm only willing to give up one slice of pizza to get two cookies. What's happening? First, I was willing give up two slices of pizza to get one cookie. Now I'm only willing to give up one slice of pizza to get two cookies. What's happening? Yeah. AUDIENCE: You don't want a cookie as much? JONATHAN GRUBER: Because of? AUDIENCE: Diminishing marginal utility. JONATHAN GRUBER: Exactly. Diminishing margin utility has caused the marginal rate of substitution itself to diminish. For those who are really kind of better at math than I am, it turns out technically, mathematically, marginal utility isn't always diminishing. You can draw up cases. MRS is always diminishing. So you can think of marginal as always diminishing. It's fine for this class. When you get to higher level math and economics, you'll see marginal utility doesn't have to diminish. MRS has to diminish, OK? MRS is always diminishing. As you go along the indifference curve, that slope is always falling, OK? So basically, what we can right now is how the MRS relates to utility function. Our first sort of mind-blowing result is that the MRS is equal to the negative of the marginal utility of cookies over the marginal utility of pizza. That's our first key definition. It's equal to the negative of the marginal utility of the good on the x-axis over the marginal utility of the good on the y-axis, OK? Essentially, the marginal rate of substitution tells you how your relative marginal utilities evolve as you move down the indifference curve. When you start at point A, you have lots of pizza and not a lot of cookies. When you have lots of pizza, your marginal utility is small. Here's the key insight. This is the thing which, once again, it's a light bulb thing. If you get this, it'll make your life so much easier. Marginal utilities are negative functions of quantity. The more you have of a thing, the less you want the next unit of it. That's why, for example, cookies is now in the numerator and pizza is in the denominator, flipping from this side, OK? The more you have a good, the less you want it. So start at point A. You have lots of pizza and not a lot of cookies. You don't really want more pizza. You want more cookies. That means the denominator is small. The marginal utility of pizza is small. You don't really want it. But the marginal utility of cookies is high. You want many of them. So this is a big number. Now let's move to point B. Think about your next decision. Well, now, your marginal utility of pizza, if you were going to go from two to one slice of pizza, now pizza is worth a lot more than cookies. So now it gets smaller. So essentially, as you move along that indifference curve, because of this, you want-- because of diminishing marginal utility, it leads this issue of a diminishing marginal rate substitution, OK? So basically, as you move along the indifference curve, you're more and more willing to give up the good on the x-axis to get the good on the y-axis. As you move from the upper left to the lower right on that indifference map, figure 2-6, you're more you're more willing to give up the good on the x-axis to get the good on the y-axis. And what this implies is that indifference curves are-- indifference curves are convex to the origin. Indifference curves are convex to the origin. That's very important. OK, let's see, they are not concave. They're either convex or straight. Let's say they're not concave to the origin, to be technical. Indifference curves can be linear. We'll come to that. But they can't be concave to the origin. Why? Well, let's look at the next figure, the last figure, figure 2-7. What would happen if indifference curves were concave to the origin? Then that would say, moving from one pizza-- so now I've drawn a concave indifference curve. And with this indifference curve, moving from point A to point B leaves you indifferent. So you're happy to give up one slice of pizza to get one cookie. Starting with four slices of pizza and one cookie, you were happy to give up one slice of pizza to get one cookie. Now, starting from two and three, you're now willing to give up two slices of pizza to get one cookie. What does that violate? Why does that not make sense? Yeah. AUDIENCE: Law of diminishing marginal returns? JONATHAN GRUBER: Yeah, law of diminishing marginal utility. Here, you were happy to have one slice of pizza to get one cookie. Now you are willing to have two slices of pizza to get one cookie, even though you have less pizza and more cookies. That can't be right. As you have less pizza and more cookies, cookies-- pizza should become more valuable, not less valuable, and cookies should become less valuable, not more valuable. So a concave to the origin indifference curve would violate the principle of diminishing marginal utility and diminishing marginal rate of substitution, OK? Yeah. AUDIENCE: What if it's like something like trading cards? JONATHAN GRUBER: OK. AUDIENCE: I mean, I mean, as you get more trading cards, you have-- you're already made a complete set. JONATHAN GRUBER: That's very interesting. So in some sense, what that is saying is that your utility function is really over sets. You're saying your utility functions isn't over trading cards. It's over sets. So basically, that's what's sort of a bit-- you know, our models are flexible. One way is to say they're loose. Another way is to say they're flexible. But one of the challenges you'll face on this course is thinking about what is the decision set over which I'm writing my utility function? You're saying it's sets, not trading cards. So that's why it happens. Other questions? Good question. Yeah, at the back. AUDIENCE: What about like addictive things, where like, the more you have it, the more you want to buy? JONATHAN GRUBER: Yeah, that's a really relishing question. I spent a lot of my research life, actually-- I did a lot of research for a number of years on thinking about how you properly model addictive decisions like smoking. Addictive decisions like smoking, essentially, it really is that your utility function itself shifts as you get more addictive. It's not that your marginal utility-- the next cigarette is still worth less than the first cigarette. It's just that as you get more addicted, that first cigarette gets worth more and more to you. So when you wake up in the morning feeling crappy, that first cigarette still does more for you than the second cigarette. It's just, the next day you wake up feeling crappier, OK? So we model addiction as something where essentially, each day, cigarettes do less and less for you. You get essentially adjusted to new-- you habituate to higher levels. And this is why I do a lot of work-- you know, this is why, unfortunately, we saw last year, the number-- the highest number of deaths from accidental overdose in US history. 72,000 people died from drug overdoses last year, more than ever died in traffic accidents in our nation's history, OK? Why? Because people get habituated to certain levels, and they get habituated to certain levels. So people get hooked on Oxycontin. They get habituated to a certain level. They maybe switch to heroin, and they habituate to a certain level. And now there's this thing called fentanyl, which is a synthetic opioid brought over from China, which is incredibly powerful. And dealers are mixing the fentanyl in with the heroin. And the people shoot up, not realizing-- at their habituated level-- not realizing they have this dangerous substance, and they overdose and die. And that's because they've got habituated to high levels. They don't realize they're getting a different product. So it's not about not diminishing marginal utility. It's about different-- underlying different products. All right? Other questions? Sorry for that depressing note, but it's important to be thinking about that. That's why, once again, we're the dismal science. We have to think about these things. OK, now, let's come to a great example that I hope you've wondered about, and maybe you've already figured out in your life, but I hope you've at least stopped and wondered about, which is the prices of different sizes of goods, in a convenience store, say. OK, take Starbucks. You can get a tall iced coffee for 2.25, or the next size, whatever the hell they call it, bigger, OK? You can get, for 70 more cents-- so 2.25, and you can double it for 70 more cents. Or take McDonald's. A small drink is $1.22 at the local McDonald's, but for 50 more cents, you can double the size, OK? What's going on here? Why did they give you twice as much liquid, or if you go for ice cream, it's the same thing. Why do they give you twice as much for much less than twice as much money? What's going on? Yeah. AUDIENCE: Since your marginal utility is diminishing as you have more coffee available to you, you're willing to pay less for it, so they make the additional coffee cheaper. JONATHAN GRUBER: Exactly. That's a great way to explain it. The point is it's all about diminishing marginal utility. OK, when you come in to McDonald's on a hot day, you are desperate for that soda, but you're not as desperate have twice as much soda. You'd like it. You probably want to pay more for it, but you don't like it nearly as much as that first bit of soda. So those prices simply reflects the market's reaction to understanding diminishing marginal utility. Now, we haven't even talked about the supply side of the market yet. I'm not getting to how providers make decisions. That's a much deeper issue. I'm just saying that this is diminishing marginal utility in action, how it works in the market, and that's why you see this, OK? So basically, what you see is that that first bite of ice cream, for example, is worth more, and that's why the ice cream that's twice as big doesn't cost twice as much. Now, so basically, what this means is, if you think about our demand and supply model, on a hot day, or any day, the demand for the first 16 ounces is higher than the demand for the second 16 ounces. But the cost of producing 16 ounces is the same. So let's think about this. It's always risky when I try to draw a graph on the board, but let's bear with me. OK, so let's say we've got a simple supply and demand model. You have this supply function for soda, and let's assume it's roughly flat. OK, let's assume sort of the cost the firm proceeds within some range. The firm-- basically, every incremental 16 ounces costs them the same. So that's sort of their supply curve. And then you have some demand curve, OK? You have some demand curve which is downward sloping, OK, and they set some price. And this is the demand for 16 ounces. Now, what's the demand for the next 16 ounces, OK? Yeah, this isn't going to work. We have to have an upward-sloping supply curve. Sorry about that. We have a slightly upward sloping supply curve, OK? Now we have the demand for the next-- so here's your price. Here's your $1.22, OK? Now, you say, "Well, what's my demand when I sell 32 ounces?" Well, it turns out demand doesn't shift out twice as much. It just shifts out a little bit more. So you can only charge $1.72 for the next 16 ounces. Probably, if you want to go to the big-- if you go to 7-Eleven, where you can get sizes up to, you know, as big as your house, OK-- they keep these curves keep getting closer and closer to each other. So those price increments get smaller and smaller. And that's why you can get the monster, you know, ginormous Gulp at 7-Eleven-- is really just not that different from the price of getting the small little mini size, OK, because of diminishing marginal utility. All right, and so that's how the market-- that's essentially how we can take this abstract concept, this sort of crazy math, and turn it into literally what you see in the store you walk into, OK? Questions about that? Yeah. AUDIENCE: So how does this [? place ?] [INAUDIBLE],, like if for example, you wanted to buy a snack that you were going to have for breakfast every day-- JONATHAN GRUBER: Awesome. Awesome question. AUDIENCE: And then every single day, it was going to be your first granola bar, right? So I think that it's going to diminish every single time, but it's still cheaper to buy in bulk than it would be to buy a single granola bar every single time. JONATHAN GRUBER: Great, great question. Yeah? AUDIENCE: I think that has more to do with packaging cost than marginal utility. JONATHAN GRUBER: Well, I mean, the risk of my going to this model is, once we get nonlinear, the order we do things in this class, we have to start talking about supply factors I want to talk to. But there's two answers. One is packaging efficiencies. But the other is, if you actually go to Costco and look at their prices, for many things, they're not actually better than the supermarket. So actually, the price of buying the giant like, 8,000 bars of granola is actually not that much more-- not that much less than 1,000 time buying eight granola bars. It's less, but it's not nearly as much less of these examples as sodas in McDonald's, which is exactly your point. Utility diminishes less, so they don't want to charge as much less for multiple packages. So you can actually-- if you compare the gap in perishable product pricing by size, it's much larger than the gap in nonperishable pricing by size. Great point. Yeah. AUDIENCE: Is there also just like a different time frame to which the utility starts diminishing for every product? Because you gave the example of soda, but it's like, would that reset later in the day, if we wanted-- were thirsty again, or-- JONATHAN GRUBER: Awesome, and that is why they don't let you walk back in with the same cup and refill it, right? That's exactly right, and that comes to this point. It's sort of like it's nonperishable as you get longer apart. But you know, it's all just really interesting. So at Fenway, OK, you can get-- you get like a regular sized soda, it's like crazy. It's like $6. Then for like $8, you get a big soda. Then for $10, you get a refillable big soda, OK? Now, the question is, can you bring that refillable soda back to additional games? Technically not, but I do. [LAUGHING] And basically they sort of understand-- so this interesting question of sort of the perishability of things and how that's going to affect things going on. It's a really-- it's an interesting question. Other comments? OK, I'm going to stop there. Those are great comments. Thanks everyone for participating. And we will come back next time and talk about the sad reality that we haven't won the lottery, and we have limited amounts of money.

MIT_1401_Principles_of_Microeconomics_Fall_2018

19_International_Trade_Welfare_and_Policy.txt

[SQUEAKING][RUSTLING][CLICKING] JONATHAN GRUBER: All right. Let's get started. Today, we're going to continue our discussion of international trade. I want to finish up discussing comparative advantage, and then we'll talk about the welfare implications of international trade, and then how that drives our thinking about trade policy, one of the hottest topics in economics discussions today. So let's finish our discussion of comparative advantage that we started last time. Remember, the key lesson from the PPF-type analysis we had last time is that if there's comparative advantage, it can yield gains to trade through specialization. So that's sort of the key lesson, is that comparative advantage yields gains from trade through the mechanism of specialization. So basically, because of comparative advantage, you can specialize, and that means you can essentially get an outward-bending PPF. You can essentially get economies of scope by specializing and combining your activities. Now, you don't literally get economies of scope, because each person's doing their own thing. But it acts as if the economy as a whole is yielding economies of scope, because basically, by specializing, you're allowing the economy as a whole to benefit from people doing what they're best at. So that's the basic intuition of the economics of why international trade can yield gains. OK? Now, that raised the natural question of, well, where does comparative advantage come from? Why do some countries have comparative advantage? We [INAUDIBLE] from me versus LeBron James. He's got better genes and he's worked harder than I have. But basically, when it comes to countries, where does comparative advantage come from? And basically comes from, roughly speaking, two sources. The first source of comparative advantage is factor endowments. Some countries, just through the nature of geography and geological history, are endowed with things that people want, and therefore gives a comparative advantage in trading those things. So if you think about Canada, Canada is endowed with enormous amounts of forested land, and they've become a major exporter of lumber and paper products over time. That's their comparative advantage because they happen to have all this forested land around. So for example, that's why most clothing today comes from China and less developed countries. Because their factor endowment is cheap labor. It's not that they have more natural textile threads or whatever. It's just they have a cheap labor, which is the primary factor which goes into producing textiles. So one source of comparative advantage is factor endowments. Now, once again, some country could have more of everything. Canada could have cheap labor and lumber. But the key point is in the simple two-by-two world we're thinking of, it's all about relative, comparative advantage. So it's the one that have the most of relative to other countries. OK? And the second source of comparative advantage, the second reason why some countries acquire advantage is technology. So if you look at Japan and cars, Japan has no comparative advantage of producing cars. It's not like they have more of the raw products that make steel than, say, the US does. The difference is that Japan was a leader in developing the technology of the modern automobile. And that technological boost gave the comparative advantage. So factor endowments are kind of natural. Technology is a created source of comparative advantage that essentially, by developing the technology, you give yourself really sort of a first mover advantage, and are able to then have leadership, have the relative comparative advantage in producing that good. That's why, in some sense, technology policy really becomes trade policy. And we'll talk about that at the end of the lecture when we talk about trade policy in the US. OK. So that's really all. I just want to finish up on comparative advantage. The main thing's the intuition we developed last time, but just want to highlight also what competitive advantage means and where it comes from. What I really want to focus on today is the welfare implications of the international trade, and why economists are such big fans of international trade. So the welfare. Welfare and trade. And why do economists, in simple diagrammatic form, such strong proponents of free international trade? So to consider that, let's go back to our example of roses, and look at figure 19-1. We have here a simple representation of the domestic market for roses. You've got a supply curve and a demand curve. And you yield your consumer and producer surplus. So let's say this is for the US. You've got US supply of roses, US demand for roses, consumer and producer surplus. Now let's ask, what happens once we allow international trade? Once we allow for the importation or exportation of roses. Let's go to the next figure, which is pretty complicated, so let's walk through it. Now we look at the rose market with imports. Now, the presence of imports doesn't actually change the nature of demand and supply. So the domestic or US demand and supply curves are not shifted. What imports do is effectively increase the supply curve of roses available to US consumers by adding roses from other countries. So what we do is to make this work, what we basically-- the trick we use in these models is we essentially assume that in international markets, there's a perfectly elastic supply of the good. So what we say is international trade creates a perfectly elastic supply of rose at the price P sub W. That is, the price used to be P sub A for autarky. That's the price-- where domestic supply hit domestic demand. We're going to assume that what international trade does is lower the price to P sub W make supply perfectly elastic. That is, the US can get as many roses as it wants at a price P sub W. Now, for a small country, this sort of makes sense. For a small country, essentially, they're like a small producer in a perfectly competitive market. For the US, it might not make so much sense. The US is big enough that it's hard to imagine that we're price takers with respect to anything. We're the US, we don't take any prices. But in principle, the basic intuition we get from this model works, so we're just going to treat US as a small country for these purposes. And for something like roses, it's probably not bad to think of the US as a price taker in the rose market. So it's a bit of a cheat to say the US faces perfectly elastic supply in the world market, but it's not a bad cheat, and it doesn't hurt the important intuitions we'll derive. So we're going to assume the US faces a perfectly elastic world supply at price P sub W. Now, what does this do? What this does is, suddenly, at that lower price P sub W, consumers now want C sub T. They want a lot more roses, because the price has fallen, demand's downward sloping. But domestic producers want to produce a lot fewer roses, since marginal cost is upward sloping. If you're going to force the price down to P sub W, they don't want to produce as many roses. They only want to produce Q sub T. How do you resolve this? You resolve this through imports. That the difference between Q sub T and C sub T is imports. So the total amount of roses consumed does rise from Q sub A to C sub T. The total amount of roses consumed does rise. But the amount produced domestically falls. So both things are happening. Both US consumers are consuming more roses, and US producers are producing fewer roses. And the gap is made up by imports. Kind of a complicated diagram. Are there questions about this? Because I'm going to build on this diagram going forward. Any questions about how this works? OK. Now let's ask, what are the welfare implications? Let's go to figure 19-3. What are the welfare implications to international trade? Well, the welfare implications are, for consumers, their surplus used to be-- their surplus used to be W. They used to get Q sub A roses at a price P sub A. So their surplus used to be the area W. What is the surplus of domestic consumers? Let me-- an important point for welfare. We're going to look at domestic welfare only. We'll come back later to caring about the rest of the world, but for now, we only care about the US. So when I talk about welfare, unless I say otherwise, I think about US welfare only. US consumers, US producers. Very important to keep in mind, because otherwise this can get confusing. So for now we're talking about domestic welfare only. So for US consumers, what has happened? They used to get a surplus of W. What's their surplus now? Yeah. AUDIENCE: X, W, and Z. JONATHAN GRUBER: Yes. Z, by the way, is the entire light purple area. It's confusing, because there's this dotted line down the middle. Z's either side of the dotted line. The entire triangle is Z. So consumers now get W plus X plus Z. Why? Because they're consuming C sub T at a price P sub W. Don't let the other complications in the graph confuse you. Remember, consumer surplus is just the area below the demand curve, above the price, at the equilibrium quantity. The new equilibrium quantity consumed is C sub T. The new price is P sub W, so they get that giant triangle. Producers used to have a surplus of X plus Z. X plus Y, I'm sorry. Producers used to sell Q sub A at a price P sub A. So the area above the supply curve below the price was X plus Y. Now, X has been transferred to consumers. So producer surplus has fallen to Y. So what's happened? What's the opposite of the deadweight loss diagrams we saw before? When we did things like impose a tax, we transferred from one group to another, creating deadweight loss. International trade is the opposite effect. We've transferred from one group to another and created social gain for the US. US consumers have gained. And on net, US society is better off by the area Z. And this, in the terms of this course, is why economists like international trade. That expands the opportunity set. By allowing consumers to get these cheaper goods, you raise consumer surplus more than you lower producer surplus. And that's why we like international trade. It's a net win for society. Now, different parties might feel differently about it, and we'll come back to that. But let's just focus right now on total social surplus. And for total social surplus, which are defined as producer plus consumers, we are better off. And that's why we like international trade. OK? Question about that. Yeah. AUDIENCE: What is the new producer surplus? JONATHAN GRUBER: The new producer surplus is just Y. It used to be Y plus X. So producers are losing X. Consumers gain that X, but consumers also get Z. So it's sort of the flip side of deadweight loss we talked about before. You're transferring for producers and consumers, but you're also creating new gains for consumers along the way. OK? OK. So now, that's imports. What about exports? Let's flip and think about computers. Let's go to figure 19-4 and talk about computers. For computers, once again, we start at point A. We start with computers under autarky without trading. The price is where demand equals supply, a P sub A, and the quantity's Q sub A. And that's the initial equilibrium for computers. Now we open up to trade. Well, what happens now, is now, the rest of the world wants US computers, so the US can sell at a higher price. Before, the US could only sell to domestic consumers. But now, their demand for their goods worldwide-- we don't show world demand here. But world demand is way above US demand. If you flip back to 19-3, the reason that that P sub W line is below the P sub A is that world supply-- or 19-2 is easier-- world supply is above US supply. Since supply is higher, the price falls. Now, in 19-4, world demand is higher than US demand, so the price rises. So we end up with a P sub W at the higher level. What does this mean? With a higher P sub W, domestic consumers want fewer computers. They used to be able to buy computers at P sub A. Now they have to pay P sub W, because they're competing with consumers around the world. So they have to pay more. So they want fewer computers. Their demand for computers falls from QA to CT. What about domestic producers? Well, now they're getting a higher price, so they want to make more computers. So their production rises from QA to QT. So you now have consumers consuming much less than producers are making. So the producers send the rest abroad, and that's exports. So we have the flip side. Before, we had-- in figure 19-2, if you flip back and forth, in figure 19-2, we had consumers wanting more than producers domestically willing to make. So you had imports. Now we have domestic consumers wanting less than domestic producers are willing to make, so you have exports. What are the welfare implications? Go to 19-5. Now what you get is consumers are worse off. They used to have a consumer surplus of W plus X, where X is the entire dotted area. Now, the consumer surplus has fallen to just W. Remember, we're doing domestic consumers only. There's some people in Colombia who got computers. They're happy, we're ignoring them. We're just doing US consumers. They used to get W plus X. Now they only get W. Below the demand curve, above the new price. But producers, who used to get Y, now get Y plus X plus Z. So you've transferred X from domestic consumers, domestic producers, but you've also given domestic producers this extra bit Z. So once again, surplus has gone up. So this is the crazy thing about trade. Imports raise welfare, and exports raise welfare. Either way, whether the goods are coming in or the goods are going out, society as a whole is better off. Why? Because of comparative advantage. Because we're better off as a world where we can share our goods across nations, because we can then rely in the more efficient producers lowering the prices for roses in places like the US, and raise the-- lowering the prices for goods US consumes, and raising the price for goods US produces. And same with every other country. It's literally a win-win. Every country, by the need to specialize, gets to see lower prices in the goods they weren't good at making and higher prices at the goods they are good at making. So this transfers across producers and consumers, but overall, welfare goes up. So basically, this is the bottom line. These two graphs are why economists unambiguously-- traditionally-- traditionally, economists unambiguously like free trade. So you get into subtleties, like caring about producers versus consumers separately. Basically, the logic of last time, which is a comparative advantage allows specialization, means that by trading, we can allow countries to specialize, and thereby getting higher prices for things they send abroad and lower prices of the things they bring in from abroad. OK? Questions about that. And we can see from this why the notion of a trade deficit as something that we should care about is sort of ridiculous. Because it's all just about your endowments. At the end of the day, you want to sell stuff that you have a comparative advantage in and import stuff you don't have a comparative advantage in. If the rest of the world has more compared advantage than you, then you want to import more than you export. Doesn't mean you're worse off, it just means your consumers are benefiting from the fact the rest of the world is good at making stuff like sweaters. So that segues us naturally to our important policy discussion, and one of the main focuses of public policy today, which is trade policy. How do we take this framework and apply it to thinking about government policy over international trade? And that's what I'll spend the rest of today talking about. So let me just ask just before we go there, any questions about the economics here, before we start talking about policy? OK. Now, let's start once again with trade policy with the standard economics view. Which is that you'll often hear people say, well, imports are a job killer. And in some sense, they're right. That's what figure 19-3 is showing. Producers are the ones with the jobs. Producer surplus is falling, so there's less profitability in the corporate sector, so there may be less jobs there. Now, many people say we should react to that by imposing restrictions on imports. They say, look, exports are great, but what about imports that are killing jobs? Why don't we restrict those? And there's two fundamental forms of restrictions on imports. There's quotas, which is literally a limit on how much of a certain good you can import. Those aren't used so much anymore. The more typical form of import deterrents is tariffs. Is tariffs. OK? And basically, what these are are taxes that are levied only on imports. A tariff is just a name for a tax that only applies to imports. That's what a tariff is. It's a certain kind of tax that only applies to imports. So we could, for example, levy a tariff on roses coming in from Colombia. If we levy that tariff, making roses expensive enough, then folks might go back to buying it from the US producers, and we can reopen those greenhouses and the jobs that come with them. That is true. What that misses the fact that US consumers then suffer from paying higher prices for roses. And on net, we're worse off as a result. To see that, let's go to figure 19-6. So basically, what figure 19-6 does is we now start from the position of being in international trading. So before, the other figures, we started from autarky and added trade. Now I want to start from position with trade and add a tariff. So initial equilibrium is at P sub W, with consumers consuming C1, producers producing Q1, and initial level of imports denoted by imports before tariff. That's where we start. What the tariff does is essentially raise the price back up. Now, I can't raise the price above the domestic price, because if the tariff's big enough, you just go back to buying what you want for domestic producers. But it raises it back towards the domestic price. So let's say, for example, the tariff is high enough that the international-- that the world price-- the price paid in the US rises from PW to PT. The gap being the amount of the tax. So with untaxed trade, the US would pay PW for roses. Now, when we tax trade, that price goes up to P sub T. OK? So now, let's ask, what does that do to the market? Well, it's pretty straightforward. You just say, well, what is demand and supply at that new price? Well, at that new price, there's lower demand for roses, so rose demand falls from C1 to C2. There's more domestic production of roses, so domestic production rises from Q1 to Q2. And the imports after the tariff shrink massively. The tariff had its desired effect. It shrunk imports. But what does this do to welfare? Let's go to figure 19-7. Now, we can just compare this area between the two horizontal lines. We used to be at the bottom horizontal line. Now we're at the higher one. A, the area A is the new producer surplus. Producers used to get the area-- that low triangle below area A. We used to call it Y. Now they've added A. Producers have now added a new producer surplus A. But consumers have lost A, B, C, and D. That entire area is lost to consumers. Yeah. AUDIENCE: Is this [INAUDIBLE] the comparison to when there was-- JONATHAN GRUBER: Yeah, that's what I said. Comparison-- when there was free trade, no tariff. So it's a comparison to the second situation in figure 19-3. So 19-3-- and this is confusing, I'm sorry about that, but it's too hard to put it all on one graph. 19-3 is what happens when we move from no trade to free trade. 19-7, 6 and 7, I'm going from free trade to free trade with a tariff. By the way, if the tariff was-- brought us exactly back to the no-trade situation, it'd just be the flip of the previous diagrams. Just makes it a little more interesting by making the tariff somewhat lower than that. So what's happened to consumer surplus? Well, look at this way. Remember consumer surplus. It's the area below the demand curve, above the price. So it used to be a huge triangle. Now what's happened is fallen by the entire trapezoid, ABCD. Can folks see that? Look at the new consumer. Ignore the fact that it's four different areas. Just think of it as one trapezoid. The new consumer surplus is the area under the main curve above PT. The old consumer surplus is the area under the demand curve above PW. That's followed by the trapezoid ABCD. Now, why do we split that into four pieces? Well, first of all, because A is gained by producers, so we want to call that out. That's not a net loss of welfare. There's also the area C, the green area C. C is also a gain. What is C? That's the government revenue from the tariffs. The government's gained something from the-- we get tax revenue. We can give that back to consumers if we want, so that's not lost surplus. The tax revenue's exactly the amount of new imports times the tariff. So it's complicated. Now we have this third player which is the government, It used to be just either producer or consumer, or society. Now we have this third player, which is the government. So the deadweight loss from the tariff is B plus D. What we've lost is B plus D. So let me go through the math again. Consumers lost A plus B plus C plus D. Producers gained A, and the government gained C. So the net loss on the tariff is B plus D. That's sort of hard, but do people see that? And that's why economists don't like tariffs. Because the amount producers gain, plus the amount the government raises, is much less than what consumers lose. Yeah, we've produced some jobs growing roses. Yeah, we've gotten a little tax revenue. But we've really screwed producers who now have to pay much higher prices for their roses. And if you add that up, it's worse. Why is it worse? Because you're taking less advantage of specialization in comparative advantage by forcing us away from the most efficient point. The most efficient point's where we can take-- everyone can specialize. You force that away. You've put the US, which shouldn't be growing roses, back in business of growing roses. And Colombia, which should be growing roses, gets out of that business. Yeah. AUDIENCE: The tariffs-- where are tariffs applied? Are they on US-- I guess importers receive goods, or are they on [INAUDIBLE] exporters [INAUDIBLE].. JONATHAN GRUBER: Yeah. So that's a great question. I am not expert on the actual logistics, but roughly speaking, when it comes into the US, into Customs-- so any goods shipped in the US for sale into Customs, at that point, there is a tax levied on it. Now, I don't know logistically who gets the bill. Does the exporting company get the bill, or the importing company get the bill? When we talk about taxes, I'll show you it doesn't matter who gets the bill. It's the same either way. So hold that thought. But that's basically what happens. Good question. OK. So with that in mind, so that is the fundamental reason, in graphical terms, why economists don't like tariffs. But that's not the only reason. In fact, there's two other reasons that aren't even in this graph why tariffs and restrictions on trade policy more generally are bad. The other reason is that they cause trade wars. So let's say we impose this tariffs for Colombian roses. Well, Colombia will be like, screw you. If you're going to tax our roses, we're taxing your computers. Well, what would happen if there was a tariff that Colombia placed on our computers? Well, I don't have that figure, but you should be able to show yourself that that is exactly the opposite effect. It raises US consumer surplus, because now computers are cheaper in the US because we can't sell them in Colombia. But it lowers our welfare, and now we don't get the government revenues. Colombia does. So we lose almost all the trapezoid in that case. If you flip this around, we would get a small rise in consumer surplus but a huge loss of producer surplus, and we wouldn't even get the government revenues to make it up. So that's really bad for us. And of course, it's a natural response. Why wouldn't Columbia do it? So this first problem is, this understates, yes, we might create some jobs in roses, but we're going to destroy jobs in computers, because we're not going to sell as many computers as we used to. So it's not even job-creating. Once you take in the fact that other countries retaliate, in fact, we make consumers worse off and producers worse off. Because we make the price for roses much higher, and we lose jobs in the computer sector. Yeah. AUDIENCE: Is there a way to diplomatically eliminate this if you're like a world monopoly? I feel like a world monopoly-- JONATHAN GRUBER: So basically, one way to do it is to come to trade deals, where you essentially cartelize countries. So this is much-- if you could think of this in very parallel ways to a non-cooperative oligopoly, if you can cooperate, what do you do? You just get to free trade. And we'll talk about trade deals in a few minutes. That's essentially what those are doing. Yeah. AUDIENCE: Why are current economic advisors in the government [INAUDIBLE] tariffs [INAUDIBLE]?? JONATHAN GRUBER: So let me get to that. OK. So basically, what we've done-- and let me answer the first question. What we've done is, because of the archetype made here, over time, we've created essentially co-operative oligopolies around the world. The one you've heard of a lot is-- OK. So I'm sorry, trade wars is the first thing. Let me [INAUDIBLE] the second reason. The second reason is that, actually, as decent human beings, we might care about people in other countries too. And the fact is that there's-- when you-- these both import restrictions and trade wars hurt other countries just like they hurt us. It's not only we worse off, but other countries are worse off. Even without the trade war, we've made ourselves worse off, and we've hurt Colombia, because they can't sell the roses to the US. So the reason we don't like-- so the other reason besides this figure is that basically, if we place any weight at all on the utility of people outside the US, which we should, it's bad as well. Now, I'm not saying you have to weigh the Colombians' welfare the same as US welfare, but as long as it's nonzero, then that's a second reason to oppose it. And that's why there's been a huge growth in agreements, the most famous of which we've heard of is NAFTA, the North American Free Trade Agreement, signed in the early 1990s under President Clinton, which essentially set up a cooperative oligopoly between the US, Mexico, and Canada. Essentially, let's just get rid of these trade barriers so we can have freer trade within our regimes. And basically-- but this was actually you know quite popular at the time, economists liked it. But it's become very unpopular over time. Was opposed by President Trump, and in fact, he recently ripped up NAFTA, although replaced with something that's pretty much the same, just renamed it as USMCA. So why are people so opposed to free trade? It's a great question. Why are Trump's advisors-- why in fact are the majority-- the majority of Americans, depending on how you ask the question, don't like free trade. It depends how you ask the question. But it's certainly not universally popular, like I said it should be. So what's going on? Well, there's really a couple things going on. The first is that we can't-- we don't. I wouldn't say "can't." We don't compensate the losers. This is the most important part. We don't compensate the losers. What do I mean by that? Let's go back to figure 19-3. What's happened when we've put international trade? Producers have lost X. Consumers have gained X and Z. So it's a simple economic policy. If we did international trade, and then just took X and gave it back to producers, no one would be sadder. Even if we give a little more than X. What if we did international trade and gave producers X plus 20% of Z? Then everybody wins. That's the [INAUDIBLE]. The problem is we don't do that. All we do is let consumers have these super cheap sweaters. So I get to go buy this incredibly cheap sweater, and people in North Carolina's livelihoods are destroyed. We don't compensate the losers. And the winners don't notice they're paying $5 or $10 less for a sweater, but the losers notice they don't have a job. So the problem is, yes, there's more winning. But first of all, it's winning among many more people by a small amount, whereas losing is by fewer people by a large amount, and that always gets more political attention. And we don't have mechanisms in place, or don't put in-- I started "I can't," but it's not "can't." We could. We could easily address this. There'd be a simple policy. What if we simply said, there'll be a tax on all consumers of international-- not just a tax on consumers, not of international goods-- that would be like a tariff-- but just a general tax on consumers of consumer goods that tend to come in international trade. So it's a tax on clothes. And we'll take all that money, and we'll redistribute it to people who lost their jobs in the textile sector, help retrain them for new jobs, or help pay their bills while they find a new job. Then we could literally deliver some of X and even maybe some of Z to the producers. But we don't do that. And that's the main reason, to answer your question, why does opposition to free trade? Is because people see the cost, they don't see the benefits. And well-off people like me who don't need-- who could happily afford to pay twice as much for a sweater, we just get these benefits we don't even pay attention to, whereas typically lower-income workers, because they compete in lower-income countries, lose their jobs and they notice it. So that's one big reason. The other big reason is that there can be socially-damaging routes to comparative advantage. There can be socially-- that sounds like socially-damaging routes to California. Socially-damaging routes to comparative advantage. That is, there can be ways that countries get to comparative advantage that are not so happy. So why does China have low labor costs? Partly because they got a billion and a half people to work. But partly because workers are massively exploited there. Work conditions are terrible. And it's a horrible life being a worker in a Chinese factory. Moreover, there's terrible environmental conditions imposed by Chinese production. In the US, we have restrictions that try to minimize, in some ways, the environmental damage done by our production. They don't have those in China. Or India. India is home to three of the top five polluted cities in the world. They don't have those. So by creating their comparative advantage, they may be doing damage that we don't like. Just like we care about the fact that consumers-- that we care about other countries' welfare, we might also care about the fact that this free trade is hurting other countries' welfare. So a great example-- I've got a great example. A very relevant example was a recent story about lead poisoning in China. There was this battery factory-- batteries are big business now-- that made lead acid batter for motorcycles and electric bikes. And basically, they operated in flagrant violation of environmental law. They would just dump the mercury and stuff they were using in the rivers and lakes all around. Flagrant, everybody knew it. And 233 adults and 99 children were found to have lead in their blood up to seven times the safe amount. And they're basically are going to be-- lives will be destroyed by this. And this is a big issue in the negotiation of NAFTA, and a big reason a lot of people on the left don't like international trade, is because they fear for the welfare of people who are being affected. So when NAFTA was negotiated in the early '90s, there were some protections for workers. There were some protections. Basically, the idea was, Mexico, if you want to sign this, you got to put in some protections to raise the standard of living of your worker and improve your environment. So once again, there are ways to deal with this by saying, look, free trade, where you impose some restrictions, is still better than no free trade. So we might go in and say, look, once again, it's all about how you spend Z. You got a bunch of money to spend. How are you going to spend it? Some of the way you can spend it is by compensating the producers. Another way you can spend it is by saying, look, we'll be willing to make it all smaller, have the world price be a little bit higher, but make sure you have decent environmental conditions and decent wages for your workers. As long as we don't impose too high restrictions, we could still all gain. So that's another reason people don't like international trade. On the other hand, we taught the story of the child workers in Vietnam, and how free trade in rice was better for them, because the parents got richer and had them work less and put them in school. So it's not clear which way this goes, but that's certainly another concern. And then finally, there's the issue which is, I think, paramount in this administration, which is trade policy-- trade policy as a tool of foreign policy. Trade policy as a tool of foreign policy. Look, consider where we are with China today. There's a host of reasons to be angry with China. Now, some are irrational. The Trump's administration obsession with trade deficits, as I've explained repeatedly, is irrational. It's the Pikachu fallacy. But some of it's rational. China-- we operate under something called the World Trade Organization. Once again, coming to the other question, it's another chance to try to cartelize the country and set a-- cartelize the world and set a set of fair trading rules. The World Trade Organization. China repeatedly violates the rules set up by the World Trade Organization. For example, under those rules, China should allow much freer sales of US goods in China, but they implicitly restrict them. Not on paper, but in practice. They set up lots of practices which makes it harder to sell US goods. They also do a lot of significant industrial espionage and stealing our industrial secrets. So one way to do that, for example, is China has a rule that you cannot have a solely-owned subsidiary in China. If you want a subsidiary in China, it has to be jointly owned with a Chinese company. Which then will promptly take the ideas, and pass them on, and kick you out of the country. So China's engaged in some nasty practices which are bad for the US-- which are certainly bad. They may be good for China. They're certainly bad for US self-interests. And that's true. That is unambiguously true. The question is, what's the right response? And the Trump administration argues the right response is tariffs. The right response is to punish China through tariffs for their bad behavior. And economists-- virtually all economists-- say that's the wrong response. There is literally less than 1% of economists who agree with this policy. Just a few of them happen to work with the Trump administration. But basically, at the end of the day, basically virtually all economists agree that it's a bad idea. And that's for three reasons. The first reason is our standard reason that we think trade is good. We think we're hurting ourselves by imposing tariffs on Chinese goods. All we're doing is hurting ourselves. We're making people pay more for stuff. Why would we want to do that? Moreover, we're causing a trade war. China's now imposing tariffs on our goods, which hurts our producers. So our farmers, for example, are going to suffer because it'll be harder to sell our farm goods in China because of the new tariffs they've imposed. So the standard arguments we have here still hold. We like free trade because it expands opportunities, and if we try to limit it, other countries will respond in a way which further restrict opportunities. So that's the standard reason economists are mad about this. The second reason is it's sort of hard to do this. It's hard to use trade police [INAUDIBLE] tool of foreign policy because it's not even clear what's made in China and what's made in America anymore. So if we build the car in America, but the parts are made in China, is it an American car or a Chinese car? Oftentimes, we'll have things which are shipped to China, back to America, then back to China, or vice versa. So it's not even clear what's a Chinese import or export and what's US import or export. These are blurry lines now. Now, once again, that's all because of the efficiency of production. Basically, what we've done over time is we've sort of disaggregated production to more and more efficient components. It used to be one guy made the whole car. Now we recognize there are certain parts that are made more efficiently in China, certain parts made more efficiently in Cambodia, more efficiently in the US, more efficiently in Germany, and we put them together. What that does is it makes international trade super messy to understand. But it makes the world better off. It's just further increasing comparative advantage and specialization. But it makes it hard to decide what's a Chinese good and what's a US good. So that's the second problem. And the third and probably the single most important criticism economists make is, this is a really silly thing to do by yourself. The US is big, but we still represent a minority of Chinese exports. We obviously [INAUDIBLE] a large share, but still a minority. As long as China can still sell to other people, and they're making plenty of money off their practices against us, they might not stop. Now, they prefer to sell to us. But if you think on the one hand, they're making a ton of money ripping us off. On the other hand, they'll still get to sell 70%, 80% as much to the rest of the world once they adjust things. Then maybe they'll be like, fine. We're happy to let you do that. You're just cutting off your nose to spite your face. But if we get the whole world to coordinate and say, you violated the norms of the World Trade Organization, we're now all going to set up trade restrictions on China, then they feel the pain. So the other reason economists oppose this is moving unilaterally on trade just doesn't make sense. If you want to try to deal with these problems, you need to through coordinated response. Now, let me be very clear. This is a very, very hard issue. Economists try to make it very simple with our surplus diagrams and stuff like that, but it's a super hard issue. Nonetheless, it's an issue where your bias in thinking about it should be towards the basic bias of economics, which is that if we can expand opportunities, that's a good thing. And the real challenge is twofold. How do we expand opportunity sets in a fair way? And two, how do we deal with compensating the losers as opportunity sets advance? And this is something we'll spend a lot of time on after Thanksgiving, is how we redistribute society from one group to another. OK. Now let me get some questions. I know, this is sort of a top [INAUDIBLE]---- yeah. AUDIENCE: So on an international level, when a country behaves badly, sometimes the UN will suggest that sanctions are imposed against the country. And I was wondering if you know anything about how that tends to work in terms of welfare? Like, do the sanctions tend to make a country behave better, and then ultimately the world is better off, or are those also a bad idea? JONATHAN GRUBER: That's a really interesting question. So let's think about sanctions in this framework. So sanctions-- let's say one sanction would be-- it depends on the form of the sanctions. It depends-- et cetera. So one sanction would be, literally, you have to pay a bunch of money. Well, then, that's just like a tax on the country. We just think about that, think of the country individual. They're worse off. The UN gets some money. In some sense, it depends on whether the sanction works or not. So the easy case is just simply the trade-off of, do you encourage a behavior you want to encourage, versus what pain do you impose on the country, and how do you trade those two things off? If they're a really bad-acting country, you might not carry imposing pain on them. As long as there's some chance you get the change in behavior you want, you're happy. But what if the sanction's limiting trade? Well, then you're hurting other countries too. Then it becomes a bit trickier. So then once again, the trade-off is three-piece. I hurt the country I want to hurt. But I hurt countries I don't want to hurt. But I also may get a change in behavior I want. So you have to weigh-- you have to add those three pieces and put them together. How do you put them together? That's exactly what we'll talk about after Thanksgiving. You put them together by thinking about a weighting function, which we call social welfare function, which weights the well-being of all these different parties and puts them together. A very related issue we haven't talked-- yeah, I'm sorry, go ahead. AUDIENCE: So do you think that the current administration is putting in these policies to try and gain popularity because so many people understand trade? JONATHAN GRUBER: You know, I think there's almost no one who makes any money by trying to understand what's in the mind of the current administration. So I have no idea. I think China is engaging in bad practices, but I also think it's good politics, and it's hard to know what's the mix. But let's actually relate-- [INAUDIBLE] take the current administration, this raises another issue that's related to all this which we haven't talked about, which is what about immigration? Another-- I'm going to stop before I get to abortion. What about immigration? Well, immigration, actually, the framework is very much the same. If there are people who can contribute to our society by coming here, then we are better off by letting them in by the same logic of specialization and comparative advantage. If there are people-- for example, let's say US folks don't want to pick our crops, and we don't want to clean toilets. But picking crops and cleaning toilets at a low wage is a terrific opportunity for someone from another country. Then we could be better off by letting them in. Or let's say-- as it's turned out to be true historically-- the kind of person who wants take the risk to immigrate to America is the kind of person who often becomes an entrepreneur and thinks up new ideas. Much of the entrepreneurial ideas in America have come from our nation's history of immigrants. Because who picks up and leaves? It's kind of a risky thing to do. To pick up, leave your home, and leave, and come to this unknown place. And those risk-loving people are often the kind of people who think up new ideas and want to start new businesses. So for that reason, immigration has traditionally been an enormous benefit to the US economy. And I think today, the general consensus-- this is more controversial than free trade. But I think the general consensus among economists is that immigration on net is good for the country, but there's redistributional consequences. It's good for rich people because they get low-pay people to pick their fruits and do their lawn work. But it's bad for people who used to pick the fruits and do the lawn work. So once again, if we can figure out a way to compensate the losers, immigration is probably [INAUDIBLE] good. Now, it's more complicated. When we import a car, it doesn't get welfare or commit crimes. Immigrants might get on welfare or commit crimes, so it's more complicated than just free trade. So what you have to do in that case is you have to look at the evidence. And the evidence is that immigrants commit crime at a much lower rate than comparable US citizens and collect welfare at a much lower rate than comparable US citizens. So you can't compare an immigrant to me. An immigrant is more likely to commit crime or collect welfare than I am, but immigrant's also less educated than I am. I'm not the comparable person. Compare an immigrant to the person they're replacing in the labor market. A low-skilled US citizen. Immigrants are less likely to collect welfare and less likely to commit crimes than are those people. So as a result, those arguments don't really bode well for restricting immigration. On the other hand, I think there's very few people who say we should have a totally porous and open border. Because of illegal movement of goods, and because we don't want people objectively criminal in other countries that are also risk-loving, we don't necessarily want them here. So I think this is a hard issue. Unlike international trade, where I think economists would say just more is good, I think that immigration, there are some difficult trade-offs, because people come with a set of baggage that goods don't come with. But I think economists would generally say a lot of the same arguments apply. We might need a more comprehensive policy than we have when it comes to trading goods.

MIT_1401_Principles_of_Microeconomics_Fall_2018

1_Introduction_and_Supply_Demand.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: This is 14.01. I'm John Gruber, and this is microeconomics. Today, I want to cover three things. I want to talk about the course details. I want to talk about what is microeconomics. And then I'll start the substance of the course by talking about supply and demand. Couple of the points about the course-- the course will have a distinct sort of policy angle to it. I sort of do economic policy, government policy is my thing. So I think it's what makes economics exciting and it sort of offers, I think, an interesting angle to understand why we're learning what we're learning. I think sometimes in an intro class, it's sort of hard to understand why the heck you're doing things. However, that's just sort of a slight flavor. If you're really more interested in this, I teach a whole course called 1441. I'm not teaching it this year, but it will be taught by a visitor in the spring, Kristin Butcher from Wellesley. And I'll be teaching next year. That dives much more into these policy issues. So I'm going to use government policy as sort of an organizing theme, but it won't be the dominant theme of the class. Finally, three points about my teaching style. I don't write everything on the board. We're not in high school anymore. You're actually responsible for what I say, not what I write. Partly that's because my handwriting is brutal, as you can tell already. So what that means is, please, please do not be afraid to ask me what the hell I just wrote on the board. There's no shame in that. Don't just lean to your neighbors, and say, what the hell did he just write in the board. Ask, me, because if you can't read it, I'm sure someone else can't read it, so feel free to ask. And in general, please feel free to engage with questions in this class. The other point of my teaching style is I talk way too fast. And the longer I go-- there's a mathematical function, which is the longer I go without interruption, the faster I speak, until I just spin off. So basically, please ask questions. If anything is not clear, or you just want to ask questions about some related tangent or whatever, please feel free to do so. You might think, how would that work in a class this big? There's always way too few questions, even a class this big. So never be afraid that it will slow me down or whatever. Ask me questions. We have plenty of time on the class. And you'll be doing your classmates a favor, because it'll slow me down. Finally, last point, I have this terrible tendency to use the term "guys" in a gender neutral way. So this class, I like to see, looks like it's a fairly healthy representation both males and females. When I say "guys," I don't mean men. I mean people. I mean people. So women, don't take it personally. "Guys" means economic agent. It means people. It doesn't mean men. Just the way-- just a bad tendency. It drives my wife crazy, but I've decided better to just apologize up front than try to fix it throughout, which is impossible. So let's talk about what is microeconomics. So fundamentally, microeconomics-- how people took AP high school Econ? How many people-- for how many people was it taught really well? That's about right. That's why I did my high school online class. That's the answer I wanted to hear. So tell your friends still in high school who are taking high school Econ, if your high school teacher isn't great, tell them to go on EdX and take the class. And help out your friends still in high school. So what is microeconomics? Microeconomics is the study of how individuals and firms make decisions in a world of scarcity. Scarcity is what drives microeconomics. Basically, what microeconomics is is a series of constrained optimization exercises, where economic agents, be they firms or individuals, try to make themselves as well off as possible given their constraints. Yeah. AUDIENCE: Will this cover irrationality? JONATHAN GRUBER: I will, but not as much as I should. Essentially, we have another course in the department called 1413, Behavioral Economics, which gets into that much more. I will sprinkle it throughout, but not as much as I actually believe in it. In other words, the way we think about economics is it's best to sort of get the basics down before you start worrying about the deviations. Find it's better to climb the tree before you start going out in the branches. So basically, what this course is then about is it's about trade-offs. It's about given that you're constrained, how do you trade off things to make yourself as well off as possible? And behind this notion of trade-offs is going to be-- I'll say about 100 times this is the most important thing in the course, so just ignore that. But this is one of the most important things. I'll say "one of the most important" things in the course, is the notion of opportunity cost. Opportunity cost is a very important concept that we teach, sort of the first concept we teach, which is that every action or every inaction has a cost in that you could've been doing something else instead. So if you buy a shirt, you could have bought pants. If you stayed at home and watched TV, you could have been out working. Everything you do has a next best alternative you could have done instead. And that is called the "opportunity cost." And that's a critical concept in economics, and that is why, in some sense, we are referred to casually as the "dismal science." Economics is referred to as the dismal science. First of all, I'm flattered we're considered a science. But it's called the "dismal science" because our whole point is that nothing is free. There is always a trade-off. There's always an opportunity cost. Anything you do, you could be doing something else instead. And your constrained optimization means you're going to have to pass up one thing to do another. Now, some may call it "dismal," but as a former MIT undergraduate, I call it "fun." And this is why I think MIT is the perfect place to be teaching economics, because MIT engineering is all about constrained optimization. That's what engineering is. And economics is just the engine. It's just the principles you learn in engineering applied in different contexts. So if we think about the 2.007 contests-- that still exist with the robots, 2.007? Yeah, the 2.007 contests, those, as you know, are contests where you're given a limited set of materials. And you have to build a robot that does some task, like pushing ping-pong balls off a table or something like that. That's just constraint optimization. It's got nothing to do with economics, but it's constrained optimization. So just think of microeconomics as like engineering, but actually interesting. So think of microeconomics as engineering, but instead of building something to push a ping-pong ball off tables, you actually build people's lives, and businesses, and understand the decisions that drive our economy. So same principles you could think of for your engineering classes, but applied to people's lives. And that's why, in fact, modern economics was born in this room, this room or 26.100 by Paul Samuelson in the 1940s and '50s, who wrote the fundamental textbook that gave birth to modern economics. Because he was here and applied the kind of engineering principles of MIT to actually develop the field of modern economics. What we'll learn today was developed at MIT, so it's a great place to be learning it. Now, with that as background-- any questions about that, about what is microeconomics? With that as background, let's turn to our first model we'll talk about this semester, which is the supply and demand model. Supply and demand-- now, the way we're going to proceed in this course is going to drive you crazy, because we're going to proceed by teaching, as the very first question pointed out, by teaching very simplified models. We're going to essentially-- what is a model? A model is technically a description between any two or more economic variables or any two or more variables. But unlike the models used in all your other classes, these aren't laws, by and large, they're models. So we don't have a relation between energy and mass which you can write down. It's a law and you're done. We have models which are never 100% true, but always pretty true, "pretty" being somewhere between 10% and 95% true. So basically, the idea is to make a trade-off. We want to write down in our models a set of simplifying assumptions that allow us, with a relatively small set of steps, to capture relatively broad phenomena. So it's essentially a trade-off. On the one hand, we'd like a model that captures as well as possible the phenomena in the real world, like E equals Mc squared. But we want to do so in the most tractable possible way so that we can teach it from first principles, and don't need an arrow to teach every single insight we have. So basically in economics, we tend to resolve that by erring on the side of tractability. That is why I can teach you the entire field of microeconomics-- which is really sort of-- macro is kind of a fun application. Micro is really economics. I can teach you the entire field of microeconomics in the semester, because I'm going to make a whole huge set of simplifying assumptions to make things tractable. But the key thing is that you will be amazed at what these models will be able to do. With a fairly simple set of models, we will be able to offer insights and explain a whole huge variety of phenomena, never perfectly, but always pretty well, generally pretty well. And so that is essentially the trade-off we're going to try to do this semester. So the line I like is the statistician George Box said that all models are wrong, but some are useful. Now obviously, it doesn't apply to models in the hard sciences, but in the social sciences, that's true. And basically, I'm going to write down a set of models like that. Now, with every model I write down, I'm going to try-- my goal is to have you understand it at three levels. The first and most important level is the intuitive level, the level which you sort of understand. I call it "passing the Mom Test." You can go home and explain it to your mom at Thanksgiving or at the end of semester. No offense to dads, just called it "the Mom Test." So basically, that's the intuitive level. You really understand it in a way that you could explain it. The second is graphical. We were going to do-- most of our models here were developed in a graphical framework using x/y graphs that really in economics, we think delivers a lot of shorthand power. And the third is mathematical. The mathematical is probably the least important, but it's the easiest to test you on. So we're going to need to know things mathematically as well. So let's start by considering the supply and demand model by using the famous example brought up by Adam Smith. Adam Smith is sort of considered the father of economics. If Paul Samuelson is the father of modern economics, Adam Smith is the father of all economics. His 1776 book, The Wealth of Nations did an incredible job of actually laying out the entire core of the economics field-- no math, just words, but he just nailed it. And one of his most famous examples was the water diamond paradox. He said, think about water and diamonds. He said, start with water. Nothing is more important for life than water. It's the building block of all of life. Even when we look for life on other planets, we always start by looking for water. Now think of diamonds, one of the more frivolous things you can buy, certainly irrelevant to leading a successful or happy or productive life, or any life. Yet for most of us, water's free and diamonds are super expensive. How can this be, Adam Smith asked. Well, the answer he posed is that what I first described was just demand. That is, we demand lots of water. We demand fewer diamonds. But we have to match that with the concept of supply. And the supply of water is almost infinite, while the supply of diamonds-- maybe not naturally, maybe it's through decisions of various businesses-- but it's somewhat limited. So basically what he developed is what we call the "supply and demand scissors"-- that you can't just think of supply or demand in isolation. You have to put them together if you want to explain the real world phenomena we see, like the fact that water is cheap and diamonds are expensive. So let's just about an example. So there's one graph that was handed out in the back, which is, let's talk about the market for roses. So in the market for roses, we have a demand curve and a supply curve. So what we have here-- this is the kind of x/y graph we're going to look at all throughout the semester. On the x-axis is the quantity of roses. On the y-axis is the price of roses. The blue, downward-sloping line is the demand curve. Now, what I'm going to do here, I'm just giving you a overview. We are going, over the next five or six lectures, dive into where this demand curve comes from. We'll go to first principles and build it back up. But for now, what we know of a demand curve is it simply represents the relationship between the price of a good and how much people want it. Therefore, we assume it is downward sloping. At higher prices, people want less of the good. And we'll derive where that comes from shortly, starting next lecture. But for now, I think it's pretty intuitive that if the price of roses is higher, people want fewer of them. And that's why it's downward sloping. Basically, as the price of roses goes up, people want fewer roses. The yellow curve is the supply curve. Now, after we've derived the demand curve, we'll then go and spend about 12 lectures deriving the supply curve. That's a bit harder. But once again, we'll start from first principles and build it up. For now, you just need to know that's how much firms are willing to supply, given the price. So basically, as the price goes up, firms want to produce more roses. The higher price means you make more money, so you want to produce more of them. This is slightly less intuitive than demand, but we'll derive it and explain how it can be. But for now, just go with the basic intuition that if you're making something, and you can sell it in the market for a higher price, you're going to want to make more of it. And that leads to the upward sloping supply curve. Where the points meet is the market equilibrium. Where supply and demand meets is the market equilibrium. And that is the point where both consumers and producers are happy to make a transaction. Consumers are happy because on their demand curve is the $3 and 600 roses. That is, they are willing to buy 600 roses at $3. Producers are happy, because on their supply curve is the same point. They are willing to supply 600 roses at $3. That is the one point where consumers are happy and producers are happy. Therefore, it's the equilibrium-- highly non-technical, but that's the basic intuition. The point at which they're both willing to make that transaction, the point at which they're both satisfied with that transaction, is the equilibrium, which in this case is $3 per rose and 600 roses. Now, this raises lots of questions. Where did the curves come from? How does equilibrium get achieved? Why the heck do we give roses? These are a bunch of questions. We will come to all these questions over the next set of lectures. But the basic thing is to understand this intuition of Adam Smith's supply and demand model. Questions about that? Now, this model also raises another important distinction that we'll focus on this semester and is easy to get mixed up. So I want you to, if you're ever unclear, I want you to ask me about it. And that's the distinction between positive versus normative analyses-- positive versus normative. Positive analysis is the study of the way things are, while normative analyses is the study of the way things should be. A positive analysis is the study of the way things are, while normative analysis is the study of the way things should be. Let me give you a great example, which is eBay auctions. Auctions are a terrific example. They're like the textbook example of a competitive market. You can see it in your head-- demand comes as a bunch of people going on and bidding. People who want it more bid more, so you actually get a demand curve. The higher the price, the fewer people you're getting to bid. Supply is how many units of it are for sale on eBay. You bid until those two meet. And then you have a market equilibrium at that bidded price. Now, one example of an eBay auction that got a lot of attention a number of years ago, early in the days of eBay, was someone offered their kidney for auction. They said, look, I got two kidneys. You only need one to live. There are people out there who need a kidney. I'm putting my kidney on eBay for auction. And what happened, bidding went nuts. It started at $25,000. It climbed to $5 million before the auction was shut down, and eBay decided they wouldn't allow you to sell your body on eBay, bodily parts on eBay. So this raises two questions. The first is the positive question, why did the price go so high? So what's the answer to that? What's the answer to the positive question? AUDIENCE: Somebody wanted a kidney. JONATHAN GRUBER: Good answer, but let's raise hands and give answers. That's part of it. Yeah. AUDIENCE: Low supply, high demand. JONATHAN GRUBER: Low supply, high demand. Demand is incredibly high, because I'd die without it. Supply is low, because like not a lot of us are willing to sell their kidneys on eBay So low supply, high demand led to a high price-- Adam Smith at work. That's the positive analysis. But then there's the normative question, which is, should you be allowed to sell your kidneys on eBay? That's the normative question. The positive question is, what happens if you do? The normative question is, should you? Now, the standard economics answer to start would be, of course you should. We're in a world where thousands of people die every year because there's a waiting list for a kidney transplant. and these are people who would happily pay a lot of money to stay alive, I presume. Meanwhile, there's hundreds of millions of people walking around with two kidneys who only need one. And many of these people are poor. And lives could be changed by being paid $1 million for their kidney, and might be happy to take the risk that one kidney will be fine, as it is for most everyone for most of their life, in return for having a life-changing payment from a stranger. So economists say, look-- here's a transaction that makes both parties better off. The person who gets the kidney gets to stay alive, and they are willing to pay a huge amount for that. The person who sells the kidney in most probability is fine, because almost all of us can make it through life fine with one kidney, and create a life-changing amount of money that could allow them to pursue their dreams in various ways. So that's the standard argument, would be, yeah, you should be able to sell your kidneys on eBay. So the question is, why not? Why would we want to stop this transaction? What are the counter-arguments to that? Let's raise our hands. Yeah. AUDIENCE: Potentially, I think maybe the issue is because on eBay, there's no way to regulate it or you don't necessarily know. People could be like selling fake kidneys, per se. JONATHAN GRUBER: Right. So the first type of problem comes out of the category we call "market failures." Market failures are reasons why the market doesn't work in the wonderful way economists like to think it should. So for example, this answer puts up there could be the problem of fraud. People might not be able to tell if they're getting a legit kidney or not. There could be the example of imperfect information. Do you know what the odds are that you can spend the rest of your life with only one kidney? I don't either. We ought to know that before we start selling our kidneys. There could be imperfect information. This is one type of problem, which is the market, maybe the market may fail. Yeah. AUDIENCE: Well, the current system also holds people who are poor and have a failed kidney-- and which are people who would be completely screwed otherwise in the [INAUDIBLE] system. JONATHAN GRUBER: A second problem is what we call "equity" or "fairness." Equity or fairness, which is we would end up with a world where only rich people would get kidneys. Currently, there's a bunch of voluntary donors and people who are in accidents who have kidneys left over. And those go to people on the basis of where they are on a waiting list. It's actually a prioritized waiting list. It's kind of a cool-- one of my colleagues, Nikhil Agarwal, if you think about-- I'll talk a lot this semester about the imperialistic view of economics, all the cool things we can study. So he actually uses economic models to study the optimal way to allocate organs to individuals. now it's just done based on a waiting list, but it may be that someone further down the waiting list needs it more than someone higher up the waiting list because they're more critical or whatever. So there's various optimal ways to allocate. But certainly, the optimal way to allocate wouldn't be the rich guy gets it first. That would be unlikely to be what society would necessarily want. So there's an equity concern with that. What else? What other-- yeah. AUDIENCE: In that situation, since you know you can make money off of selling kidneys, and you take advantage of people, it's very bad, the black market for kidneys. JONATHAN GRUBER: Right, so there's sort of a third-- it's related to fraud, but there's sort of a third class of failures that gets into the question about behavioral economics that was raised earlier, which we could just call behavioral-- it's called "behavioral economics," for want of a better term, which is essentially, people don't always make decisions in the perfectly rational, logical way we will model them as doing so this semester. People make mistakes. That's a word we hate using in economics. We hate saying "mistakes." Ooh, boo, mistakes-- nobody makes mistakes. We're all perfectly economic beings. But we know that's not true. Increasingly over the past several decades, economists have started incorporating insights from psychology into our models, to not just say people make mistakes, that their lackadaisical, but to rigorously model the nature of those mistakes and understand how mistakes can actually happen due to various cognitive biases and other things. In this world, you can imagine people could make mistakes. They could not really sit down and quite understand what they're doing, and they could have sold their kidney when it's really not in their own long-term interest. Yeah. AUDIENCE: Would another example be if there's a family that is in extreme poverty, even though they only have one kidney, they might sell the other one, just to get more money for the family, per se? JONATHAN GRUBER: Well, in some sense that would be, once again-- if we took this factor out, if the market works well with its behavioral effects, we'd say, you know, that's their decision. If they otherwise they starve, who are you to say? But once you choose this, say, wait a second, maybe they're not evaluating the trade-offs correctly. Even if there's no fraud, even if there's perfect information, they may not know how to process that information correctly. But that is not standard economics. That's not what we'll spend a lot of time on in the semester, but it's obviously realistic. So those are a bunch of good comments, great comments. And yeah. AUDIENCE: Also, in inelastic demand, such that people always need kidneys-- JONATHAN GRUBER: That won't turn out to be a problem. That doesn't turn out to be a problem. We'll come back-- that's a great comeback that we talk about the shape of demand curves. We want to return to that question in a few lectures, but that doesn't actually cause a problem. It's just that's more of a positive thing about why the price is so high, but it's not a normative issue about whether you should allow it or not. So basically, these are exactly-- to me, honestly, I spend my life thinking a lot about these things. I think these are really interesting issues. But you can't get to the normative issues without the positive analysis. You do the positive analysis to understand the economic framework before you start jumping to drawing conclusions. That's no fun. We all want to jump to draw conclusions, saying this should happen, this shouldn't happen. You can't do that. We have to be disciplined. We have to start with the fundamental economic framework. And basically, the bottom line-- I said I'll teach this course with a policy bent, but you have to recognize that economics at its core is a right-wing science. Economics at its core is all about how the market knows best, and that basically governments only mess things up. That's sort of the basic, a lot of what we'll learn this semester. As the semester goes on, we'll talk about what's wrong with that view and how governments can improve things. Indeed, I teach a whole course about the proper role of government the economy. But the standard of economics is, "the market knows best." And that leads us to the last thing I want to talk about, which is basically, how freely should an economy function? Let's step back to the giant picture. Let's step back from a market for roses to the entire economy. How freely should a market, should an economy function? We have what's known as a "capitalistic economy." In a capitalistic economy, firms and individuals decide what to produce and consume, maybe subject to some rules of the road set by the government. There's some minimum rules of the road to try to avoid fraud or misinformation, but otherwise, we let the dice roll. Firms let consumers decide sort of what to do. Now, this has led to tremendous growth. America was not a wealthy nation, was not a very wealthy nation 100 years ago, or 150 years ago. Led to tremendous growth, where we are now the most powerful, still the most powerful and wealthiest nation the world, largely driven by the capitalistic nature of our economy. On the other hand, we are a nation with tremendous inequality. We are by far the most unequal major nation in the world. The top 1% of Americans has a much higher share of our income than in any other large country in the world, any other large developed country in the world. The bottom 99% has less of our income corresponding with anywhere else. So it's led to major inequality. And it's led to other problems. It turns out that the government can't appropriately set the rules of the road to avoid things like fraud, as we saw with Enron, if you remember back to that, or a lot of what happened in the financial meltdown. It turns out it's hard to get people perfect information, et cetera. So we've seen the problems. We've grown very wealthy as a nation. We've introduced a whole set of problems through this system. Now, the other extreme is what's called the "command economy." Rather than a capitalist economy, it's what's called a "command economy." In this case, the government makes all the production and consumption decisions. The government doesn't just set the rules of the road, the government owns the road. The government says, we're going to use this many cars this year. And people can get them in some way. It could be a lottery, could be waiting in line. How do we decide how to allocate them? We're not going to let the market allocate them. We, the government, will allocate them. We'll allocate how many get produced and who gets them. And this was the model of the Soviet Union that I grew up with. This was the pre-1989 Soviet Union. The government decided how many shirts, cars, TVs, everything. It's sort of bizarre to think that literally everything the government decided how much to produce. And by and large, the government decided who got it partly through corruption-- that is, the party members, party leaders got it first-- and often just through waiting in line for the remaining application. Now in theory, this ensured equity by making sure that everybody had shot at things. In practice, it didn't work well at all and actually was what dragged down the collapse of the old Soviet economy, was that the command model simply doesn't work. Partly there's just too many opportunities for corruption. When the government controls everything, that means there's no checks and balances on the opportunity for enormous corruption. The capitalist economy puts some natural checks and balances on that. And partly because it turns out that it's hard to control human nature. And Adam Smith had it right. Adam Smith talks about the "invisible hand" of the capitalist economy. The invisible hand is basically the notion that the capitalist economy will manage to distribute things roughly in proportion to what people want. And that's where folks want to be. Folks who want a certain kind of car are going to want to get to that kind of car, and if the government has it wrong, they're going to get upset. And it's going to lead to a less functional economy. So basically, Adam Smith's view is that-- the invisible hand view is that consumers and firms serving their own best interest will do what is best for society. So the fundamental core of the capitalistic view is that consumers and firms serving their own best interest will do what ends up being best for society. And that's essentially the model we'll learn to start in this course. Yeah. AUDIENCE: In that definition, are we defining the best for society as in everybody has the most money? Or everyone has the best health or the best standard of living? What is the best [INAUDIBLE]? JONATHAN GRUBER: Great question. We're going to spend a lot of the semester talking about that. For now, we're going to define "best for society" as the most stuff gets produced and consumed. That's how we're going to find it-- obviously raises a set of issues about what about pollution, what about health, et cetera. We're going to come to those, but for the first two-thirds of the course "best for society" means what we're going to call "maximum surplus," which is the most stuff gets produced that people value. So that's how we're going to do it. And in his view, the invisible hand does that. And by and large, it's a very helpful framework to turn to. However, at least it can lead to outcomes that are not very fair. So the way we're going to proceed in this course is we're going to start by talking about how Adam Smith's magic works. How does the magic happen? How does individuals and firms acting in their own self-interest, without caring about anybody else, end up yielding the largest possible productive economy? How does that happen? And we're going to talk about that. We'll start with demand, which is how do consumers decide what they want given their resources. We'll talk about the principle of utility maximization, the idea that I have a utility function that I can mathematically write down what I want. I'll have a budget constraint, which is the resources I have, and those two constrain optimization. We'll say given what I want and the resource I have, what decisions do I make? Boom, we get the demand curve. Then we'll turn to supply, and we'll talk about how do firms decide what to produce. That's much more complicated, because firms have to decide what inputs to use and what outputs to produce. And we'll talk about how firms can operate in very different markets. There is a competitive market that Adam Smith envisioned, but that doesn't always work. Sometimes we get monopoly markets, where one firm dominates. And you can actually have outcomes which aren't the best possible outcome, even with the invisible hand. So we'll talk about different kinds of markets. Then we'll put it together to get market equilibrium, and talk about Smith's principles. And then from there, we'll talk about how it breaks down in reality, different change in reality, how there are various market failures that can get in the way, why we have to care about equity and what implications that has, about behavioral economics, about a set of other factors. So that's basically how we're going to proceed this semester. As I said, the lectures are important, but the recitations are as well. Once we're sort of in steady state, the recitations will be about half new material and half working through problems to help you prepare for that next problem set. So the way the problem sets are going to work is the problem set that's assigned will cover material that's taught up to that date. So for example, problem set one is going to be assigned next Friday. That will cover everything you've learned up through next Wednesday. Therefore, in section on next Friday, we'll do a practice problem which you should understand because it'll cover things that were taught in class, and help prepare you for the problems. And we'll do that every week. That's about half the section. The other half of the section will be new material. This Friday, the section on Friday is all new material. What we do on Friday is cover the mathematics. I don't like doing math. I always get it wrong. So I leave math for the TAs, who are smarter than I am. So this Friday, we'll be doing the mathematics of supply and demand, and how you take the intuition here and the simple graphics, and actually turn it into mathematical representations, which is what you need for the problem sets. That's this Friday. Then we'll come back on Monday and start talking about what's underneath the demand curve. All right, any other questions? I'll see you on Monday.

MIT_1401_Principles_of_Microeconomics_Fall_2018

12_Monopoly_II.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: OK, so let's continue our discussion of monopolies. Last time, we talked about monopolies and talked about how they're another extreme of the market structure spectrum. We have perfectly competitive firms, where there's facing a perfectly elastic demand curve, and competing with essentially infinite number of other firms and taking prices, versus a monopolist who owns the market and sets the price, but therefore, faces this poisoning effect, which leads them to under-produce and creates deadweight loss. So today, we want to continue our discussion of monopolies with three sort of separate topics. OK, the first topic I want to cover is, where do monopolies come from? So the stork does not bring monopolies. They come from somewhere else. So sort of, how do monopolies arise? And there's really two sources of monopoly that I want to focus on today. The first is monopolies that arise from cost advantages, the monopolies that arise from cost advantages. So some markets come with a natural built-in cost advantage for one participant. So it could be there's an essential input, like there's one rock quarry in town. And whoever controls the rock quarry, they supply the rocks for the whole area. In that case, we say that this is an example of a natural monopoly. A natural monopoly is a market where, for all relevant quantities, one firm can always produce at a lower average cost than another firm can. So for all relevant quantities, one firm can always produce at a lower average cost than can other firms. OK, this basically is the same as saying that it's a market where average cost is everywhere declining, at least in the relevant production range. You can imagine where, so for some crazy quantities, average cost increases. But where average costs is everywhere declining is a market with a natural monopoly. So why would this be true? Let's consider what I think is the simplest case. Let's think about a water utility, delivering water to people's houses through pipes underground. Once those pipes are laid, it is that the giant fixed costs are paid, and the marginal costs are trivial. It's the cost of the water. The fixed costs are so large relative to the marginal costs that average cost always declines. In our typical production function, remember, average cost first declines, then goes back up. It first declines, you pay off the fixed costs, then back up as the rising marginal costs start to dominate. This is a market where that second purchase never gets big enough, say where, essentially, marginal productivity is close to flat, so the marginal cost curve isn't rising that fast, and/or the fixed costs are enormous. So we could see that in figure 12.1. Imagine a water utility, which has roughly flat marginal cost. That is, the marginal cost is the cost of procuring another gallon of water. Now once again, at some point, if you want to give every person in America one million gallons of water, marginal costs would start to rise. But for the relevant range, it's roughly flat. And yet there's enormous fixed costs. You got to lay those water pipes. So in that world, average cost is everywhere declining. It's approaching marginal cost, but it never crosses marginal cost. In such a world, once one firm has laid the pipes that's providing water, it never makes sense for another firm to enter. Entry never makes sense. Why is that? Because if another firm thinks about entering, they can say, oh, look this water utility is making all this money, this is a great market to enter-- remember, we talked about unlimited entry and exit and perfect competition that drives profits away. Well, let's say another firm says, look at all this profits, I want into this market. The first firm will say, I'm just going to tell you right now, if you enter this market, I will price at marginal cost until you leave. For the first firm, they've already paid to lay down the pipes, so as long as they're pricing at marginal cost, they're not losing money. But for the second firm, they will lose money if the firm stays at marginal cost. So they'll just never enter. Because they know the first firm has a barrier to entry. They have a natural monopoly, having laid down those pipes. Having already paid that fixed cost, that sunk, they will win any battle. They'll just price at or near marginal cost and drive any competitors out of the market. So these kinds of essentially natural monopolies arise when enormous fixed costs create a barrier to entry when enormous fixed costs create a barrier to entry. You can see natural monopoly. And that's one common way monopolies arise in the world. Think of utilities. Delivering water is a classic example of that. That's one way we get monopolies. The second way we get monopolies is through government action. Governments also create monopolies remember, today I told you we talk about governments as good guys. But still governments can do things which can create deadweight loss. And now sometimes, governments do this for a good reason, like a natural monopoly reason, like the postal service for example. Local postal delivery, delivering letters locally, has all the features of a natural monopoly. It's a huge fixed cost of building all the post offices and a small marginal cost of transmitting letters from place to place. Now however, in other cases, this is not necessarily true. In the US, we have very few government-created monopolies. The rest of the world, especially in the developing, it's much more common. Many countries have the government control the production of steel, the airlines are controlled by the government , banking is often controlled by the government. These are places where probably evidence suggests it would be more efficient to not have the government control this. There is no real natural monopoly in banking. But it's more for political reasons that they're created. But that's not really that relevant in the US. Even the post office isn't a monopoly anymore, with FedEx and UPS and stuff. So it's not really relevant in the US. The more relevant for the US, way that the government creates monopoly is by creating barriers to entry. The government could create barriers to entry, the most prominent of which is patents. The most prominent barrier to entry is a patent. This is the case where the fixed cost here isn't building a pipe. It's coming up with an idea. And that basically the government has a law which says that, if you have a new idea and you patent it, you are granted a monopoly to sell the resulting good for 20 years from the date of patent. So when you patent a good, you're granted the right to a monopoly over that market for 20 years. Essentially, the government is creating a monopoly. It's saying, once you've patented that-- now remember, very importantly-- especially for drug development-- it's not the date at which the good comes to market. It's the data at which you file. So if you file for a patent and it takes you 19 years to develop the product, then you only get one year monopoly in the market. So it's the sum. 20 years is both the development period and the sales period. You get 20 years from the date you file. Now, patents are quite interesting because the welfare implications are kind of mixed. On the one hand, a patent, by creating monopoly, creates deadweight loss. For potentially up to 20 years, you've got deadweight loss in that market, because one firm has the monopoly right to sell the good. On the other hand, why might patents be a good idea. Yeah. AUDIENCE: [INAUDIBLE] for developing drugs, the company that can't ever make any money off of its research and development costs will never be incentivized to develop it [INAUDIBLE].. JONATHAN GRUBER: Right. Because research and development is a huge fixed cost. And if you pay that huge fixed cost, and the minute you develop it someone else can just copy you, then you'd never invest those fixed costs. We'll talk later in the course about externalities. But you think of this as sort of a positive spillover from R&D. When firms do R&D, when a firm invents something, it benefits everybody who might produce that good. So if I invent something, then everybody could copy me, could benefit. So I will not invest in R&D unless I can be sure I might get some money out of it. And that's what the patent does. It reduces those spillovers by saying, OK, you get to own this for 20 years and make some money on it. Therefore, go ahead and invest. In other words, what a patent does is, essentially, the government's saying, you're going to get a reward for being first to market. And that reward is 20 years of privilege of being a monopolist in a market. Therefore, that's an incentive for you to go and invent. Yeah. AUDIENCE: What's the process like for extending your patent? Because I know Disney, for example, has [INAUDIBLE] which is giving them more years than what they had? JONATHAN GRUBER: That's a very detailed concept that I don't have enough time to get into, but there's a whole bunch of legal battles around that. But the bottom line is, there's a patent tradeoff here. The tradeoff is, on the one hand, it's what we call a static versus dynamic tradeoff, sort of today versus the future. At any point in time, there's a ton of deadweight loss out there, because monopolists who have patents are under-producing. At the same time, over time, we're getting cool new products because of patents. And that's the tradeoff. How did 20 years get picked? Ideally, you'd have some optimal period to resolve that tradeoff, the ultimate result of a tradeoff long enough to get creation, but short enough so that the gains on that creation don't exceed the deadweight loss from monopoly. And that's an example how a monopoly can arise from a government action. And the point is that, for both natural monopolies and patents, there are legitimate reasons for monopolies to arise. So what I want to say is, monopolies are not bad. They're not necessarily an inherent flaw in the system. There are legitimate reasons to have monopolies, both because of natural monopoly and because of patent. But they do, at any point in time, create deadweight loss. Yeah. AUDIENCE: Is there a way to measure how the impact of [INAUDIBLE] something that receives patent, like how [INAUDIBLE] that relates to the deadweight loss [INAUDIBLE]? JONATHAN GRUBER: It's a great question. There are people who do spend their lives doing that. And basically, the way you would do it is you would essentially measure the consumer surplus created from the new good, versus-- it would be the triangle sizes. It would be, how much consumer surplus would you get from the new good versus deadweight loss from that good sales being restricted. Now obviously, if it's a good that never would have existed otherwise, by definition, the patent's a good thing. Because who cares about deadweight loss if you never would have had it in the first place. The question really is about the substitutability. If you're patenting something which is only incrementally better, then maybe it's not worth it. So that's sort of the tradeoff. Other questions? OK, so now let's talk about addressing or regulating monopolies. Let's go back to think about our natural monopoly case. Let's think about our natural monopoly case. Actually, I'm going to call this section not regulating. I'm going to call this addressing monopolies, addressing monopolies. There are different ways that we can address monopolies. And we'll go back to our natural monopoly case to make it easy. The first way we can address monopoly is through government regulation. So let's step back here. The bottom line is, we have a monopoly that should exist, say a natural monopoly. It exists naturally. At the same time, that is creating a deadweight loss. It's creating inefficiency. So the brilliance, the insight of the first fundamental theorem of all for economics was, the competition through entry and exit will bring us the efficient production level and reduce deadweight loss. Here with a natural monopoly, that doesn't work. Competition can't work, therefore, we end up with a deadweight loss, because the firm that owns the market is under-producing. This is the first case, as I mentioned last time, of what we call a market failure. The market outcome is not delivering the welfare-maximizing result. And once there's a market failure, there is a potential beneficial role for government. I said last time, with no market failures, government's just the bad guy. With no market failures, all government can do, it can help with the redistribution. But from an efficiency perspective, all the government can do is muck up the market. But once there's a market failure, there's a potential role for the government addressing it. So to see that, let's return to our monopoly example from last time in figure 12.2. Figure 12.2 is our monopoly example from last time. Remember, we had demand curve that was Q equals 24 minus P. We had a cost curve that delivered the marginal cost of MC equals 2Q. And that's graphed here. And we said last time that the monopoly outcome would be at E sub M. The monopolist would sell 6 units at the price of 18, creating a deadweight loss of C plus E. Now, what if the government came along and simply mandated that the monopolist was not allowed to charge more than 16? The government says, look, I understand it's a monopoly. I'm not going to try to break up the monopoly. It's a natural monopoly, whatever. Simple rule-- price ceiling of 16. Now, when I mentioned pricing a couple of lectures ago, you should have learned to say, boo, price ceilings, bad, increase deadweight loss, terrible. But now, suddenly a price ceiling can actually get rid of deadweight loss. How's that possible? Let's take a look. In this case, the monopolist knew marginal revenue curve is a bit different. The new marginal revenue curve is the old marginal revenue curve, until you get to the point where the first dashed line intersects marginal revenue. It's the old marginal revenue curve, until you get where the marginal cost crosses marginal revenue. Then it jumps to the flat line at 16. Essentially, the new marginal revenue curve is downward sloping till the point where marginal cost equals marginal revenue. Then it jumps back up to the flat part that goes from 6 to 8. Now the exact shape doesn't matter. It's the intuition that matters. The point is the following. Think about the monopolist's decision. He decides, based on the logic of the last time, to sell six units. That's the point where the poisoning effect offsets the benefits of selling another unit. Now let's say the government comes and says, monopolists, you can't charge more than 16. Well, once a monopolist is forced to charge 16, now think about her decision to sell the seventh unit. Well, before she didn't want to sell the seventh unit, because selling the seventh unit meant she had to lower the price to 17. Well now she has to lower the price to 16 already, so why not sell the seventh unit? Indeed, why not sell the eighth unit? We've essentially gotten rid of the poisoning effect by pre-poisoning them, by already telling them, look, you can't charge more than 16. There's no way to charge more than that, so you might as well sell eight units. So by telling the monopolist they have to charge a competitive price, you will force them to sell the competitive quantity and you get rid of deadweight loss. That's our first example of how the government has improved things. The government has gotten rid of the deadweight loss by setting a price ceiling. Price ceiling, in a perfect market, is only bad, but a price ceiling here can actually improve things. So that actually sounds pretty good. So what's the problem? Why not, in reality, just say, hey, we can solve all our monopoly problems by just having the government regulate the price at the competitive level? What's the problem, in reality, with that solution? Yeah. AUDIENCE: How would the government know the competitive level? JONATHAN GRUBER: How the hell does the government the competitive level? It's great with this chart. But in reality, the government doesn't know where the competitive level is because there's two difficult points. What does the government need to collect? First of all, the government needs to know what the demand curve is. Well, it turns out demand curve's aren't written on our foreheads or out there on the dark web. Demand curves are something which you have to estimate. You have to gather information from people. And it turns out that it's pretty hard. Because if you want to gather information on what people are willing to pay for apples, you can just go to stores and look at the prices charged for apples. But if you want to gather information on the willingness to pay for getting water delivered to your house, there's no market to turn to. It turns out to be pretty hard to figure out how you value a lot of goods, which is natural monopolies. Now, you might say, well, why do you just ask people what it's worth to them? You just say to people, look, tell me you demand for water. It turns out that's really hard because people aren't very good at thinking about how to assign prices to things they don't shop for. If I say to you, what's an apple worth, the first thing in your head is, you think about being in a supermarket looking at apples and what they cost. You don't think about the inherent value to you of consuming an apple. But when you think about what water is worth, you have to think about sort of the inherent value. And so you have to rely on something that we call contingent valuation, which is the economist's fancy word for just asking people what it's worth to them. The problem is, people aren't very good about that. So for example, a classic case was the rise in environmental issues. How much is it worth to have clean air? How much is it worth to save the Grand Canyon? So it turns out people give sort of crazy answers that violate all the rules we set up in the first lecture by utility functions. Like for example, if you asked folks what it's worth to save the Grand Canyon as a single question, and then you ask them as the third question on the list, it's worth 1/5 as much to save the Grand Canyon was the third question on the list, which doesn't make any sense. It doesn't matter what the other questions are. If you ask people, how much they would pay to save seals and whales, in that order, they'd say saving seals is worth $142 and whales $195. So whales and seals are about the same. But when you reverse the order, whales are worth about twice what seals are worth, just by reversing the order. Well, more relevantly, if you ask people what it's worth to save a bird or 10 birds or 1,000 birds or a million birds, they give you pretty much the same answer, which doesn't make any sense. So basically, the problem is these contingent valuation methods don't really give sensible answers. And it's very hard to get sensible answers. In my other course, Public Economics, we talk about lots of interesting clever methods for trying to get people to reveal their preferences for these public goods. But it turns out to be hard. So that's one problem, is it's hard to measure the demand curve. The other problem is, it's hard to measure the supply curve. Because where does supply curve come from? It's sort of a firm's marginal cost function. You as the regulator do not know what their marginal cost function is. So how do you find out? You ask them. You say, hey, business I'm going to regulate, I'd like to know what your marginal cost is. And by the way, the lower you tell me it is, the lower price I'm going to let you charge. So what's your marginal cost? Oh, god, our marginal cost is horribly high. It terrible. You wouldn't believe it, can't complain enough about how high our marginal cost is. And basically, unless the regulator perfectly knows the production function and perfectly knows the input prices, none of which are perfectly known by anybody but the firm, they aren't going to know what the supply curve looks like and what the marginal cost looks like. Question. AUDIENCE: Couldn't they also, say, in a public company, basically inspect their factory and manufacturing [INAUDIBLE]?? JONATHAN GRUBER: You could absolutely do that and you can try to collect data. But ultimately, they could, that day that you show up with the inspector, have a really expensive-looking setup in their factory or whatever. It's basically very, very hard. So basically, the problem is that regulation-- and this will generally be a feature when we talk about the benefits of government intervention. We'll say, market failure makes it possible that government intervention can make things better, but not definite. So the way to think about the logic is, in a perfectly competitive market with no market failure, government intervention only makes things less efficient. We're leaving redistribution aside. It only makes things less efficient. When you move to markets with failure, you open up the possibility that government intervention can make things better, but not for sure that it will. You open up the possibility. It's a necessary but not a sufficient condition for saying the government improves efficiency. So for example, imagine that the government came in to this market and said, we think the competitive price is 10. We think the competitive price is 10. Well, if the government set the price equal to 10, we know that firms are going to produce where marginal revenue-- which is now just 10, because it's regulated-- equals marginal cost, so where 10 equals 2q. Little q and big Q are the same because of the monopolist. Where 10 equals 2q, or where q equals 5. So if the government comes in and says, we're fixing your price at 10, we know the firm produced five units. Well, with five units, the deadweight loss is even bigger. The key trick with deadweight loss triangles, deadweight loss is essentially proportional to how far you deviate from the optimal point. So the more you move to the left, the bigger the deadweight loss triangle is going to get. So deadweight loss with 5 units sold is bigger than deadweight loss with 6 units sold. So the government's actually made things worse. By coming and setting this price of 10, the government's made things worse. Indeed, the government could actually set the price so low you shut down. The government could set the price below the shutdown point and wipe the whole business off the map. So here we have the tradeoff. The government can potentially make the market more efficient through price regulation, but it won't necessarily. It depends on its level of information and how well it does setting the price relative to the competitive price. So basically, this is a very sort of interesting and difficult problem for the government. So one way is through regulation. Questions about that? The other way to try to address monopoly is by saying, is there some way to introduce competition? That is, even in what feels like a natural monopoly market, is there some way to choose competition? So for example, think about broadband delivery. Now we're still wired. In the future, it's all wireless. This is different. But it's still wired. The way it works is, broadband delivery has features of a natural monopoly, which is, someone has to lay the wires to your house to deliver it. At the same time, once those wires are laid, it's a 0 marginal cost to allow you to connect to the internet. So what governments do in other countries, but not the US-- in the US, we have competition. Everybody lays their own lines down. In Europe, they recognize that's inefficient. It's a natural monopoly. What they do is, they have one set of publicly laid lines, and you compete to deliver content over those lines. So there's competition where the monopoly is not natural, which is over the speed and the quality. It's the quality of the connection. But the lines themselves, since that's a giant fixed cost, they realize there can't be competition over that. So that's one way you could try to deal with a natural monopoly. In other words, the government could control the pipes underground delivering the water and have firms compete over delivering the water to you. So that's one way to deal with it. Another example is to think about the public sector and think about education, education delivery. Now, for those of you who grew up in the US, you are evidence of the success of the US educational system. But you are the exception, not the rule. The US educational system, by international standards, performs very badly. Probably out of the top 30 countries, we're like 15th in terms of things like math scores and other things like that. Yet, we spend more money per pupil than any other educational system in the world by a large amount. So if you look at figure 12-3, figure 12-3 shows, in the blue, primary school spending per pupil, and in the red, eighth grade math scores. And you can see the US spends way more than other people, but our scores aren't any better. Indeed, it doesn't look like there's a very strong relationship here at all between how much countries spend and what you get in return. But at least it's quite striking how low the US is given how high our spending is. Now, why is that? One reason is because we have given local schools a monopoly. We have said to local schools, you have a monopoly where you get to deliver the education to anyone who lives within a certain radius of your school. We've created that natural monopoly. And once we created that natural monopoly, we reduce pressures on firms to produce efficiently. We've essentially said to firms, you have a monopoly. There's no reason you have to produce efficiently. Because there's no entry and exit, which can put those pressures on you. So in some sense, that's a government-created monopoly. Your local public school is a government-created monopoly. So what can we do for that? What can we do? Well, we can actually introduce competition. So for example, we can have public school choice, which many cities now have, where, in fact, you don't have to go to the school in your neighborhood. You can try to go to any school in the district. You can essentially enter a lottery and try to move around to other schools in the district. And that introduces competition. Because then the schools that want kids will have to be better and have to produce more efficiently. And then a further mode of that is, you can have charter schools. Charter schools are publicly funded, but not city provided. They're in between public schools and private schools. Charter schools are separate schools, which get funding from the government. But they aren't under the local government's control. Regulatory control, but they aren't delivered by the city as education. And that provides more public school choice for individuals. And there's a lot of evidence, much of it by my colleagues here at MIT, which shows that there's been enormous benefits to these movements, that public school choice and charter schools have delivered-- now I know some of you might have seen the John Oliver segment a couple of years ago where he ragged on charter schools. I mean, I love him in general, but he was wrong on that one. There's a couple of bad actors. But by and large, charter schools have been an enormous benefit to the education system by introducing competition and allowing students an option to improve their educational outcomes. Now, the furthest out option, and something you'll hear a lot of discussion of, is vouchers. The most radical option is what we call public school vouchers, or just vouchers. Here's how this would work. This is a very popular idea on the conservative side of the spectrum. Here's how this would work. So I live in Lexington. What that means is, if I send my kid to a Lexington public school, it's free. But the minute I pull my kid out and send him to a private school, it costs me the entire amount. So imagine a Lexington public school education's worth $10,000. I'm essentially getting $10,000, conditional on sending my kids to a Lexington school. The minute I choose to send them elsewhere, I literally give up that $10,000. A voucher system would say, the way it would work is, everyone in town would get a check or would get a voucher for $10,000 to be used at any school they want. So if you want to go in Lexington, you just hand it in. Life's never changed. But now if you want to go to private school, we're allowing you to take your money elsewhere. That puts competitive pressure not just on schools within the district but on the whole district. The whole district now says, wait a second, if we don't do a good job, people are going to leave and take their money elsewhere. So the idea is to actually set up broader competition to put pressure on districts to improve their performance. So this has been a longtime attractive option to many economists. And I've spent a whole long time talking about the pros and cons of this in my other class. But briefly speaking, the pros are all the reasons we like competition in markets. It'll cause more production efficiency because schools would have to compete. The cons are numerous, though. One con is, you then have to have the public sector that has a vested interest in making sure private schools are delivering a public quality education. So basically, you could set up a private school that is just a football training academy. And suddenly, people could take their voucher and go there, and suddenly they wouldn't get education. That's one con. Another con of these systems is that they're expensive. So take me. I live in Lexington. I'm a pretty rich guy. I sent my kids to private school. I sent them and I paid. Now, imagine Lexington had a voucher system and they give me a check for $10,000. Why should Lexington give a relatively rich guy $10,000? That's sort of silly. I was already sending my kids to private school. So who would be the big winner from a system like this? Well, some winners would be kids who then get a better education. But a lot of winners would be rich people who already sent their kids to private school. We'd just be handing them checks. That's not a great outcome. So there are definitely tradeoffs inherent in a system like this. I know there's a few questions. Let's do a couple of questions, then we'll move on. Yeah, in the back. AUDIENCE: So say you have these vouchers, and instead of going to the public school, students are leaving to the private school, and then the fixed cost of having the infrastructure of a public school, that's then spread out over less students. So how do you account for, the price of teaching one student might go up if there are these students that are leaving? JONATHAN GRUBER: That is an awesome question. There's two elements to that. Question one is, there is an actual natural monopoly feature to local public education. So remember, competition doesn't make sense when it's a true natural monopoly. There is some natural monopoly element. So as you shrink the number, you suddenly are raising the average cost. And that is a potential problem. There's an efficiency issue. There's also an equity issue, which is, who uses the vouchers and leaves? The people with motivated parents who are with it. Who gets left behind? People's parents who don't give a shit about him. There's an equity issue, too, of, suddenly, you're pulling all the smart kids, all the motivated kids out of the public schools and leaving the kids behind who aren't motivated or whose parents aren't motivated. There's an equity issue there, too, so that's another tradeoff. Yeah. AUDIENCE: You got a good education. JONATHAN GRUBER: How do you tell if people get a good education? Well, that's [INAUDIBLE]. AUDIENCE: Well, if you're going to get a good education. JONATHAN GRUBER: Well, that's another problem. Which is, competition, one of our fundamental assumptions of perfect competition was perfect information. That's fine when you're looking at apples or maybe even computers. It's not so easy looking at schools. Now, schools publish test scores. You can look at the test scores of kids who go to that school. But once again, that's not that reliable, because if smart kids are going to the school, they'll have high test scores, even if teachers suck. For example, Harvard-- sorry, I couldn't resist. [LAUGHTER] So basically, you can't really tell that. In other words, there's a number of failures of the private education market, that are a problem with this kind of voucher solution. But it doesn't mean it's wrong. It means it's interesting. There's a tradeoff. On the one hand, we'd introduce competition, which would maybe increase efficiency in the market. On the other hand, there's a lot of market failures which might get in the way of this functioning properly. Fascinating topic, and if you want to learn more, I urge you take 1441. We spend a whole lecture talking about it. So that was the second topic I want to talk about. Once again, we've started down the road of questioning the wisdom of the market, so the road of market failures. The first market fail is monopoly. We've talked about the pros and cons of trying to deal with monopoly. But I want to talk about one other topic before we leave monopolies, which is this topic of what we call contestable markets, which is sort of an informal term. But I really like it as an intuition, so I'd like to spend some time on it. Contestable markets are monopoly markets without market power or without much market power. That is, we talked about monopolists as having lots of market power. Remember, we said the markup was essentially proportional. We said that monopolists had a markup, price minus marginal cost over price, which is equal to minus 1 over the elasticity of demand. That was their markup. And so you know some markets, different monopoly markets will have different levels of market power. You can be a monopolist, but not have much market power if consumers are very elastic. But there's another reason why monopolists face pressure besides the elasticity of demand, which is, in some sense, the size of the barrier to entry. So one reason monopolists are constrained in their pricing is because demand's elastic. Another reason is because the barriers to entry might not be that large. So you could think of it, roughly speaking, for a given elasticity of demand, the larger the barrier to entry, the more market power you have. Because the larger the barrier to entry, the more you can be sure no other firm's going to come in. Or in other words, the amount to which you can charge above marginal cost for a given elasticity is proportional to how severe the barriers to entry are. If there's no way a second firm can get in, then essentially, you get to just obey this formula. But if a second firm can get in easily, then you might not even be able to have this big a markup. Because if you try, someone's going to come in and steal your profits. So in other words, there's essentially an important issue, which is that the market monopolists can get as a function both of elastic demand and the size of their barriers to entry. This is an important issue that came up in the area of airline deregulation, which is what I want to talk about for a few minutes. It's an important application of the kind of stuff we've been talking about. Now, how many of you have seen Mad Men. Jeez, really? Wow, OK, it's a pretty good show. I mean, it's not outstanding. How many have seen Breaking Bad? OK, good. That, you have to all see. Mad Men's pretty good. But basically, what's nice about Mad Men, it shows what flying was like when I was a kid. When I was a kid, when you flew, it was like luxurious. You went on, there was plenty of leg room. You had free meals, free movies, free drinks. And the reason was because-- I sort of skipped ahead in the story. When I was a kid, airlines were regulated. What that meant was that, essentially, the government viewed airlines as a natural monopoly. They said, look, planes are very expensive. There's a big fixed cost, so it's a natural monopoly. We don't think competition will work well here, so we're going to have regulation of airlines. What that meant was, the government set the price of every flight and regulated where planes could go. So airlines every year would submit a set of routes they wanted to fly and a set of prices. And the government would approve or turn them down. It was a regulated market. And it was generally viewed that the government basically sort of screwed this up and set the rates too high. That was sort of the general view, that, basically, government, because, essentially, they didn't really understand the supply curve, they were being fooled by the airlines to think costs were higher than they were and were setting prices too high. But economists said, look, this is not a natural monopoly. Because in fact, it's not actually that cheap to get a used plane and enter this business. Because these companies turn over their planes all the time. And yes, buying a new plane's expensive. But getting a well-functioning used plane isn't that expensive. And you could easily create competitors in this business. They said it was what we called a contestable market, a market with very low barriers to entry, that, in fact, if you allowed competition, you would have a lot of entry into this market. And it would function fine, that it wasn't a natural monopoly, it was a contestable market. And they argued, in fact, that if you lacked competition, price would fall very close to marginal cost. Price could fall very close to marginal cost. So in fact, economists carried the day. In the 1970s, we deregulated the airline industry. The government stopped setting prices and regulating routes. What happened? OK, three things happened. The first thing is, price fell enormously. The cost of flying fell by about 1/3. The best example was the airline I took home from MIT in 1984 called People Express. People Express Airlines introduced-- this was sort of shortly after deregulation. I could fly from Boston to Newark for $29. Now, how did that work? It was crazy. You showed up at the airport. There was no reservations. You just waited on line. They let people on until the plane was full. Then you waited for the next one. You paid on the plane with a credit card. I still to this day don't know what happened if you didn't pay. Did they throw you out the window? I don't know. But you paid on the plane. And it was incredibly competitive. And this is what happened. Flying got incredibly cheap. The second thing that happened is you had many more routes. It turned out that the government thought that flying from point A to point B wasn't profitable often when it was. So the government wouldn't let airlines flight from point A to point B, when, if fact, airlines could make plenty of money doing that flight. So you had cheaper flights and more routes. The third thing is, flying sucks. When I was a kid, flying was awesome-- meals, booze, big seats. Now it's terrible-- no meals, no booze, tiny seats. Why? Why did this happen? Flights got cheaper and there are more of them, but why were they suddenly crappier? Yeah. AUDIENCE: There was more competition so they were trying to bring down the marginal cost so bring prices lower. JONATHAN GRUBER: Right, that's one way to put it. But the point is, what this example points out is that there's always competition. Yeah, what were you going to say? AUDIENCE: I was going to say price discrimination. They could discriminate based on what people [INAUDIBLE] JONATHAN GRUBER: It actually wasn't. They still price discriminate. It wasn't quite that. But yeah. AUDIENCE: People were willing to fly [INAUDIBLE] JONATHAN GRUBER: They were willing to fly. But what were they doing before? Yeah. AUDIENCE: If you couldn't compete on price, you could compete on luxuriousness. JONATHAN GRUBER: Right. So there were multiple airlines before. The government said we'll only compete on price. But they said, great, we'll compete on other stuff. Remember, economic actors want to be economic actors. They want to use economic tools. So if the government says to airlines, we're not going to let you charge a higher price, they're like great, we'll compete by having better food, better drinks, bigger seats. Once the government said, now you have a new mode of competing, they realized people weren't willing to pay for better food and better drinks and bigger seats. They'd rather fly cheaper. So they switched the mode of competition from quality competition to price competition. They used to use quality competition. Now they use price competition. This means that all of us who complain that flying sucks should just shut up. Because if we really cared that much, then there'd be a good airline that charged more and gave you better stuff. But there's not. I mean, JetBlue's a little better. But by and large, there's not airlines that charge more and give you better stuff. And that's because, at the end of the day, we would rather fly cheaper than have this extra stuff. It wasn't worth the money that we were paying for it. Which really says that regulation was failing here. Regulation was forcing airlines to compete in an inefficient way. They're competing by giving us nice meals when we would rather have the money. So it was an example that regulation failed. So that's all really good news. But there's one piece of bad news, which is that, economists aren't all-knowing. And the economists, there's a fourth outcome we missed, which was the rise of the hub and spoke system. Which is that what we missed is that airplanes aren't a natural monopoly, but airports are. And what airlines started doing is essentially taking over airport slots, and then saying, we are going to have all the flights. So for example, my wife is from Minneapolis. We used to go to Minneapolis, the only option was Northwest. Northwest owned all the slots in the Minneapolis airport. And they'd say, if you ever fly anywhere on Northwest, we're going to route you from Minneapolis to wherever you go. It was called the hub and spoke system. You always went to a central point and went out. So US Air had Pittsburgh. Northwest had Minneapolis. American had Dallas, et cetera. They had these hubs that you'd go through. What that meant was, they essentially got a monopoly on the route into the hubs, because they had monopoly on the slots in the airport. Now you might say, well, that's easy to fix. Don't give monopoly slots in the airport. But that's harder to fix than you think. Because whenever the airport in Minneapolis would say to Northwest, we want to allow other airlines, they'd say, great, we're moving our headquarters out of Minneapolis. See ya. And they'd say, OK, fine, we won't do that. So essentially, there became political equilibrium where these local airlines were such big employers, they'd bully the governments into letting them dominate the airports. And they essentially recreated a monopoly. But the monopoly wasn't on planes. It was on airport slots. And actually, flying got much more expensive again. So the price of flights really came down until airlines figured out this hub and spoke system, and they've gone back up since. Now, they're still cheaper than they were under regulation. But roughly speaking, the price fell by more like more than 50% initially, and then sort of risen back up. It's still lower than it was under regulation. But it's risen back up because there's this sort of new natural monopoly problem that we didn't see. So that leads to crazy things, like a price of a nonstop to San Francisco today is something like half as much from Boston as a flight to Minnesota. Now, I don't know how your geography skills are, but San Francisco's a lot farther from Boston than Minnesota is, so it's clearly not a marginal cost issue. So that's an example. It's sort of a nice case study of kind of, our motivation was right, by and large economists got it right. By and large, we're better off in a deregulated world. But not as well off as we thought, because we missed this other element of natural monopoly. Why don't I stop there. We will continue next time by talking about oligopoly.

MIT_1401_Principles_of_Microeconomics_Fall_2018

18_Increasing_Savings_Introduction_to_Trade.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: Today, we're going to finish our discussion of savings, continue my nagging on you guys about how you should be saving money. And then we're going to move on and talk about international trade. So let's finish our discussion of savings. Now, savings turns out to be a critically important element of growth in the economy. And we've now traced through why that happens. Our basic story is as savings goes up, that means that the capital supply shifts out. So the capital supply curve shifts outward as people save more. That's an increase in the capital supply curve in the capital markets. That means that interest rates fall. Because there's more supply for given demand, the price is going to fall. The price of capital is the interest rate. So as people save more, that's more supply into capital markets. It's increasing the pool from which firms have to borrow. For a given pool, firms have to pay less money to borrow from that. If the interest rate goes down, that means the NPV of investment goes up, because remember, the net present value of investments is a negative function of the interest rate. The lower the interest rate, the more firms will say I might as well buy a new machine, because the bank's not paying anything. That means that NPV investment goes up, which means investment goes up. So the bottom line is, the way we get firms to invest more and grow the economy is by saving more. We save more. We increase capital supply. That lowers the interest rate. That raises the NPV of investment, which leads to more investment. So that's the link by which savings lead economic growth, is that savings leads firms to invest more. Is that clear from the sort of structure we built in the last couple lectures? If it's not, just go back through and work through the math. But you'll see we talked about the capital market. The interest rates, the price, and the capital market-- last time we talked about the lower the interest rate, the higher the net present value of any given investment. That means firms will invest more. So that means a critical public policy concern is getting people to save enough. Now, obviously, that varies with economic conditions. In a deep recession, we don't naturally want as much savings as when we're not in recession. But in general, in the long run, we will grow more as an economy if we save more as a society. And the US has an incredible low savings rate. Our savings rate in the US, depending how you define it, is maybe 3% to 5%. In Europe, in Japan, it's like 15-plus percent. So we have a very low savings rate. And as a result, that has led public policymakers to try to think about tools they can use to encourage savings. And the major tool we use in public policy to encourage savings is the tax subsidy to retirement savings. The tax subsidy to retirement savings is the major tool that we use to increase savings in the US. Now, how does this work? The basic logic is the following-- When I put my money in the bank and I earn interest, that interest gets taxed. Just like my labor supply gets taxed, my capital income gets taxed. So when I put money in the bank, I don't just earn the interest rate. We'll use our-- we don't care about real interest rate. [INAUDIBLE] earn interest rate. I earn r times 1 minus tau, where tau is the tax rate. That's all I take home. So the bank pays me 10%. Say inflation is zero, so nominal interest rates are the same. If the bank pays me 10% and my tax rate's 50%, I only take home 5% on that. If we assume substitution effects dominate-- which is a big assumption, but typically one we make-- this means the taxation, by lowering the return to savings, will lead to less savings. So by taxing people's savings, we lead to less of it, because we sort of assume substitution effects dominate. To offset this, we say, well, I'll tell you what-- if you save for retirement, we won't tax it. So if you save it, and you pull it out next year, we're going to tax it, but if you put it in special accounts which we've labeled as retirement savings, we won't tax it. So these have a number of forms. One form is employer sponsored pensions. These are things where your employer takes some of your pay, puts it aside in an account. And when he does that, you're not taxed on that pay. So if MIT is going to pay me $100,000, and they pull $10,000 aside and put it in a pension, I'm only taxed on $90,000. That $10,000 isn't taxed. And likewise, the interest that's earned on that isn't taxed. There's also 401(k)s. A 401(k) is like a pension, but where you control the money. So when you get a job, you might get offered a 401(k). That's something where some of your money gets pulled out of your salary. It doesn't get taxed, and it gets saved instead, and you control where it goes. And then we also have individual retirement accounts, with the unfortunate name IRA. It's not the Irish Republican Army, but individual retirement accounts, which are similar features, where you could take your money and save it on a tax-free basis. Now here's the trick about how all these things work, is they're not actually tax free. They are tax deferred. And what do I mean by that? What I mean by that is that when you take the money out eventually, it does get taxed. So the way, say, your 401(k) would work-- you get a job at whatever, Google. Google offers you a 401(k). You put money in it. That money then accumulates. And when you take it out-- you put money in. That money is not taxed when you put it in, but when you take it out, it is taxed. So it is eventually taxed. So what good does that do you? What good does that do you if it's going to get taxed eventually anyway? Yeah. AUDIENCE: If you don't get taxed right now, you get more money now, which will then accumulate more money through compound interest. JONATHAN GRUBER: Right. It takes advantage of present value. Remember, we talked about this last time. With present value, money in the future is worth less than money today. By that same logic, paying taxes in the future costs you less than paying taxes today, because you have the money. You can earn all this interest on it, and you have to pay taxes on it off in the future. So let's see an example of that. Let's look at figure 18.1. Let's say you have two types of accounts, a regular account, or an IRA account, which is a tax-deferred account. And let's say the tax rate is 25% and the interest rate is 10%. And let's say you're just going to put the money in-- to make life easy, imagine put the money in for one year. Imagine you're retiring next year. This is a super easy example. You're retiring next year. The question is, for this last year of work, should you put your money in a tax-deferred account or in a regular account? If you put it in a regular account, you will take your $100 of earnings to pay $25 of taxes right away. You only get to put $75 in. On that $75, you'll earn $7.50. And when you take the $7.50 out, you'll pay $1.88 in taxes. So you pay $25 on your earnings. You pay another $1.88 on the interest you earned. And you end up with $80.62. Now let's say instead, you set up an IRA. There, you get to put the whole $100 in. That $100 earns $10. And when you take out $110, then you pay tax on it, so you pay $27.50, then, in the end. But if you put it all together, you end up with more money-- $82.50 rather than $80.62. Why is that? It's because you delayed paying taxes by a year. By paying taxes one year later, you got to earn the interest on that money during the year. Think of it this way-- if you pay taxes now, the government gets the money and they get to earn interest on it. If you pay taxes next year, you keep the money and you earn interest on it. So it's much better to pay taxes later, and that's why these accounts matter. It's a simple example. It's a simple example. If you have a 30-year account, if we kept the interest rate at 10% and the tax rate at 25%, and did a 30-year calculation, you would have twice as much money after 30 years, if you put it in an IRA rather than putting it in a regular bank account. So even though it eventually gets taxed, it's a big advantage. It's a big advantage. And that's why public policy introduced these tools-- 401(k)s, pensions, IRAs-- because they're trying to encourage saving by raising the return to saving. And they hope that by encouraging saving, they'll get all this good stuff coming out of it. So that's why these things come up. So the first lesson is, just like my continued nag on you, is save early and save often. Things like 401(k)s, by putting your money away, not only do you get the compounding I talked about before, but you also get the tax benefit of paying taxes later rather than sooner. You get the compounding on the money and the compounding on the taxes. So these kinds of retirement accounts, when you get your job and you're thinking about why would I worry about retirement, worry about retirement. There's a big advantage. Then you ask, that's fine, John, but when I get my 401(k) paperwork, I've got like a million things I could invest in. What I do with the money? So let's talk for a couple of minutes about investment strategy, about what you do with the money. So let's think about three different ways you can invest your retirement money, three class of options a 401(k) will typically have. They'll typically be a money market fund, a bond fund, and a stock fund. And there will be various combinations of these, but these are the categories. What do these mean? Money market means the money is invested in government bonds. The money is invested in what are called "Treasury securities," government bonds. These are things which are ultra safe, as long as the US doesn't default on its debt, which we've never done in our history and, the good Lord willing, won't do in the near term. Because the US doesn't default on its debt, you get paid back. These are super safe, the safest place you can put your money. But they pay a very low interest rate. Right now, they're paying interest rates, a typical government bond fund right now would be paying interest rate of around maybe 2%, maybe 1% to 2%-- very low interest rate, maybe up to 3% now, but very, very low single digits, but totally safe. A bond fund invests your money in corporate bonds. Basically, this is making loans to corporations. Instead of loaning money to the government, you loan the money to a corporation. You're not actually buying a bond in GM. That's too hard. So what these 401(k) accounts do is they say, we're going to buy a bunch of bonds. We're going to put them together, and let you own a small piece of that entire set of bonds. That's what a bond fund is. So you own a small piece of bunch of corporate bonds. The difference is, unlike the government, corporations do go out of business all the time. So these are riskier. These are riskier, but they pay a higher interest rate. Bond funds typically paying 4% to 5% right now. Then finally, the last thing you can do with the money is you can put in stocks. You could literally own corporate equity. You could own a piece of companies. You could own a piece of the companies, and then what you get is you get something, it doesn't really matter. With these two, it only matters-- you get paid as long as the government or the company doesn't go out of business. Here you get paid in proportion to how well the company does. If it does really well, you make a lot of money. Stock prices go up. If it goes badly, you lose money. It goes bankrupt, you lose everything. So similar to these, in the bankruptcy, it costs you everything. The difference is these, either you get paid or you go bankrupt. Here, it's much more variable. Things do well, you get more. Things go badly, you get less. So this is the riskiest option of all, but it also pays the highest rate of return. Traditionally, stock investments pay a long rate of return about 7% a year. So what we see here when we compare these is what economists call the "risk-return trade-off." The riskier things are, the higher you earn by investing in them, but the riskier that earnings is. The more [INAUDIBLE] invest in them, but the riskier that earnings is. That in other words, people are willing to accept a lower interest rate for a safe investment than they want for a risky investment. And we'll talk about, in two lectures from now, why that is, why people's preferences are that way. It's because people are what we call "risk averse." People don't like risk, and we'll talk about why that's a natural way to be. So basically, that is your set of choices. And what economists recommend is that the key in all of these investments is to diversify. The key recommendation of economists is diversification, that you spread your money across these options to get the right balance of risk and return. So that basically, if you're someone who-- now, how you balance depends on your taste for risk. For someone who really hates risk, you put most of your money here. But it's silly to put all your money in money markets, because you're losing a lot of return. If you're someone who's risk loving, you might want to put most of your money in stocks. But still don't put all your money in stocks, because you have a much higher risk. You want some safety behind you as well. So an economist would say you should diversify. Now, that's a general. I can't tell you what percentage is the right percentages. I can tell you to diversify. But I can give you one specific piece of advice-- that when you go work for Company X, the one thing you do not want to do is put your money in the stock of Company X. Why is that? Why is the last thing you want to do when you go to work for Company X to put a bunch of your money in the stock of Company X? Yeah. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: No, this would be through a 401(k) [INAUDIBLE]. It wouldn't be like that. Yeah. AUDIENCE: If you lose your job, or if the company shuts down, you can lose both your job and the money. JONATHAN GRUBER: That is the maximally risky strategy, because you are tying your risk of your investments to your risk of your salary. That's the most risky thing you can do. So absolutely what you would not want to do is go work for Google and say, good, I'm going to put all my retirement in Google stock. You might think Googles safe, but safer companies than Google-- companies thought to be safer than Google at some point have gone bankrupt. Google is bankrupt, you're totally screwed. You've lost all your savings and your job, so the last thing you want do is that. Yeah. AUDIENCE: If you're in a more corporate position, and they gave you bonds as promises. JONATHAN GRUBER: Well, I mean, certainly a lot of companies give you stock options, where you're sort of stuck in the stock of that company. And my lesson there would be, that's part of the conversation, but as soon as it's best, sell it and get out. That doesn't mean you shouldn't-- you should value-- it's not saying stock options in the company are worthless. They're worth something. It's just they're worth less than if they just gave you the cash, because if you got the cash, you'd go invest it in somewhere that wouldn't be tied to your job. And this came to a head with the example of Enron, which is sort of before your time. That's sort of when you were young. But basically, Enron was an energy company that got into some really shady dealings to try to essentially prop up its stock price. To make itself look valuable it essentially created shell companies that were the same company it sold to them. So it looked like it was generating a lot of sales, but it was just selling to itself. It was illegal activity. But it caused their stock price to go through the roof. And Enron, in order to have enough money in the company, encouraged their employees to invest all their retirement savings at Enron. Indeed, the Enron 401(k) said, we'll give you extra money in your account if you invest your money in Enron. So most people at Enron had their retirement savings invested in Enron. When the whole thing-- and Enron was a very successful company, in the Fortune 500, doing quite well. When the whole thing collapsed, these people not only lost their jobs, they lost their entire savings. That's the problem with non-diversification. And the ultimate form of non-diversification is to invest in the company you're working for. Have I scared you enough? Any questions about that? Now, let's move on. That finishes our discussion of savings. Let's move on to a totally new topic, which we'll talk about this lecture and the rest of next lecture, which is international trade. And let's start with what is international trade and why the ruckus? This is the great time to be teaching on international trade, probably the best, most exciting time of any year I've ever taught this topic. It's front and center in a way it hasn't been, probably, in decades. So what's the big deal? Let's talk about a simple example. How many people have ever given someone roses for Valentine's Day? Raise your hands. MIT, man-- that's OK, when I was your age, I hadn't either. Now, here's the interesting problem with Valentine's Day-- Valentine's Day falls in the winter. Roses don't grow outside in the cold in the winter. So what do you do? For many years, what we did was we had hot houses where we grew-- basically were dedicated, their reason for existing, was essentially to grow Valentine's Day roses. So it's an industry around growing these indoor roses in the winter so that you could have them available. But what happened over time is we realized it was a lot cheaper to actually by the roses in Colombia or countries like that, and fly them up, than to actually have them grown here. So what happened was all the roses we get now in the winter come from South America. And we don't grow roses in America anymore in the winter. So is that a good thing or a bad thing? Well, on the one hand, roses are a ton cheaper now than when I was a kid. So that's a good thing for the romantics among us. You can get a dozen roses for 25 bucks. It was like 80 bucks when I was a kid or something. It was crazy. On the other hand, a bunch of guys who used to grow these roses are out of jobs. There was an industry devoted to growing these roses and these people now have no job, because they're grown somewhere else. So how do we think about-- this trade-off is sort of a microcosm of the larger debate around international trade. The larger debate about whether trade is good or bad comes down to this rose example. It sort of summarizes that issue. So now, when we think about international trade, we think about three concepts. We think about exports, which are the amount of goods that we sell to other countries, and imports, which are the amount of goods that we buy from other countries. So a country exports to other countries. It imports from other countries. Currently, the US exports about $1.6 trillion of goods a year. That's trillion with a T. It imports about $2.4 trillion of goods a year, leaving us with an $800 billion trade deficit. You may have heard the President mention this once or twice-- an $800 billion trade deficit. So the question you want to ask is, how big a problem is this that we have a trade deficit? And the answer economists give is none problem. No? No Spinal Tap fans here? OK, whatever. So basically, let's explain this in a simple example. Imagine that you have two Pikachus and your friend has two Jigglypuffs. And you want to have a more diverse set of Pokemon, so you go to your friend and you say, I will trade you one Pikachu for one Jigglypuff. So you send your friend a Pikachu. He sends you a Jigglypuff. You've just created a massive Pikachu deficit. Think about it-- you used to have two Pikachus. You sent one away. You sent one away. You didn't get any Pikachus back, did you? So you've created a huge deficit of one Pikachu. Is that bad? No, you got a Jigglypuff. You're happy. You wanted to make the trade. You made the trade. Calling the trade deficit, define trade deficit in terms of-- if you define trade deficit, instead, in terms of Jigglypuffs, you've got a huge trade surplus. You used to get no Jigglypuffs, now you get one. So basically, a trade deficit, trade surplus, is all about an arbitrary definition. The bottom line is America then spends $800 billion more. You're sending money to other countries, but we get $800 billion year more stuff for it. There's no bad or good. It's just the way trade works. So any time you've had a trade in your life of anything, with a friend or whatever, you've created a deficit and a surplus. But that doesn't mean anyone is worse off or better off. Whether you're worse off or better off depends on the indicators of the trade, depends on how much each party values what's going on. So basically, if I said, oh, good Lord, if I had a headline like, "Good Lord, Gruber Trades Pikachu Deficit, Must End Trade with Jigglypuff Land," we must shut down trade, because good Lord, we're creating this huge Pikachu deficit, that would be bad. Because I wanted to trade my Pikachu for a Jigglypuff I was happier. That would be a bad thing to shut that off. And that, in a stupid example, is why economists like international trade. It's the same reason we like trade in general. Think about this whole course-- the whole course is a bit about trade. It's about trading our money for cookies and pizza, about trading a firm's money for workers and machines, about trading my time for a wage. Life is all about trades. International trade is just another example of that. Just because it's got this bizarre label of a trade deficit, we think about it differently. So that's sort of basically, if you think about-- now let's replace Pikachus and Jigglypuffs with the US and a poor country. Instead of US having two Pikachus, the US has tons of money. Instead of the poor country having two Jigglypuffs, the poor country has really underpaid workers who can make stuff really cheaply. So I, in the US, say, wait-- I have a lot of money. You've got a lot of sweaters. I'm going to send you my money and get your sweaters. I'm better off because I was happy to buy those sweaters cheaply. You're better off because you can't eat sweaters. You need money, so you get some of my money. We're both better off, but I've created a trade deficit. That is not inherently a problem. Now of course, the reality is more complicated. The reality is that when I send the money to those countries to buy the sweaters, and I bring the sweaters in, my consumers of sweaters are much happier. And I keep focusing on sweaters because it's just amazing. This sweater I just got at Old Navy, $15. Pretty nice, decent, it's not cotton, whatever. $15. This would have been a fortune when I was your age. In those day's dollars, it would've been $30, and that would be like $70 today. Why? China. Because we used these to make these in North Carolina, now we make them in China. So the good news is, I went to Old Navy, I got two pairs of pants, and like four sweaters, like $100. That's the good news. The bad news is the guys in North Carolina lost their jobs. That's the bad news. So there's a trade-off. Now, I'm going to argue over the next lecture and a half that that trade-off is worth it in aggregate, but we can't ignore the fact there's a trade-off. Because that's what leads to the very intense debates around international trade, is the fundamental tension between the consumers of goods that are made cheaper by international trade and the producers of goods that get wiped out by international trade. So that's our setup. That's our big picture. Now I'm going to dive in, and I'm going to teach you the nitty gritty of how we think about this with models, but that's the big picture I wanted you to have in mind. So let's dive into the models. And to do the models, I need to introduce a new tool to you, which is the Production Possibility Frontier, the PPF, the Production Possibility Frontier. What the PPF shows you is the maximum combination of outputs you can produce for any given set of inputs. So basically, let's think about that. Let's go to figure 18.2, because it's easy to see this graphically. Let's go to figure 18.2. Let's think of you as a firm. And you as a firm produce two things-- problem set points and exam points. Sorry, it's all you're good for at MIT here. You produce problem set points and exam points, but you can produce either. Those are the two things you produce. And to make life easy, let's say given your innate intelligence, the points are produced as a function of time. Literally the more time you put in studying for an exam, the better you do, the more time you spend on the problem set, the better you do. And you have one scarce input, which is time. This production possibility frontier tells you if you devote your scarce input across different activities, what will the result be in terms of what's produced. So what's this line saying on the left-hand side? It's saying, if I put all my time into studying for exams, I'll get 200 points on exams and 0 on problem sets. If I spend all my time doing problem sets, I'll got 200 points of problem sets and 0 on exams, and some combinations in between. That is a production possibility frontier. It shows the combination of output you get for any given fixed level of inputs. Now, what is wrong with the graph on the left? And why is the graph on the right a probably more realistic description of what your actual production possibility frontier looks like? Yeah. AUDIENCE: You know, if you can get more points on the problem set, and you know more, you're probably more well prepared for the exam. JONATHAN GRUBER: Yeah. It's wrong because in fact, what this system features is something we call "economies of scope." We talked about economies of scale a number of lectures ago, the notion that if you double inputs, you double output. This is economies of scope, which is that by producing more than one thing at once, you may get better at both. By studying for an exam, you get better at problem sets. And by doing problem sets, you do better on the exam. So what that means is, doing some of both leads to a better outcome than doing all of one or all the other. You get a concave to the origin production possibility frontier. A concave to the origin production possibility frontier demonstrates economies of scope. Doing some of both is better than doing all of just one and all the other. And that makes sense for you when doing problem sets and exams. It makes sense you'd be better off doing some of both than devoting all your attention to just one and just the other. So that is a production possibility frontier. Now, there can also be dis-economies of scope. Another great example for MIT is that when I was here as an undergrad, I came in as a tennis player. And I was on the tennis team. Now tennis was only in the fall and spring. In the winter, they had squash. So I thought I'd play squash. Well, it turns out that the key with tennis is keeping your wrist firm, and the key with squash is keeping your wrist flexible. And by playing squash, I screwed up my tennis game, and by playing tennis, I screwed up my squash game. So I had a dis-economy of scope. I was worse off at both things by playing both, rather than just playing one or the other. So there can be economies of scope, which is when doing both things make you better at both, and dis-economies of scope when doing both things make you worse at both. And that's the nature of the shape of production possibilities frontier. An economy of scope is when doing both makes you better off. A dis-economy of scope is when doing both makes you worse off. Questions about that? Now, I just introduced that tool because that tool is going to be the fundamental modeling feature that allows us to demonstrate why international trade is beneficial. So now, we're going to take that tool and we're going to go and talk about the concept of comparative advantage, which is the core concept in international trade economics. This is the core of how it all works. Let's think of a particular example to understand this. Let's take the US and Colombia and the rose market. And as I said before, let's say it's expensive for the US-- and the Valentine's Day rose market. It's expensive for the US to grow Valentine's Day roses. It's cheap in Colombia. On the other hand, let's compare roses to computers. Roses are cheap to produce in Colombia and expensive in America. Computers are the opposite. We have giant factories, and skilled labor, and giant machines that can quickly crank out computers. In Colombia, they'd have to like assemble the computer by hand. So it's a lot cheaper to produce a computer in the US than it is in Colombia. It's a lot cheaper to produce a rose in Colombia than it is in the US. The way to think about this is through the lens of opportunity cost. What we're saying is the opportunity cost of making a rose in terms of computers is higher in the US, In other words, the amount of computers you have to give up to make a rose is high in the US, because computers are cheap and roses are expensive. The amount of roses you have to give up to make a computer in Colombia is higher, because roses are cheap and computers are expensive. So we say that Colombia has a comparative advantage in roses. It is relatively cheap-- in a world of computers and roses, it is relatively cheap for them to produce roses compared to computers. The opportunity costs in terms of forgone computers is lower. Whereas the US has a comparative advantage in computers. It is comparatively cheaper for us to produce computers. So international trade is all about what you're relatively better at. Now, let me emphasize "relative" for a second. This is an important concept. The reason we don't just say "advantage" and we say "comparative advantage" is it doesn't actually matter if you're better. It matters if you're relatively better. Let me explain. This is hard. I would say of the main economics concepts in the world that we need people to understand, this is one of the top three least understood concepts in all the world in economics. [INAUDIBLE] it's like in our blood, and we can't understand why regular people can't understand it. It's because it's hard. Let me give you an example. Take me and LeBron James, and imagine there's two activities in life, mowing the lawn and playing basketball. That's all life consists of. Now, LeBron James is better than me at both playing basketball and mowing the lawn. But he's much, much, much, much, much, much better than me at basketball and only much better than me at mowing the lawn. So LeBron James has an advantage in everything, both activities, but he only has a comparative advantage in basketball. He only has a comparative advantage in basketball. The comparative advantage is about opportunity costs. In other words, if LeBron James mows his lawn, the amount of basketball he's giving up is ungodly. It's crazy to have to mow his lawn, given how much basketball he could've played. If I mow my lawn, I'm not giving up much basketball, because I suck at basketball. So I have a comparative advantage in lawn mowing. You say, you're not at lawn mowing than LeBron James. I'm not. But I have a comparative advantage in lawn mowing, because the opportunity cost to me is much lower. The opportunity cost to LeBron James of mowing his lawn is high, because he could be playing basketball. The opportunity cost to me is low, because I can't play basketball. Yeah. AUDIENCE: [INAUDIBLE] the only one in the world that can mow lawns. JONATHAN GRUBER: That's a weird edge case. I won't do that. Then there'd be no reason for-- that makes trade things hard. We can come back to that. Now of course, the example doesn't stop there. Should I mow my lawn? No, I should not mow my lawn, because there is someone who doesn't have as much education as I, who can't earn as much money than I at work. And they have a comparative advantage in lawn mowing. I have a comparative advantage in office work. So in fact, not only should I mow-- do I have comparative advantage over LeBron in lawn mowing, but some less-educated guy has a competitive edge over me in lawn mowing. So just as LeBron would be better off letting me mow his lawn and I'm better off letting him play basketball, I'm better off letting some less-educated guy mow my lawn, and letting me go sit in front of my computer all day. Comparative advantage is all about what you're relatively better at. And the key insight of international trade is that in a world of comparative advantage, people should always specialize. In a world of comparative advantage, people should always specialize. You should do what you're best at. You should do what you're best at because otherwise, it's simply silly for LeBron to spend any time mowing his lawn. As much as my wife thinks I'm just trying to get out of it, it's simply silly for me to spend any time mowing my lawn. I have a comparative advantage sitting on my computer and doing that stuff, so I should hire someone else to mow my lawn. So to actually see that, let's go to figure 18.3. This is a confusing figure, so I'm going to go through it slowly. On the top, we have the US production possibilities frontier. And to make life easy, let's assume that there's no economies of scope between roses and computers. Makes sense there wouldn't be. Let's assume there's no economies or dis-economies between roses and computers. It's a linear production possibility frontier. So we're showing each graph as a production possibility frontier between roses and computers. And we're assuming it's just linear, which makes sense. You don't make better computers by making roses and vice versa. On the top, we have the US. Let's say that the US's production possibility frontier is the following-- given the resources we have in the US, we can produce 2,000 computers or 1,000 boxes of roses, given our resources, our skill level, our capital intensity, et cetera. Now go to Colombia. Say that Colombia, given their skill level, resources, the sunshine, the beautiful weather-- any Colombians here? Sounds perfect. It's like 75 all the time. They can make 2,000 roses and 1,000 computers, or 1,000 computers, or some combination in between. So the first thing I want to make you guys understand is why these are sensible production possibility frontiers for each country. They're sensible production possibility frontiers because they basically show the US has a comparative advantage in computers. That is, the trade-off in terms of roses foregone to make a computer is lower in the US than Colombia. And Colombia has a comparative advantage in roses. The trade-off in terms of foregone computers to make a rose is much lower in Colombia than in the US. So comparative advantage is about opportunity cost. Let me say it again-- the US has a comparative advantage in computers because to make a computer, we have to give up less roses than Colombia does. To make one computer, we give up half a box of roses. That's the slope of this line. Colombia, to make one computer, has to give up two boxes of roses. So we have a comparative advantage in computers. Yeah. AUDIENCE: Would it be possible to have comparative advantage in both? JONATHAN GRUBER: You cannot. That's the term "comparative." You can have an absolute-- in this simple two by two model. You can have comparative advantage in multiple things in life. From the simple model, you can have the absolute advantage of both, but you can't have a comparative advantage in both. That's absolutely right. In this model, with two goods and two countries, each country has a comparative advantage in one thing or the other. Or there could be weird edge cases, but generally that's right. So now, I did computers. Now let's flip and do roses. In roses, for the US to produce a box of roses, they have to give up two computers. For Colombia to produce a box of roses, they only give up half a computer, so they have a comparative advantage in roses. People understand the setup here, these graphs on the left? Yeah. AUDIENCE: Does the price of what you can sell each unit for factor into what you actually produce? Because I can imagine that yes, maybe you can produce more roses-- JONATHAN GRUBER: But they're worth less. AUDIENCE: --more easily, but they-- you can't sell a rose as much as you can [INAUDIBLE] JONATHAN GRUBER: Right. So for right now, we're ignoring prices. We'll come back to prices later, but for right now, we're ignoring the prices. Right now, we're just sort of-- we're going to basically have prices come in later, but for right now, we're doing prices. So now, let's imagine that tastes are such that consumers in the US want 1,000 computers and 500 boxes of roses. That's point C US. And consumers in Colombia want 500 computers and 1,000 box of roses. I just made this up. C US and C CO, I just made up tastes. I just said, let's just make it a case where people in Colombia like roses better, people in America like computers better. And imagine we do not allow international trade. This is the best case. Actually, international trade would be even more valuable with a different assumption. I've made assumptions which make trade less valuable than it would otherwise be. I said look, Colombia produces roses and people there like roses. American produce computers and people there like computers, but not totally. Some US guy still wants some roses. Some Colombian guy still wants some computers. Now imagine we don't allow trade. What's the outcome? Well, in the US, we will produce-- like I said, we want 1,000 computers and 500 roses, so we'll produce 1,000 computers and we'll consume 1,000 computers. So on the chart on the right, you've got production and consumption. For the US, we'll produce 1,000 computers, consume 1,000 computers. We have to consume what we produce. There is no trade. And same with roses-- we'll produce 500, consume 500. Colombia's the flip. So you end up in the world with 1,500 computers being produced and consumed, and 1,500 roses being produced and consumed. Now, you might say that's silly, computers cost more than roses. We'll come back to prices. For now, we're leaving prices out of it. The point is, given the tastes I've suggested with C US and C CO-- I just made those up-- given those tastes, we'll end up in a situation where every country just consumes what they produce, because there's no trade, and the world as a whole will produce 1,500 boxes of roses, 1,500 computers. Now let's say that we allow trade. What trade does is introduce economies of scope. Why? Because trade allows specialization. Flip to the next figure. The next figure is the figure for the world. Imagine the world only wanted computers. That's the point on the y-axis. Then the US would produce 2,000, and Colombia would produce 1,000. We'd have 3,000. Imagine the world only wanted roses, the flip side. We have 3,000. But if the world wants both, you get more. Why? Why is it outward bending when you allow trade? This is not a technological change, change in technology. All we've done is take these two PPS and combine them and allow trade. Why does that suddenly introduce economies of scope? Because what do countries get to do compared to what they were doing before when they couldn't trade? What do they get to do now? AUDIENCE: Specialize. JONATHAN GRUBER: Specialize the US specializes in-- and Colombia specializes in roses. By specializing, we can make more, because that's what we're good at. So by trading, we allow specialization. Without trading, we can't specialize, because some guys in the US want roses. We've got to make some roses. But that's stupid. We shouldn't make roses. We should let Colombia make roses. If LeBron can't hire anyone to mow his lawn, he's going to mow his lawn. That's stupid. LeBron should hire someone to mow his lawn and play basketball all the time. By allowing trade, we allow people to specialize and take advantage of their comparative advantages. Without trade, you can't take advantage of comparative advantage. Without trade, it doesn't matter if the US is better at producing computers or roses. What's produced is just determined by people's tastes. But with trade, you can specialize. And that yields the outcome we discussed in figure 18.5. Figure 18.5 takes the previous figure, 18.3, and adds trade. So let me go through this. The basic, the blue, the PPFs are the same on the left, same PPFs as before. And if you look at point CO and C US [INAUDIBLE] before, we could have 1,000 computers and 500 roses in the US or 1,000 roses and 500 computers in Colombia. But now suppose we allow trade. What happens now? Well, what happens now is the US goes to producing only computers and Colombia goes to producing only roses. Now let's assume, just to make life easy, let's just assume that people's tastes are proportional. So if there's more computers and more roses, they just want proportionately more of each. So the US, as there's more of both, wants two computers for every rose. And Colombia wants two roses for every computer. That's just their tastes. What happens now is we shift out to these new levels, C prime US and C prime Colombia. At the new levels, the US produces 2,000 computers. All they do is computers. Colombia produces 2,000 roses. The world as a whole ends up with more. Flip back two pages and look at the table on the right. The world as a whole ended up with 1,500 roses and 1,500 computers. Now the world as a whole ends up with 2,000 roses and 2,000 computers. What happened? Nothing changed technologically, same PPS here as before. All the change is by trading. We allowed people to exploit their comparative advantage through specialization. Without trading, the US was inefficiently producing roses and Colombia was inefficiently producing computers. With trading, we've now opened up the possibility that they can specialize in what they're good at. And the world as a whole ends up better off. And look at both countries. Both countries end up with more consumption. The US gets more computers, so flip back and forth between 18.3 and 18.5. The US goes from 500 roses and 1,000 computers to 750 roses and 1,250 computers. Colombia goes from 1,000 roses and 500 computers to 1,250 roses and 750 computers. Now, the split between roses and computers here is not determinate. That's just made up. I made up what tastes look like. The point C and C prime, that's just made up. What you care about is the bottom panel. What you care about is fact that the world as a whole now has more roses and more computers in it. And that is the mechanics of why international trade expands our production possibility frontier. We have more goods in the world once we can trade, not because-- I mean, you still have the same, where Pikachu and-- when I allow trade between you and your friend, that doesn't create any more goods. You each have two cards. But in the real world, it creates trade, because the production allows people to specialize. And that's the beauty of trade. We'll come back next time and do a welfare analysis, talking about why international trade makes people better off.

MIT_1401_Principles_of_Microeconomics_Fall_2018

7_Competition_I.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: Why don't we get started. We're going to start by finishing up our lecture on costs, with a concept I didn't get to cover last time. Then we'll move on to talking about competition. So I want to start by talking about one concept on costs we didn't get the cover last time, which is fixed versus sunk costs. And sunk costs are an important term in economics, so I want to make sure people understand what they are. Sunk costs are essentially costs that cannot be changed, no matter what action you take from this date forward. So in some sense, some costs are long-run fixed costs. I know we said in the long run, nothing's fixed. But a sunk cost is the idea of an investment that once made, can never, ever be changed. So the thing is, if we think about it in the short run, let's imagine you're a doctor. In the short run, your variable costs are how many hours you work, or the nurses you employ, or the physician assistants you employ. In the long run, your fixed costs or how big your office is. You can get a bigger office or smaller office. But your sunk cost is having gone to med school. You can ever undo having gone to med school. Having paid that cost, it is sunk. It is gone. You're never going to-- you could go to more med school. You could take supplementary classes, but you can never undo having gone to med school. So that's the sense in which some things are sunk costs. They're investments that are made that can never, essentially, be undone. And essentially, we can think of these as long-run fixed costs, which I know is really confusing, because the long run is when costs aren't fixed. But that's essentially what they amount to. Now, sunk costs turn out to have a very important place in economic lore, because the thing about sunk costs is it's hard to think about them. It's hard to remember the rule that sunk costs are always sunk. So let me give you one example-- it's in one of my videos for the 1401X videos that supplement this course. But if you've seen it, I apologize. We'll go through it again, which is literally a case where I got this wrong two years ago. So about two and a half years ago, I made the decision that I take my wife to go see the band Journey. So I went and I bought tickets, and the tickets were $240 for the pair-- pretty good tickets at a big stadium, $240 for the pair. Then about a month before the show, I was looking, and realized I didn't actually like Journey that much. I like-- I love a couple of their songs. I looked at the set list. I'm like, I don't really like that many of their songs. There's a couple I love, but I don't want to sit there. It's a couple hours to get there. I don't want to sit there all night for two songs that I like. I want to sell them. So you can sell tickets through StubHub or other secondary mechanisms, but you have to decide what price you're going to ask. And I said, I gotta at least make back my $240. That's my goal. And then I realized I was thinking about it completely wrong. I didn't realize immediately, but I was [INAUDIBLE] about it completely wrong-- that the $240 was already gone. The fact I spent $240, given that I'd already spent it, is irrelevant. How should I think about it? I should only ask how much am I still willing to pay to see Journey? And as long as someone else is willing to pay more than that, I should sell them my tickets. And if they're not, I should go. So let's say, for example, I decided, I'd still pay $100 for my wife and I to go see Journey. Then I should say, as long as the tickets sell for more than $100, I'll sell them, but if it's less than $100, I won't. And the fact I paid $240 is irrelevant-- that's gone. That is a sunk cost. So a common [INAUDIBLE] is the sunk cost fallacy-- the notion we pay attention to things that don't matter anymore. Having spent that money was irrelevant. All that mattered was the decision looking forward, which was, do I want to go or do I want to sell the tickets? And that decision simply depends on how much I was willing to pay now. Now that's confusing. I've explained this. If you guys get it, you're a rare breed. I've tried this story on many people, and they're, like, no, of course it matters how much you paid. It doesn't. AUDIENCE: Does this kind of [INAUDIBLE] apply to the housing market crash, and homeowners are trying to sell their houses for way more than they're worth, because they're trying to get some of their money back? JONATHAN GRUBER: The sunk cost fallacy finds its name very much in the housing market. Many homeowners set as a benchmark what they paid for their house. They'll say, I don't want to sell less than what I paid. That's silly. Doesn't matter what you paid. What matters is how much you can sell it for and whether or not you want to sell at that price. And what you paid is irrelevant. Now, it might matter because there are borrowing constraints we'll get into. You might have a mortgage you have to pay back, and you can't afford to pay it back unless you sell for a certain price. But aside from that, let's imagine you just bought your house outright without a mortgage for $1 million. Then that's irrelevant. Whether you sell your house for $900,000 just depends on whether it's worth $900,000 for you to stay there or not. The million you paid is irrelevant, except if you have a mortgage and you've borrowed it. That can make it complicated. So basically, sunk costs are sunk. Now, I ended up selling the tickets for $220, so I did OK. But if I thought about it incorrectly, I would've said that's a bummer. I lost $20. And in the big picture, it is a bummer. I never should have bought them in the first place. But having made that decision to buy them, the fact I was only willing to pay about $100 at that point to go and sold for $220 meant I did well, not badly. Questions about that? It's a confusing example. None of your friends understand this. You could totally mess with your friends' minds by explaining this to them. Yeah. AUDIENCE: [INAUDIBLE] more I might make [INAUDIBLE] JONATHAN GRUBER: What I mean is, think of them-- I have a set of Journey tickets. I have a choice-- I can sell them or go. So the only question is, how much is it worth to me to go to Journey? I decided it was worth $100 to go to Journey. So I simply put them on StubHub, and said if they sell for more than $100, I won't go. If they sell for less than $100, I'll go. The fact that I paid $240 is irrelevant. Yeah. AUDIENCE: So, basically you've lost [INAUDIBLE].. You've lost overall you've-- [INTERPOSING VOICES] JONATHAN GRUBER: I lost $20. AUDIENCE: --by making the mistake of buying them. JONATHAN GRUBER: Yeah, ex ante, there was a mistake made. But having made the mistake, that's irrelevant. So that's an important concept to keep in mind, is sunk cost. Let's go on now and move on to the next topic, which is the focus of the next two lectures, which is perfect competition. Perfect competition. Now, stepping back, as we highlighted, a lot of what we did for producer theory is the same as what we did for consumer theory. We have isoquants and isocosts instead of indifference curves and budget constraints, but the same damn exercise, same math, same graphics. The difference was we have a series of tangencies between the isocosts and isoquants, and we didn't know which one was the right one to choose. With consumer theory, we pin that down by the fact our parents gave us a certain amount of money. For producer theory, there was nothing pinning down q. There's nothing pinning down our total costs we can afford to spend. What pins that down is an additional constraint we bring into the system, which is the market. So now, we're going to take that next step of actually deciding what a firm produces. What a firm produces partly comes from all the math we do before developing the cost curve. But it also comes from the fact that firm exists in a market. And we focused on the cost side. Now, we turn to the revenue side. The markets can determine what the firm can make from selling that good, and that's and that's going to pin down how much they produce. Now, we're getting set of three different market settings. Today, we'll talk about perfect competition, which is the market with many, many firms competing to sell a homogeneous good. I'll talk about it more precisely later. Then, we're going to talk-- that's one extreme. That's the extreme of markets that economists sort of dream about is a perfectly competitive market. The other extreme is monopoly, where there's one firm, only one firm that sells the good. And then in between, we have my favorite word in economics-- oligopoly, which is when there's several firms competing to sell a good, not as many as perfect competition, but more than one. That turns out to be super complicated. So what we do is we start with the two extremes, develop a set of intuitions and rules, and then we sort of hand-wave a bit around oligopoly. And that's where I introduce the notion of game theory, which we'll talk about some in a couple of lectures. So today, we're going to start with one extreme-- perfect competition. What is perfect competition? Basically, the technical definition of a perfectly competitive market is where producers are price takers. A perfectly competitive market is one where a producer doesn't have any influence over the price that they sell their good at. They are price takers. So I can't actually, as one-producer market, can't actually affect the market price. The market price is given to me. When will this be true? This will be true when the demand for a firm's output is perfectly elastic, when the demand for a given firm's output-- little q, not the market elasticity, but the firm elasticity-- when you have an infinitely elastic little q. So let's think about that case. Let's turn to the first figure. So we have little q2 and-- so we basically have little q, because it's the firm's output on the x-axis. The y-axis is the market price. The firm faces a perfectly elastic demand curve. That means that they can't change the market price that's charged. So what does that mean? That means that no matter what their cost function is, whether their supply curve 1 or supply curve 2, they always sell at price p. So shifts in the cost function-- that is shifts in the supply curve-- only affect how much you sell, not the price you get. That would be true in a perfectly competitive market. So what conditions make a perfectly competitive market? There's basically three conditions. The first is identical products. A market will only be perfectly competitive if all the firms are selling at least what consumers perceive to be identical products. They don't have to be technologically identical. But from a consumer's perspective, they have to be viewed as identical. The second condition is there's full information about prices. That is, consumers know what every firm is charging. So I go to market, and I know what every firm is charging. I have full information. And the third condition is that there's low transaction costs, or what you might call "search costs," that it's very costless for me to search across opportunities. It's very costly for me to care-- everybody's price is perfectly posted, and it's very costless for me to search across them. Sort of two and three are kind of related. So this is obviously never true, much like many of our assumptions are never true. But it's a useful benchmark for thinking about a lot of what we're going to think about. So you want to think about markets like this-- so for example, the classic case you might think of is eBay. If you go on eBay and you search-- I just bought a pair of 72-inch red shoelaces. There's 72-inch red shoelace or [INAUDIBLE] It turns out not quite-- some are flat, some are oval. But within oval 72-inch red shoelaces, there's literally no variation. They're 72-inch, red shoelaces. Now, they could be of different quality, and that might be unobservable. That's why it's not perfect, but it's pretty close to identical. I go on eBay, and all the prices are listed there. So I have full information on my prices, and they're easy to compare. Now, it's not perfect because, a, there could be unobserved quality differences. Some shoelaces may be made by cheaper manufacturers. Others are easier to break or fray. And the second reason is that I might not have full information about prices, because at least in the old days-- eBay's changed this-- they can price compare on the non-shipping cost price. Now eBay's fixed this. You get the price including shipping costs. So in the old days, you could shop, and think you had the cheapest deal, but it turned out when you add shipping costs, it wasn't. So even eBay, which is sort of the economist's dream platform, doesn't quite meet these conditions, but it's about as close as you can come. Now, the other example I like to point to is like buying little knickknacks in a tourist area. That basically, if you've ever gone to a tourist area and tried to buy a little knickknack like a replica of the Eiffel Tower around the Eiffel Tower, there's a bunch of guys with blankets, out selling them. And it's pretty easy to get see they're all the same replicas. And it's pretty easy to ask the guy, what do you want for your Eiffel tower replica? So that's another market. Now, I taught this as a casual example. Someone watched this video from when I did this four years ago, and went to the Eiffel Tower-- some guy in France-- and actually did the exercise of marching from blanket to blanket. And he found that when you got further from the Eiffel Tower, they're more expensive, because there weren't many people selling them. But once you got close to the Eiffel Tower, all these guys selling them, he found that everyone charged the identical price. And he sent me a little package of 24 little Eiffel towers. It was really cute. So that was sort of this exercise in reality. Now, the important thing to remember, though, is this is a perfectly elastic, firm demand. So I want to talk for a second about demand for a firm's good versus demand for a market's good, so firm versus market demand. These are two different things. And the way to think about this, think about the concept of residual demand. So you could think about the demand for a market, demand function for a market, as being some Q of p. That's a demand function of the market. That's this demand curve we've been looking at since lecture one. It's a downward-sloping function of the price. And we can then think of any given firm, little q's demand, q of p, as equal to big Q of p minus S0 of p, where S0 of p we call "residual demand." S super 0 of p is residual demand. It's what everyone else is supplying. So the demand to my firm-- once again, under these conditions, perfectly competitive market-- the demand to my firm is simply the total market demand minus what everyone else is selling. So if I'm going to set up my little tchotchke blanket by the Eiffel Tower, my demand is going to be the total demand for little Eiffel towers minus what everyone else is selling at on their blanket. The key point is, even with fairly inelastic, big Q, you can get really elastic little q. So let's do an example. Let's first differentiate this. If you do dq dp, what do you get? You get d big Q dp minus d S0 dp. Now, the first term is negative, because demand is downward sloping. This term is positive, dS0 dp. The higher the price, the more suppliers are going to be in the market. So it's a positive minus a negative. So already, we see d little q dp is going to be bigger in absolute value than d big Q dp, because dS0 dp is positive, and you're subtracting it. But we can actually go further if we assume firms are identical. Let's assume firms are identical, so that little q equals big Q over N. Assume there's N identical firms. And what that means is that S0 equals N minus 1 times little q. The supply of everyone else is the number of firms in the market-- that's an N, big N-- number of firms in the market minus 1, because that's your firm, minus little q, because you're all identical. Then we can rewrite this-- you can do the math at home-- we can rewrite this equation as-- under these conditions, we can rewrite this as the elasticity of demand facing firm i is equal to the number of firms times the market elasticity minus N minus 1 times the market supply elasticity. Nu is the market supply elasticity. eta is the market demand elasticity. eta i is the firm demand elasticity. So the firm demand elasticity is equal to N-- I keep mixing my little n's and my big N's. It's all the same thing. N times e-- so firm demand elasticity is equal to number of firms times market demand elasticity minus number of firms minus 1 times the market supply elasticity. We know this is positive. Supply curves slope up. We know this is negative. So we know it's a negative number. What's good about this formula is it gives us examples showing how much bigger firm elasticity can be than the market elasticity. So for example, suppose, as a simple example, suppose N equals 100 suppose it's 100 people selling Eiffel towers on their blankets around the Eiffel Tower. And suppose that the market elasticity is minus 1-- that is, it's an elastically demanded good. It's not super elastically demanded. Basically, we call that an elastically demanded good, minus 1. So it's sort of a 45-degree line demand curve, a 45-degree line demand curve sloping down. And let's also imagine that the elasticity of supply is 1. That's a 45-degree line, upward-sloping supply curve. These are pretty standard assumptions. It's what I drew in lecture 1, a 45-degree demand curve, a 45-degree supply curve, with 100 firms. That gives you that the firm specific elasticity is minus 199. That is virtually flat. That is approximately negative infinity, as far as these things go. So basically, even with just a regular-looking, downward-sloping demand curve, if it's 100 firms of independent marketing, you get a virtually flat firm specific demand curve. So it's not crazy that firms in a really competitive market would face essentially perfectly elastic demand. And that's the key thing that's going on in a perfectly competitive market, is firms themselves-- not the market demand, the market demand can be sensible-- but firms themselves face virtually perfectly elastic demand. If you're all selling little Eiffel towers next to each other, then if you try to raise your price by one euro, you're gone. No one buys from you. If you lower it by one euro, everyone buys from you. That sort of makes sense. Questions about that? Yeah. AUDIENCE: What was the difference between big Q and little q? JONATHAN GRUBER: Little q is the firm and big Q is the market-- very important to remember. Always remember. That's a middle of the night thing. When I wake you up in the middle of the night, you have to know that one. Little q is the firm. Big Q is the market. Now, I'm not guaranteeing I always get that right on the board. But I am guaranteeing if I get it wrong, one of you guys will correct me. So armed with this-- any other question about this? Armed with this, we now turn to how firms maximize profits, short-run profit maximization. This is what we've been heading for. With consumers, our goal was to model how they maximize their utility. With firms, our goal is to model how they maximize profit. Now, what is the key thing in the short run? The fact is that short run means we're going to make-- we talk about short run being capital is fixed, but there's one other assumption we're going to make about the short run. We're going to assume no firm entry or exit. It's sort of a complement of capital being fixed. Firms are you're in. You make your capital investment and you're in. You could stop producing, but you still already made your capital investment. Or in other words, in the short run, capital is a sunk cost in the short run. You're in. And once the market starts, no one new is coming in. So you sort of roll in this market. People have announced to begin. They've set up their blankets. No one else has come to set up a blanket and no one's rolling up their blanket and going home. So now, let's ask first question-- what is profit? Well, that seems sort of easy. Profit, as I wrote down earlier, is just revenue minus costs. And if you're taking one of those goddamn boring counting courses in course 15, that's where you'd stop. But you're not. You're take an interesting course 14 course instead, where we tell you that while this may be the correct definition of accounting profits, accountants simply say, you add up the revenues you make minus the costs you incur, and that's the profit. But economists say, wait a second-- that's not right, because we also have to account for opportunity costs, as well as cash costs. Let's do a simple example. Let's say you're going to start a website design firm when you graduate. And all the firm's going to be is you, some slave programmer you're going to hire for $40,000, and the computer they're going to use. And moreover, let's say you already have a computer sitting around. That's your computer. It's in pretty good shape. So you could just have the slave work on that computer. And you can just basically say, look, I'll pay this guy $40,000. I'll think up all the ideas and supervise them. And then that's the way the firm's going to work. Let's say you do that, and at the end of the first year, you have sales of $60,000. Then your accounting profits are the $60,000 that you earned minus the $40,000 you paid your employee. So you've made a $20,000 accounting profit. Why does that make no sense. In fact, why are economic profits, the true profits from this enterprise once you account for opportunity cost negative. What do the accountants miss that economists get? Yeah. AUDIENCE: [INAUDIBLE] something better. JONATHAN GRUBER: The opportunity cost of your time. You just spent a year doing this. You could have been off making a zillion dollars, like you guys all will. So we missed the opportunity cost of your time. Let's say you could have graduated and gotten a job at $80,000 a year. Well then, actually, the cost of the year is not just the $40,000 you paid the programmer. It's the $80,000 that you forgoed by working that year on your company. What else? What's the other opportunity cost? Much smaller, but still relevant. What could you've done with the computer? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Well, you could have done what? AUDIENCE: You could have sold it. JONATHAN GRUBER: You could have sold the computer, right. Now, used computers aren't worth that much, but let's say you could have gotten a grand for it. Well, that's an opportunity cost. If you would just gone on and used your-- now, you might get all the utility from the computer, whatever. Let's put that aside. But you just gave it to this guy. You never touched it, literally got rid of it. It was like you didn't have the computer that year. Then you could have sold it. Let's say you could have sold it for another $1,000. So now, you've paid him $40,000. You've given up $80,000 of earnings and $1,000 on the computer, so your true costs are actually $121,000. So actually, you haven't made $20,000. You've lost $61,000. And that's why we don't care about accounting profits. We care about economic profits, the difference being economic profits account for opportunity costs-- very important to remember. Yeah. AUDIENCE: The profits, economic profits will be higher than financial profit or typically [INAUDIBLE]? JONATHAN GRUBER: It all depends on-- it depends on there's no way to assign it. yeah. AUDIENCE: You've been talking about sunk costs and opportunity costs. Is it like [INAUDIBLE] economics to consider time to be a sunk cost, like if you spend a lot of time on something? JONATHAN GRUBER: Well, no, it's not. I mean, in some sense, a sunk-- basically, sunk costs is sort of an irreversible fixed cost. If you spend time on it, yeah, in that sense, from today's perspective, it's sunk. But the point is, the more important concept is that time is money, that if you spent your time running this business, that's time you could've spent doing something else. Now, one thing we'll come to in a couple lectures, you might say, well, hey, I would've taken the year off and screwed around. So in fact, there was no $80,000 cost. What if instead of running this company, the only thing I want to do in my life is run the company. My second choice is watching TV. So then there's no opportunity cost. That's wrong. And we'll teach you why in a few lectures. I don't want you to tell me why now. I want you to think about why, even if you would have spent the year watching TV. No, I don't. I said, no. Put your hand down. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: I knew you're going to try. I love, I love that willingness in this class to answer questions. This one I want you to just think about, because we're going to come back to this in a few lectures. Now, let's go on and talk about maximizing profits. Maximizing profits-- now that we defined what profits are, how do you maximize them? We know how to maximize a function. If pi equals R minus C, then the maximum d pi dq-- you can't control the price. All you can control is how much you produce. Your only control variable is q, little q. So d pi dq equals dR d little q minus dC d little q. Know what dC d little q is. We defined that last time. What's that? What do we call that, the change in costs with respect to an increment in quantity? AUDIENCE: Marginal cost. JONATHAN GRUBER: Marginal cost. So we know this is the marginal cost, but for a competitive firm, what is marginal revenue? So that's marginal revenue, the amount you work on the next unit you sell. For a competitive firm, what is their marginal revenue? What is the amount? AUDIENCE: Price. JONATHAN GRUBER: Price, which is given to them, because they're a price taker. Firms, they don't have to think about a complicated concept here. This is all complicated, with tons of math. I hated that last lecture, all the math. This is hard. This is easy. It's price. You're given-- if you're a price taker, you're given a price by the market. So marginal revenue for a perfectly competitive firm is just price. So you maximize profits when price equals marginal cost is the profit maximizing point. Profits are maximized when price equals marginal cost. That is, you want to produce until what you get from the next unit equals what you spend to make the next unit. Yeah. AUDIENCE: What if you make-- what if you move it to cost a bit less and you sell it for below the market. JONATHAN GRUBER: Let's go to an example. So let's go to figure 7.2. Here we have the cost function that we derived last time. C of q equals 10 plus 5q squared. You remember that from last time. And then we have a revenue function, and I assume the price per unit-- we have a revenue function where I'm going to assume the price per unit is 30. For our example, I'm going to assume P equals 30. I've just made that up. Once again, that comes from God in these perfectly competitive firms. They have no idea where this comes from. It's just a price. We'll talk later where it comes from, but for now, it's just a given thing. Let's say it's 30. So the firm's cost function is of the form-- is graphed on the left-hand side here. The revenue function is simply a straight line, which is 30 times q. So if they sell one, revenue is 30, two, revenue is 60, et cetera. And the profits are simply the difference. So when we maximize, how do we maximize profits? Well we want to graph-- what we want to do is graph the profit for each additional unit sold. So for example, when you go from selling no units to selling one unit, at no units, your cost is what? If you sell no units, what's your cost? Somebody raise their hand and tell me. Yeah. AUDIENCE: 10. JONATHAN GRUBER: 10, not 0, because you have the fixed costs. Those are paid no matter what. Remember, those are fixed in the short run. We'll come back. This is very important. So if you zero, your costs are 10, your revenues are what? 0, so your profits are negative 10. If you produce one unit, what's your cost? 15. What's your revenues? 30. So you make a profit of 15. If you go from one unit, now you're producing one. You want to know, should I produce the second unit? What's the marginal cost of the second unit? Well, we know what marginal cost is with this function. Marginal cost, we just differentiate this with respect to q and we get 10q is marginal cost. So we know the marginal cost of the second unit is what? 20. What's the revenues from the second unit? Well, that's linear. It's 30. So you're still making-- you're making a profit of 10 on that unit. Now, we go to the third unit. So right, now after two units, you are still making profit. Now we go to the third unit. It's hard, because we're discretizing a continuous example, but let's go to the third unit. The third unit, what's the cost? What's the marginal cost? 30. What's the marginal revenue? 30. So you're at the profit maximizing point. You have climbed the hill. You have maximized your profits. What's your total profits at that point? Well, your total profits is 3 times 30, which is 90, minus 10 plus 5 times 3 squared. Minus 10 plus 45 is 55. So your total profits are 45. That's your maximum profits. That's the most you're going to earn. Now, coming to the question that was asked a minute ago, or a related version-- wait a second, if I sell one more unit, I still have positive profits. So my profits were 45 selling three units. The next unit, I'm still in the black. Why not sell more? Why not sell one more unit? Why not sell that fourth unit? Yeah. AUDIENCE: Because it's going to cost you more than it's going to give you. JONATHAN GRUBER: Right. Because we always make marginal decisions in economics. We always ask what's the next step I should take. And the next step is a losing step, because the fourth unit, what's your marginal cost? 40. What's your revenue? 30. So the fourth unit has a negative 10 profit, so you don't want to make it. So profit maximization is a hill-climbing exercise. I like to think of it, and I describe in my videos, I think of it as it's like you're climbing a hill blindfolded. All you know is whether you're stepping up or stepping down. And you need to figure out when you get to the peak. If you step up, if your next step is upward, you must be short of the peak. If your next step is downward, you must have passed the peak. It's just keep going till your next step leads to a flat part. Your next steps, right in front of the other. That's what profit maximization is. The key thing is you can-- the key thing about the blindfold is you don't need to see the big picture. All you need to know is, is my next step increasing my profits or decreasing my profits? If it's increasing, I'm doing it. If it's decreasing, I'm going backwards. That's all you need to know, is just think about putting one foot in front of the other. Does that next unit make me money or lose me money? So that is how we think of profit maximization. Yeah. AUDIENCE: So what's the next [INAUDIBLE]?? You just stop it or-- JONATHAN GRUBER: Well, that is the optimal production. That's pinning down how much you want to produce. Remember I said at the biggest lecture, the problem with producer theory is we have some set of relationships between how much we produce and costs. It doesn't tell you how much you should produce. This tells you how much to produce. You have now solved the firm's problem. This one extra-- this imposition of the perfect competition constraint has allowed us to finally solve the problem. We have now solved it, and we've said that the firm, given this cost function and given a price of 30, a firm should produce three units. Done. Just like before, we said, we had this many pizzas and cookies. Now we're pinning down q. Once we pin down q, we can, of course, go back and pin down l and k, because l and k are a function of q. But this is where it's one step harder. So this extra step we have to take is we have to impose the market condition to get to the little q we want to produce. And in this case, that's 3. Questions about that? Yeah. AUDIENCE: So you know how in the beginning, you said that in a perfectly competitive market, the producers are price [INAUDIBLE]?? JONATHAN GRUBER: Yeah. AUDIENCE: So I'm having trouble understanding why all of the guys who are selling Eiffel towers wouldn't get together and [INAUDIBLE].. Let's just make all of the Eiffel towers more expensive. JONATHAN GRUBER: Great. Great point. That's exactly-- we call that a monopoly or an oligopoly, depending how you want to think about it. They essentially could monopolize. And we're assuming that doesn't happen here. And I'll talk later about why that's way harder than you think. So we're starting-- once again, in economics, we always start with simplifying assumptions to draw general lessons. Then we'll make some more, and then we'll come back to the more complicated, real-world examples. But it turns out virtually everything we learn here is still going to hold. Yeah. AUDIENCE: So if-- JONATHAN GRUBER: We can go back to it. So now let's ask, how big is the firm's profit? How do we measure the size of the profit? Profit, if we think about profits per unit, profits equals revenue minus costs. So profits per unit, profits per q, equals revenues per q minus costs per q. What is this term, cost divided by quantity? What do we call that? Average costs. And how much do you get per unit? You just get the price. So dR dq and R over q are the same. In a competitive market, marginal price and average price are the same. It's the price. So this just says profits are equal-- profits for q are equal to price minus average cost. So your per-unit profits is price minus average cost. And we can see that in figure 7.3. Figure 7.3 shows our cost curves for this 10 plus 5q squared function we derived before. And you can see we have an average total cost curve, which first declines and then increases. We have a marginal cost curve. And then we have an average cost curve, average total cost curve. And then we have average variable and average fixed. Now, we already announced that the optimal production level was three, where price hit marginal cost. What's our profit at three? Our profit is the difference between that price and the average cost of three units over all the units we produce. So the profit is actually 30. So there's a discreteness problem here. I have to work this out. The problem, I said 45 before-- it's 45 mathematically. It's 30 in this because of the continuity problem of the sort of discreteness between two and three. So the bottom line is, in a continuous example, this would be the same. And the bottom line is the profit, is the difference between the price and the average cost times the number of units sold. That's going to be the bottom line profit. Average cost here is 10 over 3 is 3. Let's see, let's make sure we've got the math right here. Average cost is 10 plus 5q squared. So average cost is 10 over q plus 5q. q is 3, so it's 3.33 plus 15, which is 18.33. Yeah, no, my math was right with 45. This graph is wrong. So our average cost is 18.33. What's the revenue at that point? It's 30. 30 minus 18.33 times 3 is 45. So it is 45. Is that right? 30 minus 18, 12 point-- no, that's wrong. No, 30, no, I got that wrong. You're right. It's 35. So it all comes from the discrete-- it's all about the problem of a very discrete function versus a continuous function. The bottom line is this-- did I get right? Yeah, 11.67 times 3 is 35. So the bottom line is, we got 45 from a hill-climbing exercise and 35 from this graph is basically a discreteness problem, that in some sense, in this continuous function, they give you the same thing. The bottom line is the profit is equal to-- we'll be clear on this in the problem sets and exams, what we're looking for. The bottom line is the profit is the difference. The profit per unit is the difference between the average costs, and the price that you get for that unit. Now, with that in mind, let me ask you the following question-- what if I imposed a tax of $1 per unit? What if I imposed a tax of $10 per unit? What if I imposed a tax of $10 per unit? A $10 per unit tax, what would that do to the cost curve? Somebody, without looking at the handout and cheating, somebody tell me what's the new cost curve if I impose a $10 per unit tax? What's the new cost curve? What was the old cost curve? The old cost curve was 10 plus 5q squared. What's the new cost curve? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: That's the answer that almost everyone always gives the first time, but it's not right. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: 10 plus 5q squared plus 10q, because I said per unit I charge $10. If I'd said a fixed tax of $10, you would've been right. It would've been 20 plus 5q squared, but it's not, because it's a per unit tax of $10. Yeah. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: We'll see that in a second. It's just an easy way to explain it. So that's the new-- with a tax of $10 per unit, the new cost function is 10 plus 5q squared plus 10q. Oh, good, we didn't show that in 7.4. So what does that do? Well, we can go to Figure 7.4 and see that. Here, the marginal cost and the average cost have both risen by 10. The old marginal cost was 10q. Now, it's 10q plus 10. So marginal cost is now 10q plus 10. Average cost used to be 10 over q plus 5q. Now it's 10 over q plus 5q plus 10. So the bottom line is both marginal and average cost have shifted up by 10. What does that mean? Well, first of all, it changes optimal production. If marginal cost is 10q plus 10, and our optimization is set price equals the marginal cost, now we're setting 30 equals 10q plus 10. 30 equals 10q plus 10, which says the optimal q drops to 2. Subtract 10 from 30, divide by 10, you get the optimal q. Now that falls to 2, so now, you only want to produce two units, because the marginal cost is higher but the price is the same. You're going to produce less. So now, you're only going to produce two units. And you could see that in the graph, that your marginal cost hits the price now at two units, rather than hitting it at three units. You see that in Figure 7.4. Your average cost is also higher, so you're going to make less profits per unit. So you're producing fewer units and making less profit per unit. So the entire rectangle is shrunk. The rectangle has shrunk. Yes, the entire rectangle shrunk. It shrunk because you're selling fewer units. And it shrunk because you're getting less surplus per-- you're getting less profit per unit. Remember, profit per unit is price minus average cost over quantity or price minus average cost. Since average cost is higher, you're getting less profit per unit. You're selling fewer units at less profit per unit. So that tax has significantly lowered your profit. Questions about that? Doesn't have to be a tax. If I'd said suddenly, whatever you're producing, there was a change in what it costs to produce, any change in what something costs to produce in the production function can ultimately affect how much you sell and you profit. So one excise you can show yourself at home, imagine that I had said there was a fixed tax of 10. Actually, let me ask you this question-- imagine I had said there was a fixed tax of 10. How would that have affected your profit? How would that affect the amount you sell and your profit? It's a fixed tax of 10, not 10 per unit. How would that affect how many units you'd sell and what your profit is? Yeah. AUDIENCE: Both [INAUDIBLE] would have less profit. JONATHAN GRUBER: Exactly. You'd sell-- why would you sell the same amount? AUDIENCE: Wouldn't affect your MC. JONATHAN GRUBER: It wouldn't affect your marginal cost. Remember, profit maximization is where marginal cost equals price. If I have a fixed tax, that's an effective marginal cost, but it does lower your profits, because you've still got to pay that extra tax. So here, your profits get doubly hit. You sell less, and you make less per unit. With a flat tax, you would sell the same amount, but you'd make less per unit. That's an important distinction. I think I'll add that for next year. Now, one other important point-- the firm has one other decision to make here that we haven't covered yet, which is that it has to decide whether to shut down. Now, remember I said there's no entry and exit. That means you can't literally leave and go somewhere else, but you can just walk away. You can just literally say, look, I'm not in this market anymore. I'm not going to go set up shop somewhere else, but I'm not in this market anymore. So the question is, when would you do so? So I'm going to get to it. Hold on. Is it a question or answer? If it's an answer, I don't want it yet. If it's a question, I'll take it. So basically, suppose the price in this market fell to $10. Should you continue to participate in the market? Well, if the price falls to $10, what is the optimal level of production? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Zero. So does that mean you should walk away? No. What are your profits? If you sell zero, what are your profits? Negative 10. If you shut down and walk away, what are your profits? Negative 15. What's profits? Profits is revenues minus costs. Costs at zero-- I'm sorry, you don't produce zero. That's bad. You produce one unit. I got the answer wrong. If price equals marginal cost-- yeah. AUDIENCE: [INAUDIBLE] with the tax imposed, or are we-- JONATHAN GRUBER: No, without the tax. I'm sorry. That's what I confused. We're back to without the tax. The price in the market drops to $10, so you produce one unit, not zero. I shouldn't listen to you. I should look at my notes. One unit, not zero. You produce one unit. At that point, what are your revenues producing one unit? You get 30. Yeah, this example is messed up. I'm going to have to come back to this. We made this-- this example is wrong. I'm going to have to come back to this one. So the key point with the shutdown is basically, you only want to shut down if you're actually going to lose more money by staying in the market than you get by exiting the market. So let me stop there. I'll come back and fix this. I made an error on this. I'll come back to fix it at the beginning of the next lecture. And then we'll talk about finishing up short-run profit maximization. So let's pause there and come back.

MIT_1401_Principles_of_Microeconomics_Fall_2018

25_Health_Economics.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: So today, we're going to have sort of a different kind of class since it's the last class. Today, I'm going to talk about essentially how we bring to bear the set of issues we've talked about this semester to a real-world topic, and actually, how it plays out in policy and practice. And I'll draw on some of my own experience, having applied the kind of tools we learned in 14.01 to the field of health care economics for 25 years, and how that has led me to be able to help in the development of health care policy in the US, and talk about sort of where health care policy stands at this point. So let's get a little bit of background about health care in the US. Basically, when we're talking about health care in the US, we have to recognize that the US spends, by far, the most money on health care of any developed nation in the world. We spend about 17 and 1/2% of our gross domestic product on health care. That amounts to almost $10,000 per man, woman, and child-- every man, woman, and child in America. That dwarfs the rest of the world. The typical European nation spends about 2/3 as much as a percent of GDP on health care. England spends less than half as much on health care. So basically, we spend a lot on health care as a share of our economy. And what do we get for it? Well, the evidence here-- the first fact is clear. The evidence that we get for it is a little bit mixed. So if you look at the typical thing on the web, you know, US health care is terrible. Our money's wasted. You'll see that on things like infant mortality, we rate, like, 20th in the world. Or life expectancy, we're, like, 20th in the world. So by those metrics, we don't do very well. But in fact, those metrics are misleading because we also have-- we have the most unequal health care system in the world. So the right way to think about it is to think about the haves and the have-nots. The haves, which is us and most people in America, people who are well-insured in the system, actually get probably the best health care in the world, Now that might be disputed by many people. But I think about this like an economist would think about it, which is, how would you decide whether you would prefer product A versus product B, whether they buy product A versus product B? Every year, one million people come to the US to get treated for their health care problems. No one leaves. No one's going to England for surgery. No one's flying from the US to England for surgery. They're coming here. If you're in the system, we have the best health care in the world. Unfortunately, if you're out of the system, we have some of the worst health care in the world. So a white baby born in America today, there's roughly a slightly more than 0.5% chance the baby will die in their first year. That's comparable to northern Europe. If you look at a black baby born in the US, the odds they die in the first year are about twice that, which is worse than Barbados. So the problem in the US is not that our outcomes are bad. The problem is they're very unequal-- that we're spending all this money. We're delivering good, but not exceptional, outcomes for people in the system and bad outcomes for people out of the system. So clearly, we're not getting a lot of value-- it's not like we deliver exceptionally good outcomes to people in the system. We're slightly better, despite spending a lot more, and we're worse for many Americans who are left out of the system. So that's sort of the setup of where we are, which is really, you have two fundamental problems in health care in the US. Our spending is too high, and our access is too unequal. Now so I want to focus today's lecture on those two aspects and think about how can we bring the kind of lessons we've learned in this course to thinking about addressing those problems. So I'm going to focus on the access problem and the cost problem. Let's start with the access problem. Now in America, before 2010, we had about-- or before 2014, we had about 50 million uninsured Americans. 50 million people who did not have health insurance in the US. We're the only nation in the world-- only developed nation in the world with a significant uninsured population. Now the fact that 50 million people are uninsured, is that a problem? On its face, if I just said here's a fact. 50 million people in America don't have health insurance. Based on that fact alone, can you tell me whether there's a problem or not? You shook your head no. Why not? AUDIENCE: Because it might be better for you not to have-- JONATHAN GRUBER: Yeah. You know, many more people than that don't have flat-screen TVs and don't own homes. Why do we think that we should care if people don't have something? The answer would be, we would only care if what? Under what condition? When do we-- yeah-- AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: If-- well, they'd be better off if they did have it. Now they could be better off because they could be richer, but that's not our problem. Given their budget, they're not buying it. What-- under what condition is the market not-- under what type of conditions would the market not deliver the best outcome? AUDIENCE: If there's a failure, like-- JONATHAN GRUBER: If there's a market failure. So the fact that people aren't insured doesn't matter except A, if there's a market failure, or B, for redistribution purposes. Remember, that's the two reasons we want the government involved. So if health insurance markets were perfectly functioning and people who were uninsured were roughly equally distributed in income as everyone else, there'd be no cause for worry. But in fact, that's not true. We've talked in this class about why markets like health insurance won't function well, which is a problem of adverse selection. The problem is information failures which will lead health insurance markets not to function well. And the people who are uninsured tend to be much poorer than the people who are insured. It's also redistributional concern. So the reason we care about the uninsured are both because of market failures and for redistribution, that they tend to be lower-income. What's interesting is the uninsured don't tend to be the poorest in society. They tend to be the near poor. So here's the way sort of health insurance coverage works in the US. For the vast majority of Americans-- 60% of American-- 60% of Americans have what's called employer- sponsored insurance. So like your most of your parents, like me, they get health insurance from their employer. The typical upper-income American gets health insurance from their employer. The typical average-income American does. About 60% of Americans. Then-- and I'm going to do this sort of pre-ACA. So before 2014, before the big change that was put in place by the Affordable Care Act, you had about another, maybe, 6% that bought into what we call individual or non-group health insurance. That is, they went out on their own and bought insurance. But that's a tiny market compared to ESI. And the reason is because of exactly the adverse selection problem we talked about. Think about yourself as an insurer. And you're worried about yourself as an insurer. And think about what your goal is. Your goal as an insurer is to essentially absorb risk in a way that allows you to make a profit. So what you want is you want to live off the law of large numbers. You know that with a large enough group, you could be able to predict what their costs will be. And therefore, you can just make a profit on top. So insurers love-- when MIT comes to an insurer, they're delighted. They're like, look, you got-- between MIT and Lincoln Labs, you've got about 10,000 employees. I, with great certainty, can predict what the costs will be next year for a group of 10,000 employees. And so I, as an insurer, can know I can just charge that, plus X percent, and I'm golden. But when Jon Gruber walks in the door, they're like, why are you coming to me, individual? Maybe because you know you're sick, maybe because you love skydiving. I don't know. But I'm wary of you, so I'm to charge you a lot of money to get health insurance. As a result, most-- very few people bought health insurance on their own. And in particular, the reason they didn't is because insurers would not offer health insurance to people if they were at all sick. They would do things like having what we call pre-existing conditions exclusions. These were features of insurance contracts which said, look, you walked in the door, Jon, and you want health insurance. But I know, in the past, you've had cancer or asthma or knee surgery. I'm going to tell you, I'm going to insure you, but not for any expenses that might arise from recurrence of those past injuries. So you had cancer in the past. Anything that comes up in the future because you had cancer, I'm not going to cover. Anything that comes up in the future because you had knee surgery, I'm not going to cover. Anything that comes up in the future because you had asthma, I'm not going to cover it. So I'm going to give you, essentially, partial insurance. So it's going to be a market failure. I'm going to insure you, but only for part of what you need. Alternatively, they could use pre-existing condition solutions-- they could use what is called medical underwriting, which was basically saying, OK, Jon, come in. I'm going to give you an exam, and if you look sick, I'm going to deny you insurance. Or if you look sick, I'm going to charge you 100 times more than someone else. So these were not illegal or even immoral. These were just ways insurers came up with to try to deal with the adverse selection problem. As a result, this was a market that did not function very well. Question about that? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: No, totally legal in every state-- virtually every state, totally legal and not immoral. I mean, this is just they're maximizing their profits. It's what companies do. And the point is that when they did this, what this meant was if you didn't have employer-sponsored insurance or insurance from the government, which I'll come to next, then you were subject to the fact that if you got sick, you might not be able to get insurance, which is sort of weird. Insurance is supposed to cover if you're sick. But in fact, if you were sick, you might not be able to get it. So that was the fundamental market failure we had here through adverse selection. Now we also-- that was employer-sponsored insurance, so that was about 2/3 of the population. You also had on the order of 15% of the population had government-sponsored insurance-- probably more like 20%. 20% of the population had government-sponsored. Insurance. The two big programs here are called Medicare and Medicaid. Now if you ever take my 1441 class, I will only hold you responsible for one thing if I ever meet you 10 years later, which is remember the difference between these two programs. Medicare is health insurance for the elderly. Medicaid is health insurance for the poor. And those are our two big public insurance programs. And about 20-- and if you're in those programs, you're also set. They don't have any of these features. If you're in, you're covered for everything. So about 20% of people are there. And then finally, if you add up the numbers, we had about 15-- the numbers don't quite add up, but you had about 15% of the population was uninsured. 15% uninsured. So you had about 2/3 private, about one fifth public, and about one sixth uninsured. And those are individuals who typically were not the poorest because the poorest people got Medicaid. The typical uninsured person is, like, what we call the working poor, someone who's got a job, but it's a crappy job that doesn't offer health insurance. But they make enough money that they can't qualify for being in the low-income program. So your family is struggling at, like, $40,000, $50,000 a year, high enough income that they're not qualifying for Medicaid but not in a good enough job they're getting health insurance. That's your typical uninsured family. 2/3 of the uninsured are in families that are headed by a full-time, full-year worker. They're not typically the unemployed down-on-their luck people. They typically are the people who are trying to play by the rules, as they say in politics, but typically can't get a job with health insurance. So that's your basic landscape. And what we know from that landscape is that a lot of the access problems were because of this group and this group because the people who couldn't get in this market, and as a result, were often uninsured. That was a lot of what drove the access problems. So that was sort of the first-- one of the two big problems that faced our system. And for many, many years, we knew we had that problem. And for about 100 years, we've tried to reform health care in the US to deal that problem. And probably about every 17 years, on average, there was a big attempt to reform health care, and they always failed. And they always failed because they failed between two extremes. There were two extreme views that could never quite meet in the middle ground. And they come to what I talked about last time, which is how do we solve the problem of market failures in insurance markets? Well, one version of solving that, I described, was subsidization. You could-- remember, with my MIT program, if I paid the healthy guys $500, they'd all buy two, and I'd solve the problem. So one version was subsidization. The problem is subsidization only works if it's big enough to overcome these problems. And no one ever proposed subsidization big enough to overcome these problems. In my MIT example, I was going to give $400 to every healthy-- first of all, it means giving money to healthy people, which is sort of politically difficult. Like, hey, the healthier you are, the more money you get. It seems a bit weird. Also, it's just hard to solve these problems by just subsidizing people. Insurance companies are still too good at trying to get rid of the sick people. And even if you subsidize people who come in, insurance companies will always have an incentive. They'll say, great, healthy people come, we'll subsidize. They'll still want to avoid the sick. So it doesn't solve the problem in insurance companies. I didn't talk about this last time, but as MIT's insurance company, I should try to shed the sick people. And that problem still existed under this solution. The other extreme, which is sort of back in style again, is the single-payer model, which is saying, look, let's just have the government provide health insurance to everyone. We have the government provide Social Security to everyone. The government provides health insurers to every elderly in America through Medicare. Everyone over 65 in America, boom, gets government-provided health insurance. Talk about socialism. Every American gets that. In Canada, everybody gets government-provided health insurance. Why not just do it here? Let's get rid of all the crap with insurance companies we don't like. After all, insurance company administrative costs are about 15% of medical spending. So, boom, we could lower 15% of medical spending. That is, you know, that's like $500 billion a year. Boom, it's gone. So basically, why not-- so single payer is something a lot of people have advocated for. Let's just have one giant universal health insurance program. Now the problem with this-- the problem with the single-payer approach is largely-- there's pros and cons to the economics perspective. But the problems here are largely political, which is that to make single payer happen, you have three enormous political barriers, which come back to economics. Everything comes back to economics, but they play their way out in the political system. The first problem is paying for it-- paying for it, which is that single payer-- to have the government give everyone health insurance means a massive expansion in the government, which means a big increase in taxes. And we know taxes have deadweight loss. We know taxes are politically unpopular. Now here's what's misleading about that. Here's the fundamental thing. So I worked for the state of Vermont. The state of Vermont wanted to do their own single-payer plan. If any place can do it, it's Vermont. They're, like, super lefty. They essentially have one insurance carrier, which is Blue Cross anyway. They're a small state. It seemed like if anyone was going to do it, Vermont was going to do it. So I worked with them to put the numbers together, what it would cost them. And I had good news and bad news. The good news was, I said to Vermont, if you do single payer you will lower the cost of health care in Vermont total by at least 10%, at least. That was conservative. The bad news is to pay for it, you're going to have to more than double the entire amount of taxes collected in state of Vermont. And that second sentence just killed everything. What's the problem? The problem is that right now health insurance in America is paid for by essentially a hidden tax. What's the hidden tax? It's the fact that when your employer gives you health insurance, they pay you less wages as a result. Remember our tax incidence discussion. And we said that essentially taxing the employer falls on the employer-employee depending on basically elasticities. Well, you can think of health insurance the same way. When your employer gives you health insurance, he doesn't just eat the whole cost. He says, look, I'm paying you a total set of compensation, part of which is health insurance. So I'm going to pass the cost of that health insurance on at least partially to your wages. That's essentially a hidden tax. So at MIT-- right now, I have a health insurance plan through MIT, which costs about $18,000 a year for my family. I pay about $6,000 a year out of my paycheck. MIT pays $12,000. But the truth is, MIT pays me $12,000 less. They don't just give me that health insurance out the goodness of their heart. They take it out of my wages, or least partially out of my wages. That's essentially a hidden tax that Americans pay every year to finance health insurance. If we went to single payer, that hidden tax would go away. I would get a $12,000 raise. That's great. But I'd also face a high new taxation to pay for the government-sponsored plan. Now given that the total cost would fall-- we should be able to net this out in a way that most people win. The problem then becomes the politics, which is you're tracing a hidden tax with a non-hidden tax. And that's very ugly politically. So people don't believe their employers will pay them more if you don't make the employers provide health insurers, like, oh, the employers will just pocket it. And I could teach them tax incidence till my face is blue, but they just won't believe it. They'll say employers will just pocket it. But I have to pay this new tax for single payer. So that's the first problem single payer faces is that people don't really understand that trade-off between getting rid of the hidden tax and adding a new non-hidden tax. That's problem one. Problem two is the problem we talked a little bit about, behavioral economics, and about loss aversion. There's a general feature, what we call status quo bias in human thinking. Status quo bias, which is, essentially, it is harder for me to give up what I'm used to than to grab something new. We talked about the mug example. Remember, I talked about mugs. So basically, you had to pay me more to get the mug away from me than I was willing to pay to buy it. That once you have something, you value it more than if you didn't have it yet. Well, right now, 60% of Americans have employer-sponsored insurance. And if we say to them, give that up for Berniecare, they're going to be, like, eh, I don't know. I kind of like my employer-sponsored insurance. You know, yeah, you might tell me Berniecare is going to be better, but that's just you talking. I know what I have right now, which I have employer-sponsored insurance. I don't want to move away from that status quo. So status quo bias makes it hard, in general, to do radical changes on an economic system. And this is a perfect example. It's going to be hard to get people to give up what they have for something that they don't really know about yet. That's the second problem. The third problem is, once again about money, but really beyond the scope of this course, which is the problem of the insurance companies and lobbying, which is that the insurance business is big business in America. Health insurance companies make about $900 billion a year. If you said to them, hey, health insurance companies, would you guys mind just giving up your $900 billion to begin a single-payer health care, they'd actually say, yeah, it's been a good run. Go for it. No. They're going to lobby and fight that because they want to keep their business. And that's going to be a pretty hard force to overcome. So single payer has always struggled with dealing with these kinds of political problems. And that's why we've been stuck. We've been stuck between one alternative, which is subsidization, and the other alternative, which is single payer. And that's where economists have come in-- came in the 2000s, folks like myself, to talk about a new alternative way to do it, which was essentially to try to bring in some of the best features of these two approaches. And the solution we proposed-- so if you want to read more about this, I've actually written a comic book to explain it. It's a graphic novel, technically. It's called Health Care Reform. It's, like, $9 on Amazon. And so I like to think of everything in terms of images. Now I'm not going to draw one. I'm not going to try to draw anything. But the way I like to think about this is the solution we came up with, which we first pioneered here in Massachusetts and then brought to the whole country through the Affordable Care Act, is what we call a three-legged-stool approach, three-legged-stool approach. Leg one is deal with this problem. Deal with the insurance discrimination problem. And so leg one is ban insurer discrimination. No more pre-existing conditions, no more medical underwriting. That is, if I walk in the door, and you have offered anyone-- you have to offer me health insurance at the average price for my age. And you have to offer it to me. So any 40-year-old who walks in the door wanting insurance, you have to sell it to them, and you to sell it to them at a fixed 40-year-old price. You can't say, you're sick. I'm not going to sell it to you. So the first step is to ban insurer discrimination, to try to solve that problem. Now the problem this raises is you have simply-- if you do this alone, you've created a new problem, which is if you tell insurers they can't discriminate against the sick, you don't solve the adverse selection problem. You're just making insurers go bankrupt. Now here's the way I like to think of it. I'm sure none of you ever gambled on sports. But if you had gambled on sports, you might know the way sports gambling works is that there's a guy in the middle, called the bookie. And the bookie's goal is to not-- is to get exactly the same number of bets on either team. So they take no risk, and just make their profits off the top. So what bookies do is they set point spreads. So the Patriots played the Dolphins this past weekend. I am-- sadly, I'm a Dolphins fan. The Patriots played the Dolphins. The point spread was something like-- does anyone know what the spread was in the Patriots' game? I think is was, like, 8 points. So that spread was chosen. The Patriots were favored by 8. What that meant was your bet was either the Patriots win by either more, or the Dolphins win, or the Patriots win by 8, or by less than 8. So one side is Patriots win by 8 or more. One side is Patriots win by less than 8, or Dolphins win. And the reason you have that bias thing is because people think the Patriots are better. They are better. And as a result, you want to get-- if you set an even bet, Patriots win, Dolphins win, everyone would bet on the Patriots. You'd lose money. So you want an equal distribution of risks. So what you want is you want to set the point spread so the distribution of risk is equal. Then having done that, you just make your money off the top. Now imagine I passed a law which said all sports books have to reopen at halftime and make the same bets available they made before the game started. Well, for those of you who watched the exciting game this weekend, you realized at halftime, it became pretty obvious the Patriots weren't going to win by 8, that it was a lot closer game than people thought. So if they reopen that, a bunch of people would suddenly bet against the Patriots. The Patriots ended up losing, and the insurers would have gone bankrupt-- the bookie would've gone bankrupt. Insurers are just bookies. That's all they are. They just want a predictable distribution of risks. So if you tell them, you have to offer health insurance to everyone for the same price, but only the sick are going to buy, they're going to lose money. So that's why we need the second leg of the stool, which I talked about last time, which was the individual mandate. The individual mandate, which is to say, OK, insurers, if you offer health insurance to everyone at a fair price, we will, as our part of the deal, make sure everyone buys health insurance. So when the 40-year-old walks into your office wanting insurance, you can know it's not because they're sick. It's just because they have to. So we say to insurers, you price insurance fairly, and in return, we'll make sure you get the fair distribution of risks. So you say to me-- my MIT insurance, you price insurance at $1,500 and don't try to keep out the sick, I'll make sure everyone buys. And you'll make your $100 profit. So that's-- the mandate was essentially trying to bring-- was trying to allow-- get rid of discrimination by bringing in the entire pool of people so insurers could fairly price. The problem with that is you can't mandate something people can't afford. So in Massachusetts, where we were creating this plan in the mid 2000s, the typical family health insurance policy was about $12,000 a year. The poverty line for a family was $22,000 a year. We couldn't exactly mandate people that they spend 55% percent of their income on health insurance. That was not really feasible. So the third leg of the stool we came up with is subsidies to make health insurance affordable, saying, if you're low-income, we will offset the cost of your insurance just like the subsidy approach here. We'll offset the cost of your insurance to make it more affordable. We'll do it on an income-related basis, so it doesn't cost so much. So we're not going to have to pay for everyone's insurance like single payer. Remember, single payer, essentially taking someone like me, who's happy with my insurance, swapping it out for new government insurance. This is saying, no, if you're happy with your insurance, stick with your insurance. But if you're low-income and can't access the employer market, this gives you a new place to go. And that was the idea that became Romneycare, the plan here in Massachusetts, and eventually then became Obamacare, or the Affordable Care Act. So this is essentially the idea of that plan. Now, did it work? Unambiguously, yes. Now you won't find anyone more biased than me on this question. But I think what-- I think if I've tried to teach you one thing in this class, it's that we need to rely on real facts wherever possible. And if not, we could turn to theory. But here we have a set of real facts that we can turn to, which is that essentially what we did in Massachusetts with this law is we covered about 2/3 of the uninsured population. At the federal level, we covered about 45% of the uninsured population. It was a lower number because the federal law did not apply to undocumented immigrants, which are about a quarter of the uninsured. It's not an issue in Massachusetts, but a big issue in other places. That's about a quarter of the uninsured because the federal law did not apply to undocumented immigrants. So as a result, the share cover was lower. But a large number aren't covered. Yeah. AUDIENCE: If there was an initial mandate, then how was there anyone who was left uninsured? JONATHAN GRUBER: Great question. So there are three reasons why people were left uninsured. The first reason was a quarter of the uninsured were undocumented immigrants, and the law didn't apply to them. So right now the upper bound was 75%, just to start. The second reason is that the individual mandate contained exemptions to make it both a little more humane and, quite frankly, politically feasible. So if you could not get-- if your income was below the poverty line, you were not subject to the individual mandate. And if you could not get insurance for less than 8% of your income, you were not subject to the individual mandate. So there were exemptions. And the third thing was the individual mandate was not, like, we're going to throw you in jail. It was a tax penalty. And many people decided they'd rather just pay the penalty than buy health insurance. So for those three reasons, a number of people did not get health insurance under the Affordable Care Act. Now there's a bunch of interesting questions, like should the mandate penalty be bigger? How should we handle that? There's a lot of-- I could go for hours on this. But that's basically the structure of what we had. So basically, that worked. It didn't get us to universal coverage. It wasn't as effective as single payer would have been, but it was the largest single insurance expansion in American history. And the evidence is clear. It brought many people into insurance. It improved people's use of health care. It improved health. So basically, that was kind of the step forward on access. Now the problem with this is it's only a step. There's still many uninsured, and this has been politically really challenging, because these two answers are quite simple. Just give people money or just have single payer. This is super complicated. I can talk about these in about 15 seconds each. These took five minutes to go through. And people thought it was just too complicated. It didn't make sense. Lots of reasons-- we could talk lots reasons people didn't like it. So it's never really been as politically successful as people like myself, who helped develop it, would have hoped. And it's left a lot of people uninsured. So we haven't solved the access problem. We made a big step forward, but we haven't solved it. And that's the ongoing debate today we see, particularly in the Democratic Party. The Republican Party really doesn't focus much on insurance coverage. But the Democratic Party does. And they're-- that's why there's a lot of energy behind single payer right now is like, look, you tried the kind of halfway ground. That kind of worked, but didn't work all the way. So let's just go all the way to single payer. Yeah. AUDIENCE: The initial mandate, does that-- I guess, for the more people living in poverty, does that work together with Medicaid or-- JONATHAN GRUBER: Yeah. Basically, a lot-- actually, it's quite interesting. It worked quite well with Medicaid. A lot of people who aren't insured, actually, are people who are already eligible for free Medicaid coverage and just don't take it. Now we don't quite know why. It could be language barriers. They don't understand. A lot of people-- a lot of even legal immigrants just don't understand they're eligible. It could be people just don't want a government handout. They're embarrassed taking help from the government. It could be people think, I don't need it. I'm never going to be sick. We don't know why. So part of what the mandate did was say, look, you already have free health insurance. Just pay attention and take it. That's part of the effect it had was bringing people in. So a large part of the coverage increase is actually bringing in people who were already eligible, just weren't taking it up before. So that's kind of where we are. So where we stand now in coverage is we've taken a giant step forward. We've covered probably about now, probably, between a third and 40% of the uninsured in America. But we're sort of right now kind of stuck at that point. And the question is, do we just sort of stick there, or do we try something more aggressive? With the political problems, I don't know. But that's going to be the challenge going forward. So that's-- questions about that-- because that's where we are on problem number one, which is access in coverage. Now let's turn to problem number two, which is cost. Cost is way harder. What I just did was the easy part. It's way harder to get your health care costs, and here's why. Two facts that are seemingly contradicted if you think about it. Fact one. Since 1950, US spending on health care as a share of our economy has quadrupled. We've gone from-- more than quadrupled. We've gone from 4% of our GDP being health care to over 17%. And it's been worth it. If you look at the improvements in our health, and you value them in the way economists do, which is we have statistical values of life we apply, or statistical values in improvement in health, the improvement in our health has been worth the money spent on health care. You guys don't realize it. Health care totally sucked in 1950. Babies born in 1950 were four times as likely to die before they reached their first birthday. If you had a heart attack in 1950, you were four times likelier to die within the first year. To put it in terms all young healthy people care about, if you hurt your knee skiing in 1950, tore your ACL in 1950, or tore your cartilage, you were in the hospital for a week. You were on crutches for six weeks and had arthritis the rest of your life. Today, you go to an outpatient center. You get arthroscopic surgery. You're back on the slopes a couple weeks later. Health care is just way better, and our health is way better. America is a much better off nation, spending 17% of GDP on health with how healthy we are than we were in 1950. And once again, do the economists tests. No one ever advertises, hey, would you like 1950s health care at 1950s prices? No one out there is offering that because it's worth it. That's fact one. Fact two is we waste a huge amount of money on health care. By some estimates, about a third of what we spend on health care is totally wasteful, does nothing to improve our health. Now how can those two facts be consistent? It's worth it, but it's wasteful. Well, the answer is that the other 2/3 is super awesome, that basically the increase in health care, where it's been productive, has been amazing. But we dragged along all this unproductive spending too. So it's good news and bad news. So the good news is, well, that's great. We just cut out the one third that's unproductive, we've solved our problem. Literally, if we could just simply cut out the one third that's unproductive, we'd spend the same amount as Europe does on health care. We'd solve our entire long-run fiscal problem. The bad news is that it's easy to look back and see what the one third was. It's hard to look forward and say what it's going to be, that health care comes with a huge amount of uncertainty about what's going to work and what's going to be worth it. And as a result, it is very hard to say, OK, fine. We'll cover this. We won't cover that, because it's hard to know what's going to work and what's not. And so, essentially, you're in this very difficult spot. So what are the kind-- that's the sort of fundamental trade-off that we face. So what are the potential solutions to this problem? So essentially, there's a couple of different solutions to the problem, two different paths we can follow. Path one is the regulatory path, which is basically the path that Europe follows. What Europe does is they just much, much more heavily regulate the delivery of health care. And they do that in two ways. One is they actually have regulations about what health care you can get. So for instance, England has the euphemistically named NICE, the National Institute for Health and Care Excellence, which actually tells people they can't get some things. It literally rations. So for example, for many years-- it's no longer true-- in England, if you're over 75, you could not get a transplant. They said, look, we got a limited number of kidneys. You're going to die soon anyway. Let's give the kidney to a young person. Actually, kind of makes sense. The idea is, look, we have some limit on our kidneys. Why should it be determined by some random fact, like when you got on line? It should be determined by who gets the most value from the kidney. It's going to be someone who's 30, not someone who's 75. So one regular route is to literally have regulations like that. That's actually pretty rare. Most countries don't actually regulate in that way. Most countries kind of let you get what your doctor says you should get. There's three routes. So one route is sort of regulatory. The other route of regulate-- so one route is sort of what we call sort of regulating, you know-- I don't want to call it access-- sort of technological regulation, regulating which technology you can get. The second kind of regulation in Europe is supply regulation. So they basically don't let there be many doctors. And there are not many doctors and hospitals. So there are as many MRI machines in LA as there are in Canada. Basically, just not many place to go get an MRI in Canada. So if you' hurt your knee in the US, you go, you get an MRI, like, the next day. In Canada, you get it six weeks later. So the only way to control it is to actually regulate the supply of medical care. Just give people less stuff they can use. And the third way to control-- the third regulatory mechanism, and the most important, is price regulation. We are the only nation in the world which essentially lets the free market determine the price of health care services. Every other nation regulates the prices that people pay for their health care services. Now the question we have to ask is why? Why does that make sense? Well, the answer would be that we think-- it would make sense if we think there's a fundamental market failure in the determination of health care prices. And in fact, it turns out there are numbers of market failures in determination of health care prices. So one market failure, for example, is imperfect information. I don't know-- I can't shop effectively-- when I'm in the back of the ambulance dying from a heart attack, I can't be, like, you know that hospital looks expensive. Take me over there. I want to shop there. You can't really shop. It's a hard market to shop. And if you could, prices aren't posted. You don't really know what it costs to get your heart attack treated in different places. So imperfect information. There's also imperfect competition, which is if you have your heart attack on Cape Cod, there's, like, one-- or Nantucket, which is an island, with no way off but a ferry, there's one place to go. There's one hospital. They have a perfect monopoly. You can't get off the island. You're going to die otherwise. So it's imperfect competition. There's even imperfect competition where you think the competition might be perfect. So take Boston. There are so many hospitals in Boston, you cannot literally fall down without hitting a hospital. Yet there is an enormous dispersion in the prices hospitals charge. In particular, the very famous hospitals, like Mass General Hospital, charge multiples of what less famous hospitals charge, even though less famous hospitals are really nearby. Why? Well, because they have essentially what we call a reputational monopoly, that even though they don't have an actual physical monopoly, people are like, I want to go to MGH. They're the best, even if they're not necessarily the best. They just have this view of being the best. And they can charge higher prices as a result, even if their outcomes aren't necessarily better. In other markets, we think perfect information would allow us to get rid of these kinds of inefficiencies. It doesn't exist in health care. As a result, perfect competition simply does not work in health price setting. And as a result, all other countries regulate health care prices-- and then not other countries-- even the US can regulate health prices. So the Medicare program has regulated prices. That covers millions of Americans. It's just for the non-government, private health insurance in the US, there's non-regulated prices. Now I am not, despite my tone, saying that regulating prices is the answer. It's not clear. Regulating prices comes with a huge number of additional problems like we talked about. We talked about regulated monopolies, which is the government may not know the proper price to set. The government may do a terrible job. They may get lobbied. They may be corrupt. Indeed, in the US, the 1970s, virtually every state did regulate hospital prices. And every state went away from that because they thought the system was broken. So it's not like there's any-- it's not like the European solution's an easy answer. That's why the other route that people have been pushing lately is a different route, which is the incentives route, which is basically to say, look, we don't want to regulate supply or prices. What we're going to do is we're going to say, doctors and hospitals, you get together and form these units we call Accountable Care Organizations, ACOs. This is a big innovation of the Affordable Care Act of Obamacare, set up these ACOs. These are hospitals and doctors all get together to be basically, like, soup to nuts, all the health care you need in one group. And we say to them, we are going to pay you one flat amount of money to care for Jon. And then within that, you decide what he gets. You decide what prices everybody pays and makes. You figure all that out. But we're going to give you a flat amount. In particular, that flat amount is not going to rise much. And that's going to bring the costs of health care under control, where basically every ACO will get an amount that's a flat amount, and it just won't rise much. And that's how we'll bring health care costs under control. That has a number of wonderful features. First of all, it's much less evil sounding than things like not letting [INAUDIBLE] rise or regulating what prices. Second of all, there's much fewer regulatory tools. We just say, here's a flat amount we're giving you per person, and we're done. So that sounds great. The problem is we haven't been able to get it to work. And that's because it turns out doctors and hospitals aren't very good at figuring out how to set prices and set supplies. They're just not-- they don't know how to really figure this out. And the ACOs so far have not actually performed very well. They've not saved much money. So really, we're stuck between a route which seems a lot easier but we haven't really figured out how to make work, and a route which has worked all around the world but seems politically nightmarish. And that's kind of where we are right now in terms of controlling costs. And that difficulty is what we find ourselves in. But let me be clear. This is not like, oh, that's very interesting, Jon. I'll go home and forget about it now. This is the entire future. Health care costs are the key to determine the entire fiscal future of the US. As I mentioned last lecture, the US is currently estimated about $75 trillion in deficit over the long run. $70 trillion of that is health care. Health care is the single determinant of the US fiscal balance in the long run. Literally, it's the single most important government problem facing-- health care cost is the single most important government problem facing your generation and the next generation. I like to say that all that matters when we think about the future is health care cost and global warming because either way we're under water. Basically, those are the two big issues we have to face going forward. So this is a serious issue that your generation is going to have to struggle with-- sorry-- as you go on. So that's health care in the US in 40 minutes. So this class-- you know, there's a famous skit from Saturday Night Live, which is what you remember five years after college. And it's five minutes, and 3 and 1/2 minutes of spring break. I don't expect you to remember the formula for-- if you're not going on in economics, I don't expect you to remember the formula for deriving cost function. What I expect you to get out of this class is A, an interest in economics. And I hope you'll go on. I sincerely hope that. And I'm available to anyone who wants to talk about the pros and cons of going on in economics. Obviously, I'm more pro. But I'm happy to talk about it. So always feel free to reach out about that. But B, even if you don't go on in economics, I want this to make you a more educated consumer of the newspaper. This is-- we are in an era, as I said in my very first lecture, where truth and facts and the scientific method are, themselves, under attack. And MIT is the last bastion of fighting this war. We are the place that explains the scientific method, that uses the scientific method. And we need to use the methods you've learned here to think intelligently-- whatever your conclusions-- but to think intelligently about these economics topics. And fundamentally, that means being annoying. And to illustrate that, I'd like to end with a joke that some of you may have heard. Sorry, I apologize if you have. So the joke is a doctor, a priest, and an economist go golfing. They get on the golf course-- and they hit the golf course, and they're behind someone going incredibly slowly. I don't know if there are any golfers among you, but the idea is if you're very slow, you're supposed to allow the people behind you to play through and get ahead of you. This person won't let anyone play through. And he's, like, 50 shots a hole. It's disgusting. And there's, like, 50 people lined up behind this guy. And these folks are so disgusted, they quit after nine holes. They go back to the clubhouse. They're pounding their beers like, what an asshole. I can't believe he wouldn't let us play through. It ruined our day. And someone comes up to them and says, excuse me, are you new to this club? And they said, yes, we are. He said, well, I can tell you're new to the club because if you weren't new, you would have known the person you were playing behind is blind. And actually, it's a miracle he can get the ball in the hole at all. And usually, it's an honor to be on the same course as he is. And the person walks away. And there's, like, a deadly silence. And the people at the table are like, wow. I feel terrible. And the doctor goes, I can't-- I feel terrible. I can't believe I'm-- myself, a man of healing, would be so insulting towards someone who's blind. I'm going to dedicate a wing of my hospital to the blind. And he turns to the priest. And the priest says, I can't believe myself, a man of the cloth, and that I'm supposed to care for the less able in society, would do this. I'm going to set up a free soup kitchen for the blind. And they turn to the economist. And the economist says, well, if he's blind, why doesn't he just play at night? And-- makes sense, right? And basically, the point is that the job of the economist is to sort of be annoying and look for the basic flaws in arguments, to understand them, to ask the difficult questions, but to have responsible answers. And that's what I hope you'll get out of this course that I hope you'll take forward with you. So thank you very much for sharing it with me. And good luck on the final. [APPLAUSE]

MIT_1401_Principles_of_Microeconomics_Fall_2018

4_Demand_Curves_and_IncomeSubstitution_Effects.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: All right, let's get started. Today, we are going to complete our discussion of consumer choice by actually coming back and deriving the demand curve that we started the semester with, actually showing you how from the limited set of tools we've given you we can actually derive the underlying demand curves that we see in this class. I'm going to spend the rest of lecture then talking about the elasticity of demand-- what determines the shape of that demand curve. I'll talk about how changes in income affect demand. And then, we'll come back and talk about the effects of a price change, and we'll talk about the theoretical concepts underneath how you analyze a change in price in economics model. So let's start with deriving demand curves. So we started in this class with the demand supply curve, and I said we'd tell you where they came from. So, basically, we're now going to talk about how do you derive, from the tools we've learned so far, the relationship between price and quantity demanded-- the downward-sloping relationship between price and quantity demanded that we showed you in the very first lecture. So to do so, let's return to our example from last time. Remember, our example from last time is utility function was of the form u equals P times c. Your parents had given you an income of $72 with which you could buy pizza and cookies, pizza at a price of $12 and cookies at a price of $6. That was the parameters in are example from last time. And once again, remember, all I've done is giving you stuff here. Obviously, this part is non-controversial as a price for pizza in the market, the price of cookies in the market. There's an income that's the amount of money your parents give you. That's all non-controversial. And this is just a sensible assumption of what someone's preferences might look like. All I'm saying is with these four things, we are done. With these four things, we can now derive a demand curve. And how's that? Well, let's start by looking at it graphically in figure 4-1. Figure 4-1, on the left-hand side is exactly the kind of a difference curve analysis that we did last time. So we start with budget constraint bc1. That is something where you can either get up to 12 cookies and no pizza or up to 6 pizzas and no cookies. And we know, from our analysis last time, given this we showed you last time, and you went home and practiced and were so excited you told your mom and everything, we told you how you could show that you'd want 6 cookies and 3 slices of pizza at a point like point a. We showed that last time. Now, we also talked about how when the price changes. So, for example, imagine that the price of cookies rises to $9. Well, we know if the price of cookies rise to $9 is that you could still, on your budget, afford 6 pizzas. So the y-intercept does not change, but the x-intercept changes. Now, you can only afford 8 cookies, so you move to bc2 sub 2. You used to be able to afford 12 cookies, now you can only afford 8 cookies. And most importantly, the slope of the budget constraint has steepened. Remember, what is the slope of the budget constraint? The slope of the budget constraint-- the slope is minus P c over Pp. That's the slope. And that steepened, because the price of cookies has risen. So you have a steeper budget constraint. The slope used to be minus 1/2, now it's minus 3/4. So now, that's gone from minus 1/2 to minus 3/4 with this new higher price for pizza-- higher price of cookies. I'm sorry. The price cookies rose to $9. That budget constraint has steepened. What does this do to demand for cookies? Well, we find the highest tangency of your indifference curves with this new budget constraint, and you could solve mathematically to show that that will occur at point b, where you will choose to continue to have 3 slices of pizza but to only have 4 cookies. That is what happens if you take this, you do constrained optimization-- if you change this price to 9, re-optimize the way we showed you last time, you'd end up finding that you would want 3 slices of pizza and 4 cookies at point b. Now, what if instead the price of cookies fell to $4? So cookies got cheaper. Well, in that case, now, you could still only afford 6 slices of pizza. So the y-intercept once again doesn't change. But the x-intercept now moves out to 18, because now you can afford 18 cookies. So your new budget goes straight is flatter. It's bc sub 3. It's the flatter, outermost budget constraint. Your opportunity set is much larger because cookies are cheaper. You have the same amount of money from your parents, but it can buy more stuff, and you can buy more cookies. so your opportunity set is larger. So you now can re-optimize with a now flatter budget constraint. The slope, instead of being 1/2, has now fallen to 1/3. When the price of cookies is $4, the price of cookies over the price of pizza is now 1/3. So an absolute value, the slope is lower. So you can now-- and we say that if you take the same utility function and optimize at the new price, at a price of cookies of $4, you will find that you will choose point c. Still 3 slices of pizza, but now 9 cookies. So all three of those points we simply did by redoing optimization I did last time-- by taking this utility function, maximizing it subject to a budget constraint dictated by these parameters. And all I did was change this parameter twice. Once I changed it up. Once I changed it down. Questions about that? Well, you've now derived a demand curve. Oh, yeah, I'm sorry. Question. AUDIENCE: Will this also change the indifference curve? JONATHAN GRUBER: No, your indifference curves are determined by utility function. Your difference curves do not change. It changes which indifference curve you end up choosing, because the tangency point changes. But the only way to change the indifference curve would be to change utility function. Indifference curves purely come from this part of it. So indifference curves come from here, budget constraint comes from these three pieces. Yeah? AUDIENCE: Is it possible [INAUDIBLE] logical for different indifference curves in a quite long-range component, is that possible? JONATHAN GRUBER: Well, no. But it is possible to take a bunch of demand curves and conglomerate them to get one demand curve. And I'm going to talk about that later. But, no, indifference curves you can't really add up. Because, remember, utils are not a thing. So you can't add up indifference curves. But if we want to add up people, you would add up demand curves. And we've now derived the demand curve. How have we done that? Well, what's a demand curve? It's the relationship between price and quantity. Well, I just showed a relationship between price and quantity. When the price goes down, quantity of cookies goes up. When the price goes up, quantity of cookies goes down. Well, you just shift over to the right-hand side a figure 4-1. there's a demand curve. I just graphed these three points, I just graphed, for each of the different price of cookies I gave you, how many cookies are demanded. Well, at a price of cookies of 6, our initial price-- geez, this is really small-- 6 cookies were demanded. When the price of cookies goes up to 8-- I'm sorry, when it goes up to 9-- only 4 cookies are demanded. When the price of cookies falls to 4 at point c, then 9 cookies are demanded. So literally, we've just derived the demand curve. Starting with these primitives-- your tastes, your preferences-- and your budget constraint, we've derived a demand curve. And That's it. That's where demand curves come from. And that's essentially what is underneath the demand curve. What's underneath the demand curve is the fact that the reason to demand curves slope down is that as the price goes up, you want less of the good. Because with the given utility functions, that price goes up, you want less of the good. And that's why demand curve slopes down. So we've just derived that. Questions about that? Yeah? AUDIENCE: Is it always true that regardless of your utility function, you will always want three [INAUDIBLE].. JONATHAN GRUBER: Great question. Did you peek at my notes? No. Because if you had, you would have seen that's exactly the question I would have asked you. So since you asked me, I'll answer it. No, it is not always true. In fact, that is a feature of this particular utility function I've chosen. This particular utility function I've chosen gives the feature that the demand for a good is a function only of your income and that good's price, not of the other good's price. That's a feature of this. That's why we like this utility function. It has that nice feature. In general, that's not true. In general, when the price of one good changes, it will affect the demand for all goods, not just that good itself. So with a more general utility function, that would not be true. With this utility function, it gives what's called the flat price consumption curve-- not a term you need to know, but just if you want be more technical. Which is basically, with this utility function, demand for any good is a function only of its own price and your income. Therefore, when other prices change, it doesn't affect the demand for that good. But that is not generally true. In general, when you change the price of one good, demand for all goods changes. Questions about that? Yeah? AUDIENCE: Is this why it shows 3 as a constant, and just-- JONATHAN GRUBER: Oh, it wouldn't have mattered what number I'd chosen. But the bottom line is as the price of cookies changes, the number of pizza slices would never change. Demand would never change. That's a feature of this particular utility function. It's a nice feature of it. Once again, that comes to this trade with modeling. I've chosen a simplified utility function with this nice feature that's almost certainly not true. But once again it allowed us to derive a very sensible demand curve without introducing other complications. Now, this demand curve's particular shape, what determines the shape of this demand curve? And that leads us to the second topic I want to cover today, which is the elasticity of demand. What determines the shape of a demand curve is what we call the elasticity of demand, which we define the elasticity of demand, epsilon, as delta Q over Q over delta P over P. It's the percentage change in quantity for a percentage change in price. So, for example, if quantity falls by 2% for every 1% increase in price, that's an elasticity demand of negative 2. So elasticity of demand, as long as demand curves are downward-sloping, is less than 0. Because the price goes up, quantity falls. So elasticity of demand is less than 0-- I'm sorry, less than or equal to 0. Yeah? AUDIENCE: Is this the change in quantity over the new quantity or the the old quantity? JONATHAN GRUBER: Great question. When we're doing this, if we did it in calculus, it would be infinitesimal epsilon change. So it wouldn't matter if it was new or old quality. When you do it discreetly, use the old quantity. So I should say, Q0 and P0. Once again, if you did this in calculus, which is the way we're sort of thinking about in our underlying intuition, it's an epsilon change. So Q0 and P0 are epsilon. It doesn't really matter. But if you do it discretely, you'd want to use Q0 and P0. These are great questions. Keep them coming. Now, whenever we think of constant elasticity, it's always useful to think about the extremes. So what are the extremes of this measure? One extreme would be if the elasticity demand was 0. We would call that perfectly inelastic demand. It would be epsilon equals 0. Now, extremes never exist in the real world. But there are cases that come very close. So can someone, without flipping the page of the handout-- don't do it, because before you flip the page of the handout-- give me an example of a good that might have perfectly, in a very, very, very inelastic demand, where no matter the price, you'd still want the same quantity. Yeah. AUDIENCE: Water. You have to consume it to live. JONATHAN GRUBER: Water-- something that's essential to life like water. What else? Yeah, in the way back. AUDIENCE: Insulin. JONATHAN GRUBER: Insulin is the classic example we use. Yeah? AUDIENCE: Sewage removal. JONATHAN GRUBER: Sewage removal is interesting. Although, not quite, because we see a lot of variation around the world in sewage removal. But something like that-- basic essentials of life. Insulin's a class example we use. Indeed, that's the example I do you use in the next-- oh, I guess I didn't list it ion here. So you could have turned the page. We think about perfectly inelastic demand-- someone, once again for those of you who don't know about the medical science here, basically, if you're diabetic, you have trouble controlling your blood sugar. Insulin is a medicine you need to take to help you control your blood sugar. Without it, you die. So you'd think that's pretty inelastic demand. Like, I want to live, so I'm going to want to have my insulin. And if the price goes up, I'm not going to say, nah, I'll just die. You're going to want to still have the insulin. So we think in that case, quantum would be fixed at some Q, which is insulin you need to live, and it wouldn't really matter what the price is. So your elasticity would be 0. You have a perfectly inelastic curve. Quantity would not change with price. Basically, that would happen when there is no plausible substitute. The reason water is not as good as insulin is because I can drink something else. And sewage removal, I can do other things to deal with the sewage in my house. But insulin, there is no substitute. I die. So basically, the bottom line is, when there's no plausible substitute, demand will be perfectly inelastic. Likewise, we consider-- yeah? AUDIENCE: Can there be multiple companies that sell insulin? So if one, say, increased the price, they would shift to another. JONATHAN GRUBER: OK, but this is a market-- that's a great point. I'm just doing this as-- let's actually image just one company for now. You're right, I'm not doing choice across companies. So that would be inelastic. We could also do the other extreme, perfectly elastic. So what's an example? Perfectly elastic demand is that epsilon equals negative infinity. So what would cause perfectly inelastic demand? So people raise their hand and tell me. Perfectly elastic demand-- I'm sorry. What would cause perfectly elastic demand? What would cause it? I'm just going to try to spread it around a little bit. You two have been-- Yeah AUDIENCE: Would it be something that's very unnecessary, like diamonds? JONATHAN GRUBER: Well, that's interesting. I mean, that's not-- I'm going to come to that. But that wouldn't-- that's something where demand would be less elastic, but not close to perfectly elastic. Because, basically-- well, let me ask the question this way-- what would drive a good to very, very elastic demand in general? Yeah? AUDIENCE: When it has perfect substitution-- [INTERPOSING VOICES] JONATHAN GRUBER: Exactly. Perfect substitutes. Diamonds don't have perfect substitutes. I mean, we're increasingly making fake diamonds that are better and better substitutes, but a real diamond still doesn't have a perfect substitute. So what does? What's an example of something with a good-- with a very good substitute? AUDIENCE: I don't know. JONATHAN GRUBER: Yeah? AUDIENCE: A spork. JONATHAN GRUBER: A spork. [LAUGHTER] There is no sub-- I am insulted that you would suggest that the great spork! You need two different things to replace a spork! AUDIENCE: You're never using both at the same time. So if the price increases-- you never-- JONATHAN GRUBER: Yeah. But then you have to buy a spoon and a fork. No, I reject your spork suggestion. [LAUGHTER] I'm just joking. No, but what else? What else? What is something that has really good substitutes? Yeah? AUDIENCE: Off-brand medication. JONATHAN GRUBER: Yeah. Something which is an off-brand medication, or I like to think of a fast food, like fast food burgers have pretty good substitutes with fast food pizza. Basically, once again, it's hard to think an extreme example. Nothing's ever perfectly elastic. But things which have very good substitutes-- typically this works well when we think across brands. If you think about different brands of gum. Jeez. I mean, who the hell cares? Or different brands of fast food. Yeah? AUDIENCE: What about something like a $5 gift card, where after $5 nobody's going to buy it. JONATHAN GRUBER: That's a separate issue. Let's come back. That's sort of a separate-- that's like a weird, kinked budget constraint. That's not really about substitution. But the bottom line is-- we don't need any more examples-- the key element-- and we show that in figure 4-3. That's going to be a horizontal demand curve. With a perfectly elastic demand, you have a horizontal demand curve. And therefore, price never changes. It's sort of a weird way to think of perfectly elastic demand. The way it works is if I ever charge a price 1 epsilon above someone else, I'd lose all my business. And if I charge a price 1 epsilon below everyone else, I would have the entire market. Because if it's perfectly elastic, like if one pack of gum-- I don't what-- these stupid gum things-- Orbits and Ellipse or whatever, if they charge a penny more than someone else, no would buy them. They'd go out of business. That's the idea here. And that's why the price can't change. The price is fixed. So if anyone deviates from that price, boom, they lose the whole market. Yeah? AUDIENCE: So wouldn't the market for dollar bills be perfectly elastic? Like-- JONATHAN GRUBER: There's not really a market for dollar bills. [INTERPOSING VOICES] AUDIENCE: --dollar bills. JONATHAN GRUBER: There's not really-- I mean, you could say the demand for cash in general might be fairly elastic, because you have credit cards and things like that. That's an interesting idea. Yeah. So, basically, when you have perfectly elastic demand, you end up with sort of this constant price, and quantity changes but price doesn't change. So that's kind of the extremes. Now, in general, we have goods that are more and less elastic. In general, we end up with this range between perfectly elastic and perfectly inelastic. The bottom line is-- here's the intuition I want you to have-- what determines elasticity is substitutability. The more substitutable goods are, the more elastically demanded they are. So now, I want to go on to another topic which is related. Which is, we talked in our example about what happened when I change prices. What about when I change income? So the third topic I want to talk about is what happens when income shifts, and how does that affect demand curves? When income shifts, how does that affect demand curves? Well, we could do the same exercise we did before. We did an exercise before. We said, well, let's just use the tools we used before to solve for you new choices at different prices. Let's just solve for you new choice at a different income. So let's go to figure 4-4. Figure 4-4, once again we start with bc1. Same parameters as before-- same setup as before. We choose point a. Now, let's say I raise your income from $72 to $96. You've done well. Your parents are giving you more money-- $96. Well, in that case you will choose to have both more pizza and more cookies. Given this utility function, you will choose the point b. You will choose point b. Likewise, if I lower your income from $70 to $48, then you will choose point c. So as your income goes up, you'll choose more of both. As your income goes down, you choose less of both. Now, once again, you'll notice this-- well, let me come to that. So basically, what that says is I can trace out the relationship between how your income changes, how you demand for cookies change. I can then graph that on the next graph and generate what's called the Engel curve. The Engel curve is the relationship between income and quantity demanded. And we'll come back to why this matters. The Engel curve is the relationship between income and quantity demanded. Now, here the Engel curve is linear. Once again, that's just a feature of this utility function. In general, it doesn't have to be linear. But here the Engel curve is linear. That's just because of the way we structured this utility function. And the slope of the Engel curve is what we call the income elasticity of demand, gamma, which is delta Q over Q-- Q0 once again, if we're doing it discreetly-- over delta y over y0. That's the income elasticity of demand. Now, let me just say one comment here about this. Because there's sort of a big cheat that I'm doing here with all this that I need you guys to be aware of, which is constant elasticity versus linear curves. We know a constant elasticity curve will not be linear. If it's linear, it's not constant elasticity, because if it's linear, the elasticity will change along the curve. So the demand curve. I just drew in figure 4-1, that was a constant elasticity demand curve. That's why it was curved. This Engel curve I drew here would not be constant elasticity, because it's linear. You can calculate for yourself that the percentage change, the income elasticity will shift as you go along this curve. And you can show yourself that. What we're going to do in this class is we're going to draw linear constant elasticity curves, which is, of course, technically wrong. Just think of them as blown up versions of a large demand curve. Everything is locally linear. Once again, for an epsilon change, everything is linear. So the truth is we'll cheat a little bit and often draw a linear constant elasticity curves. And I want to own that that's a cheat. But if everything is really local, it's not that bad a cheat. So that's kind of how we're going to-- that's a cheat we're going to do constantly through this course. And we'll be very clear-- we make clear any problems or anything if we need you to deviate from that. Now, what we have here, in addition to a linear Engel curve, is we have an upward-sloping Engel curve or positive income elasticity. We call goods with a positive income elasticity normal goods. Goods where the more money you have, the more of them you want, we call normal goods. Because that's sort of normal. However, it is also true that a number of goods in the world actually have gamma less than 0. And we call those inferior goods. Why would a good be inferior? Why, when your income goes up-- yeah? AUDIENCE: [INAUDIBLE]. JONATHAN GRUBER: Exactly. Any examples anyone can think of? Yeah? AUDIENCE: Maybe at a fast food restaurant. The minimum amount of food you have to eat. And then, like, [INAUDIBLE] your fast food, and then I have to get more money-- [INTERPOSING VOICES] JONATHAN GRUBER: Exactly. Great example. Yeah, in the back. AUDIENCE: Omega watches versus Rolex watches. JONATHAN GRUBER: I wouldn't think of either of them as inferior. Relative they may be inferior. But, obviously, when your income goes up, you're not going to suddenly-- you're going to want more of both. Whereas fast food you actually would want less of it as your income goes up. Literally, you will eat at McDonald's less if you're richer. You're not going to have fewer watches if you're richer. So the bottom line is that inferior goods, you're actually getting-- it's a bit subtle. You have different goods of the same kind. Let's think about classes of goods rather than brands of goods. Once you get across brands, you're right. Let's think about classes of goods. Watches-- luxury watches, in general-- are clearly not inferior. Fast food may be inferior. Literally richer people may eat probably less fast food than poorer people. Yeah. AUDIENCE: Would something like your refrigerator count, where after you buy one you don't really need more of them. JONATHAN GRUBER: Well, that's like a quan-- that could be right. But yeah. I think not, because the bottom line is rich guys are much more likely to have two refrigerators than one. So it's a discreteness problem. It's too discrete to really use as an example. But I think fast food is a great example. The class example we use is potatoes. Where in the old days, before fast food, that was the cheap, filling, shitty-tasting food stuff that guys sort of ate all the time. And now, when they have money, they say, I'm going to move on to steak, and they eat less potatoes. So something where essentially you'd rather shift to something else if you have more money is inferior. Now, moreover, within that, within normal, we're going to draw a distinction between what we call luxuries and necessities. Luxuries are going to be good where gamma is greater than 1. And necessities, gamma is less than 1. So we're going to say-- essentially, the question is, proportional to your budget, what happens if your income goes up? In other words, do you spend more and more-- they're both normal goods. The richer you are, the more you buy of it. But a luxury good is you spend even larger share of your budget on that good as you get richer. So that would be luxury, like watches, boats, maybe refrigerators, et cetera-- things where the richer you get, the more of your budget you spend on it. Necessities are things like food, where clearly rich people spend more on food than poor people. But they spend a smaller share of their budget on food than poor people. So it goes up, but doesn't go up proportionally with your income. It goes up less in portion with your income. Yeah? AUDIENCE: [INAUDIBLE] for as some sort of human necessities, in the sense of if you're more wealthy, you might buy more name brand food or something like that, instead of more-- JONATHAN GRUBER: Yeah. Once we get into brands, you can absolutely see that. You can think of luxury brands and necessity brands. Absolutely. But if we stay with categories of goods, then let's think of jewelry as the [INAUDIBLE] example here and food as the canonical example here. Yeah? AUDIENCE: Do you have an example of something that's near the border between luxuries and necessities. JONATHAN GRUBER: You know, there is a huge industry in estimating these elasticities of demand. So I'm sure there's an answer to that. I don't have it off of my head. All right. So now I've given you the underlying tools of consumer demand theory. I've told you how to decide what quantity a consumer wants. I've told you how you can use that to derive demand curve. I've explained the shape of the demand curve. I then talked about what happens as your income changes and talked about the shape of the income elasticity. Now, I'm going to put this together to come back to revisit something you think you already know the answer to. Which is, what happens when a price changes-- the effects of a price change. Now, you might say, well, that's sort of silly. We already did that once this lecture. We already did, when we derived the demand curve-- the effects of price change-- we did some price changes, right? I showed you what happened as the price of cookies change. But, in fact, we cheated a little bit. We didn't cheat-- we gave you sort of the bottom line but didn't get into the elements of why people react the way they do to price changes. Now, we're getting in some sort of deep theory. Now, I'm going to talk about something-- it's deeply theoretical in the sense that in some sense if all you care about is what happens in the real world, they just care about when a price change happens how does quantity change, I'm going to use some theoretical concepts, which are going to become very powerful later on in the course, which are important understand now. Which is, how is the underlying decision calculus changed by price-- changed when the price changes? And the way we're going to do that is that we're going to decompose your response to a price change into two effects-- the substitution effect and the income effect. We're going to separate your response into two effects. So the separation is going to become very important later on. The substitution effect, we're going to define as the change in the quantity of a good when the price changes holding utility constant. So it's delta Q-- d Q d p for shorthand-- but holding utility constant at some fixed level u bar. The change in quantities price changes-- so it's the elasticity of demand-- but at a constant level of utility. The income effect is the change in quantity of a good as income changes. Change in quantity, dy. Which is the income elasticity we talked about-- the change in quantity as the income changes. And this is actually multiplied by the initial level of income. We'll learn about this in a section. But that's technically how the income effect is defined. And you'll come into a section about why that is. But we're going to decompose this. So that's sort of confusing. So let me start with graphically to understand it. So let's go to figure 4-5, one of our more complicated figures. We start at budget constraint 1-- same parameters as always. All the math here-- all this graphic stuff follows from the math using that utility function and these price and income-- same as before. So we start, as before, at point a. Your tangency is the best package you can have, given that utility function, is 6 cookies and 3 slices of pizza. Now, let's imagine the price of cookies rises to $9. The price of cookies in our example goes from $6 to $9. That's the example we're going to analyze. Now, we know from before that will ultimately move you from 6 cookies to 4 cookies while holding pizzas constant at 3. So we know where you'll end up. You'll end up at point c. We did that before. But actually two things are happening to get you there. The first thing that's happening is the substitution effect, which is the change in prices with utility constant. And how do we measure that? We want to ask, given that the price changed but the utility is constant, what's the new quantity you choose? What does utility constant mean in this graph? What does it mean to hold utility constant? Let's get some other folks involved here. What is it mean? Yeah? AUDIENCE: Same indifference curve. JONATHAN GRUBER: Same indifference curve. So what we want to do is ask, given the new prices but the old indifference curve, what quantity would you choose? Well, the way we do that is we find the tangency between the new slope of the budget constraint and the old indifference curve. And we do that by drawing sort of an imaginary budget constraint bc prime. bc prime is a sort of imaginary budget constraint. It's not a real budget constraint, but it has the slope of the new budget constraint, but it's tangent to the old indifference curve. That's the key thing. bc prime, the imaginary budget constraint, the dashed line, has the slope of the new budget constraint, same as the new price ratio. So the slope is the new price ratio. The slope is at the new price ratio, but it's tangent to the old indifference curve. So bc prime is basically going to the tangent [INAUDIBLE] indifference curve at point b. So what we're saying is the substitution effect moves you from point a to point b. That is holding utility constant but at these new prices, you would choose to have fewer cookies and less pizza. We call this notion compensated demand. That is, I'm compensating you. I'm holding utility constant. I'm saying price of change sucks for you. You're worse off. Your opportunity set's restricted. But I'm going to compensate you by holding your utility constant. So call this compensated demand. Your compensated demand would mean that when the price goes up, you would choose to reduce your consumption of cookies from 6 to 4.89. Now, here's the key thing about substitution effects-- we can sign them definitively. They are always negative. The substitution effect is always negative. The income effect could go either way. We'll show that. The substitution effect is always negative. We can see this in two ways. Graphically, think about it this way-- you have to be tangent to the same indifference curve with a higher sloped line, so you have to move to the left. If you get a tangent to the same indifference curve with a line with the higher slope, it's go to be to the left. So that's a graphical intuition. Mathematically, it's worth writing out the steps, because it helps. That's why we teach this, it helps remind us of our consumer theory. Step 1, you're at this new tangency. Step 2, we know that at any such tangency that's optimized, the marg utility of cookies over the marg utility pizza equals the price of cookies over the price of pizza. We know that's true with any tangency, because that's the optimal choice. Step 3, we know P c over P p is up. I just said that. And this assumption is the price cookies up. Therefore, that leads to step 4. Which is that M Uc over M Up must be up, because it's still equal. Well, how do you raise the ratio of the marg utility of cookies to marg utility of pizzas? How do you accomplish that? By having fewer cookies and more pizza. So that means-- that implies-- that cookies are down and pizzas are up-- and/or pizzas are up. How do you get that ratio to be higher? Well, remember, this is why do this math here. Remember the key intuition-- more cookies means lower marg utility of cookies. The more cookies you have, the less you care about the next cookie. So I want the marg utility of cookies to be lower. If the marg utility of cookies, I'm sorry, to be higher, I've got to have fewer cookies, or I've got to have more pizza, or both. So the substitution effect is always negative. If I'm going to hold utility constant, and change prices, and raise the price, you'll always want less of that goods on the substitution effect. Question about the graphics of the math? Now, let's come to the income effect. The income effect says, holding prices constant-- so I shouldn't put this here. The income effect is this at a constant price. I should have added that. So the income effect is the change in quantity demanded as income changes holding prices constant. So now we're saying-- the income effect is saying, look, the price changed. So therefore I shifted my consumption away from cookies. But the other thing that happened is I got poorer. Maybe I'd say you didn't get poorer, your parents didn't send you less money, but remember your opportunity set restricted. You effectively are poorer because this price went up. How do we represent that? Well, we can exactly represent that by the shift from bc prime to bc2 Because bc prime and bc2 have the same slope. That shift is just the income effect. So that's holding the prices constant. The price ratio for bc prime and bc2 is the same. But you're now effectively poorer, because at the same income you can afford fewer cookies. So you're effectively poorer. So now your income's effectively fallen. And at that new budget constraint with that new lower effective income and higher price ratio, you choose point c. So we go from a to c, just like I told you before, but we actually get there in two steps. One step is sort of the relative change in prices caused you to say, ooh, I want to get away from cookies. The other say, I'm poorer, so I want less of everything, including cookies. So one is the price effect, one's the income effect. And these two effects matter. Now, in this case, they don't matter, because they work together. In this case, you might say, well, look, why do I care? The bottom line is, the number of cookies fell by 2. Substitution effect work together, why do I care? Well, you might care, because if the good is inferior, then the income and substitution effect work against each other. If the good is inferior, then the income substitution effects work against each other rather than with each other. To see this, let's go to figure 4-6. Now, we're going to totally change our example. Now, we're going to be choosing between steak and potatoes. Totally different example-- steak and potatoes. You start a budge constraint bc1. Potatoes are $1, and steak is $5. So in our new example-- I'll put this up here-- we've got the price of steak is $5, the price of potatoes is $1, and your income is $25. That's our new example. In that case, you will choose-- and this is a different utility function. This is a totally new example. I'm not going expect you to understand the underlying math here. I'm just showing an example of something that might be true. So you choose point a. You choose 7 and 1/2 potatoes and 3 and 1/2 steaks given those prices. Now, we're going to say, what happens if the price of potatoes goes from $1 to $3? The price of potatoes goes up. Well, two things happen-- first of all, the change in compensated demand moves you from point a to point b. How do we know that? Because you draw a new imaginary budget constraint that has the new price ratio, but it's tangent to the old indifference curve. So you're point a to point b. So the substitution effect lowers your demand for potatoes from 7 and 1/2 to 4. But the income effect raises your demand for potatoes. Now, the income effect means you actually move back from 4 to 5. On net, you're still having fewer potatoes, but the substitution effects went opposite ways. Why? Why did the income effect cause you to want to have more potatoes? The substitution, you wanted have to less. Yeah? AUDIENCE: Because the price of potatoes went up [INAUDIBLE].. JONATHAN GRUBER: Close. Price of potatoes went up. So you're effectively what? Poorer, right? And when you're poorer, how does that affect your consumption of inferior goods? You want more of them. So price of potatoes went up. So effectively you're poorer. Now, when you're poorer, in our previous example with cookies you wanted fewer cookies than a normal good. But potatoes are an inferior good. So as you're poorer, you want more of them. The income effect goes the opposite way of the substitution effect. That's where this starts to get interesting. If the income effect always goes the same way as substitution effect, this is sort of just a purely useless theoretical exercise. It gets interesting when income effect goes the opposite way of the substitution effect. Yeah? AUDIENCE: Isn't it like the [INAUDIBLE]?? JONATHAN GRUBER: Hold on. I'm getting there. I'm getting there. OK. Yeah? AUDIENCE: [INAUDIBLE]. JONATHAN GRUBER: Utility-- well different utility functions will give you-- you have to have a different utility function to get the good to be inferior. So you utility of steak and potatoes is not going to square root of-- you won't get inferiority with square root of p times s. So it's a different utility function, and I mentioned that I think. It's a different utility function. Yeah? AUDIENCE: It's the different functions that makes potatoes be an inferior good mathematically. JONATHAN GRUBER: Exactly. So basically, your intuition-- I give you intuition why they're inferior. But, mathematically, it would occur because we'd have a utility function. The utility function would generate an inferior would be a different looking utility function. AUDIENCE: [INAUDIBLE]. JONATHAN GRUBER: Not off the top of my head. But we'll do it in section. So the bottom line is-- yeah? AUDIENCE: When you're setting that price of steak and potatoes, is it that they both provide similar nutritional value? So one potato does not necessarily equate to one steak. But the price that are setting, $5 worth of steak is equivalent, in terms of my health, and will fill me up as much as-- JONATHAN GRUBER: No, no, no, no. I'm not saying that at all. I'm not writing down a health function. I didn't write it down. This is a utility function. So it's about filling up, it's about taste. Steak may leave me hungrier, but it's way better. So it's about what fills me up. Literally, all I'm saying is, given my utility function-- and I don't have the utility function written down-- I was choosing a balance of mostly potatoes and some steak. Now that the price of potatoes goes up, I end up wanting fewer potatoes, but not as few as you might think from the substitution effect, because potatoes are inferior. Yeah? AUDIENCE: [INAUDIBLE] the price is more constant [INAUDIBLE]?? JONATHAN GRUBER: No, no. What I'm saying-- the income effect is holding the price constant. What happens to your demand? So holding the price constant-- what I mean by that is moving from bc prime to bc2 is the income effect, holding your price constant. So the price change is reflecting the substitution effect. That's reflected in moving for bc1 to bc prime. The income effect is holding the price constant that is, given that same slope of that imaginary new budget constraint, you're now poorer. So that's why. So once again, it's a hard thing to wrap your mind around. It's theoretical. But the notion is when a price changes two things happen-- it changes the relative desirability of two goods, and it changes your income-- your effective income, your opportunity set. Two things are happening. Yeah? AUDIENCE: First, when the price first changes, you needed a substitute, so you keep-- JONATHAN GRUBER: Oh, no. They both happened at the same time. Let's be clear. This isn't sequential. This happening in real time. It's not like you said the price went up, I'm going to compute my, you know-- it's happening in real time. It's just we're decomposing into two effects. And the reason we're doing that is because once goods are inferior, this decomposition becomes interesting. So I like to think about this in sort of a simple table to help remind you how to think about this. So let's think about a simple table. Here we have the price change. so the price can go up or the price can go down. Up and down. Here we have the substitution effect, here we have the income effect, and here, we have the total. Well, in the case of a normal good-- if a good is normal-- then we know the substitution effect when the price goes up leads you to want less of a good. The income effect also leads you want less of a good. So you definitely want less of the good. These are all equals. There's always corner cases. Likewise, when the price falls, the substitution effect makes you want more of the good. When the price falls, you're effectively richer. So the income effect makes you want more of the good. So you clearly want more of the good. That's the easy case. The more interesting case is, what if it's an inferior good? Now, if the price goes up, the substitution effect is the same. Substitution effect is always negative. Higher price means you substitute away from the good. But the income effect is now positive. So the net is unclear. Likewise, if the price goes down, the substitution effect is always positive. You always want more of the good if the price goes down. But the income effect is now you want less of the good. Why? Because you're richer. If the price of a good falls, you're richer. Richer means you want less of that inferior good. So the net effect is unclear. So the interesting case becomes inferior goods. Questions about the table? So this raises the question, are there goods-- what does this imply? If the income effect dominated the substitution effect, you could get what? Yeah? AUDIENCE: The higher prices, the more you buy. JONATHAN GRUBER: Yes. You could get an upward-sloping demand curve. You could actually, theoretically, get an upward-sloping demand curve. And we call this a Giffen good. A Giffen good is a good where you get an upward-sloping demand curve, where actually the inferior income effect is so large it dominates the substitution effect, and you actually get an upward-sloping demand curve-- that a higher price leads people to want more of the good. Now, in fact, this is named after some guy, Giffen, I guess. And it probably was a guy, because it's old. But, in fact, it's convenient, because it's close to where gryphon. And gryphons are imaginary, and so are Giffens. It's actually hard to find examples in reality of Giffen goods. It's actually pretty hard to find examples from reality. But there was one interesting experiment that was run, which is sort of the first convincing evidence that in some situations Giffen goods could exist. So I ran the follow experiment. They run a study in China, where very, very poor households-- most households in China are poor, but they divided it into super poor versus moderately poor households. And they basically gave them coupons which lowered the price of rice, which was their staple good. Basically, they eat rice, and that's sort of their basic good they eat. And they basically lowered the price of rice. What they found is that for families that weren't super poor, they found a typical downward-sloping demand curve. Giving them the coupon off rice meant they bought more rice. But for the very, very poor families, they actually did find an upward-sloping demand curve. Giving them a discount on the price of rice actually caused them to have less rice. Because, literally, that's all they ever ate, was rice. So by definition they could've had more, because they didn't eat anything else. Now that you're essentially saying, look, you used to eat all rice, you'd spend your whole budget right, now I've basically giving you extra money. Because all that rice you used eat, you can now have cheaper. Are going to spend that to buy more rice? No, you're buy something else. And that's a Giffen good. So if you think about sort of a corner solution where people are buying all the good that's inferior, then by definition if you give them more money, they're going to move on to another good. Or there's at least a possibility. And that's the Giffen good example. But we have to search pretty darn hard. Typically in the demand context, we think that demand curves are downward-sloping. However, when we get to other contexts, we're going to find it's more normal to find that these income and substitution effects fight against each other and lead to strange-shaped curves. And we'll come back to that later in the course. OK, I'll see you all Wednesday.

MIT_1401_Principles_of_Microeconomics_Fall_2018

11_Monopoly_I.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: OK, so let's continue. So this is actually, in some sense, a key breaking point in the course. Which is, in some sense what we've done so far is give you a set of tools to understand how to think about how consumers and producers make decisions, and then to understand how you compare positive and normative implications of economics. What we're going to do for the rest of the class, for the rest of the semester, is we're going to start to talk about how you apply those tools to more realistic situations. Let me answer your very first question, what you learn today is not on the midterm, OK? So that's the question that's all on your mind because it is. So the midterm tomorrow night will cover everything through what we covered through last week, but what we learn today will not be in the midterm. But don't leave because it will be on the final and it is interesting. So what we're going to do today is talk about taking the tools we developed so far in this class and applying to more realistic situations. And the more realist situation we're going to start with is the case of monopoly. And we don't talk about monopoly profit maximization. Now so far on the producer side we've been discussing one extreme of how the market's organized, which is perfect competition. And perfect competition is a wonderful sort of theoretical benchmark but never actually exists in reality. No market is perfectly competitive. The other extreme is not widely applicable but still does exist in reality, which is monopoly, OK? Monopoly is a situation where a market only has one firm in it. So a market with one firm is a monopoly market, OK? So that's basically a case where you have only one firm providing the good, OK? Now in reality, most markets fall between perfect competition, which never truly exists, and monopoly, which rarely exists. Most markets are in between, we call them oligopolies. That's markets with several firms but not perfectly competing. And we'll get to that. But that's actually one of the harder things we'll cover, so we're going to start with this other extreme of monopoly which gives us a lot of the insights we need for oligopoly but in a much simpler case. Now, the key thing with monopoly, the key difference from what we've done so far is that now firms are going to be price makers, not price takers. That's going to be the key change from the perfectly competitive situation. The perfectly competitive situation, from any given firm's perspective the price is something given them by the great market. OK, we talked with our two diagrams side by side about how the market gives the price. But we sort of-- from any given firm's perspective they were price taker. They couldn't affect the price. Monopolists however, when you're the only firm you get to set the price. So monopolists are going to be price makers, not price takers. And that's going to change the dynamics of everything we do, and that'll be our focus today, OK? So that's going to be the big change, we'll go from competitive markets to monopoly markets. Now, for the first 2/3 or 3/4 of the lecture we're going to make one other assumption. We are going to assume there is one price in this market, that is no price discrimination. This monopolist sets the price, but he sets one price for everybody. So if a monopolist is selling a good, they sell that at one price to everybody buys the good. Now once again, this is not super realistic, monopolists often have different prices for goods. But it's a very important, helpful extreme. And you've got to remember that we're imposing this constraint because it's going to drive the intuition I'm going to come to next, OK? So you're a firm selling a good at one price. You offer it up, anybody can buy it at that price, OK? Now, what we're going to do-- if you remember, a mon-- the fundamental part of producer that holds is the goal of monopolists is to maximize his profits, OK, which is revenues minus costs. And as we showed before, profits are maximized when marginal revenue equals marginal cost. That's the profit maximization point. Now, nothing on the cost side is going to change. So monopolish monopolist, the way we get cost curves doesn't change. That's all about-- remember, because that just comes the technology production and that comes from input prices. So nothing on the cost side is going to change. All that hard and somewhat boring work we did deriving cost functions and stuff, that all is-- that's done. The only place monopoly gets exciting is on the margin revenue side, because before we said marginal revenue was just price for a price taker. But that's no longer true. Marginal revenue is no longer just price. Now marginal revenue's going to be more interesting. So the monopoly difference, if you will, or the imperfect competition difference, doesn't come at all from the cost side. That side is done, doesn't matter what kind of market you're looking at, OK? All the interesting action here is on the revenue side as we think about monopoly and oligopoly markets, OK? So to think about that, let's go to figure 11-1. Think about-- let's start by think about a competitive firm. This is a diagram I could have shown earlier. It just sort of is a way to think about a competitive firm. Think about a competitive firm's profits. So imagine that this firm-- to make life easy let's imagine that we have a firm with marginal cost of 0, OK? So you've got demand curve, and so therefore any money you make is profit. And we're in the short run, OK? So there's profit in the short run, even for this perfectly competitive firm. But a perfectly competitive firm faces a horizontal demand curve. So if they sell q units, OK, they make profits of area a, which is simply q times P1, little q times P1. If they sell q plus 1 units, they make an extra profit rectangle of B. So literally, the marginal-- the extra profit they make is just the amount P. It's 1 times P1. The horizontal distance from q to q plus 1 is 1, the vertical distance is P1. So the marginal profit is just the price, OK? Marginal revenue, I'm sorry, it's just the price. Yeah? AUDIENCE: [INAUDIBLE] competitive [INAUDIBLE].. JONATHAN GRUBER: Once it's in the short run there can be, right? In the short run there can be profit in a competitive market. In the long run there's no profit, OK? So basically, their marginal revenue, what they make-- that next unit they make is the price, OK? Now let's think about a monopoly firm. A monopoly firm does not face a perfectly elastic demand curve because a monopoly firm's demand curve is the whole market. Notice little q has become big Q on the x-axis. The monopolist no longer faces only their perfectly elastic firm demand. They face a market demand, which can be any degree of elasticity but typically is somewhat elastic. It's downward sloping. Now let's say you're a monopolist and you're selling big Q units at a price P1. Your profits are C plus A. Once again, same marginal cost of 0 to make this easy, or C plus A. Now let's say you want to sell one more unit. Well, what's different? What's different is now you're facing a downward sloping demand curve. So if you want to sell one more unit you have to lower the price. That's the difference. If you want to sell another unit, you can only do so by lowering the price. And you're a price maker, so you have the right to do that. So if you want to sell another unit, two things are going to happen. A, you are going to sell a second unit and make money on that unit. That's the area B. But b-- I should say one and two. One, you sell the second unit. That's the area B. Two, you will lose money all the other units you sold because now you have to lower the price. Because remember, there's only one price, OK? So basically, you used to make A plus C. Now you make A plus B. You've added B, but you've lost C. So what is the marginal revenue? The marginal revenue for a monopolist is now the area B minus the area C. Or P2, which is area B-- 1 times P2 minus P2 minus P1 minus P2 times Q0, the original quantity. OK? This is the area C, but this is the area B. The original quantity times the change in price. So your marginal revenue is not this simple just priced more and more. Now it's this more complicated term, OK? More generally, marginal revenue can be defined as-- the marginal revenue is defined as-- marginal revenue is defined as P plus delta P delta Q times Q. That's marginal revenue. The first term is positive, that's the money you make on the next unit. The second term is negative because, by definition, the demand curve's downward sloping. There's no given goods anymore in this class. Demand may be on a test but not in reality. The demand curve's downward sloping, OK? So this term is negative. Now once again, I'm writing delta but really it's derivative. So this is all increment-- this is all-- this is sort of epsilon changes. This is Q0, OK? This is sort of epsilon changes. But the bottom line is, for an epsilon change in price you get-- for an epsilon change in quantity you get the price on the unit you sell minus the money you lost. And then you too can no longer sell, OK? This is the key intuition of monopoly math. The k-- yeah? AUDIENCE: What's that equation equal to? That [INAUDIBLE] JONATHAN GRUBER: This is your marginal revenue. I just did it graphically. That's your marginal revenue. You just see that from the graph, OK, and I just rewrote it here, OK? So basica-- or for those of you into calculus, if you just differentiate, just-- if those of you who have your calculus, you could just write, you know, dR dQ, d revenue dQ, OK, what do you get? You get P plus dP dQ times Q. That's just the derivative of revenue with respect to quantity, OK? So let's think intuitively about what's going on here because this is very, very important. What's going on here is to sell another unit you have to lower your price. You're working your way down your demand curve. And therefore, it offsets the benefit you get from selling another unit, OK? I like to call this the poisoning effect. I mean, it's not original to me but it's a term that I've heard use that I like, the poisoning effect. The idea is that to sell another unit, I have to poison myself a little bit by lowering the price on all the previous units I sold. If I want to go out in the market and force consumers to buy one more unit, I've got to lower the price on all the units which offsets money I used to make. So that's why we call it a sort of poisoning effect. Now, why doesn't a competitive firm face this issue? Why are we just bringing this up now? Why does a monopoly firm face a poisoning effect and a competitive firm doesn't? Or alternatively, another way to ask the question is, in what situation would a monopoly firm not face a poisoning effect? Yeah? AUDIENCE: Actually, in a [INAUDIBLE] in a competitive firm [INAUDIBLE] JONATHAN GRUBER: Right. They're not-- that's [INAUDIBLE] right. The competitive firm's the price taker. But in particular, even though the price taker, why when they sell more quantity is there no poisoning effect? Yeah? AUDIENCE: Demand is perfectly inelastic? JONATHAN GRUBER: Demand it's is perfectly elastic. The firm faced a perfectly elast-- so likewise, a monopolist facing a perfectly elastic demand would also have no poisoning effect. Think of it this way, what is dP dQ with perfectly elastic demand? Zero. The price doesn't change, you sell more. So the reason the perfectly competitive firm face no poisoning effect is because it's facing a-- it doesn't, the price doesn't have to change to sell more units, OK? But a monopolist faces a poisoning effect because it's got a downward sloping demand curve. OK? So we can actually see this-- so what we want to do in figure 11-3 is we graph the monopolist's marginal revenue curve. So let's actually work out the math here. Let's imagine there's a demand curve. Let's imagine that I've got a demand curve-- and I'm just making this up-- demands of the form A equals 24 minus p, OK? Let's just say that's the demand curve, OK? Now the first step is to flip the way we express this and write this-- write price as a function of quantity because now the monopolist is choosing his price. So we can write that p equals 24 minus Q, OK? Just inverting it, writing price as a function of quantity. So in this case revenues, which are equal to p times Q, are equal to 24Q minus Q squared. I just multiply through by Q. So marginal revenues differentiating are just 24 minus 2Q, OK? So once again, this is a new sort of mathematical trick you'll have to get used to. Flip the demand curve, so you express price as a function of quantity. Multiply through by Q and then differentiate, and you get that marginal revenue is 24 minus 2Q. So you can see that in figure 11-3, the marginal revenue curve is the demand curve shifted in, OK? Now in fact, this nice relationship will not always hold. This sort of picture of a marginal revenue curve being the demand curve shift [INAUDIBLE] will not always hold. It only holds for certain functional forms. Mostly we'll use functional forms for where it holds. The main lesson is the marginal revenue curve has to be at or below the demand curve. That's the proof, OK? Whether it has this nice sort of shifted in relationship, that depends very much on functional form. But what is absolutely true is the marginal revenue curve is always everywhere at or below the demand curve. Why? Because the demand curve-- because by selling a next unit you're going to make less through this poisoning effect. So the marginal revenue curve is always below the demand curve, all right? Questions about that? OK, so now let's go on and let's talk about the critical relationship which I just developed-- intuitive, let's do it mathematically-- between marginal revenue and the elasticity of demand. OK, so take this-- we've got our marginal revenue expression, marginal revenue equals p plus delta p delta Q times Q, OK? Now take that expression and multiply and divide by p. So I'm just going to take this expression, I'm going to multiply and divide by p, OK? Then I'm going to get p plus delta p over p delta Q over Q times Q over p-- oh, p plus p. I'm sorry, no, I did this wrong. My bad, don't write that down. I've got to look more at my notes. I'm going to write it as-- I'm going to multiply and divide by p. p plus p times, yeah, delta p delta Q times Q over p. I skipped a step. So I just multiplied and divided by p, multiplied and divided by p. p plus p times delta Q, delta p over delta Q times Q over p, OK? Now if you look at this expression, delta p over delta Q times Q over p, that is the inverse of the elasticity of demand. So you can rewrite this as p plus p times 1 over the elasticity of demand. Or rewriting one more time, that marginal revenue equals p times 1 plus 1 over the elasticity of demand. Marginal revenue equals p times 1 plus 1 over the elasticity of demand. So think about this for a second. This gets the intuition we just talked about. If e is-- if the elasticity is negative infinite-- is negative infinity-- OK I had it wrong with 0. It's the elasticity of demand is negative infinity then you've a competitive market. Perfectly competitive market is elasticity of demand negative infinity so you get a competitive market, OK? As the elasticity of demand-- but if the elasticity of demand is 0, if the elasticity of-- if the elasticity of demand goes like-- the elasticity of demand is negative 1. Let's do that case. So negative infinity is perfectly competitive, OK? What about negative 1? What's special about the elasticity of demand of negative 1 in this case? What's the marginal revenue? 0. That's the case going back to figure 11-2 where B and C exactly cancel out. So you probably thought about this when you looked at this graph. You probably quickly asked yourself, well, should the monopolist try to sell more or not? Well the answer is, the elasticity of demand is minus 1, they're indifferent. So this is sort of the important insight. With an elastic demand of minus 1, me monopolist is indifferent about selling another unit because what they gain from selling the unit they lose on the previous units. If the elasticity of demand is greater than minus 1 absolute value, then they're going to lose money by selling additional units. If the elasticity of demand is less than 1, then they're going to make money by selling additional units. But let's-- I'm skipping ahead. Let's go on and talk about profit maximization, OK? So let's go on and take the next step which is the monopoly profit maximization. Imagine a monopolist's cost function curve is of the form 12 plus q squared. Let's take the same monopolist and write down cost function of 12 plus q squared, OK? So with this cost function, marginal cost equals 2q. That's marginal cost with this cost function, OK? So what's the profit maximization rule? It's that marginal revenue equals marginal cost. Well marginal revenue, we wrote down here, is 24 minus 2Q, so it's where 24 minus 2Q equals marginal-- I should-- equals marginal cost. Now with monopolists here's the trick, little q and big Q are the same. So I wrote a little q here, but if you're the only firm in the market little q and big Q are the same, right? There's only one firm in the market. So 24 minus 2Q equals-- it's a big Q now-- equals 2Q. So the optimization point is where 24 equals 4Q or Q star equals six. That's the optimal-- that's the profit maximizing sale on quantity for the monopolist is where 24 minus 2Q, where marginal revenue equals marginal cost. We derive marginal revenue, OK? Marginal cost we know how to derive from our cost function. We set them equal and we get the optimizing quantity, OK? And you can see this in figure 11-4. Figure 11-4 shows what's going on here. So I've driven all the va-- I've drawn all the various cost curves for this cost function, OK? I've driven all the various costs first for this cost function. You can see the marginal cost curve and then you can see it intersects the marginal revenue curve at a quantity of 6. Intersects the marginal revenue curve at a quantity of 6, OK? What is the price? Someone raise their hand and tell me, tell me why. The quantity 6, what's-- what price has the monopolist set? Yeah? AUDIENCE: 18. JONATHAN GRUBER: 18. Why-- you're supposed to get that wrong. You didn't-- you didn't, you didn't follow my instructions, you didn't get it wrong. You got it right, I'm just joking. How did you know it was 18 and not 12? Usually people guess 12 because that's the point where the curves intersect. Why is it 18? AUDIENCE: You can go up to the demand curve because that's what people are willin-- JONATHAN GRUBER: Because even the monopolist, as powerful as he is, has to respect demand. So the monopolist if they're going to sell 6 units, they have to choose a price such that people want to buy 6 units. So your intuition, which is my fault-- I've always said, look where the curves intersect, do a quantity and a price-- your quick intuition, which was to look and say the price was 12, which was the wrong answer I usually get, OK, is wrong because you need to actually respect the demand curve. So a monopolist solves for the optimal quantity, but then to get the optimal price he has to plug this back into the demand curve. Well, what's demand? Demand is 24 minus p. So-- I'm sorry, it's p equals 24 minus Q, that's our demand. So if Q star is 6, then price is 24 minus 6, or 18. P star is 18, and that is what you will get wrong when you do this. If you're going to get anything wrong in monopoly problems this is what you're going to get wrong, OK, which is remembering you can't just-- at the end you have to do an extra step here. To get the price you have to solve for quantity, but then respect the demand curve to get the price. There's a question somewhere. OK, yeah, question? AUDIENCE: [INAUDIBLE] of the intersection of the [INAUDIBLE] JONATHAN GRUBER: It's pretty much-- yeah, it's pretty much meaningless. So that intersection used to pin down the quantity, but price has to come from the demand curve because you have to respect-- you can't sell something consumers don't want to buy, OK? So you've got to respect that demand curve. And in fact, you can show yourself-- so basically-- OK, so basically that's the profit maximization except this stupid goddamn shutdown rule still holds. In the short run we still have to respect to shut down rule, so you still have to check, is price less than average variable cost? Now once again, in this function OK, in this f-- you have to check with the price. So even if profits are negative you're still going to have to check if price is greater than average variable costs. Now here at 6 units, the average variable cost is 6. You see-- we could see-- you see the dashed line. If we sell 6 units, the average variable cost is 6. So clearly price is greater than the average variable cost. You wouldn't shut down, you're making positive profits, in fact. You're making profits of 60, OK, so you wouldn't shut down. But you always do have to check the shut down rule, OK? So that's how you do a monopoly problem, OK? Just to go back, how you do a monopoly problem, you'll be given a cost function-- you know what to do those in your sleep-- and a demand function. The demand function gets turned into a marginal revenue function simply through these couple of steps. So the demand function gives you a marginal revenue function. If you have marginal revenue and you have marginal cost, then you know how to solve for optimal quantity. If you have optimal quantity and you respect the demand curve, you know how to solve for price. Once you have a price, OK, and an average cost curve you can both check the shutdown rule and compute profits. Profits are simply the price you get minus average costs. So we can compute the profit and check the shutdown rule, OK? So the only thing we did here that's new is this sort of interesting quirk that we have, this new marginal revenue function. Otherwise, it's the same sort of analysis we did before. So this should be doable with some practice. Now, the key-- a key concept here that we need to think about, that comes to question, you might have asked yourself at some point-- you might have said, well, if monopolists are the only firm in the market, why don't they just charge whatever they want? Why do they have to-- why are they sort of constrained by the sort of mathematics we've done before? And to answer this, let's turn to the concept of market power. The market power of monopolists we will define as their ability to charge price greater than marginal cost. Your ability to charge price greater than margin cost is your market power. In other words, competitive firms have no market power. They have to charge a price that's the same as their marginal cost. And that's why they make no money. Monopolists have market power, OK? So now let's return to the condition for profit maximization. We said that marginal revenue can be rewritten as price times 1 plus 1 over the elasticity of demand, right? That equals marginal cost. So we can rewrite this as marginal cost over price equals 1 plus 1 over the elasticity of demand. And this is the monopolist's market power condition. So if we define the market-- so we can define something we call a markup. Casually we can call it profits, but technically it's the markup, OK, as the percentage markup a monopolist can make as p minus MC over p, how much of the price is actually a markup over cost? It's sort of an intuitive concept. So basically, how much of the money you get for the unit is a markup as a share of the money you get? Then that is simply-- that is equal to minus 1 over the elasticity of demand. So the monopolist markup is equal to 1 over-- minus 1 over the elasticity of demand. So if the elasticity of demand is negative infinity, that is a perfectly competitive market. Then the markup is what? 0, just like a competitive firm. Competitive firms have to charge marginal cost. But as demand gets more inelastic the monopolist gains power to mark up the price and make money. And this answers our question of why monopolists aren't infinitely powerful. What is the limiting factor of monopolists? It's not other firms producing their good. What's the limiting factor on monopolists? Yeah? AUDIENCE: How much more will they compete with [INAUDIBLE] JONATHAN GRUBER: Yeah, it's other products. So think about a monopolist in insulin. They can charge whatever the hell they want, right, because basically there's nothing else to buy. Well, maybe multiple insulin products, but imagine one insulin product, OK? So nothing constrains the monopolist. He should charge an infinite price up to, like, what Congress will put up with, OK? But if you think about a monopolist in a good where there's a substitute-- so if I'm the monopolist in gum, if I'm a gum-opolist, OK, then I'm constrained by how much people want to-- eventually people just substitute the candy, OK? Anything-- so I'm constrained not by competitors in my market, but by the fact that consumers can substitute to other goods. So what's the limit on monopolists? Consumers, we're the limit on monopolists. Our willingness to put up with that good relative to other goods, i.e. Our elasticity of demand, is the only thing limiting monopolists, OK? So basically, there is market discipline to monopolists. Even though they're the only provider of a good they're still subject to market discipline. It's just the discipline doesn't come from other firms, it comes from consumers, OK? So a great example of this is let's look, if you go on-- at least as of last year, if you go on Amazon and look at the prices of two goods that you've probably had to consume in high school, Huckleberry Finn and Great Gatsby, two books that many of you had to consume in high school. You had to read Huckleberry Finn, you had to read Great Gatsby. They're both comparable lengths-- Great Gatsby is a little shorter I think, but they're both comparable lengths. The production cost-- the marginal cost to produce them is probably pretty comparable, OK? But if you go on Amazon, Huck Finn costs $4 and the Great Gatsby costs $16. Now why is this? Why is this, anyone know? Why does Great Gatsby cost four times as much? AUDIENCE: [INAUDIBLE] it was banned [INAUDIBLE] JONATHAN GRUBER: That's interesting. That's not enough to explain. It's not banned in enough places, fortunately, to explain that. What else is going on, anyone know? Yeah, in the red shirt, yeah? AUDIENCE: [INAUDIBLE] competitive-- it has, like, has no copyright infringement. JONATHAN GRUBER: Exactly. The Great Gatsby still has copyright protection. That is, only people who get permission from the great-- from William Scott Fitzgerald's descendants-- F. Scott Fitzgerald, I'm sorry, descendents get to produce it. They have a monopoly on the good that is The Great Gatsby. Huck Finn, it turns out that 75 years after an author's death copyright protection expires. F. Scott Fitzgerald hasn't been dead for 75 years, Mark Twain has. So Huck Finn can now be produced by anyone. You can go out tomorrow and produce a copy of Huck Finn and sell it. So we think probably Huck Finn should be produced in a pretty competitive market. That is, basically the $4 you pay for Huck Finn should be roughly the cost of producing a book. OK, now no market's perfectly competitive, a little markup but not much. Whereas The Great Gatsby, it's not limited by competition. But it is limited by the fact that if they try to charge $500 for Great Gatsby teachers would assign something else, OK? There's still a limit. It's only $16, that's not a whole lot for one of the great works of literature, OK? It's still only $16. So why is it only $16? Because it's limited by the fact there are other great works of literature people can turn to. So Huck Finn is limited by the competition within the market for the-- across producers producing some homogeneous good. It's probably a pretty close to the competitive market, right? It's pretty easy to just set up a shop and produce Huck Finn, whereas The Great Gatsby has this extra effect which is copyright protected, so its only limited by the elasticity of demand. Yeah? AUDIENCE: [INAUDIBLE] be like, [INAUDIBLE] like, the effect of, like, when you increase price again, and even more with piracy, in a sense? Like, if, like, when you have, like, a really, like, expensive good it leaves, like, more incentive to steal? JONATHAN GRUBER: That's a great point. So the elasticity of demand, I should say the elasticity of demand for that good, where one of the substitutes could be an illegal substitute. That's a good point. OK, so let's go on and talk about the next topic I want to cover today, which is how do we-- because now we have a new set of tools we've developed that's really cool which lets us ask the question, not just what do monopolies do, but how do we feel about it? That is, we can now turn to talking about the welfare effects of monopoly. What are the welfare effects of monopoly? How do we feel about monopoly? Well, let's start with the standard case, that is the case we've covered so far as opposed to a case I'll cover in a minute, OK? And let's look at figure 11-5, which is the example we just solved for, the example where demand equals 24 minus Q and cost is 12 plus Q squared. So here we sh-- once again what do we do? We set marginal cost equal to marginal revenue. So they sell 6 units. We then say, what price permits them to sell 6 units? The price of 18. So the equilibrium is at a price of-- is that e little m, little m for monopoly, OK? They sell 6 units at a price of 18. Well at that equilibrium, what's consumer surplus? It's A, the area under the demand curve, above the price. So consumer surplus is area A. What's producer surplus? It's B plus D, the area below the price above the supply curve, but only for the units that are sold. So the consumer gets A, the producer gets B plus D, OK? But C plus E is not our units that are not sold for which the marginal-- for which the willingness to pay is above the willingness to supply. And what do we call those units? Units that are not sold-- someone raise their hand and tell me-- units that are not sold for which the willingness to pay is above the willingness to supply? Yeah? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Loss. Those are units, that's an inefficiency. Those units that are not sold that efficiently should be. That is, we've broken-- this our first example ever. This is a super exciting moment of breaking the first fundamental theorem of welfare economics. The first fundamental theory of welfare economics was that the equilibrium, the competitive equilibrium maximizes welfare. Well, that's no longer true. Here we have an equilibrium that comes out of competition. It's what the market delivers, but it doesn't maximize welfare. There's a dead weight loss, and that's because the market's imperfectly competitive. This is our first ever example of what we call a market failure. And that's what, from my perspective, makes economics fun. If there were no market failures, if all markets functioned the way we said they function, then basically we could have been largely done with the course by now. What makes this all interesting is markets don't function the way we said they function in that extreme case. And therefore, a market failure is defined as a case where the market equilibrium does not maximize social welfare. So whenever the market equilibrium does not maximize social welfare, you've got a market failure. In the perfectly competitive case we didn't have a failure because the market equilibrium maximized welfare. That's not true here. Now the market equilibrium does not maximize welfare. Therefore, it's a market failure. Hint, what's exciting about that to me is that means there might be a potential role for policy, OK? So far in this course, the government's just been a bad guy. Its done nothing but set mean minimum wages and things like that, and terrible price ceilings. But in fact, as we'll see next time, this starts to introduce the role for the government as a good guy, OK? Now, what makes-- to drive this situation home, make this more interesting, let's contrast that to the more realistic case of price discrimination, or potentially more realistic case of price discrimination. Now what happens if we don't force the monopolist to only charge one price? What if we allow the monopolist to charge different prices to different consumers? Here's a cool conclusion actually, kind of crazy conclusion. If monopolists can perfectly price discriminate, that is that they can sell a separate price for every consumer, then monopoly is welfare maximizing. If monopolists can set a perfect price for every consumer, then monopoly does maximize social welfare. To see that, simply look at figure 11-6. Now let's take a perfectly price discriminating monopolist. Let's think about what-- this prices they've set. Well, essentially what they're going to do is set price where? A price discriminating monopolist is going to set the price for each unit to what? Yeah? AUDIENCE: The demand. JONATHAN GRUBER: The demand, the willingness to pay. If you're perfectly price discriminating, you will screw consumers to the max. And how do you do that? By delivering them no surplus. By taking all the surplus for yourself. So for the first unit, you charge 5. For the second unit, you charge 4. And for the sixth unit you charge 18. Now, the [INAUDIBLE]---- now the previous monopolist stopped at 6. Why do he stop at 6? Because the sale of the seventh unit would have lost him money. He would've had to lower the price on all previous units. But the price [INAUDIBLE] monopolist has no poisoning effect because he doesn't have to lower the price on the previous units. He can say for that seventh unit, I'm going to sell that one at 17. I'm going to make 17, and I'm not going to lose any money because I can keep the other prices the same. So with a perfectly price discriminating monopolist, you get rid of the poisoning effect. They can just march their way down the demand curve, charging every consumer exactly their willingness to pay. Therefore, they will continue to produce until the competitive equilibrium. They will continue to produce until willingness to pay equals willingness to supply. But they will capture all the surplus. So the new equilibrium will be EC, the competitive equilibrium, but the entire surplus goes to the monopolist. So the fascinating case, there's-- we maximize social welfare, but only because we define social welfare in this particular way, which is the simple sum of consumer producer surplus. Here, producers get all the surplus, therefore welfare is maximized. So a perfectly price discriminating monopolist, OK, gets maximized social welfare, OK? Now, there's two interesting points to come out of this. Point one is this is a cool way to understand why there's a dead weight loss for monopoly. It's cool to understand because you can see- you can focus on that point E sub M and think about why the perfectly price [INAUDIBLE] monopolist gets to sell another unit and the regular monopolist doesn't. Because the regular monopolist has the poising effect and the perfectly price discriminating monopolist does not. So it's a good way to sort of think about that intuition of the poisoning effect. That's lesson one. Lesson two is, gee, we may want to talk about definition of welfare that's not just the sum of consumer producer surplus. A model of welfare that delivers the fact that a producer that can screw every single consumer out of any of their surplus is the best possible outcome might not be a model we're so happy with, OK? But that we can come back to. But for now it's a nice extreme. And in fact, in reality no firm is perfectly price discriminating, OK? Amazon tried to be. There was a controversy a number of years ago where Amazon would set your price according to your-- what's the little, the string of numbers address, IP address. They literally would set prices by IP address. They'd literally say, well you know, you're coming from an IP address that's, for example, you're in a high-high income area. I'm going to charge you more. You're in a university, therefore you need this book for a course. Therefore, I'm going to charge you more, et cetera. That, they got busted and that was found illegal. But just because you can't perfectly price discriminate doesn't mean we don't have lots of examples of partial price discrimination. So what are examples in the real world of price discrimination? What do firms do? And what's the general-- let me ask, tell me what firms do. And also I wonder, what's the general principle? What's the basic idea that firms-- if you want to price discriminate, what do you want to figure out? What is your goal to figure out? Yeah. AUDIENCE: Isn't [INAUDIBLE] make airplanes? JONATHAN GRUBER: [INAUDIBLE] the what? AUDIENCE: Airplanes. JONATHAN GRUBER: Airplanes. So explain. AUDIENCE: Because if know somebody is buying a ticket, like, two days before a flight, it's probably for a business trip so they can pay a lot more money. JONATHAN GRUBER: And why are they willing to pay more money? AUDIENCE: Um, because the elasticity is-- JONATHAN GRUBER: The elasticity is low because? AUDIENCE: Because they have to do it. JONATHAN GRUBER: They have to go. If you're going to buy two days before, you got to go. If you're buying six months before, you know, you might have to go that day, you might not have to go that day. You might have to-- but if you're buying last minute, you've got to go so your elasticity's lower. So price discriminating monopolists look for signals of elasticity. That's their search. They can't literally know your willingness to pay. Amazon tried, but even Amazon couldn't perfectly know your willingness to pay. So they're looking for signals that are correlated with your willingness to pay. One signal is whether you want to fly at the last minute. That means you have a low elasticity. You're screwed, you've got to go. What's another thing airlines do? What's another signal of elasticity the airlines use? How else-- yeah. AUDIENCE: How long have you been on the website? JONATHAN GRUBER: They don't-- I don't know if they actually use that, how long you been on the website. They could, that would be kind of interesting. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Yeah, I don't know. But that, that's pretty subtle. What's a more blunt one they used even before websites? Yeah? AUDIENCE: Search history? JONATHAN GRUBER: No, even before search-- I know you guys grew up with internet but there was life before the internet. And even before the internet, airlines price discriminated. What did they-- what did they do? In the back, yeah. AUDIENCE: You have, like business class and-- JONATHAN GRUBER: Yeah, different levels of flight, right? Why is a first class ticket more than a coach ticket? Because rich guys are less elastic than poor guys. So basically, another way of price discriminating within the same flight is by having different quality. So what they do is they say, we're going to give you a higher quality product, but look, first class business-- so I flew business class for the first time in my life. I was very exciting. I went to India, I flew business class. It was awesome. But it raised the price of the ticket from $1,000 to $2,500. Now, it was better but it wasn't $1,500 better. Or [INAUDIBLE] put this, let me rephrase that. It cost more, but it's certainly didn't cost $1,500 more. I got a bigger seat, and like, you know, people slaved over me. But like, it certainly did not cost them $1,500 to provide that service. They clearly made more of a markup on the business class seat than they did in the coach seat because I was less elastic. Why? Because someone else was paying for my flight, OK, so as less elastic. And that's-- if you talk to people in business class, none of them are paying for their own flights. They're either super rich or their company's paying, OK? So the bottom line, that's another way airlines [INAUDIBLE]. Let's get away from airlines. What are other examples of price discrimination? Yeah. AUDIENCE: Isn't that a common thing, like, Google, like, [INAUDIBLE] actually use your search history and cookies to, like, figure out how much money you make and how much you're willing to spend on certain items? JONATHAN GRUBER: I think-- I have to look this up. I think they're not allowed to do that. They're allowed to do that in targeting advertising, but I don't know if they're allowed to setting the price. I don't know if a company's allowed to that setting the price. Yeah? AUDIENCE: Sometimes chains will be more expensive in cities and less expensive in rural areas. Like, the same meal at McDonald's can cost a different amount depending on-- JONATHAN GRUBER: Exactly, or-- that's exactly right. Restaurants or supermarkets. Now here's what's interesting, if you look at a McDonald's, OK, its prices-- no, it's true for both. Basically, if you look like, like if I buy-- when I buy McDonald's on the highway out in the suburbs it's more expensive than I would buy here in Cambridge. Why is that? Why is-- why is a big so-- yeah? AUDIENCE: Not a lot of other options. JONATHAN GRUBER: Yeah. I'm like, I'm driving. Where am I going to go? I can't say, oh, it's too expensive, I'm going next door. So that's an example. Inner city price-- supermarkets charge much higher prices in the city than out of the city. Now you might think that's sort of strange, OK? But that's because in the city people have to walk to get there. They don't have a lot of options. There's not a lot of shopping. Outside, they can drive from supermarket to supermarket, OK? So there's all these complicated things that go into it. One of my-- let me sort of tell you a couple of my favorite examples. One of my favorite examples is early bird specials. Now you guys might not know about this, but you might've hung out with your grandparents and known that, like, if you go to a restaurant sometimes before 5 o'clock or 5:30, it's cheaper. Or if you go to a matinee movie, a movie during the day is cheaper than a movie at night. Same movie, why? Why is a movie during the day cheaper than a movie at night? Yeah. AUDIENCE: Because in the night time you probably want to go to the movies more because you're off work and you have things to do. And so there's like-- like, if you went early you're purposely looking for a cheaper option. So you [INAUDIBLE]. JONATHAN GRUBER: You're more elastic during the day because basically you're an old retired person's who's got nothing but time in their hands. It's like, you can shop around movie theaters, you decide to go to movie or take a nap, et cetera. Same thing with early dinner specials. People who have more time to shop are more elastic. At night you're going out to see the movie. These are people, you know-- in some sense it's sort of interesting. You think midnight movies-- I don't know if this is true, midnight movies should charge and most of all. Because that's a bunch of insane fans, right, who have to see the movie first. It seems like the midnight showing should charge the most. I don't know if that's true. That would be interesting to look at. OK, that's one example. Another example I love is Disneyland. Disneyland and Disney World charge less if you live within 20 or so miles of the park. Why? Because you can go whenever. So when I took my kids to Disney World, if I went up and said, I'm sorry kids, it was $10 more than I thought, we're not going, I would have had a riot on my hands. Once you're there, you're inelastic. You're going to Disney World. There's nothing else to do in Orlando. I mean, but there's other parks, I guess. But like, but if you live locally you can decide whether to go or not. And then probably the most interesting recent example is Tesla. So during-- I feel like, did I tell you guys the Tesla hurricane story yet? I don't think so, right? OK, so during the hurricane in Florida-- Irma, I guess it was? In Florida, Tesla-- so Tesla so two cars, two models, the cheaper model and the more expensive model. And the more expensive model had some nice doodads, but most importantly the battery lasted longer. The cheaper model was like 300 miles, the more expensive model was like 500 miles. During hurricane Irma, Tesla as a gesture of goodwill said, hey guys who drive the cheap car, you can now drive 500 miles. And they're like, what the hell, it's the same battery? Tesla said, well it turns out, it's the same battery. The only difference is a piece of software that we can turn off or on. So why did Tesla do that? Why did Tesla have a piece of software they turn off or on and turn it off for some people to make them drive less? Why did they do that? There's someone else involved. Can anyone tell me about what Tesla thought, meaning besides the, oh, must be an asshole. What else? What's the other-- what's the kind of-- what's Tesla's thought process? Actually, it was perfect economics. Yeah? AUDIENCE: Price discrimination. JONATHAN GRUBER: Right, it's price discrimination. But how? How are you price discriminating by making some people-- yeah? AUDIENCE: You can make a 500 mile car more expensive. JONATHAN GRUBER: Exactly. It's like first class. You're basically saying, I want to charge more for a better product. I can't charge more for the same product, so if I give everyone 500 miles I can't charge more. But by making one product better I can separate demand. I can sell the expensive product to the low elasticity of demand consumers and the less expensive to the high elasticity of demand. But it's actually the same damn product, I just did it as a way to price discriminate. So Tesla, in an effort to be nice, screwed themselves by revealing that they had been actually falsely screw-- you know screwing consumers. It's not [INAUDIBLE]---- it's good economics. It makes total sense. OK, this actually follows one last example. When the first laser printers came out, OK, it turned out that you could buy one for home that was like half the price one for office. And someone took them apart and found the one for home was the one for the office plus an extra piece that made it go slower. And they did that simply because they knew the office guys were less price elastic than the home guys. So they wanted an expensive one-- so they said, the office one's faster. Isn't this cool? It was faster because they didn't add an extra piece that made to go slower, OK? And that's the same thing. So let's stop there, and we'll come back and talk more monopoly. Good luck tomorrow night and we'll talk more about monopoly on Wednesday.

MIT_1401_Principles_of_Microeconomics_Fall_2018

9_Supply_and_Demand_ConsumerProducer_Surplus.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: OK, why don't we get started? Today, we're going to come full circle back to the first lecture. So in the first lecture, we talked by-- we started by drawing a supply and demand graph. We've now spent the last few weeks explaining where supply and demand curves come from. And now, we're going to talk about the supply and demand curves. What do they know? Do they know things? Let's find out. So, no one? No one on that? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: OK, thank you. All right. So let's start by talking about shocking the supply and demand curves. Shocking the supply and demand curves. That was a BoJack Horseman reference for those of you who missed that. OK, let's talk about shocking the supply and demand curves. So let's start with a review of the supply and demand framework that we introduced in the first lecture. So let's go back to figure 9-1. We've got the market for gasoline, OK? On the x-axis is big Q. Quantity of gas is the market-level diagram. On the y-axis-- the price of gas. And as we said, the first lecture-- the supply curve that's upward sloping, representing the fact that higher prices call forth more supply. We now know where that comes from. We know that what happens is when there's a higher price, firms can now afford to move up the marginal cost curve, which is the supply curve. So we know where that comes from. We have demand curve, which is downward sloping. Higher prices lead to less demand. We know where that comes from. We know that as the price of a good rises, through both income and substitution effects for normal goods, consumers will want less of it, so whenever that comes from. So we now have derived these. And we're back where we started in equilibrium. So let's actually start by asking what happens. Let's start by asking, as we move forward, how do we want to think about these curves? And the way we think about them is we want to think about the demand curve, want to think about these as willingness to pay and willingness to supply curves. So think about the demand curve as a willingness to pay curve. How much are you willing to pay to get that next unit of the good? Or how much is the market willing to pay to get the next unit of the good? OK? And the supply curve is willing to supply, OK? An equilibrium is the point where consumers' willingness to pay for the next unit of the good meets the suppliers willing to supply the next unit of the good. When those are equal, we're in equilibrium. So that's where we start. Now, let's ask, what happens as these curves shift? So, for example, let's take this market and imagine the tastes change. Suddenly, everyone wants to drive big cars. Everyone wants to drive SUVs, OK? What does this do to the market for gas? Well, so what does this do? Well, what it does-- yeah, go ahead. AUDIENCE: SUVs require a lot more gasoline, so the demand goes up. JONATHAN GRUBER: Yes. SUVs are what we call a complement as opposed to substitute-- are a complement for gasoline. When demand for SUVs goes up, demand for gas goes up. So the demand curve would shift out. So we would end up in a situation like figure 9-2. But let's talk through the dynamics. All you would see in the market is quantity of gas sold would go up from Q1 to Q2. And price of gas would go up from P1 to P2. Well, let's talk about underneath, how we get there. What happens is demand shifts up. People want more gas, because they want to drive these gas-guzzling cars. So demand shifts from D1 to D2. What does that mean? That means at the previous equilibrium price-- if the price didn't change, if the price stayed a P1, what would happen? Well, we'd no longer be in equilibrium. Because people would-- firms would still be happy to supply Q1 units of gas. But people would want way more than that. We would have excess demand. If the price didn't change, there would be excess demand. People would want more than the Q1 units of gas. Suppliers will recognize this and say, well, if people want more, we're happy to produce more. But remember, we have to respect the marginal cost curve and marginal cost of rising. If we're going to produce more, we're going to have to charge more. We're going to have to move up the supply curve. So a shift in the demand curve makes firms move along the supply curve. Want to keep shifts and movement along curves separate. A shift in demand curve, meaning people are saying to gas producers, we want more gas. Gas producers are like, great. We want to give you more gas, but we're going to charge you more to do it. Because our marginal cost curve is upward sloping, which is our supply curve, as we learned. So the price rises. And we need to reach a new equilibrium at E2. So we don't see these steps in practice. In the end, we just see the price change, but think about it as two steps. Demand shifts out, creating excess demand. Providers, to meet that excess demand, have to produce more. And to produce more, they're going to charge a higher price. And that moves you from E1 to E2, OK? So we have a shift in demand, which caused a slide up the supply curve, OK? Now, let's think about a different example. Imagine war breaks out in the Middle East. Not too hard to imagine, unfortunately. And as a result, the quantity-- so suppliers need to pay more to get the oil that they use to make gasoline, OK? What does that do? We see that in figure 9-3. Now, what happens is for every unit of gas, suppliers need to charge more. Their underlying marginal costs have gone up, because they have to pay more to get the oil. That's a variable cost of production of gas. So their marginal costs have gone up. Their marginal costs going up mean their supply curve has shifted upwards, OK? For every unit of production, their marginal cost is higher, because their variable costs have gone up. Therefore, they're going to need to charge a higher price to break even. OK, we're still in perfectly competitive markets where nobody is making any profit, OK? They're going to charge more to break even. So now, let's once again talk about the dynamics of what's happening. The dynamics are the costs and the input to the suppliers went up-- oil, OK? Their marginal costs shifts up to S2. So they want to charge a higher price. So if we kept the price the same as it was before, suppliers would say, we don't want to sell Q1 anymore. We're not interested in selling Q1 anymore at that old price. OK? That doesn't interest us. Therefore, consumers want more than providers are willing to sell. And we once again have excess demand. So in both cases, we get excess demand. In the first case, we got excess demand because consumers wanted more. The lower-- the consumers' tastes shifted, so they wanted more gas at every price. Now, we have excess demand not because taste shift, but because costs go up. So providers don't want to provide as much gas at every price. So what happens is providers are going to say, fine, we're going to charge a higher price, OK? And we'll slide up the demand curve. Because as providers charge a higher price, people want less gas. At a higher price, you want less gas through the substitution effect. Because you'll buy other things instead and for the income effect. Because you're effectively poorer, because the price of gas went up. For those reasons, you're going to shift up the demand curve and reach a new equilibrium at E2. So that's the underlying dynamics of how shifts in supply and demand lead to changes in quantity and price, OK? So that's basically what we're seeing. Questions about that? Yeah? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Great, great question. So what's the answer? What's the substitution effect with gas? AUDIENCE: [INAUDIBLE] not driving. JONATHAN GRUBER: Well, you've answered yourself. It's not driving. It's taking the bus. It's driving less. It's walking or taking your bike. So once again, when everything about substitute effects, you want to think about the next opportunities you could use instead, OK? Good question. Other questions? OK, so here's an interesting point. Look at figure 9-2 and 9-3. In both cases, the price went up, OK? In both cases, the price went up. So we can't tell. If a price goes up, you can't tell from that alone whether there was a shift in demand or supply. So if I, for example, asked you on an exam or your mom came home. Your mom asked you, hey, if the price goes up, does that mean demand shift or supply shifted? You say to your mom, I don't know. I can't tell with just that information. I need to know what happened to quantity, too. OK? And then you say your mom, good question. OK, so let's go through the reasons why the supply and demand curves shift. So why do curves shift? OK? Well, on the demand side, there's at least six reasons why demand curves would shift. So why do demand curves shift? OK, one reason is tastes change. I just used that reason-- tastes change. OK, people want different things. OK? A second reason is that income changes. Second reason-- because people are richer or poorer. And so that makes them want different quantities, even with the same tastes. A third reason is the change in the price of a complementary or substitutable good, OK? Now, that's different. I should separate. The actual example before was this. Taste change is slightly different. So change in price [INAUDIBLE] is what I talked about. Taste change would be literally for everything held being equal, I just wake up one morning psyched to drive. That'd be a taste change, OK? So really, the example I used was a change in the price of complimentary-- no, no, price didn't change. No, I go back. I go back. The example I used was a taste change. People wanted more SUVs. But at the same time, imagine a different change. Imagine that we're looking at the demand for babysitters. And the price of movies goes up, OK? Well, movies are complementary with babysitters. You guys don't worry about this. You don't have kids yet, but trust me. Movies are complement to babysitters, that basically the more you go to the movies, the more you need babysitters. So if the price of movies goes up, that's going to lower my demand for babysitters or vice versa. Imagine that how a change in the price of movies affects the demand for Netflix, while those are substitutable. As the price of movies goes up, I'm going to want more Netflix and less babysitters. So change in price of complementary substitutable goods will also affect my demand curve. Another thing that could affect the demand curve is a change in the market size. So we will talk in a couple lectures about international trade. If suddenly you're selling goods to a much larger market, that will affect the demand for your good. So preference haven't changed. Price haven't changed. You just suddenly got a bunch of new customers. That will affect demand for your good, OK? And the last thing that could change, the most subtle way demand could change is expectations of the future. So for example, imagine you expect the price of gas to go up tomorrow. You might buy more gas today. And that'd be weird. [INAUDIBLE] look, nothing changed today. Your taste didn't change, prices-- nothing changed, but people buy more gas. What's going on? It's that they expect the price to change in the future. So expectations of the future can actually drive demand today, OK? We've all-- experiences in various aspects of our lives, OK? So those are the reasons why the demand curve can shift. There's a lot of reasons why the demand curve can shift. For the supply curve, why the supply curve shifts is much simpler. There's really only two reasons, OK? One reason is changes in input costs. And the second is a shift in the technology and production. So the production function changes or input costs change. That's pretty much why supply curves shift, OK? So that gives you a catalog of how to think about these curves shifting. I have a fun example in the videos that go with this class, which is that we all know Kim Kardashian is-- you may or may not know she has more Instagram followers than there are people in France. She got 80 million. It's up to about 100-plus million Instagram followers. Kim Kardashian, a few years ago, tweeted out a picture of herself in an exercise corset, she called it. She basically claimed-- a corset is this thing they used to wear back when we didn't care about women much at all. And we just made them wear these incredibly constrictive things to make them look skinnier, OK? They're basically like a brace you'd wear to make you look skinnier back in the old days. And Kim Kardashian said, actually, if you wear a corset when you exercise, it helps you lose weight. Well, actually, she's totally fucking wrong, OK? It doesn't, OK? There is no-- it does not help you lose weight, but she tweeted this out. And there was a massive increase in demand for exercise corsets, OK? And the one company that made them made scads of money. There was a huge demand shift based on this Kim Kardashian tweet, OK? So tell me what happened next. Yeah? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: More companies entered, OK? So what happened was profits were being made on exercise corsets. So more companies started making exercise corsets. And they came in and drove those profits down, OK? So that's a classic example of how demand shift and how the market in the long run will respond to return us to zero profits. Zero profits in the long run-- in the short run, some corset companies made a lot of money. They should-- they owe, Kim, OK? But in the long run, profits go to zero. Yeah? AUDIENCE: With the expectations where the demand curve shifts, is that when companies-- you're like, oh, there's these coupons for limited kind of sales? Would that be an example of demand? JONATHAN GRUBER: Yeah. And anything where you basically-- well, no. But once that's on, that's a price change. A limited time sale for good is literally just a standard to price changed, OK? It's you think the sales are going to happen in the future, so you buy less today. That's the expectations, OK? So that shifts in demand supply curves. Let's now talk about what determines the shapes of supply and demand curve. Now, what determines the shapes of supply and demand curves? OK? So basically, the effect-- not what determines the shapes. We already talked about what determines the shapes. We want to talk about the role, the shapes the supply and demand curves play. Let me rephrase that. We already know what determines the shapes. We covered that in the last 10 lectures or whatever. Now, we're talking about the role that shapes play as demand curves shift. So for example, let's think about a [INAUDIBLE].. Figure 9-3 shows what happens with a supply, the figure we're just looking at, OK? Figure we're just looking at, figure 9-3, shows what happens when the supply shift with a standard downward, sloping demand curve, OK? Which is that the price goes up, quantity falls. However, imagine, instead, we had perfectly inelastic demand. So, for example, for insulin. Then what would happen? Well, figure 9-4 shows if demand is perfectly inelastic, quantity won't change. So if there's a supply-- if there's a shock that shifts up the supply curve like war in the Middle East. So this is the question here. Why wouldn't gas just be perfectly inelastically demanded? In fact, in the short run, gas is actually pretty inelastically demanded, OK? It's not perfectly inelastic, but is pretty inelastic, OK? So in that case, you would see just prices going up, and quantities wouldn't change. Now, in the long run, do we think the elasticity for gas will be higher or lower in the short run, the demand elasticity for gas? AUDIENCE: Higher. JONATHAN GRUBER: Higher. Somebody raise their hand and tell me why. Somebody raise their hand and tell me why. Somebody else besides people who always answer questions. Yeah? AUDIENCE: People can shift towards electric cars. JONATHAN GRUBER: Exactly. In the short run, all you can do is drive less. And we got to drive to work and stuff like that. And in the long run, I can buy a different car. So this is an example of long run versus short run, how it can affect these elasticities, OK? Now, let's think instead about a perfectly elastic demand, the demand for-- I don't know-- chachkies in a market or something like that, OK? Perfectly elastic demand in figure 9. It's always hard to think of markets with perfect elastic demand. It's easier think about firms that have perfectly elastic demand. It's hard to think about markets. But think about a market for a certain kind of candy with another kind of candy that's just as good, OK? So those are markets, which are fairly elastically demanded, OK? There you see when the supply shifts, price doesn't change, only quantity does. And why is that? That's because demand is probably elastic. You can change the price. If you try to raise price by one penny, you'll lose the entire market. If you lower the price by one penny, you gain the entire market. And then your profits will go away, because your marginal cost will be through the roof, OK? So with perfectly elastic demand, you're going to get prices fixed, but only quantity changes, OK? So basically, that's how we think about these extremes. The bottom line is that's how the shapes of supply and demand will affect the response to shocks, OK? The more elastic is demand, the more price shock will come through in quantity and less in prices. The more inelastic is demand, the more supply shock will come through in prices and not in quantity, OK? Any questions about that? OK. So now, let's go on to what we can do with these supply and demand curves. So now, we're the masters of supply and demand curves. We know where they come from. We know why they shape the way they do. We know what happens when they shift. And we know what happens how that shift depends on their shapes. So we own supply and demand curves. Now let's go, what can we do with them? And what we can do with them is use them to take the next step in this class from positive to normative economics. So far, this class has been completely focused on positive economics. Why do firms behave the way they do? Why do consumers behave the way they do? And we haven't talked at all about whether it's a good thing or a bad thing. Well, we need a new set of tools if we're going to move from positive economics about why things-- the way things are to normative economics about the way they should be, OK? And those set of tools-- we're going to derive from supply and demand curves. And this is critically important. Because, for example, let's take where we ended the last lecture-- or the last lecture, I think, talking about how with perfectly-- or in the middle of last lecture, talking about in a perfectly competitive market under a set of assumptions, all firms-- a zero profit in the long run, OK? So you buy that. But you have to ask yourself, is that a good thing or a bad thing? Is zero profits in the long run good or bad? Well, on the one hand, firms are cost minimizing. That's good. On the other hand, why would anyone start a business? In the long run, they're going to make no money. That's bad. So how do we think about trading those things off? How do we think about whether it's good or bad to have long run zero profits? OK? This is the question. This set of questions is what we turn to with the notion of welfare economics. Welfare is going to be used in two senses in this class. Mostly when I say welfare, I'll mean as a measure of well-being. Mostly when I say welfare, I mean welfare is well-being. Sometimes we say welfare. We mean cash payments to poor people. That's welfare payments. That's not what I mean, usually, when I say welfare. I'll try to distinguish when I mean the other thing. When I say welfare, I don't mean the way it's used in the political debate, meaning cash payments to poor people. I mean welfare is a measure of well-being. And welfare economics is the tools of normative analysis. The tools of welfare economics are the tools of measuring well-being. And we're going to start by talking about the concept of consumer surplus. It's going to be the first thing we're going to use when we talk about welfare economics is consumer surplus, OK? Now, if we want to measure well-being, however, we have a problem, which is, how do you measure how happy I am? My utils But utils don't exist. So we've got a fundamental challenge here, which is our indicator of well-being is utility function, which isn't a real thing, OK? We use it to derive decisions, but we don't actually have a measure of well-being that gives real meaningful inputs. So what do we do? We do a clever thing economists thought of a long time ago, which is to use the concept of compensating variation. The concept of compensating variation. What does that concept-- means? That means instead of asking you how happy you are, I ask you, how much would I have to pay you to become less-- to become sadder? Or how much would you be willing to pay to be happier? So I can't measure margin utility in dollars. But I can measure how many dollars you would pay to buy the next good or how many dollars you'd pay me not to be punched or whatever, OK? I can basically measure those things by essentially asking you, how much would you pay to be better off? Or how much would you be willing to pay not to be worse off? And those are what we called a compensating variation. We measure your well-being by the money equivalent that you give to us in expressing your preferences. And what we can then define consumer surplus-- we'll define consumer surplus, which is our first measure of normative welfare economics, as the benefit that a consumer gets from consuming a good, above and beyond the price of that good. The benefit that a consumer gets from consuming a good, above and beyond what they paid for that good. That's consumer surplus. Surplus means extra, right? So it's your extra. It's how much more you get than what you actually pay to get the good in the first place, OK? So basically, consider my daughter's demand for songs by Kendrick Lamar, OK? And to make life easy, let's say this is pre-streaming and songs cost $1, OK? So she wants songs by Kendrick Lamar. So that's actually-- yeah, she wants songs by Kendrick Lamar, and there's no streaming. And the songs cost $1. So if my daughter is willing to pay $1 for a Kendrick Lamar song and it costs $1, then her consumer surplus is zero. The benefit she gets from the song is $1. It costs $1 to hit zero. But if she was willing to pay $2 for a Kendrick Lamar song and it only cost a $1, then she's got $1 in surplus, OK? So basically, the key thing is to define consumer surplus, we need two things-- the price and the willingness to pay. Well, how the hell do we get willingness to pay? Where does that come from? Someone raise their hand and tell me. Yeah? AUDIENCE: Demand. JONATHAN GRUBER: The demand curve. We already defined it. We already defined what willingness to pay is. It's the demand curve. So consumer surplus is simply defined as the area below the demand curve, above the price. Because that tells you. The demand curve tells you how much you're willing to pay for each unit. The price you face tells you how much you had to pay. So any gap between them is consumer surplus, OK? So let's go to figure 9-6. Let's do my daughter's demand for Kendrick Lamar songs, OK? Let's say that her demand is such that-- now, once again, the trick here is we've drawn a continuous demand curve. It's a discrete decision, so bear with me-- the numbers. Bear with me, just think about this. But roughly speaking, she's willing to pay for the first Kendrick Lamar song between $4 and $5, OK? For the next Kendrick Lamar song, she's willing to pay between $3 and $4 and so on. So this gives you-- so to make life easy, let's imagine she's willing to pay $4 for the first Kendrick Lamar song, $3 for the second Kendrick Lamar song, $2 for the third Kendrick Lamar song, and $4 for the first Kendrick, and-- I'm sorry-- $1 bucket for the fourth Kendrick Lamar song, OK? So imagine that's basically her demand curve. It's not quite that discrete, but we can make it stepwise if you want-- just be ugly looking, OK? So that's her demand curve. So what does that mean? That means when she buys the fourth Kendrick Lamar song, when she buys King Kunta or whatever, that is zero surplus, OK? Zero surplus. She was willing to pay $1 for "King Kunta," and it cost $1, so she's done, OK? However, what does that mean? That means when she bought "Humble," which was her first choice song, she gained a surplus. Because she paid $1 for that. But she was willing to pay $4 for it. So she gained a surplus. And the surplus is the difference between what she paid, which is represented by the horizontal line and a dollar, and what she was willing to pay which is the main curve, which is $4. So she gained that surplus. Yeah? AUDIENCE: Let's say as her father, you want to get her a gift-- all these Kendrick Lamar songs. And let's say it's special. I don't know-- $2, something like that. Would the consumer surplus be what you think she would want out of it or what she-- JONATHAN GRUBER: Let me come back to that. It's a great question. There's a famous article about that. And I'll come back to that in one minute. Let me finish this. The bottom line is the surplus there is between what she was willing to pay and what she had to pay, which in a continuous example is this entire triangle. Think of being able to buy fractions of songs-- little bits, ringtones or whatever, OK? Fractures of songs, OK? Then this entire area under the curve, above the price is her surplus. She was willing to pay the points on the curve. She only had to pay the flat line at $1. So the entire difference is her surplus, OK? The key point is this is all driven by diminishing margin utility. That is the reason her surplus goes to zero eventually-- is eventually gets tired of Kendrick Lamar songs, so it goes down. We have diminishing margin utility for the songs. And that's why we get consumer surplus as a triangle. It's the difference between the downward sloping demand curve and the flat price line that the consumer faces, OK? So the individual consumer surplus-- individual consumer surplus, OK? It's her demands-- that individual graph, OK? Individual graph. Her demand is downward sloping. And therefore, her surplus difference between is the area under the demand curve, above the price line. Yeah? AUDIENCE: If demand is perfectly inelastic, is it infinite consumer circle? JONATHAN GRUBER: Let's talk about that. Let's talk about-- actually, I don't have it here. If demand was perfectly inelastic, you're absolutely right. The consumer surplus would be infinite. Because the area under the demand curve above the price line would be infinity. It'd be a rectangle going up to infinity. Why is that? Why is the consumer surplus infinite if demand is inelastic? AUDIENCE: Because they'd pay anything for it. JONATHAN GRUBER: Because they'd pay anything for it. So at any price, it's a bargain. In theory, if you're an incredibly rich diabetic, you would pay an infinite amount to have insulin. So at any price, you're getting huge surplus. You're getting infinite surplus. Infinitely minus anything is infinity. Likewise, what's the consumer surplus if demand is perfectly elastic? Same person. AUDIENCE: Zero. JONATHAN GRUBER: What? Zero. Why? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: That's graphically why. But intuitively, why? Why do you get no surplus from a good where demand is fairly elastic? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: What makes a perfectly elastic demand curve? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Because-- why? Because there's substitutes that you're indifferent towards. That's what gets the perfectly elastic demand. So if I'm indifferent between Jujyfruits-- god, you guys probably don't know Jujyfruits. If I'm indifferent between-- God, I don't even know what candy is anymore. Whatever. If I'm ever eating candy A and candy B, and then I get no surplus for consuming candy A, why? Because I'm equally happy with candy B. So candy A gives me no surplus. What does the candy people eat? What do people eat? AUDIENCE: Jolly Rancher. JONATHAN GRUBER: What? AUDIENCE: Jolly Rancher. JONATHAN GRUBER: Jolly Rancher. I love Jolly Ranchers. AUDIENCE: M&M's and Skittles. JONATHAN GRUBER: OK. Well, no. But that's irrelevant, 'cause Skittles are just disappointing M&M's. Let's be honest. When you get Skittles, you're just pissed off they're not M&M's. Am I right? I mean, Skittles are just disappointing M&M's, so we can't do that one. Let's do Jolly Ranchers versus Skittles, maybe. Those are more comparable. Because M&M's are better than everything. So basically, Jolly Ranchers and Skittles-- since I'm indifferent to Jolly Ranchers and Skittles, I get no surplus eating the Skittles. Because I would equally happy having a Jolly Rancher. So surplus is zero for a perfectly elastic demand and good. It's infinite for a perfectly inelastically demanded good, OK? Now, let's go back to the question. There's a famous article in economics called the "Deadweight Loss of Christmas--" we're such an awful profession-- based about how terrible gift-giving is. And why is gift-giving terrible? Because if you gave people cash, they could get what they want the most. But if you give them a gift, it's by definition, lower surplus than the cash. Because they could always go out and buy that good with the cash. So by definition, giving someone a gift makes them worse off than giving them that same amount of cash. So this guy-- is he interviewed all the students. I think was at Penn State. And he asked them how much their parents' presents really worth to them. And he found the deadweight loss of Christmas is hundreds of billions of dollars. People would way rather have cash than the parents-- but what did he get wrong? What did he get wrong? Why is that not necessarily a bad thing? Yeah, you asked the first question, so go ahead. AUDIENCE: You like the surprise of opening a present. JONATHAN GRUBER: Maybe. But even ignoring that, what else did he get wrong? Yeah? AUDIENCE: It's an emotional connect if something my grandma bought me a-- JONATHAN GRUBER: That's like the surprise. There's emotional connections. That's all well and good, but that's not very big, OK? What's really big that he missed? AUDIENCE: Because the person who buys it-- they saw what they get from it. JONATHAN GRUBER: Yeah, he missed the fact the person who gave it gets utility from giving it. So in fact, the package may be efficient, because you like the surprise and the person gets utility. But if compare it to dollars, it's inefficient. So it's a clever, clever little exercise he did. OK, so basically, that's individual consumer surplus. But in this course, we don't care about individual consumer surplus. We care about market consumer surplus. So let's turn to figures 9-7 and think about a market. Let's see about the market for gas. Now, the mechanics is the same here. But we're actually now thinking not about the individual buying 1 gallon versus 2 gallons, but the market for gas. How many gallons in aggregate will be bought? But the analysis is the same, that basically the willingness to pay for gas is the demand curve for gas, the market demand curve for gas. The price is the price. So the difference is the area under the demand curve above the price. The idea here is for consumers all the way to the left, they have to drive to work. They have to drive. They have to drive a lot. They're truck drivers or whatever. They have to drive a lot. So for them, they have a huge willingness to pay for gas. So they make a huge surplus. The more you want something at a given price, the more surplus you get. Whereas you move to the right, that's people who need to drive less and less. Once you pass point A, why does surplus go away? To the right of point A, why is there no more consumer surplus? Yeah? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Didn't happen, because? AUDIENCE: Because [INAUDIBLE]. It's beyond. JONATHAN GRUBER: Your willingness to pay is below the price, right? So a transaction-- so consumer surplus can't go negative, but when negatives just wouldn't buy it, especially with negative consumer surplus, OK? When negative, you just wouldn't buy it, OK? But as you get closer and closer to A, then you actually do end up with consumer surplus going to zero, OK? So that's the market consumer surplus. So let's ask. Let's talk about a couple of aspects of market consumer surplus. First question-- what happens to consumer surplus when the price changes? Let's show that in figure 9-8. Let's say the price of gas goes up from $3 to $3.50 a gallon. Consumer surplus shrinks by a trapezoid. Consumer surplus used to be the entire area below the demand curve. It used to be the entire area is below the demand curve and above $3. It used to be that whole triangle. Now, it's just the empty triangle above the new price curve and below the demand curve. So the new consumer surplus is just the area above $3.50 on the main curve, so it's the area not shaded in. What you've lost is the trapezoid, that on the y-axis goes between $3 and $3.50 and then along the line, goes from A to B. You've lost that trapezoid. Why is it a trapezoid? Why is the loss-- consumer surplus a trapezoid? Because two things have happened. What are the two things that have happened that have reduced your consumer surplus? Get some more folks involved. Folks, go ahead. AUDIENCE: The quantity supplied goes down as well. JONATHAN GRUBER: Well, not just quantity supplied. Quantity sold goes down. Because you want less supply, because you are-- so the first thing is because the price gone up, you want less. That's the triangle you lost. You have given up units that you used to get surplus on, used to derive surplus in all the units from 900 to 1,000. So what happened here is the price goes up. And a hundred fewer people buy gas. That's the way I've labeled this. That could be people buy less gas. Let's make it easy. A hundred people buy gas. So a hundred people used to buy gas, no longer buy gas. They're out of the gas market. They bike instead, OK? Now, they clearly were not that sad to bike, or they would've had a huge surplus from gas. But they're a little sad to bike. It's a crappy day out. They rather be driving. And so they lost surplus from the fact that now at the higher price, they have to bike instead, but it's a little bit of surplus. It's just a little triangle, OK? So there's a little bit of surplus lost. Because some people who were close to indifferent now have to bike instead of driving. But why the big-- what's the big rectangle? Same person. What caused the big rectangle? AUDIENCE: The increase in price. JONATHAN GRUBER: Increase in price for who? For the people who were already buying it anyway. So the big losers are the people who are going to drive anyway and now just have to pay more for it. Because here's the key point. The people between A and B-- the last hundred people-- they were pretty close to indifferent. They didn't lose that much surplus from not driving. All the people to the left of person 900-- they get big surplus from driving. So their surplus simply went down by this rectangle. They used to get the difference between the demand curve and $3. Now, that's the difference the demand curve and $3.50. It's just a pure loss. So when you raise a price, the existing-- the people whose behavior doesn't change are worse off. Some of those behavior change. They're a little worse off, but not that much. So the triangle is small. The rectangle is big. The big loss is the people who like gas a lot, but now have to pay more for it, OK? Point one. Point two-- what determines whether consumer surplus is large or small? Well, we cover this. It's elasticity of demand, determines whether consumer surplus is large or small. So, for example, figure 9-9 takes the gas market with a price of $3.00 and a thousand people buying gas and uses two-- shows two different demand curves, both of which go through point A. So both demand curves yield the equilibrium price of $3.00 and the equilibrium quantity of a thousand, OK? So these two different demand curves are just two different sets of preferences, both of which yield the same equilibrium outcome. And yet, under the steeper demand curve, the consumer surplus is larger than under the flatter demand curve. And that's for the reason we talked about. That's for the reason. That's because with a steeper demand curve, the more inelastic demand, people want the good more. They basically-- they're less willing to give it up as the price goes up. Therefore, at any price, they're making more surplus off it. With a flatter demand curve, people are basically closer to indifferent with some other good. So they're not so sad if the price goes up. Their surplus is smaller from getting this good. They're seeing what they were willing to pay and what they have to pay is smaller, OK? So that's how we think about consumer surplus. It's basically the excess of your willingness to pay above what you have to pay. So if the price goes up, your surplus goes down. And surplus is larger, the more inelastic is the demand curve. Yeah? AUDIENCE: [INAUDIBLE] producers would want it, but consumers are having a zero surplus, if that makes sense? Because they're at the point where not only paying more, but they're selling as much as they can? JONATHAN GRUBER: Great question we'll talk about when we talk about monopolies. Right now, why can't producers do that? Why can't producers exploit that? Because perfectly-- yeah? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Exactly. That's a perfect answer to a perfectly competitive question. Because they're price takers, OK? So they can't do an exploiting of consumers. They don't have that choice. Starting next lecture or one lecture after, we'll talk about monopoly. Then they're price setters. Then they'll start thinking about that. But right now, they can't, because their price takers. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: I mean, ultimately, that's what-- yeah, ultimately, they'd like to-- the surplus is just extra money somebody has got. If you're a business owner, why should consumers have it? You want it, OK? So that's consumer surplus. Any other question about consumer surplus? OK. Now, let's move on. And let's talk about producer surplus. Let's talk about producer surplus, OK? Now, the idea here is the same. Consumer surplus was the difference between the willingness to pay for a good and its price. Producer surplus is the difference between the willingness to supply a good and its price. And how do we measure willingness to supply? The supply curve. So as figure 9-10 shows, the producer surplus for any given firm-- firms have an upward sloping supply curve. And the market is delivering them some price. So let's think about this firm. When they produce the first unit-- this is a gas production firm, OK? A gas refiner, say. When they refine that first gallon of gas, that costs them almost nothing. Because margin cost is upward sloping. They've already paid the fixed cost. They don't care in the short run. So all I care about is variable costs, OK? So at the end of the day, this is not expensive. They are willing to produce that first gallon really cheaply. They've already invested in this giant refinery plant. Marginal costs are tiny. So they get a huge-- but at the same time you pay them, you don't differentiate what you pay per gallons. You plug the thing into your car and you get the gas, OK? So they're getting $3 a gallon, but they're not paying much to make that gallon. However, as they make more gallons, their marginal cost increases. So the surplus they earn on each gallon produced shrinks. The surplus they earn at each gallon produced shrinks. And so eventually, they get to a point where they are essentially indifferent about producing the next unit of gasoline. That's at a price of $3 and a quantity of-- should be little q, OK? That's the point at which they are indifferent between producing gas and not producing gas. Therefore, their surplus is zero. So producer surplus is the difference between the price line. And the upward sloping supply curve is produced surplus. Now, in the long run, we have a name for that. It's called profits, OK? So our consumer surplus is this abstract, weird, theoretical concept. Produced surplus-- you can get your hands around. It's profits. Basically, remember, in the long run, marginal cost equals average cost, right? Because in the long run, you produce until marginal cost equals average cost. Therefore, the supply curve is the average cost curve. Price minus average cost is profits. Therefore, producer surplus is profits. Let me say it again, a little three-line proof for you. OK, in the long run, marginal cost equals average cost. Second, the supply curve is the marginal cost curve. Therefore, it's the average cost curve. Third, profits is defined as price minus average cost. Fourth, profits is the shaded area. Now, in the short run, that's not quite right. Because there's the whole shutdown decision, which makes things awkward. But roughly speaking, it's not terrible to think about producer surplus as being profits. That's a shorthand that largely works. If it ever doesn't work, we'll let you know. But that's the shorthand. It should largely work, OK? Now, of course, once again, we don't care about individual firm's producer surplus. We care about the market producer surplus, so let's go to figure 9-11. Figure 9-11 is basically the market surplus curve. And the idea here is that essentially to the left, you have a market supply curve where basically, remember, the individual firm's supply curve is always flat. But the market supply curve doesn't have to be. It doesn't have to be flat, OK? The market supply curve-- well, no, let me back up. A market supply curve is flat under a certain set of conditions. But now, let's imagine that those conditions aren't true. For example, let's go back to-- I talked at the end of last lecture about heterogeneous firms. Remember, we talked about the cotton example. Some firms are more efficient producers than others. If all firms are identical and it's very competitive, of course, the market supply curve is flat. So this graph would be uninteresting. But in fact, imagine that firms aren't identical. Some firms are more efficient producers than others, OK? For example, in that case, what you'll see is the most efficient producer will earn the most surplus, i.e., the most profit. They're all the way to the left. As you move to the right, you're getting to less and less efficient producers, OK? So profit is shrinking. So under the conditions we started with last time, then price would always equal supply. It'd be a flat supply curve at the price and therefore, profits are zero. That is producer surplus is zero. So we derived-- towards the end of the last lecture, we said, in the long run, a perfectly competitive market-- profit is zero. That's the same as saying producer surplus is zero. And why is that? That's because in that case, the price line is on top of the supply curve. Therefore, there's no gap between them. So in the long run, a perfectly competitive market-- there's zero produced of surplus, means zero profit. In reality, we talked about conditions why there would be an upward sloping, long-run supply curve, like firms different, how efficient they are. Or there's barriers to entry, which means some firms can't come in and drive profits to zero. Or there's an upward sloping input price curve, meaning that basically the more you want to produce, the more you have to pay workers. For all those reasons, the supply curve slopes up. And therefore, you can get a producer surplus. You can get some profits, even the long run, OK? So basically, what we have here is a situation where as long as the supply curve slopes up, you get a long-run producer surplus, which is the difference between the price and the supply curve, OK? And that is the same as profits. Questions about that? OK, let me cover one last point. Going back to last lecture-- going to have time to get to your last lecture. Remember, we talked about three reasons why, in the long run, even in a competitive market, supply can slope up. We talked about heterogeneous firms. That is firms with different levels of efficiency of production. We talked about barriers to entry. That is reasons why firms can't enter and drive profits to zero. Because it's not costless to enter. And we talked about upward sloping, input supply curves. We talked about the fact that as you produce more, you might have to pay more for your inputs. And therefore, you can't just charge when-- you have to charge higher prices as you produce more. I want to highlight something I said quickly last time, the difference between these two and this one. In these two, there are profits. In these two, there are profits, OK? Because in each of these, there are reasons why the market will not drive every firm to zero profit. Some firms remain in-- much like Pakistan made profits on their cotton sales. Some firms remain in. Likewise, with the barriers to entry, the firms that are in the market that have gotten over those barriers will make money, OK? In this case, the firm doesn't necessarily make money, OK? What it does here is it just pays. It takes that extra money and pays it out to workers, OK? So whether or not an upward sloping supply curve, doesn't necessarily mean the firm makes profit. It could just be upward sloping, because their input costs are rising, OK? So that's an important distinction to keep in mind. So let's stop there with that mind-blowing insight. Let's stop there. And we'll come back. And we'll talk more about welfare economics.

MIT_1401_Principles_of_Microeconomics_Fall_2018

6_Costs.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: All right. Why don't we get started? Today, we're going to continue our discussion of producer theory. Once again, to remember to put this in context, the first few lectures were working consumer theory to help us derive a demand curve. Now we're working on producer theory to help us come up with a supply curve. We started last time by talking about how producers profit maximizes, and the profit maximization implies cost minimization. Therefore, to maximize profits, you're going to want to produce as efficiently as possible. And basically, to do that, we need to understand how your costs vary with your output. If you're going to produce at an efficient level, you need to understand how what your costs are going to vary with your level of production. So essentially, our goal of this lecture is to develop a cost curve-- develop a curve which tells you how the cost of your production varies with how much you produce. And that's what we're after in this lecture. OK? So we're going to start with the short run and then turn to the long run. So start with developing a short-run cost curve and then turn to a long-run. OK? And to make this lecture sort of mathematically coherent, throughout the whole lecture, we'll work with our favorite functional form-- a production form of the function q equals the square root of L times K. Remember, firms produce goods-- little q-- using two inputs, labor and capital. Labor is a variable input. That means you can change it in the short run and the long run. Capital is a fixed input, which means that you can only change it in the long run. OK? So in the short run, we're going to have two kinds of costs. We're going to have fixed costs, which are going to come from a fixed level of capital, and we're going to variable costs, which are going to come from our labor. So we're going to have two kinds of costs: fixed costs-- they're fixed, because in the short run, you can't change the level of capital-- and variable costs, which are the costs of our labor. And then we're going to have total costs. They're simply going to be fixed costs plus variable costs. OK? So that's how we think about costs. The costs of a firm's production in the short run is the sum of their fixed costs-- i.e., their capital costs-- and the variable costs-- i.e., their labor costs. Now, we're going to show you how you can turn a production function into a cost function. And to do so, you simply need to recognize that cost-- the costs of firms' production-- are simply the amount of capital that uses k bar times the price of that capital, which we'll call r, plus the amount of labor it uses, times the price of that labor, which we'll call W, the wage. OK? This is the easy part. Think of the amount of hours of work you use times the wage per hour, or the amount of workers times the salary per year. In any case, this is easier to understand. Every additional unit of labor comes with a cost that is the wage of that unit of labor. Think about an hourly model. Every hour you work at your convenience store, they have to pay you the minimum wage for that hour. OK? That's the cost of that hour of labor. Capital is harder. We call r the rental rate. And the reason is because we don't think of buying capital. Don't think of buying a machine. Think of renting a machine. The reason we do that is to make the periodicity work. OK? You don't buy a worker, thank god. You rent that worker. And you rent that worker at a price, w. So when the firm uses your time, they're renting your time an hour at a time at a price, w. When we get machines, think of a firm as renting machines for a price, r per machine per time period. OK? So I understand firms usually don't do this. They usually buy machines. And we'll come back to even if you buy a machine how it effectively is like renting it, but for ease of thinking about this, you want to think about flows, not stocks. Think about the firm's decision as renting a worker at the price, w, or renting a machine at the price, r. Yeah. AUDIENCE: Kind of like also we consider the gas and electricity a machine can use in productivity? JONATHAN GRUBER: All that would be in there, and we'll come back to that. That's right. This is sort of the per period cost of using a machine. This is the per period cost of using a worker, is the wage. This is the per period cost of using a machine, which will include all the costs of running the machine as well as the costs of renting the machine itself. OK? So later, we'll talk about how to own the machine. And we'll come back to the fact that you can actually use r as a representation. It's not bad. But for now, just think of renting a machine. Or if it's a building, think of this as the rent you pay on that building, OK? Not the cost to build the building. Now, armed with our production function-- and let's also say, to make life easy-- I did this wrong on my notes. So I hope the other teachers figured it out. To make life easy, let's say that the rental rate, r, we're going to say is $10, and the wage rate, w, is going to be $5. Now, armed with a production function, if you simply have the production function and these two prices, you can derive the short-run cost function. How do you do that? Well, just look at the math. We know that q equals square root of L times k bar in the short-run. L times k bar. OK? So inverting that simply means that L equals q squared over k bar. L equals q squared over k bar. OK? And that means that cost can be written as 10 k bar-- the price the rental rate is 10, we have k bar amount of capital-- plus 5q squared over k bar. That's our cost function. I just plugged in for L and multiplied by the wage rate, w, which is 5. OK? So for example, for a fixed level of capital, this is cost. So for example, let's imagine our short-run level of capital is 1. Let's imagine there's 1 unit of capital in the short run, just to make the math easy. Then that simply says that cost equals 10 plus 5q squared. And that's our cost function. 10 plus 5q squared. We've just derived the short-run cost function. 10 is the fixed cost component. That doesn't vary with the amount you produce. So there's no q part of this. 5q squared is the variable component. That varies with how much you produce. So the cost function is a fixed part, which comes from that one fixed unit of capital, and the varying part, which comes from the fact that the amount of q drives the amount of labor we need. OK? Questions about that math? All right. Now, armed with this cost function, let's write that down again here. So C equals 10 plus 5q squared. That's going to be our short-run cost function we're going to work with. And remember, that short-run cost function came directly from that production function. To derive this equation, all I needed was that production function and those two prices, and I derived it. OK. Armed with that, we can define some of the key concepts that will drive our entire analysis of firms. And the single most important concept is marginal cost, which is what it sounds like-- the derivative costs with respect to quantity. OK? So in this, the marginal cost is delta c delta q. OK? That's marginal cost. OK. We'll also care about average cost, which is just c over q. It's very important in this class to keep our marginals separate from our averages. The average is simply over the entire range of production. What is the average cost to produce each unit? The marginal is what's the cost of producing the next unit? And since production functions are nonlinear, those will not be the same, generally. OK. Average will not equal marginal in general, because a nonlinear function delta c delta q is not the same as c/q. OK? So we can actually graph these in figure 6-1. Figure 6-1 shows the cost curves for this cost function I just wrote down, which comes from that production function. OK? So you can see that the marginal cost, as I said, is delta c delta q. Well, that's 10q. So the marginal cost, the cost of producing the next unit, rises with the number of units. Which makes sense. The cost has a q squared term in it, So obviously, the marginal cost is going to have a q term in it. So basically, the more you produce, the higher your marginal cost. The more unit you need to produce, the more the little q, the higher your marginal cost. Average cost is this sort of funky shaped thing, where it's-- which is 10 over q plus 5q-- I just divided this by q-- where it's first declining and then increasing. Why is that? Why is average cost first declining and then increasing? We've seen what-- just intuitively? Why is that? Why in general in the short run would we expect that? Average cost first to fall and then increase. Anyone have ideas? Yeah. AUDIENCE: [INAUDIBLE] start up [INAUDIBLE].. JONATHAN GRUBER: Well-- no, but it's falling first. So why is it falling first? It's about that-- yeah. AUDIENCE: They have a really high average-- or first fixed cost is not [INAUDIBLE] the more you make, the lower that is. JONATHAN GRUBER: Right. The first units are paying off your fixed costs if you think about it. Well, the first unit you sell, basically you start with this huge fixed cost. So actually, by selling two units, yes, you get the variable cost, second unit, but you get to pay off the fixed cost, the first unit. So look at here on this graph. We show average fixed costs and average variable costs Average fixed costs are 10/q. If you only produce one unit, your average fixed cost is $10. To produce two units it's $5. With every unit you produce, your average fixed cost is falling. You're paying off that fixed cost. Average variable cost rises. Every unit you produce, you're getting more and more variable cost. You put those together, and you get a function that first declines and then rises. You first pay off your fixed costs, so your average costs are falling, then your marginal cost-- then you start to rise, because you've got marginal costs that increase with quantity produced. And critically, the marginal cost intersects the average cost curve at the minimum of average cost. And that's just mathematical. If you have any function, and you take the average, then the minimum is going to be a derivative that basically-- before you get to here, 1.5 units, before 1.5 units, average cost is above marginal cost, because you're paying off your fixed costs. Once you get beyond 1.5 units, average cost is below marginal cost. So average cost hits marginal cost at the minimum of average cost. Yeah. AUDIENCE: In a relatively large company, eventually doesn't really worry about their fixed cost, because if they're having a lot more workers, their entire cost is going to be considered basically-- JONATHAN GRUBER: In the short run. It depends on this function. You said large, but large can be defined in two ways. But absolute true. Certainly, if we take this very function, for large enough q's average fixed cost asymptotes to 0. 10 over infinity is 0. So for large enough q's, the average fixed cost goes away-- in the short run. Remember, we're in the short run with these fixed costs. OK. Now, so other questions. Good question. Other questions about that? So that's our basic intuition, the short run-- is that at first our costs are super high, because you've got to pay-- you had to build the plant. But then over time, that plant cost falls, and then your only costs is basically the fact you've got to hire more workers if you want to produce more. Now, what we want to notice is that in the short run there is a really close relationship-- one was the key relationship between marginal cost and the marginal product of labor, which we defined last time. Remember, the marginal product of labor was dq dL. How much-- remember digging a hole and diminishing marginal product? That each additional worker, for a fixed level of capital, is less and less productive, right? We talked about that last time. Well, the marginal cost of production is, as I said, equal to delta c over delta q. OK? Well, we know from last time that delta q over delta L we defined as the marginal product of labor. Plugging those and-- so we know marginal cost is delta c delta q. And we can write this-- if you take the derivative of the cost function, so our general cost function up there-- see the cost function at the top there? Take the derivative of that with respect to the amount of labor. Well, the first term drops out, because it's fixed. So you can rewrite delta c delta q as w times delta L delta q-- w times delta L delta q. Right. I just rewrote delta c-- q delta w-- god. Brutal. Sorry, guys. w times delta L over delta q. OK. A little bit better. OK. I read that, because I just took the derivative of the cost function. First term drops out, we take the derivative. Second term, I just took the derivative here, or the discrete derivative, and I just said, delta c delta q is w times delta L delta q. Well, we know the marginal product is delta q over delta L. So we can rewrite marginal cost as w over the marginal product of labor. The marginal cost is equal to w over the marginal product of labor. And that makes sense. The marginal cost of the next unit will be higher the higher the wage and lower the more productive the worker. Making another unit with a super productive worker is cheap. Making another unit with a very unproductive worker is expensive. So essentially, the more you pay them for each hour of work, the higher your marginal cost. But the less they get done each hour of work, the higher your marginal cost. So that's why, roughly speaking, firms might want to pay a lot to people who are high skilled. So you might say, gee, why are they paying my friend twice what I'm paid? Well, maybe your friend's twice as productive as you. That would make-- or two and a half times as productive as you. And that would make sense. So we can't just say that it's a mistake to pay someone higher wages. We have to consider their wages relative to how productive they are. And that's a key relationship we'll come back to. OK? Question about that? All right. That's the short run. Now let's go to the long run, which gets a little more interesting. Actually, let's write in that side. I'll switch to this side today. Switch things up. This way, this side, I block you less, probably. So I should do this side. OK. Long run cost curves. OK. Long run cost. Now, here what gets interesting, is now K is no longer fixed. Now we get to choose our input mix. And now our goal is going to be, how do we choose our input mix to minimize costs? That's our goal here. How we going to choose the mix of workers and machines to produce a given quantity most efficiently? Now, that optimal mix may change with the quantity. So we're going to start. We're going to do this in two steps. First, we're going to say, for a given quantity picked out of a hat, what's the right mix of labor and capital that minimizes the cost of producing that quantity, given our production function? Then we're going to say, as the quantity varies how does that change the optimal mix of L and K? And does it? So two steps. First say, for a given quantity, what's the right L and K to minimize costs of producing that quantity? Then ask, well, as we vary the quantity how does L-- how do L and K vary optimally? OK. So basically, we want to find the economically efficient combination of L and K which is a combination that produces goods at minimum cost. And to do this, we are going to write down-- to derive this we're going to write down what we call isocost curves. Isocost curves. Remember last time we did isoquants, Which felt a lot like the difference curves? Isocosts are going to feel a lot like budget constraints. Isocost curves are essentially the firm's budget constraint. They're essentially mappings of the function c equals wL plus rK-- essentially, mappings for different amounts of K and L of the function c equals rK plus wL. So if you look at figure 6-2, here we see our isocost lines. OK. So let's talk about this for a second. So for example, take the middle one, the $100 isocost. This is saying, what combinations of labor and capital cost you $100? Well, with a rental rate of $10 and a wage of $5, that means you can have 10 machines and no workers or 20 workers and no machines, or some combination in between. This is just a budget constraints. It's just saying, given the amount of money you want to spend, given your cost, how many machines and workers can you have? The difference is, we don't start-- I didn't start this example by saying, your parents give you x dollars. That's why firm theory is harder than consumer theory. I pin down the consumer theory problem much more easily by saying, your parents give you x dollars, which told you which line to derive-- graph. I don't have that here. I haven't told you that here. So you have to graph a series of isocost curves, because you don't know what the optimal cost is going to be. That's to be pinned down later. That's what makes supply theory harder. You have to draw the series. So you draw these series of isocost curves-- different combinations that represent different amounts, different totals of cost. And of course, the slope of that isocost curve is delta K delta L, or minus w over r. That's the slope. Or in this case, minus 0.5. Now, those of you thinking ahead-- I know you guys are very insightful as a class, I'm sure many of you are thinking ahead-- might think, gee, that slope might change as the number of workers and machines change. Could you imagine the relative price of capital labor changes in different costs-- and it might, and we'll put that aside for now. For now, assuming for every relevant quantity these prices $5 and $10 are fixed-- let's ignore where those prices come from. They're just given now. We'll come back to that later. Like I said, this course is sort of like peeling an onion. We raise things, then we come back go to the next layer. Where'd that come from? Right now-- we'll tell you where w and r come from. Right now we're just going to take them fixed. And we'll assume they're always $5 and $10, regardless of the amount produced. OK. Now, here's the question. You're a firm that wants to release a certain amount of units. You have a production function and a cost function. How do you graphically figure out the right combination of capital labor to use to produce a certain amount of units? Yeah. AUDIENCE: Could it be the tension between the isoquant and the isocost curves? JONATHAN GRUBER: That's exactly right. Just as I asked you, what is the right combination of pizza and cookies, and you told me that it was the tangency of the indifference curve and the budget constraint, it's the exact same logic here. The optimal mix of capital and labor comes from the tangency of the isoquant with the isocost, as we see in figure 6-3. And ignore the, like-- somehow that curve is sort of connected at the top. It's sort of a glitch a PowerPoint. Just ignore that. It's not actually like a square or a trapezoid. It's just a curve. What would you call that-- the curve and the two sides. Is that a-- that's not a trapezoid. That's not-- it's not a polygon. It's just a line. All right. So basically-- is there a name for that? A curve and two lines? I don't think so. It's just a polygon, right? OK. So it's not a polygon, it's just a curve. So the curve is the isoquant for the square root of 12.5. What do I mean by that? I mean that is the combination of capital and labor that delivers square root of 12.5 units of production. So what that curve is all possible combinations of capital and labor that deliver square root of 12.5 units. Just like the indifference curve is all possible combinations of pizza and cookies that leaves you equally happy, this is all possible combinations of capital and labor that leads you to a given production level. And as we said, the further out the isoquant, the more you can produce. So you want to produce as much as you can given the prices you face in the market. Well, those prices you face in the market are delivered by the isocost curve. So the tangency is the best-- is the cost minimizing point. That's when you're producing the most you can given the costs you face in the market-- the most you can, given the costs you face in the market. And that tangency condition-- once again, considering our parallels to consumer theory, the tangency condition is going to deliver that the marginal product-- is going to deliver that the marginal product of labor over the marginal product of capital, which we remember called last time the marginal rate of technical substitution, is going to be equal to w/r-- actually, the negative of these is going to be equal to each other. But we'll just cross out the negatives. So the negative of MPL over MPK, which we called the marginal rate of technical substitution, is equal to the negative of w/r. The slope, the optimal point, is where the marginal rate of technical subsection equals the slope, which is the wage to rental rate ratio. Alternatively-- I don't know if anyone besides me likes this intuition-- we can rewrite this as MPL/w equals MPK/r, my bang-for-the-buck formulation that I like-- that the next dollar of wages, if you ask, should I spend next dollar on wages or machines, you should do it until the next dollar of wages delivers you the same return as the next dollar of machines. This is what you get for the next dollar of wages, MPL/w. This it what you get for the next dollar of machines, MPK/r. You want to continue to trade off machines and workers until that condition is true. So let's actually now solve for this for our example. Let's actually solve for that. If we solve for this, we know that the marginal product of labor, which is dq dL, is 0.5 times K over square root of K times L. And the marginal product of capital, which is dq dK, equals 0.5 times L over the square root-- 0.5 times L over the square root of K times L. Just taking the derivative of the-- all I did was take the derivative of the production function. So therefore, putting these together, we're going to-- we know that marginal rate of technical substitution in this example is equal to minus K over L. That's not a general formula. That's just this example. The marginal rate of technical substitution is equal to the negative of the ratio of capital to labor. We also know that the wage rate-- we also know we want to set this equal to the negative of the wage rate-- I'm sorry, the wage rental rate ratio. And we know that's negative 1/2. We know that's a 1/2, because that's 5 and that's 10. So we want to set the marginal rate of technical substitution to the wage rental rate ratio, which means we set minus K over L equal to minus 1/2. Or at the optimum, that means that your labor, your capital, at the optimum-- that means the amount of capital-- should be half as much as the amount of labor. In this example, we just solved for the efficient combination of inputs, the efficient combination is you should use capital to labor in a ratio of 1/2. You should use half as much capital as use labor. So let me pause there. And let's talk about where this is coming from. Yeah. AUDIENCE: So does that mean that at any given price of cost [INAUDIBLE] line, that is the optimal point where it will be tangent [INAUDIBLE] JONATHAN GRUBER: Exactly. Exactly. That's the graphic intuition. Let's come to the economics intuition. The economics intuition is the following. The production function delivers this relationship-- that the marginal rate of technical substitution was minus K over L. In other words, when you're producing goods, given this production function you're indifferent between the next machine and the next worker. That's just the way this production function worked out-- that one more machine delivers you the same amount as one more worker. Now, I've just told you that one machine costs half of one more worker. So which you want more of? No? One machine delivers the same return as one worker. You want more workers. Workers cost half machines, they're equally productive, so you want more workers. So the optimal amount of machines is going to be half as many as the number of workers. You want more workers, because you're indifferent-- look at that production function. You're indifferent. You don't give a shit about L versus K. They're the same to you. You're a hard capitalist, man. Machine or worker, you don't care. But the market's telling you you can get a worker for half the price of a machine. So you, as a good cost minimizing capitalist, take twice as many workers as machines. And that's the outcome that you get here. Questions about that? OK. So that's basically what we do to derive this. Now, what I want to do is take this and then derive our ultimate goal, which is, what is the long run cost function? That's sort of what we-- why we started this lecture. What is the long run cost function? Let's do the math. We'll do the math in five steps. Step one, q equals square root of K times L. Step two, we know from up there that K/L equals w/r. We derived that, leading us to the conclusion that K-- lead us to the conclusion that K equals 1/2 L. We just derived that. Therefore, we can rewrite q as the square root of 1/2 times L squared, just substituting it, because K equals 1/2 L. Therefore, we can solve for L is going to be square root of 2 over q. And K is going to be square root of 2 over 2 over q. L is square root of 2 over q, K is square root of 2 over 2 over q. I'm sorry-- no, not over q. I'm sorry. That's my bed. Error. Error. Go back. Should always look at my notes. It's square root of 2 times q. My bad. L is square root of 2 times q. And K is square root of 2 over 2 times q. OK. Therefore, armed with this L and K, we can rewrite our cost function. So step five is that the cost function-- given this stuff, the cost function equals r times square root of 2 times q-- I'm sorry, r times square root of 2 over 2 times q-- plus w times square root of 2 times q. I just plugged in the optimal L and K into my cost function. Now I can plug in the 10 and the 5 to get C equals 10 times square root of 2 times q. And I'm done. I just derived the cost function. That's what we came here for. This is what you got up this morning and wanted to see. You got up this morning, you said, I want to know how does the cost of a firm vary with the quantity it produces? And I've just told you. This tells you how the costs that you pay vary the quantity you produce. And I did that by deriving the optimal mix of L and K you want to use, and then simply imposing the prices of those two, and I get a cost function. Yeah. AUDIENCE: Wouldn't you be adding the two terms? JONATHAN GRUBER: I'm sorry? AUDIENCE: Wouldn't you be adding the two terms? JONATHAN GRUBER: Which two terms? Oh, I see. Yeah, plus. I'm sorry. You're right. My bad. That's a plus. Thanks. This is the most math I'll do in a lecture all year, you'll be pleased to know. It's why it's my least favorite lecture. Yeah? AUDIENCE: So is five generally true? JONATHAN GRUBER: You mean this particular functional form? AUDIENCE: Yeah. JONATHAN GRUBER: No. This is all dependent on that production function I wrote down. What's generally true is this-- or actually, K wouldn't be-- what's generally true is just C equals-- in the long run, what's generally true is C equals wL plus rK. That's what's generally true. I just made-- But what I've showed you is, given three things-- a production-- all I gave you was a production function, a wage rate, and a rental rate. Given those three things, you can then derive the cost function. Given those three things, you can derive the cost function. In fact, given two things you can derive the cost function. You could actually derive the cost function given one thing-- given just the production function, derive the cost function as a function of these input prices. OK. So that's a lot of results from one function, from one production function. Now, other question about this? Other math I got wrong? Sorry about that. Yeah. AUDIENCE: Sorry, could you just repeat the three inputs that you were using? JONATHAN GRUBER: [INAUDIBLE] all I used. Somebody tell me. What do you need to get the magical cost function? What three things do you need? Yeah. AUDIENCE: w and r and then q. JONATHAN GRUBER: No, w, r, and-- what about q? You need q, but what about-- what do you need? w, r, and the production function. So armed with the production function, that mathematical equation, this mathematical equation, w and r, I'm done. Everything I've done in this lecture comes from those three things. The math is hard and annoying. We will have you practice it. You will not like it. OK. It's just kind of what you've got to do. All right? I don't like it, you're not going to like it. It's just what we've got to do to get to the more interesting stuff. OK? Yeah. AUDIENCE: Is the bar on the K? JONATHAN GRUBER: It's fixed in the short run. The bar on the-- shouldn't see a bar on a K over here. That's all over here when I was doing the short run. Yeah? AUDIENCE: Can you use r and w in order to get to the fifth step? JONATHAN GRUBER: Yeah, we needed r and w. AUDIENCE: Well, in the sense that, to simplify, K equals 1/2 L? JONATHAN GRUBER: To K-- oh, you're right. That's a good point. I needed r and w back here. You're right. That's a good point. Good point. But I still could have done this whole thing as a function of r and w if I wanted to-- if I wanted to really screw up my math. All right? OK. So now, armed with this, let's talk about what happens when input prices change. We talked about with consumer theory, what happens when the price of pizza and cookies change. What happens when the price of labor and capital changes? What does that do? So let's talk about changes in input prices. OK. Let's go to figure 6-4. And let's look at, with the same production function, square root of L times K-- we're not changing our production function-- we're going to change the wage rental ratio. So line-- we have our initial line, our initial wage rental ratio, which is that basically you have a wage rate, but the budget constraint, essentially, that's flatter is our original budget constraint. The flatter budget constraint is our original budget constraint. That's the budget constraint with the price of capital of $10 and a wage of $5. And that intersects our isoquant at point x. So we chose five units of labor. Now we have a new-- we chose five units of labor and two and a half machines. That was our original. This is sort of a messed up graph. But our original intersection was at point x. The cost minimizing combination, the square root of 12.5 production, was to have five workers and two and a half machines. Now let's say the price of workers rises to $10. The wage rate rises to $10 an hour. So now, workers and machines cost the same. What is now the optimal mix of workers and machines? Well, graphically we know we still want to produce the square root of 12.5. So we want to stay tangent to the same isoquant. So based on-- what we're saying is, this is as if we said to consumers, keep your utility changed, change the price. What do we call that? Remember what we called that? Keeping utility constant, changing the price? Anyone remember what we call that? Who said that? All right. Raise your hand next. Be proud. Substitution effect. That's the substation effect. It's the same idea here. We want to know, for a given level of production, what happens as the price of the inputs change? And so we shift along the isoquant for point x to point y. And you'll see we choose a mix where we use fewer workers and more machines. And just as the substitution effect is always nonpositive, this shift as the wage rate rises, the price of the good, the x-axis rises. You will unambiguously use no more, and almost certainly less, workers. OK. And you can see that graphically-- think about graphically, you're looking for the tangency between this curve and the line. The slope of the line just got steeper. Therefore, you must move to the left on the curve. It's the same proof as we used the substitution effect, where substitution effect was always nonpositive. OK. It's the same intuition here we use for y. A rise in the wage rate will lead you to hire fewer workers and more machines. Guess what? You just entered the debate on the minimum wage. And if you follow the debate in minimum wage, what do people say? Well, if you raise the wage you have to pay workers, they'll be replaced by machines. That's this. This is the math-- this is the mathematical and graphical intuition behind the debate on minimum wage, which we'll get into later in the semester. But the basic idea of that debate is, gee, if you force firms to pay more to workers, they're going to substitute towards machines. That's exactly right, in theory. And practice, there's a lot of complications. But this gives you the theory of why people make that argument. Yeah. AUDIENCE: So in this example, only the wage for workers [INAUDIBLE] not the machines. JONATHAN GRUBER: Not the machine. AUDIENCE: So why does the isocost not have the same y-intercept? JONATHAN GRUBER: Ah, great point. Because here I'm drawing a new-- I am drawing the isocost that I would use while still producing square root of 12.5. So that's it's just the substitution effect. I'm not drawing the full set of isocosts at the new price. I'm just saying, to produce the same amount what's my new-- if I want to produce the same amount, what combination do I now have to use? OK? Yeah. AUDIENCE: The total cost of production [INAUDIBLE].. JONATHAN GRUBER: The total cost of production-- let's see. Yeah, it has to be-- no. Let's see. No, total cost doesn't have to be the same. The total cost used to be five workers at $5 an hour, that's $25. So it used to be $50. Now what is it? Now it's $70. The total cost has gone up. AUDIENCE: So it's not like the budget constraint or the-- JONATHAN GRUBER: Exactly. It's not like the budget constraint where your income is fixed. That's what's hard about producer theory. Because basically, the budget constraint was sort of asking, keeping your budget fixed. This is like asking, keeping your total production fixed. And so now you have to pay more to get that level of production. OK? Good questions. OK. Now, ultimately what does this lead to? So that gives us our change in input prices. Other questions about that? Now, remember I said at the beginning of the lecture, we are first going to solve for what is the cost minimizing combination of inputs for a given quantity? We derived that up there. It's half as much-- going back to our old prices, it's half as much capital as labor. Now we want to ask, how does your cost change as the quantity changes? And we call that the long run expansion path. The long run expansion path, which is, how do your costs expand as you produce more? And we see that in figure 6-5. In figure 6-5, we show the particular case of a linear long run expansion path. That's what you get in this example. It's a particular case. What this case says is, at any given level of production, the optimal mix of labor and capital is the same. In other words, you always want to have-- essentially, given the price of labor is half the price of capital, you always want to have twice as many workers as machines. So if you want to produce square root of 12.5, you want five workers and two and a half machines. If you want to produce square root of 50, you want 10 workers and 5 machines. If you want to produce square root of 112.5, you want 15 workers and 7 and 1/2 machines. So the long run-- given this production function, the long run expansion path is linear. You always want the same ratio of workers to machines. Yeah. AUDIENCE: [INAUDIBLE] consumer and firms, is the reason why we don't necessarily have a strict budget, per se, and then isn't the idea that if we really want increase production, we can take a loan out? JONATHAN GRUBER: This is what I tried to say. It's sort of-- I always say it over here, but it's hard, and we have to come back to it. The reason producer theory is harder is because we're not given a fixed constant we are with consumers. Consumers, we're saying, look-- you've got a resource, you've got to constrain maximization. We haven't constrained the maximization yet. There's another constraint we need. They have an extra degree of freedom relative to consumers. Now, in fact, consumers have degree of freedom, too. When you grow up, your parents don't give you money. You decide how much to make. So in reality, consumers-- you can do this-- will have the same degrees of freedom. But we started with the easy consumer theory case, where you constrict-- we took away a degree of freedom. Now we're writing it back, which is, you can choose how much to produce. Like, you being able choose your income as a consumer. That leads to long run expansion path. Let me go on, because I want to make sure I get through this stuff. OK? Now, the long run expansion path does not have to be linear. So think about-- look at figure 6-5b and 6-5c. So 6-5b is a long run expansion path for a production function such that capital becomes less productive the more you produce. I don't have the example of production function. But when you write down production functions which have the feature that the more you produce the less productive capital becomes, the less each additional unit capital helps [INAUDIBLE] additional unit of workers. So you know, we could think of this roughly as sort of like a fast food restaurant. That kind of-- you know, each addition-- that basically, there's so much stuff to do where workers can efficiently share tasks and things. Each additional worker-- that the marginal product of labor essentially diminishes less quickly than the marginal product of capital. On the other hand, in figure 6-5c we can have a long run expansion path where labor becomes less productive relative to capital. Think of it as like heavy machinery, where basically all workers can do is run the machine. So that second worker-- workers don't really do much but sit there and flip a switch. You need the worker to flip the switch. That's all they do. So the second worker, you're already flipping the switch. So really, adding more machines is a more productive way to expand the output. None of these is right or wrong. We're just saying that the shape of this expansion path can basically vary with how much-- with different production functions. But they're all the same idea. Now, so basically, that tells you-- but here's the bottom line that we wanted to come to. That long run expansion path is a long run cost curve. So ultimately, if you want to ask, how do my costs vary with how much I produce, this curve tells you. Because what it does, it says, for every level of production I'll tell you the optimal combination of L and K. Given the price of L and K, that will tell you the costs. And so you trace out the costs with every level of production. This is your cost curve. This long run expansion path tells you what the costs are for every level of production. And it tells you that, because you've made-- you're doing the efficient level of production. That's what the long run expansion path is telling you. OK. This is hard. I'm about to make it harder. Which is, we're now going to talk about the relationship between short run costs and long run costs. And the key insight is that long run costs are everywhere lower than short run costs. Without looking at the figure-- because the figure doesn't help with this-- why does that make sense? Why our long run costs-- why, if you can optimize over the long run, will you always have costs that are no higher, and in general lower, than optimizing the short run? Yeah. AUDIENCE: [INAUDIBLE] already had the right capital. JONATHAN GRUBER: Because you have an extra degree of freedom. I think that's LeChatelier's Principle. Is that right, for the chemists among us? That basically, like-- essentially, an extra degree of freedom means, the more you can optimize over, the better you can do in optimizing. In the short run you're constrained by the size of the building. In the long run, you could choose. So let's-- to see that, let's go to figure 6-6. This is a confusing figure. So bear with me as I walk you through it. OK? Consider a firm with three possible sizes of plants. They're going to build a plant. So the capital here is the building. And there's three possible sizes-- small, medium, and large. The small plant has the curve SRAC1. What does that curve mean? That means that the small plant-- I'm sorry, the small plant is SRAC2, the medium plant is SRAC2, and the large plant has SRAC3. Compare SRAC1 to SRAC3. What this is saying is, for small quantities of production, SRAC1 lies below SRAC3. For small quantities production, if you extend SRAC3 out, you see at levels of production like q1 or even q2, SRAC3 is way above SRAC1. When you go to a level of production like q3, SRAC1, if you extend that dashed line out, is going to be much, much higher than SRAC3. So the right-- and SRAC2 is in between. So essentially, for different levels of production these give the different optimal short run cost curves. In the long run, you get to choose. So the long run average cost curve is the lower envelope of the short run average cost curves. Because in the long run you say, well, here's my production level. I know in the long run-- so if I know I'm going to build a lot of things, I choose SRAC3. I choose the biggest plant. If I know my production is going to be low, I choose the smallest plant. But I can optimize in the long run by choosing the right sized plant for my production level. This is hard. And I'm almost out of time, so let me end with an example that perfectly illustrates this. Tesla. Elon Musk. Everybody's favorite guy these days. Tesla, when they came out, had to decide how big a plant to build-- how many batteries to make. Batteries are the key [INAUDIBLE] Teslas. And they expected to make-- to have demand for 20,000 cars by the year 2017. So they built a plant like SRAC1. They built a plant that was the efficient plant to produce 20,000 cars. The problem is, demand was for 200,000 cars. And as a result, there's a three-year waiting list to get Teslas. It turned out that was not the right size to produce. They lost money-- relative to the optimum. They made money. Musk is incredibly rich. But they didn't do what was most efficient given the underlying demand. But now, Musk can re-optimize. Now he's saying, wait a second. People want way more cars. Well, producing them at the tiny plant was exorbitantly expensive. I had to run it over and overtime-- pay workers overtime. To produce 200k cars in that tiny plant just was exorbitantly expensive. That's if you take that dashed line and extend it way the hell up. The SRAC1 extended way the hell up, incredibly expensive. So what is Musk doing now? Building the largest battery plant in the world. In Nevada, he is building a battery plant that can produce batteries for 500,000 cars. So he shifted from SRAC1 to SRAC3. He's now saying in the long run, I can more-- if I'm going to produce 200,000 cars, I can do that more efficiently with a giant battery plant. And that's what he's doing. So he's re-optimizing. Now, what if Musk is wrong? What if it turns out Teslas suck and people are like, I don't want them anymore? Someone else-- or, you know, Chevy finally figures it out and makes a good electric car. Then what's going to happen is he's going to have made a mistake in the long run. Then the third period, he'll go back to a smaller plant again. But he always can do what's efficient in the long run, given the underlying demand. So Tesla is an example of this sort of long run, short run dichotomy. Anyway, it's a lot of stuff for one lecture. We'll come back next time, talk more about costs. And then we'll start getting into competition.

MIT_1401_Principles_of_Microeconomics_Fall_2018

16_Input_Markets_IILabor_and_Capital.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: All right, let's get started. Today, we're going to continue our discussion of factor markets. If you recall, last Monday, we started talking about the labor market. And we talked about how workers make the decision between work and leisure. And we talked about the implications for setting the wage rate in the labor market. What I want to do today is return to that labor market equilibrium and talk about the important case of the minimum wage. So today, I want to talk about the labor market equilibrium and how it's affected by the minimum wage because it's an interesting case which allows us to introduce some complications as to how we think about the labor market. So let's go back and think about the labor market. So let's go to figure 16-1. The labor market, like any other market, has a price and a quantity. The quantity is the amount of labor supply. That's on the x-axis. The price is the wage. That's on the y-axis. The supply curve that's upward sloping-- typically we'll assume an upward-sloping supply curve. But as we discussed last time, that doesn't have to be true. If income effects dominate substitution effects, which they very well may, you could actually have a backward-bending or downward-sloping supply curve. So we talked about that last time. Having taught that interesting case, typically, we'll assume supply is upward sloping or at least not backwards bending, not downward sloping. But remember, that's an assumption. So this upward-sloping supply curve is not necessarily as obvious as a downward-sloping demand curve is. Downward-sloping demand will almost always exist unless there's a weird Giffen good, whereas upward-sloping supply is a little more questionable. So we have the equilibrium, and we have this equilibrium at L1 workers at a wage W1. So now we know where this comes from. So basically, going all the way back to producer theory where we just gave you a W, now we're telling where the W comes from. We're telling you where the wage comes from that you then plug into the firm's optimization for them to produce goods. Now, let's imagine that we have a minimum wage. So let's go to figure 16-2. So this is a regulation which says that you're not allowed to pay workers below some minimum level. And let's say we set that minimum wage at the level W2 above the market wage W1. Quick question. What would happen if we passed a law and set a minimum wage that was below W1? So there'd be a regulation which insists you couldn't pay workers below W2, but W2 is below W1. What would that do to the labor market? Nothing. And here's the key point. Markets in economics will always endeavor to avoid government regulations if they can. So if a government regulation is not binding, it won't matter. Markets will just avoid it. So the interesting case is only where the minimum wage is binding, as in the figure 16-2. So what happens? Well, if you set a minimum wage at W2, workers at that high wage would love to work a lot. That's a high wage. They're high in the supply curve. They would like to work L sub s hours. They would like to supply L sub s amount of labor supply to the market. Firms, however, if forced to pay a high wage, W2, are going to say, wait, I'm only going to pay that high wage if the marginal revenue product of labor is sufficiently high. Remember, we talked about the marginal revenue of product last time. It's the marginal product of labor times the price. So if you're going to raise the wage I'm going to have to pay workers, unless that affects the market price, I'm going to need to have a higher marginal product of labor, right? The demand equation was, I said, the wage equal to the marginal product of labor times the price. Well, if the price hasn't changed with the minimum wage going in, I'm going to need a high-- if the wage is forced up by the minimum wage, I'm going to need a higher marginal product of labor. How do I get a higher marginal product of labor? By hiring less workers because the marginal product of labor's diminishing. So if you're going to force me to pay a higher wage, you're going to force me to only hire workers until the point where the marginal product of labor justifies that higher wage, which means I'm going to hire fewer workers. So firms demand only L sub d. Well, workers can't get jobs firms don't want to give. So the equilibrium is L sub d jobs at a wage W sub 2, OK? What does this do to welfare? We can see before, before the minimum wage was in place, the market featured a consumer surplus that-- here, consumers are firms, right? But there was a consumer surplus of A plus B plus C. That is, firms were willing to pay what was on the demand curve. They only had to pay W1. So their surplus was A plus B plus C. Workers were willing to work at a wage that's given by the supply curve S sub 1. They were paid at W sub 1. So they got a surplus of D plus E. So here, the firms get the consumer surplus. The workers get the producer surplus because the workers are now the producers. Now let's say you roll in a set minimum wage. Well, two things have happened. One thing is you've then transferred some resources to workers. That's the area B. You've taken the area B that firms used to get, and now workers get it. That's the idea. You want to make workers better off. So you transferred to workers the area B. On the other hand, you've created a deadweight loss of the area C plus E. You've created deadweight loss in the area C plus E because now there are fewer jobs. There are workers who would happily work at a higher wage who are not being allowed to work by the limited demand that comes from the minimum wage. So the bottom line is you end up with fewer workers, a higher wage, and ambiguous welfare implications. Clearly, social welfare goes down. Whether worker welfare goes up or not depends a bit on the size of area B versus the size of area E. It's not clear if worker surplus goes up or not. It depends on size of B versus E. In this diagram, workers are a net better off, but it doesn't have to be true. What's clear is that social welfare has gone down. Because remember, as I talked about, the cheat, the shortcut I talked about when we talked about oligopoly, is, roughly speaking, welfare is proportional to the quantity in the market. Essentially, the further you deviate from the perfectly competitive quantity, the bigger the deadweight loss. So that's what happens if you put in a minimum wage. Questions about that? OK? Well, that seems pretty straightforward, and that's what I learned growing up as a kid in economics class. But then some empirical economists, some very famous empirical economists, started doing a series of articles that actually studied, gee, what happens when the minimum wage does change. They did things like, for example, comparing what happened when New Jersey raised its minimum wage but the state of Pennsylvania next door did not, and looked at fast food workers in New Jersey, where the minimum wage went up, compared to fast food workers in Pennsylvania where the minimum wage didn't go up. And what they found was there was no difference in employment, that jobs didn't fall in New Jersey even though the minimum wage went up. And a series of follow-on studies continue to find that, actually, higher minimum wages didn't seem to cause jobs to fall, which is directly in contradiction with this graph. So what's going on? That led to a big question and revision of what's going on in these markets that leads to that. And there's really three possibilities for what's going on. Possibility one is that the minimum wage wasn't binding. Maybe New Jersey set a minimum wage below the market wage. But actually, empirically, that's not true. We can look at what workers were paid before the minimum wage. It was well below where the minimum wage was set for restaurant workers that were studied in that most famous study. So this is not true. The minimum wage was binding. There's a second possibility that's absolutely consistent with a perfectly competitive market. What's a possible answer for why I could impose a minimum wage in a perfectly competitive labor market and have employment not go down? Yeah? AUDIENCE: Price goes up. JONATHAN GRUBER: The price that the firm charges goes up. But in a perfect competitive labor market, that still wouldn't happen. You might see some price adjustment, but you'd still see some adjustment in the marginal product of labor. But what else about this diagram? Yeah. AUDIENCE: The firm's demand for labor is perfectly inelastic. JONATHAN GRUBER: The firm's-- actually, you're close. It'd be the worker's supply of labor is perfectly inelastic. It's the right idea. If workers are perfectly inelastic in their supply of labor, then the same amount of workers will work no matter what the wage. So basically, you're just going to essentially end up-- you'd also, in fact-- that's a good point-- also get inelastic demand, the same thing. If either supply or demand is inelastic, you'll end up with no effect of a minimum wage. So that's another possibility. But in fact, we've done a lot of studies. So you could have inelastic supply or demand. But in fact, we've done lots of studies of supply and demand in these markets, and that's not true. Remember, supply was largely inelastic for men, but it was somewhat elastic for women. And these low-income markets have a good mix of men and women working in them. Demand has been shown to be somewhat elastic. So neither supply nor demand's very elastic, but they're sufficiently elastic that that rules out as zero. So the third possibility and the one economists have focused on is that we're not in a competitive labor market. They're focused on a noncompetitive labor market. Just like we discussed noncompetitive markets for goods with a monopoly and oligopoly, you can have noncompetitive markets for labor. It's the basic same idea. So now let's look at-- so when we thought about-- let's go back, think about perfect competition, the basics of perfect competition. We thought about perfect competition. The basic idea was, remember, I talked about laying out a bunch of rugs in a market where you could literally shop costlessly across all the people selling their little fake Eiffel towers, little statue Eiffel towers. And you could perfectly shop. It was easy to go from carpet to carpet. There was full information. The prices were posted. And so basically what you ended up was perfectly elastic demand facing any given firm. Any given firm, if they tried to charge one cent more for their Eiffel tower, no one would buy it. If they charged one cent less, they'd immediately run out. Everyone'd buy it. Well, when we are modeling labor markets-- and I discussed this last time, but not very well. So I want to come back to it. When we're modeling labor markets, we're thinking about the same feature of perfect competition. But here, it's not consumers shopping over where to buy their goods. It's workers shopping over where to work. It's workers saying, gee, in a perfectly competitive labor market, the idea is I know what I could earn at any firm and I can easily shop across firms, see where I'm going to work. So if any firm tried to pay me one cent less than the market wage, I'd never work there. And if they tried to pay me one cent more than the market wage, every worker in the world would want to work there. So in a perfectly competitive labor market, any given firm faces a perfectly elastic supply of labor. So we can see that in figure 16-4, which we actually showed-- and I'll let you skip this since we covered it-- 16-4, which I actually showed in the last lecture. Remember the last lecture. I was focused on this downward-sloping demand curve, but I casually threw in this flat labor supply curve and botched explaining it. Now I'm explaining it, hopefully more clearly, which is to any given firm, the labor supply curve is perfectly elastic because workers can perfectly shop across job opportunities. So if that firm tried to pay less, they'd get no workers. So they faced a perfectly elastic supply of labor. But just like, in reality, there's no such thing as a perfectly competitive product market, in reality, there's no such thing as a perfectly competitive labor market. In fact, we can't shop easily across all possible jobs and know what every job could pay. And the fact that we can't means that firms on the labor market side will have market power. Just like we talked about monopolists and oligopolists having market power over consumers through barriers to entry, firms will have market power over workers because workers can't perfectly shop across their job alternatives. So as a result, firms may be able to get away with paying you less than what you might earn elsewhere. In a perfectly competitive labor market, a firm could never pay you less than what you're worth elsewhere because you'd just go work somewhere else. But now, if McDonald's wants to pay you less than you might get at Wendy's, but it's hard to go find out what Wendy's going to pay you-- you have to go a distance down the road, and you have to ask them, and you're shy and it's embarrassing-- then McDonald's might be able to get away with paying you less than you might earn at Wendy's. So this is very much parallel to monopoly. In fact, we call this a monopsony. A monopsony is a labor market where firms have market power over workers just like a monopoly is a goods market where firms have market power over consumers. Now, this is not so crazy. And in fact, it applies very much to me. Think about my situation at MIT. I've been here 25 years. I just got my 25th year rocking chair, although actually it's not a rocking chair because it comes in the box with the rockers off it. And it arrived in my office, so it's sort of a short chair. My wife's 5 foot, and she always complains how chairs are too big for her. So she sat, and she's like, it's a perfect chair for me. So now I have a nonrocking rocking chair in my office that she sits in. But anyway, I've been at MIT for 25 years. It's going to be really hard for me to move. I like my house. I like my colleagues. I like my friends. Kind of, I like my view out the window. It's going to be kind of hard for me to move. Moreover, it'd be pretty hard for me to figure out what I'd get paid if I moved. I can't go to other universities and say, hey, what would you pay me if you hired me? That's be awkward. I can't really ask my colleagues what they make. That's awkward. So at the end of the day, MIT has market power over me because I don't really want to move and I can't really figure out what I'd get paid if I did move. And MIT will exploit that market power over me by paying me less than I might earn elsewhere. And we know this as a fact because in academia, the only way to get a raise is to go get an offer from someone else and have them say how much more they'll pay you, and then you take that to your boss and they say, match this. But if you're not willing to do this, as, frankly, MIT knows I'm not willing to do, then MIT can essentially underpay me. So basically, any responsible profit-maximizing or even nonprofit employer will exploit this market power and they'll pay me less than my market wage. And that means that MIT will earn surplus on me. In a perfectly competitive labor market, the firm earns no surplus on the worker. They pay the worker their marginal revenue product. So if you go to this figure, what am I paying the worker? What I'm paying them is exactly the marginal revenue product just like, in a competitive market for the goods, a firm is selling at exactly their marginal cost. So just like a firm makes no surplus in a perfectly competitive goods market, a firm hiring workers makes no surplus in a competitive labor market. But in a monopsony market, the firm makes surplus over me. They pay me less than they'd have to because I don't shop and find a better opportunity. Now, are there questions about how that market works? I'm not going to do all the math and graphs. It's all the same as monopoly, just flipping demand and supply curves. It's a pain in the ass. I'm not going to do it. I just want you guys to understand the intuition. So please, since I went through this, are there questions about this or how it works? OK. Now let's take this noncompetitive labor market and let's throw in a minimum wage. Well, as before, if the minimum wage is below what the firm was already paying, there's no effect. So let's assume it's a binding minimum wage. Now, let's say the binding minimum wage is above what my true market wage would be, what my wage would be in the perfectly competitive market. So in a perfectly competitive market, my wage would equal my marginal revenue product of labor, right? That's in a competitive market. In this noncompetitive market, my wage is below my marginal revenue product of labor. Firms are exploiting me because I can't effectively shop for a better job. I don't want to or it's hard to do so. Now, in this noncompetitive market, if we set a minimum wage that's higher than the marginal revenue product of labor, then the analysis is just like it's a competitive firm. Once that marginal wage is higher than the marginal revenue product of labor, it's just like a competitive firm. So it's not that interesting. The interesting case is, what if the minimum wage comes in and it's above the wage I make but below the marginal revenue product of labor? So let's say McDonald's, someone working there yields a marginal revenue product of labor of $10, but they're only being paid $7. Let's say you roll in minimum wage of $9-- so above what they're being paid now, but below their actual marginal revenue product of labor. Will the firm fire that worker? Why not? Yeah. AUDIENCE: They're still paying them-- they're still making a profit off of that worker. JONATHAN GRUBER: They're still making surplus, which is as long as the marginal product of labor's bigger than the wage, they love that worker. So before-- so let's write down the numbers as an example. So imagine my marginal revenue product of labor at McDonald's is $10, but my wage is $7. And then you come and you set a minimum wage of $9. Well, 10 is still greater than 9. So the firm has no desire to fire me. So all you've done is just given me money. And where'd that money come from? The surplus the firm earned. So all you've done is shifted the surplus from-- you've shifted producer surplus to consumer-- I'm sorry, consumer surplus-- consumers are the firms-- to producer surplus, the workers. So in a monopsony market, a minimum wage doesn't cause deadweight loss. It just shifts surplus around. And that's a really important outcome because that, once again, says the government isn't always bad here. This is just like-- if you want to think about this graphically, go back to exactly the analysis we did of regulating monopolies. Remember we talked about regulating monopolies. We talked about, if a regulator comes in and sets a price below the monopoly price but above the competitive price, it reduced the deadweight loss of monopoly. It's the same thing. And if you set a minimum wage above the market wage but below the marginal revenue product of labor, then you simply transfer surplus to workers without causing deadweight loss. Now, that raised the question, of course, is the minimum wage in between the wage of the marginal product of labor? Well, we don't know, but let's go back to the studies that motivated this. The very fact that the minimum wage doesn't seem to cause unemployment suggests we are hitting the sweet spot, suggests we are hitting the sweet spot, that we're basically managing, with the minimum wage policy, at least to date, to essentially just find a way, without the government spending any money, to shift resources from businesses to workers. So what does this mean? Well, it means that around the level of current minimum wages, we can raise the minimum wage by a small amount pretty costlessly. It doesn't necessarily mean that a $15 minimum wage is OK. So in some sense, the existing-- this is the important thing about empirical economics. You only learn the answer in the range that you study it. So for example, there've been studies that have looked at what happens if you have a $10 minimum wage, and those show no unemployment. There haven't been studies that show what happens if you have a $15 minimum wage. Now, Seattle just actually put in a $15 minimum wage about two years ago. So we actually can run the experiment. And the early evidence is the Seattle $15 minimum wage did lower employment, that the Seattle $15 minimum wage actually went above the marginal revenue product of labor. And once it's above, you're back in the competitive case. You're back in the case where you're lowering employment. Yeah? AUDIENCE: How can you increase competitiveness in the market? JONATHAN GRUBER: Well, that's the other question, is how could you increase-- so you tell me. How could you increase the competitiveness of a labor market? AUDIENCE: You make it easier to tell how much money you would get at each place. JONATHAN GRUBER: So Norway has a day every year they call Envy Day, which was yesterday, I believe, where they literally can go online and look up anybody's income in Norway. They literally make public every single person's tax return in Norway. And you can go online and look at what everybody makes. That would do it. So you could provide more information. You could make it easier to move between jobs. For example, there's a lot of restrictions in our labor market, like noncompete clauses, which say that if you work for one firm, you can't ever go work for another firm in that industry for x years. That gives some monopsony power to firms, et cetera. So we could do things which try to loosen the flow of the labor market, and that would close this gap between wage and marginal revenue product of labor. Now, let's go back to Seattle, just to conclude this. This doesn't mean the Seattle policy was a bad one. The bottom line is what we learned from Seattle was that basically, employment fell a small amount and a bunch of workers made a bunch more money. So is that good or bad? Well, it depends. If you're one of the people that lost their job, it's really bad. If you're one of the workers who got a raise up to $15 an hour, it's good. How do you weigh them against each other? That's exactly what we'll talk about in a couple lectures. So once we start talking about normative economics, about is a policy good or bad, there's typically trade-offs. And this is a classic example. What we're learning here is, is the minimum wage in the range we are now, right now, the federal minimum wage at $7.25-- the evidence suggests it could easily rise without causing that trade-off. The evidence suggest we could increase the federal minimum wage by some nontrivial amount, at least up to $9 or $10, without causing much of a trade-off. But once you get too far ahead of that, there starts to be a trade-off. Question about that? Yeah. AUDIENCE: Are there any states where it's actually still that low? JONATHAN GRUBER: Oh, yeah. Many states don't have their own minimum wage. Massachusetts is at $11, but we're pretty unusual. We're one of the higher ones. A number of states have $7.25 as the minimum wage, OK? And the evidence seems to be, from states like Massachusetts and others which are on the $10, $11 range, it doesn't seem to lower employment. It seems like we could clearly-- we'd be safe raising that federal minimum wage. We would simply be transferring resources and not causing unemployment. Yeah? AUDIENCE: Is there anything about the cost of living in areas where the minimum wage is more expensive? Is it possible that if a McDonald's worker makes more money in this state, McDonald's is more expensive in that state? JONATHAN GRUBER: That's a great question. So what I assumed was I assumed firms would just say, oh, you got me. I'm going to throw some of my profits at workers. Firms don't have to do that. Firms could say, well, if you make me pay workers more, I'm going to raise my price. Now, if it's a competitive output market, that shouldn't happen, right? Because in a competitive output market-- well, no. Marginal cost goes up. It's not clear. It's not clear whether that would happen or not, and the evidence is that it's unclear whether higher minimum wage causes higher prices or whether it just comes out of profits. We don't know yet, OK? All right, so that's what I want to say about labor markets. Now I want to move on and talk about capital markets. Now, as confusing as our discussion of labor markets was, that's easy compared to capital markets. Capital market's a lot harder to understand. And that's because capital itself-- labor's something you get your hands around. It's the time you spend at work. Capital is this sort of amorphous thing that I've kept pushing off defining. So I'll define it now. We talk about capital as this vague collection of buildings and machines and the other stuff that goes into production. And we know where labor comes from. It comes from our work. But where does capital come from? Well, capital is a harder concept, but there's one unifying thread that all elements of capital have, which is they represent the diversion of current consumption towards future consumption. Capital is about diverting consuming today towards consuming in the future. In fact, the original concept of capital came from farmers. Farmers, every year, when they would pick their grain, they had a choice. They could eat all the grain, or they could save some to plant for next year's grain. Now, the more they saved, the more they'd have next year, but the less they'd have today. So farmers faced a trade-off-- literally, consumption today or consumption next year. That's what we mean by capital. In other words, in today's market economy, the link is not that direct, but it's the same basic idea-- that firms have a choice, firms and their investors have a choice. They can take what they make and eat it now, or they can invest it in having more in the future. So basically, when we think about capital, we're not going to think about capital as physical capital. We're really thinking about capital as financial capital. What links all types of capital is their financial aspect. What links machines and buildings is all the aspect that, by putting money into them today, you have less you can spend on fun stuff today, but more you'll be able to spend tomorrow. And it's this financial aspect that links all forms of capital. Now, how do firms get the money to invest in machines and buildings and stuff like that? They get it through going to the capital market. Where do firms get this money that they invest? They get it through going to the capital market, which is basically the pool of money that firms can draw on to make their investments. So think of it literally as I'm a firm. I want to build a building and buy a machine. I literally go over, and there's a big pool of money. And I have to take the money out of there to go buy my machine or build my building. And where does the money in that pool come from? It comes from household savings decisions. So the capital market is a market where the demand for capital comes from firm's interest in investing and having more in the future. The supply of capital comes from people's decisions to save. And essentially, the money firms use to buy stuff is borrowed from people. And that's the bottom line of how capital markets work. So just as the supply of labor that determines how many workers a firm can hire comes from your decision of how hard to work, the supply of capital that determines how many machines a firm can buy comes from your decision of how hard to save. So let's look at figure 16-5, equilibrium in capital markets. Let's start with the demand. We already talked, last lecture, demand for capital. The demand for capital comes from the marginal revenue product of capital. It's the marginal product of the next machine. So the demand comes from the marginal product of the next machine times the price the firm can get for its output, which is the marginal revenue product of capital. So it's the same logic as for labor. There's nothing interesting there. Same logic as for labor. The supply's what's more interesting here. Where does supply come from? The supply comes from household savings, how much money is around for firms to actually get to get these machines. And how do they get it? They borrow. And what do they borrow at? They borrow at the interest rate I. So I represents the rate that firms pay households to get their money. So think of this as-- we'll talk about how it really works. But in theory, the idea is think of literally a marketplace in the center of town. Downtown Boston, Haymarket, there's this marketplace. And a firm comes and says, I need to borrow money to buy a machine. And a person's there with their savings and they say, well, I'll loan you some money. What interest rate you going to give me? And that's the market for capital. So where the supply of capital meets the demand of capital yields the interest rate. So basically, what this means is as the interest rate's higher, what that means is I have to pay people back more to borrow their money. So an interest rate of 10%, if I borrow $10 from you, I pay you back $1.10 next period. If I borrow $10, I pay you back $1.10 next period. If the interest rate's 20%, if I borrow $10-- if I borrow $1-- I'm sorry. If I borrow $1 from you, I pay you $1.10 next period. If I have 20% and I borrow $1, I pay you back $1.20 next period, et cetera, OK? So basically, that is essentially how the transaction works. And the key point here is the reason the supply curve is upward sloping is the more you're willing to pay me for my money, the more I'm willing to lend you. So if you come to me and say give me $1 and next year I'll give you back $1, I'm like, I don't know. Why would I do that? If you say, give me $1 and next year I'll give you back $1.10, you're like, OK, now I'm interested. $1.20, I'm very interested. $1.50, for sure. Literally, I just give you my money and, next year, I get back 50% more? Why not? So basically, the higher the interest rate, the more I'm willing to loan the firm and, therefore, you get an upward-sloping supply curve. Now, of course, in reality, people don't actually-- we don't sit in Haymarket, downtown Boston, and give money to firms. In reality, this transaction happens through capital markets. And essentially, there are three mechanisms by which implicitly I loan money to firms. The first is I could literally buy corporate debt. I could literally loan the money to firms. I could literally go and the firm could say, I, General Motors, am issuing a bond. This is through bond, issuing a bond. And the way that bond works is I promise that for every dollar you spend buying my bond, you'll get 1 plus I dollars back at the end-- or next year, say, depends on how long the bond is. So literally, you're loaning the money to the firm by buying-- you're buying their promise to pay you back. Now, a second way you can loan money to the firm is through investing in their equity. You can buy their stock. The way this works is GM says to you, buy a piece of me and you'll get paid back not some fixed interest rate, but you get paid back according to how well GM does. So with corporate debt, I get paid back something that's predetermined. When I buy stock or equity, I don't get back a predetermined amount. I get back some-- it depends on how well the company does. But it's the same basic idea. I'm giving the company some money today in return for my getting more money, I hope, tomorrow. That's the diversion of consumption from today to tomorrow. And the third thing I could do is I could put it in the bank. Now, how is that loaned to companies? Because the bank then loans it to companies. Why do banks say they'll pay you interest on your money? Why did banks going crazy-- I'll give you 1-- it used to be interesting. Now it's 1%, 2%. When I was a kid, I was like 10%, 12%. We'll give you lots of money. And we'll talk later about why it was so much higher when I was a kid. Why are banks so eager to do that? It's not out of the goodness of their heart. It's because when you give them dollars, they turn around and loan them. They add a bunch to the interest rate and loan them out to firms. So those dollars you're giving the banks and they're paying you 2% interest, they loan to firms at 6%. And that's why bankers are rich. So basically, the reason a bank exists is because it's a way-- corporate debt and equity markets are hard and complicated. It's much easier to put your money in a bank. You put your money in a bank. But when you put your money in a bank, you're essentially loaning it to companies. That's essentially what you're doing. So through these mechanisms, we have a capital market where essentially, by my putting money away and diverting from today's consumption, I'm loaning to a firm. They'll produce more, and they'll pay me back more in the future. Questions about that? OK, so let's talk about where the supply curve comes from. We know where the demand curve comes from. It just simply comes from the marginal revenue product of capital. Where does supply curve comes from? The supply curve comes from what we call intertemporal choice. As I said, economists like putting fancy names on things. That helps us get paid more money. It just means choosing over time, intertemporal choice. Intertemporal choice is essentially about how do you decide how much to save. What's going to determine that is going to be your decision of how much you value money today versus valuing money tomorrow. So for ease, let's imagine I'm considering two periods, this year versus next year. When I talk about periods, I'm talking about days and years and whatever. It's the basic logic. It's about now versus the future. Whether I say days or years, it doesn't really matter right now. The point is I'm just talking about today versus the future. So let's talk about this year versus next year. And let's imagine prices aren't going to change. I'll come back to prices next lecture. But let's imagine the price of goods aren't going to go up. There's no inflation in this economy, which is roughly true today. And let's suppose I'm going to take next year off to care for my children. Lord knows why I'd want to do that when the youngest one's 19, but imagine they still need my care. So let's say I'll take next-- this example gets dated. Let's say I take next year off to care for my children. And let's say my income is $80,000 a year. Now, here is my-- but I'm going to take next year off unpaid. So I'm going to work this year for 80k. Next year I'm going to take off unpaid. So I have a couple of choices. I could work this year, earn my 80k, spend my 80k, and have nothing next year to live on. I could work this year and eat nothing and save all of the 80k to live on, or some combination in between. And we could illustrate-- but the key difference is every dollar that I don't consume this year that I save to consume next year earns interest. And that's where the trade-off comes. So let's look at figure 16-6. This is a familiar-looking optimization diagram. Now my optimization is not over pizza versus cookies, but my optimization is over consumption this period versus consumption next period. It's a bit mind-blowing. We're a little science-fictiony here, right? We're now not talking about choosing between two goods, like leisure and consumption or cookies and pizza. Now I'm talking about two time periods, consumption today versus consumption tomorrow. But that's the key thing about the tools we learn with consumer choice. Those tools are incredibly powerful. You just need to shove your problem into that framework. And we're going to shove our problem into this framework. The problem we're facing is how do I decide how much to save. Well, savings is a bad just like labor's a bad. What do we do when we have a bad to model? We don't model the bad. We model the complementary good. So our choice is, how much do I consume today? My choice is, how much do I consume today and how much am I going to save? Well, saving is a bad, but the other way to think about it is, how much am I going to consume today versus how much am I going to consume tomorrow? Then that's two goods and I can model them against each other. And that's what I do in figure 16-6. I model consumption today versus consumption next year. So here's my choices. As I said, if I consume everything today, I'm at the x-intercept at 80,000. I have 80,000 to consume today, nothing next year. If I consume everything next year, what do I get? Well, let's say the interest rate is 10%. What that means is then I'll have $88,000 next year. Why will I have more next year? Because by saving, I earn interest. By diverting my consumption to the future, I earn interest. At 10%, that means I would have $88,000 next year. So my budget constraint is the line with the slope minus 1 plus I. My budget constraint is the line with the slope minus 1 plus I. In other words, the price of consumption today in terms of consumption tomorrow is minus 1 plus I. OK, let me think about it. Let me say that again. It's really confusing. The price of consuming today instead of consuming tomorrow, assuming no inflation-- so prices are the same in the market-- is minus 1 plus I. Think about that. I find it useful to think back to the labor case for parallel. In the labor case, what did we say was the price of leisure? What was the price of leisure? Someone raise their hand and tell me. In the labor-- yeah? AUDIENCE: The wages. JONATHAN GRUBER: The wages. Why? AUDIENCE: Just because that's the opportunity cost of not-- JONATHAN GRUBER: Right. So by that same logic, can tell me why is the price of consuming today 1 plus I? AUDIENCE: Because if you choose to save, then we're effectively richer. JONATHAN GRUBER: Exactly. The opportunity cost-- remember, we are an annoying discipline with a dismal science. We're telling you, hey, enjoy that cookie, but by the way, if you weren't eating that cookie, you could have 1 plus I cookies tomorrow. So just like we nag you for sitting around watching TV, we nag you for eating today by saying, hey, the more you consume today, the less you can have tomorrow. And in fact, that trade-off is that for every cookie you consume today, you forgo 1 plus I cookies tomorrow. So that's the budget constraint. The slope is the opportunity cost of consuming today in terms of tomorrow's consumption or next year's consumption, which is 1 plus I. That's the slope of the budget constraint, is the opportunity cost. And then, then we say, OK, well, that's the opportunity cost. That's the budget constraint. Well, how do I decide? Well, then we know how to make these decisions, which is go to utility function. You can write down the utility function, which is a function of C1 and C2. Now, what is C? C is all my pizza and cookies, but we're aggregating it up. Just like our utility function last time was a function of leisure and consumption-- we said consumption was the bundle of goods you eat and leisure is this thing. Now we're saying, OK, our utility function now is a function of this trade-off. Now, you might say, wait a second. How can both those be utility functions? And the answer is you have some meta-utility function that includes consumption today, tomorrow, leisure, pizza, cookies, et cetera. But we can think about this in sequential steps. First, we decide how we're going to split our income. Then we can decide what to spend it on each period. Then you can do a separate consumer maximization decision. But our first question is simply how am I going to split my income. Well, that's going to be a function of my taste for consumption in this period versus next period and the price the bank will pay me for delaying consumption till next period. Now, what happens? Questions about that? Now, what happens in the scenario when the interest rate goes up? What do you think happens if the interest rate goes up? Yeah? AUDIENCE: There's [INAUDIBLE]. JONATHAN GRUBER: Right. So what do you think you should-- what do you think will happen to your consumption pattern? Yeah? AUDIENCE: You should spend less today. JONATHAN GRUBER: Spend less today and save more because it's rewarded. And why is that not necessarily true? Yeah? AUDIENCE: Because you might only need a certain amount of money to live. So you don't have to save as much today because you'll make-- JONATHAN GRUBER: Because of what two effects? Income and substitution effects. You gave exactly the intuition that the substitution effect gives you. The substitution effect is exactly right. If the interest rate goes up, that's like the price of consumption today going up. And if the price of something goes up, the substitution effect says you do less of it. But if interest rate goes up, you're richer. And if you're rich, you do more of everything, including consuming today. The income effect goes the other way. It's like labor. Once again, income and substitution effects is why we bothered telling you so. Because income and substitution effects, in these cases, go against each other. Let's look at figure 16-7, OK? In figure 16-7, we start at point A. Now imagine the interest rate doubles to 20%. Now imagine the interest rate doubles. As you said, that pivots the budget constraint upwards. You could still consume only $80,000 this year, but now for every dollar you save, you get $1.20 next year. That has two effects on your decision. The substitution effect, we get by drawing an imaginary budget constraint-- that's the dash line-- tangent to the original indifference curve but at the new slope. By definition, that means you consume less today. You consume less today by definition. If the price of something goes up, the substitution effect always says you do less of it. You consume less today, which means you'll save more. Remember, savings is just income minus consumption in period one. So just as labor was 24 minus leisure-- and so if we just solve for leisure, we could get labor. Savings is just income minus consumption in period one. So if we solve for consumption in period one, we get savings. People see that? So basically, the point here is the substitution effect says, well, gee, the price of consumption in period one just went up. It's more costly in terms of future consumption. I'm going to do less, but then my savings is going to go up. Substitution effect says you save more. But the income effect says, wait a second. You're now richer. Every dollar of your savings you are doing now yields twice as much in interest. If you're richer, you'll consume more of everything, including period one consumption. So the income effect takes you back the other way. Now, whether the income effect dominates are not, we don't know. In this case, it doesn't dominate. In this case, you still, on net, end up consuming less in period one and saving more. But we don't know what's going to dominate. And in fact, the evidence here is incredibly weak. I won't spend a long time on the evidence because it's not nearly as interesting and strong as labor supply. The evidence is incredibly weak even about the sign. And let's come to the intuition that was given for why. Well, think about how people make savings decisions. Lots of people have savings goals. I want to have x by the time I retire. Typical way if you ask people about their savings-- if you ask them, they typically say I want to make sure I have x in the bank in case I'm in an accident. I want to make sure I have y by the time I retire. Well, in those models, if the interest rate goes up, savings rates go down. Because after all, to hit a target with a higher interest rate, I can save less. So it's actually not that surprising that you'd have a higher interest rate leading to less savings. It's kind of intuitive, actually. If people have savings targets, a higher interest rate would lead to less savings because they can get to their target more easily. So actually, we don't even know which way this goes. It's, I think, one of the great unsolved mysteries in economics empirically, is, once again, we typically assume-- and with a gun to my head, I would say it's probably true that higher interest rates leads to more savings. But the evidence on which that rests is pretty weak. And the key point for you is to understand it's uncertain and it depends on whether income and substitution effects dominate. Questions about that? OK. So now let's step back and put it all together and think about you making your decision about life. You can think about your decisions about your life in three steps. Step one is you decide how hard to work. Step one is you decide, how much money do I want to make? Well, that's about maximizing utility over consumption and leisure. Step two is, having decided how much you're going to make-- and that yields your labor. Step two is, deciding how much you're going to make, you decide, well, how do I want to spread that over time? How much do I want to consume today versus tomorrow? Well, that's about intertemporal choice. That's about deciding on C1 versus C2, and that's going to yield your savings. Step three is, now that I know how much I'm going to consume each period, now I want to maximize utility across all my goods I might want to consume-- x2, across all the goods I want to consume. That was our original cookies and pizza example. So you could think of it as a hierarchical set of consumer optimization problems that you're going to solve. Now, you might say, well, gee, Jon, that's sort of confusing because, in fact, the interest rate and how much am I saving could determine how hard I work, right? Let's say the interest rate goes way up and I have a savings target. I have to work less hard to hit that savings target. And I'd say to you, good for you. Take more advanced economics. More advanced economics, we recognize this is one integrated whole and we allow these systems to affect each other. But for here, just think of them as separatable steps, independent steps. But in practice, I hope you can see the steps will be integrated and they'll affect each other. Think of it. If the price of a good you really want to buy goes up a lot, not only will you buy less of that good; you might save more to buy it and work harder. So you can imagine how these things are integrated. But for now, we'll keep them separable, OK? Questions about that? OK. Next time, we're to come back and talk about all the interesting stuff in capital markets and how we make decisions about how much to save and things like that.

MIT_1401_Principles_of_Microeconomics_Fall_2018

10_Welfare_Economics.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: So let's continue our discussion of welfare economics. Just to review where we are, the first set of lectures in the course were about positive economics-- about understanding where supply and demand curves come from and what they mean. Now, last lecture, we turned from positive to normative economics and actually making judgments about whether things are good or bad. And we introduced the concept of welfare, economic welfare or well-being. And we talked about consumer surplus, which is a measure of how well-off consumers are made by a given exchange of goods and services, and producer surplus-- how well-off produces are made. We're going to start our lecture by proving what's modestly called the first fundamental theorem of welfare economics, which is that competition maximizes welfare. So this is basically taking our positive economics and meeting our normative economics. That is, we are going to talk about how the model we've derived so far, which is the equilibrium under perfect competition, happens to also be the outcome that delivers the maximum well-being to society. So let's step back. Well-being we called welfare. How do we define the well-being of society? Well, we're going to start with a simple definition, which is we're going to at least say that social welfare-- social welfare, the total welfare of society-- is simply consumer surplus plus producer surplus. That is, we're not going to put any different weights. We're not going to say we like somebody better than another. We're just going to say, look, surplus is produced by a transaction. And we just care about the total surplus that's produced. And later, we can come to help you about consumers versus producers, and we will. But right now, let's just say we care about the total surplus that's produced for society. How much benefit is produced by this transaction? Well, the benefits produced by this transaction is going to be the surplus generated for consumers by that transaction and the surplus generated for producers by that transaction. So our total measure of social well-being is going to be the sum of consumer and producer surplus. And we are going to prove-- as I said, you don't need to know this. But in [INAUDIBLE] it's called the first fundamental theorem of welfare economics, which is that under the assumptions we've made-- which are many-- under the assumptions we've made, the competitive equilibrium where supply equals demand is the point that maximizes social welfare. That is, the key insight is the point that the market naturally delivers. The equilibrium that's gained by the market naturally happens to be the point that also makes society as well off as possible-- a very profound result. That is, the positive conclusion, which is that supply and demand will meet at a certain equilibrium, delivers a normative conclusion, which is that equilibrium is the point which maximizes social welfare. Now, the best way to see this is just graphically. So let's go to Figure 10-1. Figure 10-1 has supply and demand curve. Once again, whether these are curved or linear, it's still the basic idea of supply and demand curve. So curve, supply and demand curve, are things which are more constant elasticity type curves. But that doesn't really affect the intuition. We have here the triangles of consumer surplus and producer surplus. So consumer surplus-- give me the letter. Somebody raise their hand and tell me which letters on this diagram correspond to consumer surplus, and why. Which areas denoted by which letters correspond to consumer surplus, and why? Yeah? AUDIENCE: R and v. JONATHAN GRUBER: R and v. And why is that? AUDIENCE: Because the price is-- it really [INAUDIBLE] here in between the supply and demand [INAUDIBLE]. JONATHAN GRUBER: Exactly. Everything below the demand curve and above the price consumer surplus. So, r plus v. So therefore, what's producer surplus? Same person. AUDIENCE: S, t, and u. JONATHAN GRUBER: S plus t plus u is producer surplus. So, consumer surplus-- r plus v. Producer surplus is s plus t plus u. You can see, those of you who are graphically oriented, can immediately see the sum of those two triangles will be maximized at the intersection of the curves and nowhere else. So for example, let's think about the case where I say, well, look, it's a shame the price is that high. We ought to make the price lower. Let's set a new price. So let's have the government mandate a new price at P2. Suppose the government intervenes and says, we're going to set a price ceiling. We're going to say no one can charge more than P2 for their product. And won't that be a good thing because the consumers will be better off? They'll pay lower prices. Well, what does that do? What does that do to consumer surplus? Well, consumer surplus used to be r plus v. It instead becomes r plus s, which is bigger. Consumer surplus rises. You lose the triangle v, and you gain the rectangle s. But the key point is at the price P2, yeah, that becomes the new consumer surplus. The new producer surplus is what? At that price P2, what's the new producer surplus? Someone raise their hand and tell me. Yeah? It's t. It drops to t. Producers just get below the price above the supply curve. So what has happened to total social welfare? It has fallen by the amount v plus u. Total social welfare has fallen by v plus u. So in some sense, two things have happened here. We've set this price. The first is a transfer. We have transferred the rectangle s from producers to consumers. s used to be part of producer surplus. We're now giving it to consumers. So first thing we have is a transfer. That was probably the idea of this policy-- make consumers better off. So we transferred what used to be producer surplus to consumers. That's the rectangle s. That's the first thing that's happened. So thing one that's happened is a transfer of s. But the second thing that's happened is we have created what we call a deadweight loss of u plus v-- a deadweight loss of u plus v. DWL-- Deadweight Loss. What is a deadweight loss? That is the net reduction in welfare from trades that are not made. The deadweight loss-- this is a key concept we'll come back to, and I'll expect you know it in your sleep. It's that deadweight loss is the net reduction in social welfare from trades that are not made. The intuition here is that every trade that makes at least one party better off without making the other party worse off is a good trade to do. Under the assumption we've made so far, if you ever trade that increased consumer surplus or producer surplus or both, that's a good thing to have. Right? If my daughter has any song she wants to buy by Kendrick Lamar that she values at more than $1.00-- she gets them for $1.00-- anything which stops her from buying those songs is bad. She's losing surplus. So basically, the key point is that deadweight loss is an inefficiency. We talked about how competition leads to maximally efficient production through cost minimization. Competition also leads to the maximum welfare outcome because that point is the point which makes society best off, defined as the sum of consumer surplus plus producer surplus. But once again, I cannot highlight enough the depth of this insight. That this point which before today, you knew as the outcome. I showed you in the first lecture this is what happens when you have supply and demand. You get to this equilibrium that happens to be the very best place to be. And that's why we call it the first fundamental theorem. It's very important. Yeah? AUDIENCE: But the idea then of since we're caring about social worth, even though the consumers are better off, there is not as many trades because the consumers don't get as much out of it? JONATHAN GRUBER: I want to come to that. Let's talk about that. Let's go to another example to talk about that. Let's talk about interventions. Let's do it. Let's do an example. That was just to teach you the basic idea, but let's go on to a more explicit example of a government intervention. Actually, I'll do it here. Let me do an explicit example of a government intervention. And that will address the question which was just asked because I skipped over a key point, here. I'm always not sure the right order to teach these things. So let's take example of a market. Let's consider the market for gas. So go to Figure 10-2. This is a market we talked about before, the market for gas. Imagine the market for gas is initially in equilibrium with supply curve S1 and a demand curve D. And it's initially in equilibrium at point little e1, with Q1 gallons of gas being sold at a price P1. That's initial equilibrium. Now, imagine that there is an oil crisis because, for example, the oil company decided it would be good idea to drill eight miles underground horizontally. And it busts, and there are spills everywhere-- something like that, some random example like that. And there's a supply crisis. What happens now is suddenly, it gets more expensive to produce gas. So the supply curve shifts upwards-- we talked about this last time-- leading us to a new equilibrium at point E2. And you remember, we talked about how the equilibrium works. Initially, if we think about it in steps, initially you've created an excess demand because gas companies no longer want to supply Q1 gallons at a price P1 from their new supply curve. So you shift along the demand curve to the new point E2, which is a new equilibrium. And all is well and good. Prices go up. That's what happened after Deep Water and things like that. Now, imagine the government doesn't like that. Imagine the government says, well, we don't like that. We don't like the fact consumers have to pay more for gas. Consumers vote us out of office when they have to pay more for gas. So we are going to solve this by imposing a price ceiling. We are going to announce the price of gas must remain at its old level P1. So let's go to Figure 10-3. Figure 10-3 shows what happens when the government imposes that price ceiling. Well, the first question is if the government imposes a price ceiling of P1, how much actually gets sold in the market? This comes to the question that was just asked. Well, this is sort of a new thing we've looked at, which is we have the situation where there's excess demand. At the price P1, consumers still want Q little d. But suppliers are only willing to supply Qs, Q little s. See? Consumers still are working off the same demand curve. If the government says we still want the price to be P1, they're like, great. We still want as much gas as we had before. Supplies are like, no way. We're not going to supply it. If you're going to keep the price at that level, we're going to produce less gas because we have a rising marginal cost. So if you're going to force out that same price, we're going to work our way back down the supply curve and produce less gas. So suddenly, you have a situation of excess demand that doesn't get resolved. Remember, last time, we said excess demand got resolved by moving up the demand curve. Well, you can't, here. You can't resolve that because the price is forced to be at P1. Now, what determines what actually gets sold when there's a price restriction? Here's the way I like to think of it. I like to think of the actual quantity gets set by the constrained party. So in this case, a price ceiling means that suppliers want or are asked to supply more than they're willing to, so they just say no. So you actually get the ultimate quantity in the market is Qs. It doesn't matter that demanders want Qd. They can't buy stuff that's not produced. Likewise, we'll do examples later with a price floor, where consumers want less than suppliers want to provide. Then, it's consumers that decide what gets sold. So basically, whoever wants less gets to decide because you can't force the producers to produce more. We have a private oil industry, gas industry. So you end up with Qs units of gas sold at a price P1. So what the price ceiling does is move you from E1 to E3. You would have moved to E2 without the government intervention. Instead, it moves you to E3. And as a result, you end up with consumers-- consumer surplus-- being A plus C, producer surplus being E, and relative to an unconstrained world, not relative to the world before, but relative to without government intervention, you have a deadweight loss of B plus D. Yeah? AUDIENCE: Why can't [INAUDIBLE] by producing stuff yourself? JONATHAN GRUBER: Well, that's a very deep question. As I said, that we don't have a nationalized gas industry. We just have a private gas industry. There's a separate issue of-- a larger issue about whether private or public sector should be producing things. And that's beyond the scope of what we're discussing here. But for now, assume the government just has a regulatory role, not a gas production role. The government, by the way, does have a little of a gas production role because the government actually has something called the Strategic Oil Reserve where they have millions of barrels they can actually release and release onto the market at certain times. The government does have a way to try to deal with this. But for now, let's assume that they're not going to use the Strategic Reserve, just regulate price. So they regulate price. And what they've done is they've created a deadweight loss. So basically, if we think about it, what are the costs and benefits of government intervention? The costs of government intervention are twofold. What are the costs of this price ceiling? There's two costs. Cost one is you've created an inefficiency. You've just created this deadweight loss because basically, if you didn't restrict things, there are people who would have bought gas to the right of Q sub S and to the left of E2. Those units between Q sub S and E2, where E2 intersects the x-axis, those are units where the consumer surplus plus the producer surplus is positive. Right? You look at the unit right to the right of Q sub S. That's a unit that consumers would have happily bought at the new higher price and producers would have happily sold at the new higher price, but the government isn't letting it happen. So that's a deadweight loss. So that's an inefficiency. So that's the first cost-- the cost to society of trades that don't get made. And we call this an efficiency loss because there are efficient trades. Things are efficient if they make the whole-- the joint surplus is positive, if on net, people are better off. They're efficient trades which both sides were happy to make, and they can't make. So we call this an efficiency loss. But that's not the only cost to this policy. What's the other cost? This is a hard question. Yeah? AUDIENCE: Do we have to force the price to not be [INAUDIBLE] what it'd normally be? JONATHAN GRUBER: Yeah, so there's enforcement. You have to go around and send regulators around to gas stations, make sure they're not charging the wrong price. That's true. I sort of say that's small. Let's think more of a theoretical-- not theoretical, but what's the other big source of-- yeah? AUDIENCE: Isn't it the loss to producers? There's no new producers wanting to innovate and stuff like that? JONATHAN GRUBER: There's sort of a dynamic. But once again, the producers think this may be a short-run thing. And once the Deepwater Horizon gets fixed, prices will be back down or whatever. But what's the other miracle of the market that we lose here? We talked about this the very first lecture. Yeah? AUDIENCE: It felt like, related to [INAUDIBLE] entering and exiting? JONATHAN GRUBER: No, there's entry and exit. But once again, let's rule out [INAUDIBLE] because it's just a short-run deal. Yeah? AUDIENCE: Is it like how do you ensure that the people who value it the most-- JONATHAN GRUBER: Yes. There is allocative inefficiency. Remember, one of the things we talked about in the very first lecture-- I think we did. Maybe not. Anyway, one of the most important benefits of the competitive equilibrium-- it doesn't just deliver the right quantity, it delivers it to the people who want it the most. So let's go back to figure 10-2. It's easy to see there. If you think about who gets gas at the initial equilibrium E1. And actually, no. But let's go 10-3. That's fine. So let's say we hadn't interfered, and we'd allowed the price to go to E2. Well, fewer people would have bought gas, right? The quantity would have fallen from Qd-- would have fallen all the way from E1 to E2. But the people who dropped out would have been who? The people who valued gas the least, the people who got the lowest surplus from it. However, now, suddenly, you only have Qs units of gas, and you have Qd people who want them. Well now, who decides who gets them? Before, the market did the magic. The market made sure the people who wanted the units got them. Now, all of a sudden, something else has to decide. So how do you resolve this? Well, we actually an answer to this. We had a gas crisis in the 1970s. And the government imposed a price ceiling. And how did it get resolved? How did we decide then who got the gas? Does anyone know? Raise your hand and tell me if you know. Yeah? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: People waited in line. People basically sat in their cars and waited in line. Now, so essentially, if you think about it, goods are going to get allocated somehow. If the market doesn't allocate them, there'll be another less efficient allocation mechanism. And basically, why is it inefficient that people waited in line for gas? There's actually two reasons. One is sort of cute. But what's the main reason it's inefficient for people to be waiting in line for gas? Yeah? AUDIENCE: They're wasting [INAUDIBLE].. JONATHAN GRUBER: Yes, the opportunity cost. Point little 1, you have the opportunity cost. You have the fact that that time I spent waiting in line, I could have been working or having fun, but certainly something I would've enjoyed more than waiting in line for gas. I think we'd all agree that there's something we'd rather be doing with our time than waiting in line for gas. Unless you really like the music in your car stereo, and you can't listen anywhere else. I don't know what the story would be. But I can't imagine that many of us would prefer to wait in line for gas than do something else. So there's the opportunity cost. That's an inefficiency. Society is losing out because you are not using your time in the most productive way-- in the way that maximizes your welfare. What else? What's the other small, other cost? Yeah? AUDIENCE: Isn't waiting in line [INAUDIBLE]?? JONATHAN GRUBER: Yeah, people used gas waiting in line. So there's literally the technical inefficiency. And remember, this is back when cars got like 8 miles to the gallon. So you'd get to the front line, get gas, and have to go back to the back of the line again because you used so much gas, plus not to mention the pollution and all that other stuff. So basically, this is the inefficiency from a non-market allocation mechanism, which is that people could be doing more productive things and not wasting gas waiting in line. Yeah? AUDIENCE: Is the excess of demand Qs minus Qd, or Qs minus E2? JONATHAN GRUBER: No, Qd-- because we fixed the price of P1, so it's Qd minus-- since we fixed the price of P1, people want-- what's the demand at that price? Qd, but it's supplied at that price Qs. So that's the excess demand. So now, yeah? AUDIENCE: You mentioned how the market makes sure that people who want the gas the most can get it. But isn't the one thing it involves like however much they're willing to spend on it? So does this market assume that all people make the same amount of income? JONATHAN GRUBER: No, it doesn't. But it does assume that basically, the market-- and this comes to the tradeoff. What's the benefit of this policy? Which you've raised with your question, which is equity. I've defined consumer surplus as simply being what the market delivers. So basically, rich guys have more consumer surplus than poor guys. That might not seem fair to people. As a result, the benefit is equity, which is that if you-- that everyone gets gas at a lower price. Rather than the price rising, the people who drop out may actually not be the people who don't need to drive. They're people who can't afford to drive, so they drop out. Remember, where does the main curve come from? It comes from utility maximization. That's a constrained maximization. So the people who drop out may be people we want. Maybe they're people who have to lose their jobs because they can't drive to their jobs anymore. That's unfortunate. So basically, the benefit of this is equity. And that raises something we're going to come back to over and over again in this class, which is what we call the equity-efficiency tradeoff. The equity-efficiency tradeoff, which is there are many government policies which make society more equal, but along the way deliver deadweight loss. And I'll actually tell you later in the class how to optimize across that problem-- how to optimize across the tradeoff between equity and efficiency. But for right now, I just need you to understand that there's a tradeoff. Yeah? AUDIENCE: [INAUDIBLE] equity. Like [INAUDIBLE] let the people who [INAUDIBLE] because if there's no price ceiling, then the people who are able to pay the most money will get it, right? JONATHAN GRUBER: Yeah, exactly. So the point is-- right, the people who-- was there more to your-- AUDIENCE: Yes. But if there's a price ceiling, then wouldn't it just [INAUDIBLE]? JONATHAN GRUBER: Exactly. At issue is a different kind of inequity. So if you're worried about-- so in some sense, the rich guy would say, well, that's not fair to me, Just because I've got a productive job I could be doing. And someone else has extra time. Why should they get the gas instead of me? So you're right. There's always an inequity. That's a great point. When I say "equity," I implicitly mean what we call-- ethics is a very important point. Whenever I say equity in this course, I implicitly mean what we call vertical equity. Vertical equity is rich versus poor. There are other kinds of equity you might care about like who has extra time versus who doesn't have extra time. And that's a good point. But for now, when I say equity, I'm only thinking about rich versus poor. When we're thinking about government policy and the equity-efficiency tradeoff, we're really thinking about vertical equity. Other questions about that? So let's do a couple more examples to drill this intuition home. Let's do a couple more examples because this is a very important point. Example A-- let's talk about ticket scalping. I love going to music concerts. I went to my 51st concert of the year last year-- last night, sorry. Went to my 51st concert of the year. I love going to music concerts, so I know a lot about the music business. And ticket scalping at concerts is a big deal. For those of you who don't know the term, this is the idea of buying tickets essentially on a secondary market. So let's use the example of Adele. So Adele, about two years ago, went on tour for the first time in four years to back up an incredibly successful album. And folks really wanted to see her life because she had a big fan base. She was quiet for a while. She made an album. People really wanted to see her live. She wanted to make sure her fans could afford to see her. So what she did is she said, I'm going to price many tickets for $40 and the highest tickets at $150, which is really, really cheap. So my daughter's going to see J. Cole tonight. And she's paying like $150 for a mediocre ticket, so $40 to $150 is really cheap. But she's saying, I want to deliver consumer surplus to my fans. I want to make sure my fans can really enjoy this and get surplus. But what happened? Well, what happened is that the tickets sold out instantly-- like, literally instantaneously. And the major purchasers were what we call scalpers, who are essentially professionals who buy tickets and then resell them on what we call the secondary market. You might have heard of StubHub or other sites like that, where you essentially go on and buy tickets to sold-out events. And the prices on StubHub were much higher. It was about $1,500 for good ticket-- about 10 times what she'd set the price at. So basically, who's getting the surplus? Not the fans, the scalpers-- the people who were quick enough to get online who had bots set up, so that the second it went online, they went online and got the tickets. They had thousands of bots set up, essentially. Somehow, they got around the "click this if you're human" box. I don't know how they do that. But essentially, I'm sure it's smart programmers. You guys can probably do that in an hour as an extra-credit project. They got around it, and they bought. And so essentially, Adele actually didn't create surplus for her fans. She created surplus for scalpers. Now, it didn't used to be this way. When I was a kid, we didn't have a secondary market. We waited on line. So when I saw my first concert in 1981-- can you guys believe that? In 1981, I saw The Cars. How many of you guys have heard of The Cars? Oh, god. Anyway, I saw The Cars, and I had to wait on line for hours to get tickets because that was the allocation mechanism. So let's think about whether life is better or worse. Because on the one hand, scalpers get money. On the other hand, people don't waste time waiting on line. So it really comes down to the same equity tradeoff we talked about before. Now. As a 16-year-old in 1981, I could not have afforded to pay, probably, the secondary market price. But as I was 16, I had nothing do with my time, so I was happy to spend hours waiting on line. But on the other hand, you could say someone is very productive, who is a big fan who is very productive and could be out inventing new products if they weren't standing on line. They might say that's really inefficient. I'm happy to pay $1,500 and spend my time inventing new things, rather than have to sit around waiting on line. So it's not clear which is the better or the worse system. It's hard to say. Yeah. AUDIENCE: [INAUDIBLE]. So like the demand for [INAUDIBLE] will be only changing prices and all-- JONATHAN GRUBER: Well, no, but here's the point-- that's a great point, which is that, in some sense, what the scalpers do is undo Adele's action and create a truly efficient market. So think of the scalpers as essentially undoing Adele's action, which distorted the market. So you have a market for Adele tickets which is at equilibrium at $1,500. Adele tried to set a price ceiling at $150, seemingly selfishly, saying, I'm going to give up my surplus for my fans. But unfortunately, it didn't work out that way. What happened was the market re-equilibrated at $1,500, but that extra surplus didn't go to Adele. The consumer surplus still remained above $1,500. The difference was that extra gap between 150 and 1500 didn't go to Adele. It went to the scalpers. So it's probably more efficient than waiting online, because waiting online has inefficiency, whereas instantaneous bidding is more efficient. But in some sense, the efficiency gain is being delivered not to consumers, as Adele wanted. It's being delivered to scalpers. So basically, now, there's another alternative. What could Adele have done instead? What's another way Adele could've approached this? She could have auctioned her tickets online. She could've said, look, I know that I can't manage to deliver $40 tickets to my fans. I just can't defeat the scalper system. But why should the scalpers have the money? I'm going to auction my tickets. And essentially, she could have set up an efficient market online, where people bid. And then you would have gotten the efficient outcome. And you wouldn't wait on line. You would have bid. Now, it would've been inequitable outcome, but basically, the same fans would have gotten the tickets as in the end got them from scalpers. But instead of paying $1,500 to the scalpers, they'd pay $1,500 to Adele. And probably that's a better outcome. I mean, Adele doesn't need the money, but at the end of the day, if Adele's generating that amount of goodwill from her fans, it seems like she should get the money, not a bunch of scalpers. So I guess probably I'd rather have her have it than the scalpers. Now, Ticketmasters tried to set up auction systems, recognizing this problem, and they're not really taking off. And it sort of speaks to the fact that people don't really like to think about economics. That basically, people were like, yeah, but if you auction, that's ripping off the fans. That's no fair. And Ticketmasters would say, well, you're getting ripped off anyway, just by scalpers. And they said, no, I can still get online first and get my ticket. And people just didn't like it, thought it was unfair, even though it's almost certainly a better outcome for society that Adele should have the money rather than the scalpers. Which speaks to the fact that morals matter in markets. We like the way these markets abstract amoral concepts, but morals do matter. And that basically, it sort of matters you sometimes can't do the right thing because it might not be the thing that makes consumers willing to participate. So that's one example, scalping. Another example-- food banks. Food banks are organizations which provide-- oh, yeah, go ahead. AUDIENCE: What if you force everyone, I mean, when you buy a ticket you have to put your name on the ticket, so that only you-- JONATHAN GRUBER: I've thought about that. So if I could do anything I wanted in life, I'd be a rock star. And I thought if I was a rock star, it would be cool. I'd have a concert for my mega fans, where they'd have to prove who they were, and I'd have a bracelet and stuff. It just seems like it's too hard. It seems the technology is just too hard. So actually, there's a really cool example. So I was on the phone this morning with rock and roll promoter who's doing a super cool thing. So as the election approaches, I will nag you guys to vote. Voter turnout among the young is an enormous problem the US. So she is running a series of concerts around the country where if you show a picture of yourself outside a polling place, you get into the concert for free, to try to promote voting among young people. It's super cool. It's called "I voted." You can look it up on the web. It's kind of a cool thing. There's a couple of concerts here in Boston. They're all over the country. So you could think about things like that. The question is ultimately, do people Photoshop their picture in front of the polling place? It's all just a question of enforcement. Let's talk about food banks. So basically, these organizations provide free food to the poor. And the biggest one is called "Feeding America." Feeding America has food banks all over the nation, and they provide free food to the poor. Now, their goal, then, is they have to figure out where to send the food to get it to people who need it the most. Now, a market does this naturally. Basically, if people want more turkey in location A, then all of a sudden, the price for turkey goes up. All of a sudden, the store runs out of turkey. It says, wow, I can charge more for turkey. It raises the price, and it equilibrates the market. So a market solves for sending the right food to where people want it. But the problem with Feeding America is that they didn't have that market. So they had to decide where to send stuff. And they'd screw up. They'd send potatoes to Idaho, where they're drowning in potatoes, stuff like that. So it was very hard for them to figure out where to send the food exactly where people wanted it the most, because they didn't have the market mechanism. They wanted to give it away for free. That defeats the purpose if they charge for it. But they didn't really know, they didn't have the market to tell them where folks wanted which foods. In the real market, if you want real food, you bit the price up. In their market, you couldn't. So Feeding America came up with a really clever solution. They made a virtual market. They said to each food bank, we are going to give you a fake budget of x $100,000, and you bid for which foods you want based on what the people in your area want. And we'll then allocate it according to those bids. So they got the market mechanism working without the food banks having to actually give any money. So in that way, they massively reallocated food. They suddenly said, hey, the one from Idaho is bidding really high for turkey and not at all for potatoes. Maybe we should send them more turkey and the potatoes elsewhere. So they essentially got market signals from a non-financial transaction. It was a super cool idea. And it was a huge benefit. They were able to effectively allocate about 50 million pounds of food through this mechanism, making sure that food got to the folks that needed it. So there's an example how you can use a market mechanism without actually-- while not violating equity. The food was always free, but by setting up these virtual prices, they managed to get the food delivered to where people wanted it the most. They let the market send its allocative signals. They let the market be allocatively efficient, send signals of where people wanted the food, without actually violating equity. Questions about that? Now, let's go to the hardest, but my favorite example, which is taxi medallions. This is a hard example, but it's a really cool one, and it ends with a great story. Taxicabs-- now, cast your mind back pre Uber. Go back 10 years, or even-- yeah, about 10 years. Taxicabs were the only way you could get around town if you didn't have-- if you wanted to get from point A to point B in a car, you didn't own one, you took a taxi. And this was a great example of an economist's perfectly competitive market, in theory. It's identical product-- you want to go from point A to point B. There could be lots of them riding around the streets. You can price compare, because cabs are coming by all the time. It should have been a very effective, perfectly efficient, perfectly competitive market. But it wasn't, because every city limited the amount of taxicabs that were allowed in their city. Every city had a system where to be a taxicab, you had to have what was called a "medallion." It wasn't really a medallion. It was originally a medallion. It's just a piece of paper, actually. And they regulated the number of taxicabs allowed in the city. They said, we're only going allow x many taxicab drivers in the city. And every city did this. Now, we're going to do both a positive and normative analysis of this policy. Let's start with a positive analysis. What did this policy do? Now I'm going to go to figure 10-4. Figure 10-4 will be one of the most complicated figures we do in this class. I'm going to go through it as slowly as I can, but please stop me if it's not clear. This is one of these figures we'll go back and forth between the two sides. So on the right-hand side is the market. On the left-hand side is the cab firm. We're going to start by assuming all cabs are identical. We assume all cabs are identical, so one representative firm tells us about every cab firm. Now, we start with the market. We have an initial demand, which is d. The line d, the blue line, is the demand for taxicabs. An initial supply curve s1, we're going to assume that essentially, in a perfectly competitive market, you have a flat, long run supply curve. You have perfectly elastic supply. Anybody can just grab a cab and start driving. Once again, in a cab market without medallions, anyone could throw a taxi sign on the car and start driving around, picking people up. So it's effectively perfectly entry and exit, so it's a perfectly competitive long run market, therefore, flat supply at s1. So the initial equilibrium is at point big E1. We have Q1 rides, big Q1 rides per month. And that amounts to-- so you have q. That's the equilibrium. Now, how many firms are there? Well, we know many firms there are, because we say at that price, p1, we go to the left. We know that firms will produce where marginal cost equals the average cost at the minimum of long run average cost. We proved that a couple lectures ago. Therefore, if the price is E1, we know each efficient firm will produce little q1. If each firm's going to produce little q1, and the total amount of rides is going to be big Q1, then that implies little n1 firms. So let me go through it again. Supply equals demand at big Q1. That gives you a price P1. Now we shift to the left. At that price P1, we know that each firm will choose to produce little q1 rides, because that's where price equals marginal cost equals average cost. Now we go back to the right. We know if each firm is providing little q1 rides, and you need big Q1, then there must be N1 firms. Questions about that? That's the initial equilibrium. And at that point, there are long run zero profits. Consumer surplus is a plus b plus c. c, by the way, is the gray area on either side of the dashed line. a plus b plus c, we mark it twice because it's confusing, the dashed line there. So the blue area, the green area, and the gray area are the consumer surplus. And what's the producer surplus? What's the producer surplus? Raise your hand and tell me. Yeah. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Zero because? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Yeah, there's no-- in long run equilibrium, there's no profits. In long run equilibrium, price-- remember, profits are price minus average cost, but price equals average cost. So there's no profit, so it's all consumer surplus-- all well and good. Now let's say the government comes in and says, we're going to have a medallion system. So let's say we're going to have a system where only little n2 cabs are allowed in the market. Little n2 are the only number of cabs allowed in the market. So what is the new market supply curve? Start on the right. What's the new market supply curve? Well, up to little n2 times q1, nothing's changed. Each cab company used to provide little q1 rides at a price P1. So up to the point little n2 little q1, we're on the same supply curve we used to be on. But then things change. Why do they change? Because now, let's say riders want more than little n2 q1. Well, you can't deal with that by entry. You have to deal with that by the existing cab drivers working more. But if they work more, supply curve is upward sloping, because now marginal costs can be rising. Before, the reason the supply curve was flat when you went to the right of little n2 little q1, as you moved to the right, you just got more entry. And everyone still produces their efficient level of little q1. Now, when you can't allow more entry, firms have to produce more. And suddenly, they're not at the cost minimizing point. They're riding up their marginal cost curve. So the supply curve becomes flat to that point, and then becomes upward sloping. And that's s2. s2 is the red line that is flat to the left of little n2 little q1, and then becomes upward sloping. And the new equilibrium is at point E2. With n2 firms-- by law, you can only have n2 firms-- producing little q2 rides each. And your new equilibrium is big Q2. And that new equilibrium price is P2. The new equilibrium price is P2. Now P2, if you go to the old-- P2 is the point, if you look at the price as P2, but forget the a c2 curve. Focus on the a c1 curve. The average cost function hasn't changed. It's still a c1. So if the price is P2 and the average cost curve is a c1, they're going to produce where price equals marginal cost at little e2. So let me back up. Let's go back to the curve on the right. The equilibrium is now big E2. That's at a price little p2. Now shift to the left. At a price of little p2, firms produce where price equals marginal cost. Price equals marginal cost at a quantity little q2. So firms produce little q2, because that's the point at which price equals marginal cost. They produce at little q2, if they're making little q2 units at a price p2, their earning profits of pi, the shaded area, because they are producing units at price above marginal cost. And they're producing little q2 units, so every unit they produce, they're earning profits on. So the taxi medallion workers or the taxi companies are now making money. They're making profits. Questions about that? Yeah. AUDIENCE: [INAUDIBLE] the idea that if you allowed for people, more people to go then, profits would be [INAUDIBLE]? JONATHAN GRUBER: Yeah, but they're not allowing them in. They're making profits. So imagine where we started at E2 and allowed free entry. Then you'd see, basically, more firms willing to drive the price down to P1. But we don't allow that, so it's profits. It's a barrier to entry, which creates profits, which breaks us from our long run flat supply curve. We talked about one of the reasons why that loop won't be flat. So now, let's move to the normative. Is this a good idea or not? Well, let's look at the welfare implications. What's happened? What's happened is consumers used to have a surplus of a plus b plus c. Now, their surplus is what? What's the new consumer surplus? Raise their hand and tell me. Yeah. AUDIENCE: A. JONATHAN GRUBER: Just a, because the line below the demand above price is just a. Producers used to have zero surplus. What's their new surplus? Behind you, yeah. AUDIENCE: B. JONATHAN GRUBER: B, because it's the area above the supply curve, below the price line. And c is what? The deadweight loss. Those are transactions no longer happening-- the deadweight loss. So basically, normatively, you can pose the problem as the following-- is it worth society losing the area c in order to transfer the area b from consumers to taxicab drivers? That's the way to think about this problem. Let me say it again. It's very important. Essentially what the government policy is doing is saying, I'm going to transfer b from consumers to producers, even though it's going to cost me c in deadweight loss. That's what the government's saying. That's the government's position. Now, why is the government wrong? Why, in fact, is that not the proper statement of what happens? It's very complicated. What does the government miss when it says, I've made the drivers better off? Sure, I've created deadweight loss, and I make consumers [INAUDIBLE],, but at least I made these drivers better off. They live terrible lives. There's articles in the paper all the time about suicides of taxicab drivers. It's terrible. What did they miss? Yeah. AUDIENCE: The cost of taxis goes up. JONATHAN GRUBER: They've missed the fact that the taxicab drivers have to buy the medallions, that the limited number of medallions aren't just given to taxicab drivers, they're bought. And the taxicab drivers-- what is a taxicab driver willing to pay for a medallion? They are willing to pay, in the limit, the total surplus they get from driving the cab minus $1 or minus some amount that they need to eat. So if suddenly, in a world with no restrictions, with no taxicab medallions, if you said we need a piece of paper to drive a taxi, but anyone can have it for free, what would it be worth? Zero. But in a world where you say if you have this piece of paper you can drive, and drivers earn profits pi, what will the taxicab medallions sell for? Pi minus some small amount. So actually, who wins? The taxicab medallion owners. And who are they? They are random folks who happened to get these things in 1920 when they were issued, and their descendants, and people who bought them. So this leads to my great story, the taxicab medallion king of Long Island was a guy who happened to have a bunch of these taxicab medallions. And as they went through the roof, they got worth a lot. In New York, New York had 12,000 permits, and that number did not change since they originally sold for $10 each in 1937. 1937, you could have bought [INAUDIBLE] for $10, and they issued 12,000 of them. They've never increased it. Right now, a taxicab medallion in New York is worth $400,000. The taxicab King of New York is a guy who lives in Long Island who made so much money that he actually, for his kid's bar mitzvah not only had rented out a hotel, whole basketball-themed bar mitzvah, but hired Nicki Minaj. This was paid for by area b. This was paid for-- so in fact, it's not the taxicab drivers that make out of this policy. It's the taxicab medallion King of New York. Now, here's what happens-- what happens, then, when Uber comes in? Who's the big loser? Not the taxicab drivers, the taxicab medallion owners. So everyone tells you Uber is a bad thing because these poor taxicab drivers are starving. Taxicab drivers were always starving. It was always a terrible life. The difference is pricing medallions is down like 50%. And I cry no tears for the Nicki Minaj-hiring class of America. So basically, when you think about people saying Uber is bad, it steals jobs from taxicab drivers, no, it doesn't. It steals money from taxicab medallion owners, and that is something we might be OK with. So feel fine taking your Uber. It's a great thing. So let me stop there. I'm going to come back. We'll come back next lecture. And we will start talking about monopoly, the market structure, not the game.

MIT_1401_Principles_of_Microeconomics_Fall_2018

21_Efficiency_and_Equity.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: Today we're going to move on to another topic I've kept hinting at all semester, but it's finally here. Which is to think more explicitly about equity, or fairness. So our discussions this semester have been almost solely couched in language of efficiency. We talk about maximizing social welfare, we talk about the total size of the triangle and the squares. But we don't actually talk about who gets what. We very much stayed away from equity concerns and focused just on efficiency concerns. The problem is, that doesn't lead us very far in life, because you can have many outcomes that are equally efficient, but have different equity consequences. The best example, of course, is perfect competition versus a perfectly priced discriminating monopolist. Remember, under perfect competition, you maximize welfare. But a perfectly priced discriminating monopolist also maximizes welfare. The difference is, in the latter case, the monopolist gets all the surplus, when the former case, it's shared between producers and consumers. So it's sort of weird to say we're indifferent between those two outcomes-- to one where Apple gets all our money, and one where Apple gets some of our money and we get some of our money. Seems strange to say we're indifferent between those outcomes. So now, in subsets, that's the easy case. So we're talking about equity efficiency. In some sense, the easy case is the case where there's two equally efficient outcomes, and they have different equity consequences. That's a rare case. The more common case is what we call the equity efficiency trade-off. Which is that by making distributions more equal, we are going to induce inefficiencies. That the act of making distribution more equal is going to introduce inefficiencies in the system. And that's where things get really interesting. OK? So in other words-- so the best way to think about this, I find most helpful, is due to a famous economist named Arthur Okun, and his thought example of the leaky bucket. Okun's thought example was the following-- imagine that the way the government distributed money from the rich the poor was literally they went to the rich, had the rich put money in a bucket, and they carried it and dumped out in front of the poor. Imagine that's the way distribution happened. Well, in that world, if I told you that every dollar a rich person put in the bucket got carried along and got handed to a poor person-- so Bill Gates' dollar became a homeless guy's dollar-- probably most of us would think that was OK. I think the vast majority of Americans would say, yeah, probably the homeless guy could use $1 more than Bill Gates could. But now, imagine that there was a leak in the bucket. Imagine Bill Gates put 100 pennies in, but along the way to the poor person it leaked out, and so we dumped it in front of the poor person, it was less than 100. Now, then the question is, how much leakage? Well, if the leakage was one penny out of 100, you probably wouldn't change your mind. Would it change your mind if it was 20 pennies out of 100? 50 pennies out of 100? What if it was 100 pennies out of 100? What if taking $1 from Bill Gates, by the time it got to the poor person, it was all gone? At what point would you say, you know what? I don't think that's a good idea anymore. And really, that's a great way of thinking about the equity efficiency trade-off-- how much efficiency are you willing to give up to redistribute from rich to poor? And the efficiency you give up is represented by the leakage in the bucket. Now, what we're going to do with this in this lecture and next Monday's lecture, is we're going to discuss this equity efficiency trade-off in four steps. The first step is going to be talking about valuation-- that is taking the difficult step we've already taken so far and asking, how does society feel about some people versus other people? So far, we've just thought of one generic person, and they're represented by total surplus. But in fact, we have a distribution of people, and how do economists think about taking money away from person A and giving to person B. That's a new topic for us. The second thing we're going to talk about is what do we know about the facts on inequality? What do we know about what's actually happened to the distribution of resources in the US at a point in time, and over time? The third thing is, we're going to talk about the sources of leakage. That is, why does the bucket leak in practice? Why do we typically think there is an equity efficiency trade-off? Why can't we just take the dollar from the rich guy and give it to the poor guy? What caused the leakage? And then finally, we're going to talk about some examples of transfer mechanisms. We're going to talk about what society does in practice to transfer from rich to poor and how it works, and what the ultimate leakage looks like. So that's going to be our goal in the next two lectures. It's going to be think about this equity efficiency trade-off. So to start that goal, we have to start with this first issue of choosing the social optimum. That is, how do we evaluate transfers from one party to another party? And so to rank outcomes, what we're going to do is use the same thing we always do when we want to think about trade-offs-- which is, we're going to do a constrained maximization exercise. When I want to think about your trade-off between cookies and pizza, I did a constrained optimization of your utility functions, subject to your budget constraint. When, now I want to think about your trade-off between me and Patricia, now instead, we're going to a different utility function. In fact, we're going to use what we call a social welfare function. A social welfare function basically society's utility function. How does society value different individuals? So loosely speaking, social welfare function is some function of utility of person 1, common utility of person 2, common, dot, dot, dot, comma utility of person 350 million, if it's the US. So it's some aggregation function. Just like we mathematically aggregate your taste for pizza and cookies. Now we're going to mathematically aggregate all society's utility to get a social welfare function. So for example, consider figure 21-1. Let's think of society as only two people, Homer and Ned-- let's imagine that's all of society, because once again, it's always these two by two examples are easy. What we've drawn here are what's called isowelfare curves. What these are, are basically society's indifference curves. Just like if these x and y-axis are pizza and cookies, I would draw a difference curve between pizza and cookies. Now drawing society as a difference curve between Homer and Ned. So in other words, what this says is that society's indifferent between Homer having U1 super H, and Ned having U1 super N, versus the Homer having U2 two super H and Ned having U2 super N. Those are combinations of resources across which society is indifferent. And much like any other indifference curve, further out is better. We'd all prefer both Homer and Ned to have more-- more is better. So that's easy-- the further out the isowelfare curve goes, the happier you are. And a long isowelfare curve are allocations of resource across individuals among which we're indifferent. Now, the question is, that's all well and easy to graph as a theoretical proposition. But in practice, what does a social welfare function look like? Utility functions, I just wrote down a usually function, and you took it as-- I wrote down square root, we worked with that. But there were some properties we wrote down that gave us a sense of what utility functions look like. Social welfare functions are much more open, because they don't come from introspection about your preferences. They come from introspection about society's preferences, which are much harder. So what we do here is we talk about some typical forms of social welfare functions that we use in economics. The most common form is what's called utilitarian social welfare function-- utilitarian social welfare function. This is due to the philosopher Jeremy Bentham-- if you ever visit University College London-- actually, if you did until about 15 years ago, you could see Jeremy Bentham's head, it was on display. He was a famous philosopher there. But apparently, students would take it out, use it for soccer. So they took it off display, you can't see it anymore. But he's a famous philosopher, and he came up with the idea of utilitarianism. And basically utilitarianism is, it simply says, the social welfare function is simply the linear aggregation of every individual's utility. So social welfare function is simply, U1 plus U2 plus U3 plus-- plus U 350 million. So the US social welfare function is literally, we just measure everyone's utility, add it up, and that's the social welfare function. And in subsets, it's a natural starting point. Right, you just say, look, it's just a linear-- it's like having a linear utility function. It's a linear utility function. Now, let's be clear-- what this says is that I don't care any more about anybody in society. So I don't care any more or less about Bill Gates and the homeless guy. I treat them-- I'm indifferent between them. But does this mean I wouldn't want to transfer money from Bill Gates the homeless guy? In fact, I still would want to transfer-- and why-- depending on the leakage. Why? AUDIENCE: Because of the condition of [INAUDIBLE].. JONATHAN GRUBER: Yeah, exactly. I care about the utility the same, but the next dollar's not worth anything to Bill Gates. It's worth a lot to the homeless guy. So a utilitarian social welfare function, which is a natural starting point-- I don't think it's particularly liberal, it's a natural starting point, you're just adding them up-- ends up with a very, in substance, redistributive conclusion, which is that you want to attribute from rich to poor. Indeed, what the optimum with utilitarian social welfare function is that you want to redistribute until marginal utilities are equal. So this social function calls for fairly radical redistribution. This says, you want to redistribute until utilities are equal. Marginal utilities are equal, I'm sorry. Yeah? AUDIENCE: So [INAUDIBLE] is constant [INAUDIBLE].. JONATHAN GRUBER: Well, once again, you're right, I mean, you just add people and subtract people. Wouldn't be a problem. But for now, let's just assume the copulation is because you've got one given society. OK, so if two people, it's just Ned and Homer, just add them up. All it says is, as Ned is more resources than Homer, we're going redistribute. Indeed, with this function, if we make the assumption that total social resources are fixed, that society has a fixed budget constraint that can't change, then what does this function imply would be the optimal distribution of income? So ignoring the fact that people might work less or more hard-- ignore that, imagine it's just the total amount of money a society has. If that's your social welfare function, what's the optimal distribution of income? Yeah? AUDIENCE: Are we assuming that makes everyone equally happy? JONATHAN GRUBER: Yes, everyone-- good point, fixed resources are identical utility functions. Great point. Yeah, exactly, great catch. If it's a fixed bundle of income, and everyone has identical utility functions, then we simply want everyone to have the same amount of money. Why? Because giving someone $1 would make them less happy than taking away from someone else would make them sad. It's all about diminishing margin of utility-- just like we talked about last time. So now, that might not be true, for example-- does anybody know who Scrooge McDuck is? Scrooge McDuck? Raise your hand if you know Scrooge McDuck. OK, not bad. Scrooge McDuck is this comic character from when I was a kid who used to like to dive and swim in his money. Now, he clearly had a higher margin of utility of the next dollar than I do. OK, so if Scrooge McDuck really likes money, we might want to let Scrooge McDuck have some extra money. But if utility functions are identical, and social resources are fixed, this is one equal distribution income. That's really radical. That's beyond what any country in the world does-- a perfect equal distribution of income. But it comes naturally out of this fairly plain vanilla social welfare function. Quite a striking finding, right? Now, but in fact, if we think of this as sort of our starting point-- Bentham was actually conservative, this is typically viewed as a conservative starting point, even though it has a very liberal conclusion. The more liberal extreme is what we call a Rawlsian social welfare function, due to philosopher James, I think, Rawls, who was at Harvard-- Philip John Rawls, I'm sorry. He said the goal of society is to maximize the well-being of its worst-off member. So Rawlsian social welfare function is the minimum of U1 comma U2 comma dot, dot, dot. In other words, all you care about is the worst-off person in society. OK, so Rawlsian social welfare function would say, all we care about is the worst-off person in society. Now, unless you think this is crazy, let's think about where Rawls came at this from. Rawls came at this from the concept that he called the veil of ignorance. Which he said, look, before you were born, you know nothing about what you're going to be. You could be born to rich or poor, healthy, sick, you have no idea. You're just a little embryo. From that perspective, he said, what you would want is to make sure that you're going to be OK. And so from that perspective, society should want to minimize the well-being of the worst-off member. That was his rationalization. But this has really radical implications. Not only does this say, we want an equal distribution of income, this says, we would destroy any amount of money of the rich to give some money to the poor. So imagine what this says, if I could take-- if everyone's distribution of income was equal in this class, let's say our society, except Patricia has $40,000 more than everyone else, then what that would say is, we would happily take $40,000 away from Patricia and give $1 to me. Because I'm, like everyone else, the worst-off member. But let's say I'm $1 less than the rest of you, make it easy. So you all the same amount of money, I have $1 less, she has $40,000 more. Rawlsian would say, we'd happily take away her $40,000 to give me $1. It doesn't make a whole lot of sense. But it's one sort of extreme, and it's basically this notion of-- it's in some sense, Rawlsian. Think of an extremely risk-averse embryo, gives you the Rawlsian. The idea that, I don't know what my income is going to be, but I want to make sure I'm not poor. Then I would want sort of a Rawlsian social welfare function. So that's sort of a liberal extreme. Now, there's two other views, which are harder to write down mathematically-- at least in the context of 1401, but are important. Let's take the most conservative extreme. So if Rawlsian is the most liberal extreme, the most conservative extreme would be the Nozick-- Nozickian argument-- it's not really social welfare function. His argument is that we should never redistribute income. We should only redistribute opportunities. In other words, once everyone has equal opportunities, then we just roll the dice and let things lay in where they may. So here's an example. Let's say all of us are born with the same opportunities in life, and we end up where we are today. And let's say that you guys are willing to pay me $10 every lecture to hear me lecture. Well, if that's true, at the end of the semester. I have a lot more money than you. Nozick would say, well, why should I be taxed and given to you? That makes no sense. You voluntarily payed me. Why should LeBron James, we're voluntarily paying to see him play, why should he then be taxed on the money we voluntarily gave him? So Nozick's views, as long as we all start with the equality of opportunity, let the dice roll. If someone has more talents and skills, and people want to pay them, then let them keep it. So basically, Nozick's idea is to essentially equalize opportunity, and letting the dice land where they may. Yeah? AUDIENCE: Is this just like opportunity that can be regulated? Or is it-- I was thinking that [INAUDIBLE] might mean like [INAUDIBLE] doesn't have equal opportunity. JONATHAN GRUBER: Right. So there's two problems with the Nozickian view. One is, what is opportunity? What is equal opportunity mean? And the answer is, it's not clear-- there's genetic equal opportunity. There's the fact that if I'm born in poverty, I go to lower quality schools, so I don't really have an equal opportunity. I went to a ritzy public high school in New Jersey, because my parents were well-off. Like someone who went to high school in some poor town of New Jersey didn't have the same opportunities I did. So the first problem with this argument is that, in some sense it's impossible to equalize opportunity. And so it starts with a false premise. The second problem with this argument is it ignores luck. It ignores luck. Which is that in fact, if you look at why some people are rich and some people are poor, even with equal opportunities, a lot of it's not skill or talent, it's luck. They were in the right place at the right time, had the right parents who gave them the right inheritance. They met the right person in business school and that person brought them into their company. It's luck. Indeed, if you try to explain differences in income by any measure of skill we have, you can never explain even half of the difference in income across people. A lot of it appears to be luck. Well, in that case, we would then not want to let lucky people be richer than unlucky people. That doesn't really seem to make sense. So I want skilled people richer than unskilled people, but it seems like we might redistribute against from the lucky to the unlucky. So that's the other problem with the Nozickian notion. And then finally, the fourth approach, the fourth approach is a totally alternative view, which we call commodity egalitarianism. Commodity egalitarianism. This view is simply saying, look, who cares how much money I have relative to you? All that matters, that you can live a decent life. So this says is what matters is not relative income but absolute resources. What matters is making sure everyone in society has food and shelter, and I would argue health care, et cetera. A set of base things everyone should have. And then above that, who cares? So in substance, commodity egalitarianism is a mix of Rawls and Nozick. It cares about the minimum, saying we've got to make sure everyone has a decent standard of living. But above that, let's roll the dice. As long as we're providing a decent standard of living for everyone, that if someone can make a lot of money, let's let them. So this is a very interesting view. And it basically talks about the view of should we give people money or stuff? In the sense of commodity egalitarianism view, says, look, let's worry less about money and more about stuff. Let's make sure everybody has enough stuff to live a decent life. And then we'll roll the dice from there. So let me start-- if I was not clear enough at all, none of these are right. These are all alternative views. It is harder to write down to make an assumption about social welfare function than it is about a utility function. We have less parameters to draw on. But the point is, these are all different ways of thinking about the trade-off. We can't avoid thinking about the trade-off. That's why, once again, we're the dismal science-- because nothing's free. We can't avoid thinking about the trade-off. We have to think about redistribution. And these give us different frameworks for thinking about that. Questions about that? Yeah? AUDIENCE: What-- for the Nozickian and the other one, what happens if there are individuals that just don't-- they always make the wrong decision. They'll spend all their money irresponsibly, and therefore, they can't have a producing [INAUDIBLE] JONATHAN GRUBER: Well, I mean, that's a very interesting question. So I think Nozick-- I mean, I don't know, I don't know how hard-hearted a guy Nozick is-- but if you took the Nozickian view to the extreme, as long as that person had an equal opportunity, then they should just die. You know, they had an equal opportunity, if they made a series of bad decisions, why should we care? Now, that's obviously an extreme view, but I think that would be the view there. Whereas, commodity egalitarianism would say, look, let's at least make sure that, despite their bad decisions, they don't starve. But we don't want to make them rich if they're making bad decisions. So let's just set a minimum and get them to that, and then let things go from there. Yeah? AUDIENCE: Is that where the idea of basic income comes from? JONATHAN GRUBER: That's a great segue to what I want to talk about next. Which is, let's actually talk about-- turn these into practice, and measuring inequality. And I'm going to back to your basic income point in a couple of minutes. So let's talk about actually measuring inequality. And the reason we're going to talk about this, because it highlights how important this issue is. So let's go to some facts-- these are from my textbook from 1441. So go to Figure 21-2. Ignore the last row for second, focus on the first five rows. What the first five rows show is the percent of income received by each quintile, or each fifth, of the income distribution. In other words, each row is 20% of people. So the first row is the poorest 20% of people, the next row is the second poorest 20% of people, and so on. The numbers in each cell are the share of income held by that quintile. In other words, if the distribution of income was totally equal, every one of these numbers would be 20. If distribution of income was totally equal, every quintile would have 20. But in fact, that's never been true in any society ever in history-- the richest always have more than the poorest. We have no perfect distribution of income. And you see that if you look at in 1967, when these data first are reliably collected, you see that the highest 20% of individuals had about 10 times as much as the lowest 20%. That there was a lot of inequality. Now, if you roll forward till about 1980, that gap was shrinking. So you saw that the highest 20% share was about fixed, but the lowest 20% was rising. But then, if you look since 1980, that gap has widened enormously. To the point now where the 2013 is the latest here, but the facts haven't really changed-- the richest 20% of Americans earn more than half the income in America. And the poorest 20% earn only about 3% of the income in America. So inequality has widened massively in the US. How does that put us in international terms? And I'm sorry, and the last row was the share of the top 5%-- this is particular striking. In fact, go to the next page-- this is a graph of the share of the top 1% of income holders in America. So this is the share of income held by the 1% richest Americans. You can see that in the early 20th century, that was pretty high, almost 20%. It then fell down about 10% by the early 1970s. It's now up-- and by the latest date, up to about 25%. It's higher than it was at the beginning of 20th century. So the 1% of richest Americans have about 25% of the income earned in society. So it's extremely unequal income distribution. Once again, not saying bad or good-- I'm making a positive statement. Unequal is irrefutable. Bad or good, we'll get to. But it's irrefutable a very unequal income distribution. It's also irrefutable, we have a much more unequal distribution of income than the rest of the world. So Figure 21-4, we compare the facts for the US to the rest of the OECD, which is a set of developed economies. Yeah? AUDIENCE: Is this before or after income tax? JONATHAN GRUBER: That's a great point. This all before tax, before tax. But if anything, if you add taxes, it makes us look even worse relative to other countries, because our tax are less progressive than other countries. This is before tax and before transfers. So if you look at other-- so this list is a lot of numbers. Look at the bottom line. The bottom line is the average across all non-US countries of the share of income held by each part of the income distribution. And the next to bottom line, and the bottom line is the US. So we say the bottom 10% of income distribution, on average across these countries, has 3% of income. Indeed, nowhere except for Mexico is the number lower than the 1.6% in the US. Likewise, you look at the top 10% of income earners, on average across these countries, they earn 25% of the income. In the US, it's 30%. And indeed, nowhere is the number higher than in-- except in Mexico and Turkey. So we are the most unequal country, except for Mexico, on this list. Once again, not saying good or bad, just saying the facts. We're going to come to how we think about good or bad. So those are the facts about inequality. But as the question here pointed out, it's not clear we care about inequality. Indeed, in a standard economic framework, I don't care about inequality. My utility is a function of my consumption, not your consumption. Sustainable economic framework, I don't care about inequality, I just care about what I have. And that speaks more to the commodity egalitarianism view. Which is, how are we doing in making sure people have enough to live? And to do that, we say, we move from-- so when we talk about inequality, inequality is a measure of relative distribution. Want to move to something which is the measure of absolute-- absolute income, and that's what we call the poverty line. The poverty line in the US is a measure of absolute deprivation-- what share of Americans are earning less than some minimum standard they need to live? Now, you can immediately see the problem. With inequality, it's unit free, right? I simply compare dollars to dollars. Once I start going here, I have to make a judgment, which is, what is deprivation? What do you need to live? So in substance, this makes more sense-- absolutely makes more sense, in substance it makes more sense to think about, do people have enough to live on? But it is more difficult, because you have to draw a judgment about what it is. So the judgment we've drawn is what's called the poverty line. The poverty line was invented by a civil servant in the 1960s, Molly Orshansky. She said, well, what does it take to live in America? She said, well, the typical person spends about a third of their budget on food in the 1960s. So let's cost out the cost of a nutritionally adequate bundle of food, multiplied by it three, and call that the poverty line. She did that, and that's still the poverty line. All we've done is taken that and updated it by inflation. Remember, we talked about the CPI, the inflation rate? All we've done is taken Molly Orshansky's poverty line and updated it by inflation ever since. And what do we get? Well, if you look at table 21-5, this shows you the poverty line in the US today. It varies by family size, because you need more resources with a bigger family. But not one to one, because of economies of scale in the household. You don't need twice as much money to have a household with two people, because you still only need one living unit, you can share cookware, you only have to heat-- heating for two people isn't much more expensive than heating for one person, et cetera. So there's economies of scale within the household. So the poverty line does not go up-- does not double with every person, does not double you go from one to two-- it less than doubles. And you could see this scale-- essentially we say that-- this is 2015, so it's higher now-- but basically says that one person with income below $11,170 is living in poverty. And a family of four, it's about $24,250. So it basically says, a family below $25,000-- a family of four below $25,000, is living in poverty. Now is that the right number? Well-- yeah? AUDIENCE: Is it different based on where in the country you live? JONATHAN GRUBER: There's a number of reasons it might not be the right number. First of all, it does not differ based on where in the country you live. So if you are-- you guys don't know because you're sheltered-- but if you right now tried to take $25,000 and go live in Boston, there's no way. I mean, it's just, like, impossible. Whereas, in rural Mississippi, you can probably do OK on $25,000. But it doesn't vary by area. It doesn't vary-- also the poverty line calculation is all messed up now, because when she did it, food was a third of people's budgets. It's now more like 20% of people's budgets. It's fallen enormously. And the other elements that poor people have to pay, notably housing and medical care, have gone up much faster. The poverty line hasn't accounted for that. So in fact, there's lots of reasons that this line is problematic. Nonetheless, it's very hard to change it. Indeed, I was in the US government for 14 months, and I only went to one super duper secret meeting. I went to a lot of meetings with the president, stuff like that. But one time, my secretary said, there's a meeting, and I can't tell you what it's about, and I can't even tell you where it is. When the time comes, they will come get you and bring you there. I was like, Jesus, it's nuclear war. Like, what the hell? So they brought me to this room, we're all in this room. I'm like, what's going on? They're like, we need to discuss revising the poverty line. I'm like, what? Well, why is it super secret? Because the US today distributes more than a $1 trillion a year based on the poverty line. So any changes you make is going to create winners and losers. And the losers are going to be really mad. And as a result, it's been incredibly hard to change the poverty line. Because for example, let's say we change it to recognize the fact that it should be higher in New York than Mississippi. Well, that means New Yorkers would win and Mississippians would lose. Bad news, politically. New Yorkers would be happy, Mississippians would be sad, but because of diminishing marginal utility, and the laws of politics, the guys who are sad make more noise than the guys who are happy. So it's very, very hard to change this, despite its problematic features. Nonetheless, it's still a useful benchmark. It is useful that it's fixed over time, because it allows us to essentially see how things have evolved over time. And we can see that in Figure 21-6. This shows the poverty rate over time. And it shows it for everybody, that's the red. It shows it for the elderly, that's the blue. For kids, that's the green. And for non-elderly adults, that's the yellow. So what do we see? We see a couple of things. First of all, for every group, poverty fell enormously during the 1960s. That's the so-called war on poverty, which used a number of social programs that lowered poverty enormously during the 1960s. For the elderly, it kept going down. And the elderly went from the most impoverished group in society to the least. For kids, it bounced back up. And now kid poverty rates are nearly as high as they were back in the early 1960s. For the rest of folks, it sort of went down then flattened. Yeah? AUDIENCE: How do you determined how much wealth a kid has? JONATHAN GRUBER: It's based on their family's income. This is not wealth, it's income. It's based on their-- so basically, when asked does a kid live in poverty, it's like, do they live in a household that's below the poverty line? So the bottom line is-- think of it as families with kids, is another way to think about this. The bottom line is, essentially, poverty hasn't done a whole lot. We haven't done a whole lot on poverty in the last 50 years. We sort of lowered a lot in the '60s, and then it's been bounced up and down, but it's been pretty flat-- is the result there. Now, the bottom line is-- now my take, now I'm going to draw a judgment-- my take is, along either of these dimensions, we don't look so hot. We're the most unequal nation the world. And we have-- still, if you look at all people, we have 15% of our people living below some standard of absolute deprivation. Literally saying, we are accepting that 15% of people America cannot afford to live. We are accepting that, including more than 20% of kids. So I don't think we're doing that well. But in some sense, this is even the most-- yeah, go ahead. AUDIENCE: 18 to 64 people, are those single people? Because I imagine kids would fall into-- JONATHAN GRUBER: I mean, that's like the average of all 18 to 64, including those that have kids and don't. So like a child at the age of 64, that would come down more. It's just the average. Now, this is a striking set of facts, but not to my mind the most striking. Probably the most striking fact that probably-- probably the fact that has seen the most influence on me in the last 10 years is shown in Figure 21-7. This is a figure put together in the wake of the Freddie Gray riots in Baltimore. You may have heard about those, that's when a prisoner was beat to death in the back of a police van. And he was from the area of Sandtown-- Sandtown, Winchester, which is a super bad area. Think The Wire-- if you've seen The Wire, super bad area of Baltimore. About 3 miles away is an area called Roland Park, which is a very rich area. The average life expectancy in Roland Park is 84 years. That's above the US average. That's pretty good. The average life expectancy 3 miles away is 67 years, which is below North Korea. And what it was around World War II in the US. 3 miles, and we got a 17-year difference in the average life expectancy. What's going on? Well, what you'd expect. People are way poor. In Sandtown, the average income is $107,000-- I'm sorry, Roland Park is $107,000, and Sandtown is $24,000-- the average income is below the poverty line. In Roland Park, 2.5% of kids live in poverty. In Sandtown, it's 55% of kids. So basically, you have incredible inequality in short distances. To me, this is striking, because it really brings home inequality by putting it in such sort of close geographic terms. So we have a lot of inequality. There's no question. And it certainly motivates-- I think it's hard to look at facts like this, regardless of your political stripe, and not worry, not at least worry, and consider the fact that there may be some role for government redistribution. But that's not the end of the discussion. This in some sense-- think of this part of the lecture as the benefits of government redistribution. The benefits of government redistribution is we have incredible inequality, incredible poverty, all around the country. Now, we have to come-- but of course, in economics, nothing free. Now we have to come to the costs of redistribution. The benefits are clear. What about the costs? And so now we have to talk about the efficiency costs, or the leakage. Efficiency costs-- costs of redistribution. Or in other words, how big is the leakage? So going back to Okun, I hope that these figures inspire you to think we out to be putting some money into buckets and giving it to poor people. But now the question is, should we? What if it all leaks out? How leaky is the bucket? And that's what we need to talk about now. And basically, leakages in the bucket come from three sources. The first and least important source is administrative costs. Literally got to pay someone to carry the bucket. So if you put a dollar in and carry the bucket, he's going to take some money out for the right of carrying the bucket. That's small, but non-trivial. Maybe low single digits. The second source of leakage is the efficiency costs of taxation. If you tax people and take their money away, they may work less hard. Think about the extreme case, where I'm at to say-- I'm going to go back to the utilitarian world, I'm going to tell everyone, we're going to equalize income. That means that everyone, no matter what they make, example, the same amount of money. Why would you work? I mean, you guys would, because you're tools. But why would regular people work? They wouldn't. Because no matter what you do, you end up with the same income. It's 100% tax. So why would you work? And that's extreme, but the point is, the extreme example makes the point that when you tax people, there's potential efficiency costs, in terms of them earning less income. The third issue is the efficiency costs of transfers. Which is when you give people money, and when you give people money, they may also work less. So not only might I work less when you tax me, I might work less if you give me money. Why should I go to work if you're sending me a check? So not only would equal distribution of income lead me to work less because I'm being taxed, leading me to work less because I'm getting money. So why should I bother working? As long as leisure is a normal good, remember? I don't want to work. So if you're just sending me money, why would I go to work? So to see this, we can sort of summarize this with one example. So let's go to Figure 21-8, and I want to walk through-- it's a fairly complicated example-- let's walk through-- this a simple illustration of a tax and transfer scheme of the type we have in America. We start in this example with an individual earning $20 an hour. So the slope of this line is negative $20 an hour. This person can work-- and remember, we don't model work, we model leisure. I almost made the mistake. Remember, we don't model the bad, we model the good. We don't model labor, we model leisure. Let's say that the max leisure you can take is 2,000 hours. So you can either take 2,000 hours of leisure and consume nothing, or you could take no leisure and consume 40,000. Let's say consumption is income, no savings in this model. So you earn $20 an hour, you consume your whole income, and you can take up to 2,000 hours of leisure. So your budget constraint is the dark line running from 40,000 on the y-axis, 2,000 on the x-axis. Now, let's say that we're going to put in, into this budget constraint, we're going to add two things. The first is a transfer program to the poor. We're worried about poverty. So we're going to have a new program, that says that every American gets at least $10,000. Everybody gets $10,000. But as their income goes up, we're going to take that away. So we're going to say, since we have a transfer program, the transfer program is going to be of the form-- the transfer you get is the max of 0 comma 10,000 minus your income. That's our transfer program. So if your income is 0, you get 10,000. If your income is 10,000, you get zero. We're just making sure everybody gets $10,000. We don't care about people having more and that, so we're going take it away. We're going to say, look, want to make sure you have $10,000, but above that, we don't care about you. So if you're someone earning $100,000, this program's irrelevant. If you're someone earning $5,000, we'll give you another $5,000 to get you up to $10,000. So that's based on a transfer program. This is typically the way welfare programs work around the world. Essentially, they give you money, but then they take it away as you get richer, to make sure the money is targeted to the poorest people. So that's our transfer program. That's the first thing it's going to do. The second thing-- we've got a transfer program, but we've got to pay for this. To pay for this, we have to have a tax. Now, let's say we only want to tax the rich-- you don't want to tax the poor, you want to give money to the poor. So let's say our tax program is of the following form-- anyone making more than $20,000 per year pays a 20% tax rate on the income above $20,000. So no one's taxed on the first $20,000. But on every dollar above $20,000, you pay $0.20 to the government. It's what's called a marginal tax. You have a marginal tax rate of 20%. What does marginal mean? It means you only pay on the next dollar. So it's a marginal tax rate of 20% above $20k. OK, so once you earn $20k, of $20k, you pay nothing. Once you've earned $20k, on every dollar above $20k, you face a marginal tax rate of 20%. You pay $0.20 of every dollar of the government. So that is our tax program. Now, look at the diagram. Do people understand how these programs work? Forgetting the diagram. Yeah, Manny? AUDIENCE: In other countries, is there any other data to show how this affects the number of hours people work? JONATHAN GRUBER: That's exactly what we're getting to. That's our concern, right? Is that that's my point here, is that this might lower how many hours people work. And let's talk about why. But the question about the logistics of these programs, first. Let's talk about how it affect the hours worked. Let's go to the diagram. Let's go to the diagram. Let's consider several different people. Start with person A. Person A earned-- before this program was in place, they would have chosen to be a high leisure, low consumption person. When this program's in place we tell them, look, person A, you can get more of both, more leisure and more consumption, all you have to do is quit. Stop working. You move from point A to point D. You get more leisure and more consumption. So that naturally happens. So everyone earning less than $10,000, immediately quits. Once again, assuming leisure is the normal good. Immediately quits because look, once you're below $10,000, it's 100% tax rate. Why work? So anyone who would have earned less than $10,000 doesn't work. Because you're just going to give back to the government anyway, why do you care? That's person A. What about person B? Person B earns more than $10,000. But note, their indifference curve crosses below the new budget constraint. Let me step back, the new budget constraint is the red segments plus the black segment in the middle. New budget constraint runs from $32,000 on the y-axis, it intersects the black line, the old budget constraint at $20,000. It's the old budget constraint from $20,000 down to $10,000, and it becomes the new red segment. I should mentioned that. So the budget constraint is the two red segments and the black segment in between. That's the new budget constraint. So let's look at person B. What happens to person B and why? What does person B do? Yeah? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Why? AUDIENCE: Because it's a higher [INAUDIBLE].. JONATHAN GRUBER: Right, crucially, since person B never crosses the red line, we know they must be better off at point D than at point B. Because we know that since point D is above that line, that's a higher indifference curve than they're on at point B. So not only do you cause every person below $10,000 to quit, you also cause some people above $10,000 to quit. Why? Well, they give up a little consumption, but they get a ton of leisure. So let's say someone is at $11,000, $12,000, like, wait a second, all my work is only getting me $1,000 or $2,000 over quitting. I might as well quit. So this welfare program causes a ton of people to quit. All the people who were going to earn very little quit, and some people who are going to earn somewhat little quit. But that's not all. Let's look at person C. Person C used to work a certain amount-- you know, used to work more than 1,000 hours. Take less than 1,000 hours of leisure. Now what happens, well what happens-- forget the diagram, step back. If I take someone and choose a marginal tax rate, what happens to their labor supply and why? What happens to their leisure and why? So I take you. You're working-- let's say you're taking 800 hours of leisure, doesn't matter what the exact number was. I then come and save you have a 20% tax rate on every earnings above $20,000. How does that affect your leisure? Yeah? AUDIENCE: Might as well increase it. JONATHAN GRUBER: You would increase it, why? AUDIENCE: Because then above $10,000, I guess you would get less money, but I guess maybe less utility above $10k, because you're going to give it to the government? JONATHAN GRUBER: You might increase it. But what might else you do? Yeah? AUDIENCE: The wage effectively decreases, so the price of leisure decreases. So you take more of it? JONATHAN GRUBER: Right, I think that you increase leisure, that's what he said. OK, increased leisure, but what else? Yeah? AUDIENCE: Might increase it just to cover taxes, so you might work-- JONATHAN GRUBER: Yeah, what do we call the two effects? AUDIENCE: Oh, your income and substitution. JONATHAN GRUBER: Exactly, we have substitution income effects. The substitution effect, let's say your net wage is lower, your net wage just went from $20 an hour to $16 an hour, right? So a lower net wage, you work less, but the income effect says, you work more. Because you're now effectively poorer. When you're poorer, you consume less of everything, including leisure. So when you tax my wage, I might take less leisure, i.e. work more, through the substitution effect-- I'm sorry, we tax my wage, I might take more leisure, i.e. work less, through the substitution effect, because the returns to work are lower. But I might take less leisure, i.e. work more, because I'm now poorer. So we don't know what's going to happen. But we typically assume substitution effects dominate. You can't go wrong in this class without making that assumption. We want you remember the trade-off-- but typically, some substitute effects dominate. So we typically think people will work less. We think taxing people will cause them to work less. But be clear, it's not obvious it will. But we typically think that's the result. And typically, once again, when you average men and women, as we discussed in our labor supply lecture, overall, you get an upward slope in labor supply curve. That is overall if you tax people that work less. So what that does, it lowers the hours of work for person C. This is our leaks in the bucket. We've suddenly massively reduced the amount, potentially massively, reduced the amount people who want to work. Why do we care? We care because a Figure 21-9. What have we done? We've gone from an initial point where people initially had initial supply curve of S1, and demand curve of D. And therefore, we're supplying L1 hours of labor at a wage, W1. Now they're producing less-- their supply curve shifts in. That's created a deadweight loss. There is less stuff being produced, because people are staying home rather than working. And the key point is, that's not a problem. Staying home rather than [INAUDIBLE] is not the problem. The problem is, they're staying home only because we've reduced the price of labor. We've increased the price-- we've reduced the price of leisure, I'm sorry. We've reduced the return to labor, reduced the price of leisure. As a result, people are staying home. We've distorted their behavior. We've caused them to stay home and not work. We cause them to stay home and not work, there's less stuff for the rest of us. So it's a deadweight loss. Efficiency falls. And that is the efficiency equity trade-off. This deadweight loss is Okun's leak. That's the leak it Okun's bucket. So now we have the trade-off. On the one hand, we have an incredibly unequal society, where a program like this can really make people better off. Take money from rich people who don't need it, give it to poor people who do. On the other hand, in doing so, we're going to have less stuff as a society. We're going to shrink the size of the pie in order to redistribute the slices of the pie. Is it worth it? That's where the social welfare function comes in. The social welfare function allows you to evaluate whether something like this is worth it. Without a social welfare function, you can never answer that question. You just can't, because there's one hand and the other hand. What the social welfare function does is give you a mathematical representation that allows you to answer that question. So in section on Friday, you'll work through an example of using a social welfare function to evaluate a welfare transfer program like this. And they'll come back on Monday and talk more about taxation.

MIT_1401_Principles_of_Microeconomics_Fall_2018

15_Input_Markets_ILabor_Market.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: All right, let's get started today with our lecture on factor markets. So when we talked about producer theory, we talked about input prices, that firms had prices for their wages and their capital. And we just sort of posed those as given. I just sort of gave you values for the wage and the renter rate of capital. But we never really talked about where those prices come from. Given that they may be the most important prices in our whole economy, it's probably worth spending a little time on talking about where do w and r actually come from. And that's we'll do for the next three lectures, is talk about factor markets, talk about the markets that give us the price of labor and capital. We're going to start by talking about factor demand, the general demand for labor and capital. And then we'll move on to talk about factor supply, where does supply come from. We'll then develop the equilibrium, and that will tell us where wages and the interest rate come from. So that's sort of the map of where we're going, is we're basically going to develop the markets that give us the wage rate and the interest rate. So let's start with factor demand, factor demand. And let's start, and we're going to start with the cleanest case. We're going to assume that factor markets are perfectly competitive. So unless I say otherwise, we're assuming the market for workers, or the market for machines, or capital, is perfectly competitive. OK, we'll come back and bend that a little bit later. So what that means is that there's basically many sellers and buyers, OK? So any worker is basically competing with lots of workers for jobs. Any firm is competing with lots of firms to hire the workers, OK? And we're also going-- we're going to assume a perfectly competitive input market, that is lots of firms and workers competing to match with each other. We're also going to assume a perfectly competitive output market, that is, we're going to examine this for the case not of a monopoly firm but of a perfectly competitive firm. So just think of this, you have a perfectly competitive firm competing with lots of other firms to hire workers, OK? So let's start by talking about short run labor demand in this context. Let's talk about short run labor demand. Now, in the short run, capital is fixed. So our decision is just, do we add another worker or not, or another hour of labor or not. Like I said, the units don't really matter here, but let's take in terms of workers. Do we add another worker or not? Well, as with everything else in this course, we want to consider the marginal benefits and the marginal costs of that decision. The marginal benefit of an extra worker is that one extra unit of labor raises productivity by the marginal product of labor, OK? One more unit of labor raises our output by the marginal product of labor, OK? But that's not the only part of the benefit, because we don't actually care as a firm about units of output. We care about revenues. So the benefit of a worker is not just the how many units it produces, but the value of those units. And what is the value of the next unit produced? It's the marginal revenue. So the value of the next unit of labor is what we call the marginal revenue product, MRP sub L. The marginal revenue product is the marginal product of labor times marginal revenue. That's the benefit of another unit of labor. It's not just what they make, but what it's worth. It's not just what they make, but what it's worth, OK? So that's the marginal benefit. The value of another unit of labor is it makes marginal revenue product amount more stuff, and you sell that at the marginal revenue. That's the marginal benefit. What's the marginal cost of another unit of labor? So this is the marginal benefit of another unit of labor. What's the marginal cost? Well, the marginal cost of labor is just the wage. So we simply set this equal to the wage. We set the marginal revenue product of labor equal to the wage, and that gives us our optimization condition for the optimal amount of labor the firms want to demand-- is to set the marginal revenue product of labor equal to the wage. Marginal benefits of hiring another unit of labor equals the marginal cost of hiring of the unit of labor. Now to go further, remember, I said this is a perfectly competitive output market. So what is the marginal revenue in a perfectly competitive output market? What's the marginal revenue of a firm producing-- yeah. Price. So I can write this more to say that I want to set the marginal product of labor times the price equal to the wage, OK? So basically, what we're saying here-- think about it-- is hire workers until the cost of the next unit of labor is the same as what that unit will actually produce for you, OK? The next unit of labor costs you w. It produces for you MPL times p. So you want to hire workers until that condition is met, OK? So think about that, and figure 15-1 sort of shows this, OK? We have a supply of labor. In 15-1, that's horizontal, because we're assuming competitive market for workers, OK? We're assuming a competitive market for workers, that is a perfectly competitive market. So if I try to pay workers one penny more than other firms, every worker in the world will want to work for me. If I pay workers one penny less than other firms, no workers will want to work for me. That's what a perfectly competitive labor market means, that literally, I am a price taker in the input market. I don't get to set the wage, OK? I don't get to set the wage. The wage is given to me by the labor market. So just like a perfectly competitive firm doesn't get to set the price of their product-- it's given to them by the competitive market. A perfectly competitive firm in the input market doesn't get to set the wage they pay. It's given them through the kind of process that delivered us our prices on the output side, OK? So we get a horizontal labor supply curve. And then we have this downward sloping labor demand curve. Why is it downward sloping? Someone raise their hand and tell me. Why is the labor demand curve downward sloping? Yeah. AUDIENCE: Marginal product of labor is diminishing. JONATHAN GRUBER: Exactly. The diminishing marginal product of labor means you have a downward sloping marginal benefit of labor. Each additional-- remember, holding capital fixed is only one shovel. So each additional worker add less and less to digging that hole, OK? So marginal product is diminishing. Since p is a constant, that doesn't really affect the slope. I mean, it affects the slope. It doesn't really affect the sign. Doesn't affect the sign. It's diminishing because the marginal product of labor is diminishing. So the equilibrium is where they intersect. So the bottom line-- this is complicated and new-- the bottom line intuition is to think about, as I decide whether to hire one more hour of work-- you've got a firm. You've got to decide, do I want the worker to work one more hour? You do the tradeoff of, what am I going to pay them for an hour versus what are they going to get me for an hour. What they're going to get me is their marginal product times the price, OK? Now, that-- So in other words, the wage is not just the marginal product. It's imagining if two workers were equally productive. With one more hour of work, they each make three more units. But let's say, in one case, a unit is a computer chip, OK? In another case, a unit is a potato chip. We clearly would not want to pay the same wage to someone who produces three more computer chips to someone who produces three more potato chips. We'd want to pay a lot more to the person to do more computer chips. Why? Not because computers are inherently valuable. In fact, potato chips are much more delicious than computer chips. Because they sell for a higher price. So therefore, you'd want to pay more to the worker who produces more units of a more valuable good. So let's think about a sports example, OK? And I realize we're all about baseball today, as we should be. Go, Red Sox. But let's focus on basketball for a minute, OK? Now, imagine you're a owner of a team in the NBA, the National Basketball Association, and you're trying to decide how much you pay one of your players. So basically, in that case, your goal is to-- your goal is wins. That's the goal. That's the profit you're trying to maximize, is your wins. Let's say you're probably trying to maximize your revenues from ads and stuff, but assume that's proportional to wins. OK, assume that basically, the more you win, the more money you make. So let's say the thing you're trying to maximize is wins, OK? So your labor demand, the marginal product you care about, is the contribution of the next player to your win total. That's what you care about. The marginal product of labor is how much does that next player add to my win total, OK? So for example, LeBron James, the best player in basketball, arguably the best player in history-- we could have that-- we could have the LeBron versus Michael debate some other time, OK? LeBron James makes $31 million, and that's because his marginal product is enormous. He adds a huge amount of wins to any team, OK? We'll see with the-- we'll run the experiment to watch how the Cleveland Cavaliers tank this year once LeBron has left, OK? Now, other players don't make as much. Let's compare LeBron James to Nate Robinson. You guys might not know Nate Robinson is. He's one of the shortest players in the history of the NBA at a paltry 5'9", which sounds pretty tall to you and I, but it's tiny for the NBA. He was a very exciting player. It's kind of fun to watch this little guy run among these giants. But he was just OK. He wasn't a great player. He was a fine player. He made about $2 million a year by the end of his career. So basically, you have LeBron making 31 million and Nate Robinson making two million, and that's sort of related to their marginal product. So LeBron adds a lot more to your wins. Now, what happened is Nate Robinson quit basketball in the US, and went to play basketball in Israel. In Israel, they love basketball. They have a league. And he went to Israel, and he was dominant. He was the best player in Israel, because they don't-- it's not as good as the US, OK? So his marginal product went way up. Nate Robinson went from being someone that had a small marginal product to maybe the highest marginal product in the league, and his wage went down from two million to 500,000. So this is a situation where someone's marginal product went way up and their wage went down. Why? Yeah. AUDIENCE: Because people aren't paying as much to watch basketball. JONATHAN GRUBER: Right, because the marginal product went up, but the price went way down, OK? And what we care about is the wage equals to marginal product times the price. So you have a situation where a player got better but got paid less because they got better. He moved from making computer chips to making potato chips, OK? He moved from a market where he was earning a valuable commodity to one where he was earning one that was much less. So basically, it's a situation-- that example shows why you have to care about both the quantity of the additional worker and the value of what they're producing, OK? Any questions about that? Yeah. AUDIENCE: When we talk about perfectly competitive input market, are we saying that like all of the workers-- like a single hour of work regardless of who you get it from is equal, right? JONATHAN GRUBER: No, no. A single hour of work is paid equally. It's not equal. Marginal product varies. We're talking about the market. Let's think about a perfectly competitive-- I probably went too fast with this. Let's say a perfectly competitive output market is where the firms sell the goods into a market where people have perfect information and can shop across all firms easily. A perfectly competitive input market is where firms hire workers in a situation workers have perfect information and compare across all firms equally. So basically, the point is, think about a perfectly competitive output market. People are in a market where lots of people are shopping, and all the options are in front of them. A perfectly competitive labor market where you as a worker have lots of firms you can work for, and they're all clearly in front of you, and they all offer a wage, and you can see it. AUDIENCE: OK, but we're not saying that the firms have perfect information across all the laborers, and [INAUDIBLE]. Are we saying if we have the-- JONATHAN GRUBER: What we're saying is-- we're not saying the firms have perfect information about the laborers. The firms essentially-- let me think of the best way describe this. So once again, the firms are-- from the firm's perspective, they do have perfect information. No, the wages aren't-- yes, right, the workers aren't the same. They have different marginal products. The firms know you're better than you or vice versa. But from the firm's perspective-- from the workers' perspective, is just like, think of the workers as the consumers in a perfect competitive output market. For a perfectly competitive output market, the consumers can easily shop across all the firms they might buy from. In a perfectly competitive input market, workers can easily shop among all firms they might work for, OK? That's a good question. Other questions? OK, now let's think about the long run. This is the short run. Let's think for a minute about long run labor demand. Think for a second about long run labor demand. Well, what's different? The only thing that's different is in the long run, capital can adjust as well. The only thing different about the long run-- all the intuition, everything's the same. It's just that capital can adjust as well. And what this means is that long run labor demand is more elastic than short run labor demand, OK? So we could see this in figure 15-2, OK? So the figure shows two different short run labor demand curves at two different levels of capital. So the short run labor demand when k bar equals 32 is that lower one. The short run labor demand when k bar equals 108 is the higher one. And what this says is, in the short run, you've got these two labor demand curves. In the long run, you could optimize capital. You can pick a point on either curve, depending on which level of capital you choose. And by definition, that allows you be more elastic at choosing your labor. You're more flexible because you can optimize not just over workers, but over machines as well. It's the same intuition we developed before talking about short run and long run costs, that the long run cost curve was a lower envelope than the short run cost curve. Same thing here. This applies that the long run labor demand is more elastic, because I basically am more flexible. I not only can choose a longer curve, I can choose which curve I use. And by definition, that means that the long run is more elastic, OK? Just a small sort of side point there. Now, the last thing I want to talk about here is capital demand. We talked about short run and long run labor demand. Let's talk about capital demand. It basically is the same thing. Capital demand is the exact same intuition. You want to get machines until the marginal product of capital, marginal product of the next machine, times the price you get for your good equals the interest rate. It's the same condition. So we want to hire workers so the marginal product of the labor times the price of our good equals the wage rate. We want to invest in more machines until the margin product of capital of the next machine times the price for our goods is equal to the interest rate. So it's exact same logic. Here's the marginal cost. The next unit of capital-- remember, we talked about the intuition. You're always renting things. So thinking about renting a machine, the next machine costs are to rent. Do you want to rent it? Well, it depends. What will it produce, and what can you sell that stuff for? So you rent the next machine if the marginal product of capital, if the goods it produces, times what you sell those goods for, you want to do that until that equals the interest rate, OK? Questions about that? Yeah. AUDIENCE: [INAUDIBLE] machine that you buy and own? JONATHAN GRUBER: Yes. We're going to talk about that a lot starting next lecture. Right now, I think I'll just put this down here. We'll come back to it, but I'm going to focus on labor for this lecture, OK? So let's focus on labor, and let's-- so I just put that down, and we'll back to capital, but focus on labor for a minute, and make sure to understand where labor demand comes from. Now let's talk about where does labor supply come from. We talked about, at the firm level, labor supply is perfectly elastic. So go back to figure 15-1. That was a firm level curve, OK? That was a firm level curve. That's a perfectly elastic labor supply to a firm, but that doesn't mean labor supply to the market's perfectly elastic. So now we want to derive market labor supply. So I'll call this deriving market labor supply, deriving market labor supply, OK? Now, this is basically the question of, how do we model how hard people want to work? This is, once again, getting where the economics is exciting, OK? You sort of knew that economics was involved in how much Ford charged for a car, but you might not have thought so much about that economics was involved in deciding how hard you work, but it is. And we're going to use the same tools of consumer choice. Indeed, I used to teach this as an application of consumer choice, and now I teach it here, because it's the same tools of consumer choice. But now, consumers, instead of choosing good A versus good B, are going to choose how hard they're going to work, OK? So basically, like any choice, there's a tradeoff. There's a tradeoff. On the one hand, if you work harder, you get more stuff. So you bring home more income. You can buy more pizzas and cookies, OK? Remember, we talked about income as a fixed thing your parents gave you, but in reality, sorry, kids, you're going to have to make your own money someday. In reality, you're going to make a Y. It's not going to be given to you. And so if you want to buy more pizza and cookies, you're going to have to raise your Y. It's not going to be given, OK? So the reason you want to work harder is to buy more pizza and cookies. The reason you don't want to work harder is because you're not an MIT student, OK? That is, normal people actually don't like work, newsflash. OK? Normal people actually like leisure. There's a thing called leisure, it turns out, and normal people like it, OK? So the tradeoff for regular people-- so it's a hard thing teach at MIT-- is that basically, the tradeoff is if you work harder, you get more stuff, but you spend more time doing something you don't want to do. Now, this is weird. When we talked about tradeoffs before, we talked about the tradeoff between goods, pizza and cookies. Now we're talking about the tradeoff between a good and a bad. The good is more stuff to eat. The bad is working harder, and we don't really know how to model that. So the trick we're going to use here is we're going to flip the bad into a good. Instead of modeling labor, we're going to model leisure. So to get labor supply, we're going to model leisure supply, and then just flip it around to get labor supply, OK? So that is, we're going to say, your ultimate labor supply, the amount of hours you work, the amount you work, the amount of hours you work, call them H, is equal to 24 minus leisure. Let's call it leisure, because leisure's called little l. Leisure's little l. The amount of hours you work is 24 minus the hours of leisure you take. What that means is I don't have to model the bad. I can model the good and just use this simple reflection equation to get the bad, OK? So this is the trick in economics. It's a good modeling trick. We don't model bad so we don't have to do the tradeoff between the bad and the good. We don't have to do the tradeoff between two goods. So turn the bad into a good. Don't model work, model leisure. Don't model your hours you work, model how many hours of leisure, OK? This is a general modeling trick. So what we want to ask is, now, not how do you derive the supply of labor, how do you derive the demand for leisure? How do we derive how much leisure people want? Well, once I say it that way, you know what to do, which is what I just said. There are two goods, consumption and leisure. I wonder how much of one good you choose-- of each good you choose. Well, that's a consumer choice problem. You know how to do that, OK? So basically, take figure 15-3, OK? In figure 15-3, now, instead of doing pizza versus cookies, now our decision is all consumption. So we're thinking about consumption as a bundle, OK, versus leisure. So on the y-axis is the goods you choose. On the x-axis is how much leisure you take, OK? It says N but actually it should be little l, OK? Should be little l. So let's call that little l, OK? So basically, as you go more positive on the x-axis, that's more leisure. But because this equation, that implies as you go to the left on the axis, that's more work, OK? Yeah. H is hours of work. H is hours of work. So as you go to the left, you work more. As you go to the right, you take more leisure. But we're modeling the good, which is leisure. And then we just go to our standard-- we go to our standard consumer choice equation. We have a budget constraint and preferences. The indifference curve comes from your utility function. It comes from your indifference between how much you consume and how much leisure you take. And the indifference curve comes from like any consumer choice decision. But instead of choosing between pizza and cookies, now it's how much stuff you want versus how much leisure you want to take. So it's the same sort of indifference curve. The budget constraint comes from what the market tells you is the cost of leisure. What is the price of leisure? What is the price of leisure? Someone else? Someone else got it? Yeah, AUDIENCE: Your wage. JONATHAN GRUBER: Your wage. Why is that the price of leisure? AUDIENCE: Because every hour you don't work is another hour of wage you don't get. JONATHAN GRUBER: Which we call what? AUDIENCE: Opportunity cost. JONATHAN GRUBER: Opportunity cost. Remember, prices and opportunity cost are the same thing in economics. Here's once again where it gets interesting to apply what we've learned, which is that basically, this is why, once again, they call economics the dismal science. Instead of having fun sitting around, we're telling you, you know, by the way, you could be working and making a wage. So you're actually spending money by taking leisure. By taking leisure, you are spending money. What are you spending? You're spending the money you could be earning. So the opportunity-- so leisure has a price, and the price of leisure is the wage. It's what you could be earning if you were working. So the budget constraint has the slope of minus w. So if you look at the budget constraint, you could take 24 hours of leisure and have zero consumption, OK? That's the x-axis intercept. Or you take no leisure and have 24w worth of consumption, OK? So basically, that is the tradeoff you face. One other modeling trick-- couple of them-- so a couple of modeling tricks here. Modeling trick one is modeling the good, not the bad, OK? Modeling trick two is, I wrote on the x-axis goods, but we don't think in quantities, we think in dollars. So to make life easier, I just said, let's assume the price of the average good is $1. That way you can-- that's called-- that's just a normalization, OK, which allows you to think in terms of dollars of goods rather than quantity of goods. That's another modeling trick we'll do. We call it making a numerator good, OK? You don't have to remember that term, but the point is a trick we'll do is we want to model dollars, not quantities. We just make the quantities cost $1, and then we can model quantities basically as dollars. So that's the trick we're doing. So the y-axis is dollars, but it's also quantities, because we made the price of everything be $1, OK? It's just another trick that makes life easier. OK, so two modeling tricks here, the numerator trick, which is making the price $1 so quantities become dollars, and the bad is good trick, which is model the good, and then reverse that to get the bad. Having done that, we know what to do. We get an optimum, which is the tendency between the indifference curve and the budget constraint, and we're done. And so what do you do? You choose-- we're going to call this L. We'll call it little l. You choose little l star hours of leisure, which means you choose 24 minus little l star hours of work, OK? So basically, you sat down. You made the decision, how much do I want to eat versus how much do I want to watch TV. You make that tradeoff, and that determines how hard you work, OK? Now-- yeah. AUDIENCE: Aren't there things that are kind of necessary? Like for example, if you wanted to-- like if your preference was completely to work, then wouldn't we be like an inefficient worker if we didn't sleep? Doesn't-- JONATHAN GRUBER: Well, and in some sense, that would be in your utility function, or it would be in your utility function and/or your budget constraint. That would be true, absolutely. But that would be a feature. That wouldn't change this maximization problem. It'd just change general structure of the equations that go into the maximization problem, OK? So basically, now, what's really interesting about this is now we finally understand why we learned all that shit about income and substitution effects. Remember, let's think of substitution effects. And you're probably saying like, "Why do I care? Price goes up. Quantity goes down. Why do I care?" Here's why you care, because now it gets really interesting, OK? Because when we're doing substitution effects for a good, they work together. As long as the good was normal, they work together. When the price went up, you substituted away from the good and you are poor. So it gets substituted down for two reasons. Now, a normal leisure effect is an inferior labor effect. What I mean by that is that when your wage goes up, you work more through the substitution effect, but now you're richer. And when you're richer, you buy more of everything, including leisure. So if you take more leisure, you do less labor. So the income effect naturally goes against the substitution effect. I'll go through this a couple of times. Don't worry. The income effect naturally goes against the substitution effect here. For consumption goods, the income effect naturally work together, OK? We almost never saw sort of a Giffen good type phenomenon, where the effect could sort of switch the overall effect. For labor, that's much more likely, and it's much more likely not because of any inferior good. It's because leisure is a normal good, and labor is the opposite of leisure. So once again, let me say it again. The wage goes up. The substitution effect-- think of leisure as a good. When the wage goes up, that's the price of leisure going up. When the price of a good goes up, the substitution effects says you want less of it, OK? So when the wage goes up, the substitution effect says that leisure goes down, right? Because you want to substitute-- wait, leisure just got more expensive. You now feel worse sitting around watching TV, because you could be out there making more money. Yeah. AUDIENCE: Wouldn't income-- [COUGHS] JONATHAN GRUBER: I haven't got to income effect. Let me finish, then you can ask it. AUDIENCE: Wouldn't income effect be-- JONATHAN GRUBER: I haven't gotten to the income effects. Let me ask finish, then you can ask it, OK? So the substitution effect says that leisure goes down, OK? The income effect says that you are richer, right? Your wage went up. You're richer. When you're richer, you want more of all normal goods. Leisure for non-MIT students is a normal good. So you want more of it. So here, with consumption goods, when they were normal, the income and substitution effects work together. With labor and leisure, they work opposite. So what this is, the substitution effect says take more leisure, which means work-- take less leisure means work harder, work more hours. But the income effect says take more leisure, which means work less hours. So you don't know what the net effect is. So that's why we do income and substitution effects, because in a case like this, they get much more interesting. Yes, now your question. AUDIENCE: Is this income effect in terms of income over time? JONATHAN GRUBER: No, this is your income, your actual cash income. You are now richer, and when you're richer, you spend more on everything. So think of it this way. Once again, imagine you're not an MIT student. You're a normal guy. OK, if we won the lottery, if you guys won the lottery, you would use that to do a startup. If a normal person won the lottery, they'd use it to not work, OK? That's the income effect. OK, when normal people win lotteries, they don't go work harder. They don't work, OK? So that's the point. You are now richer because your wage went up. So you work less, and that offsets it. So let's show this in a graph. Let's go back to our income and substitution effect graph that we did before, figure 15-4, OK? Now we're back to-- once again, this is just applied consumer theory, OK? Let's go back to the income and substitution effects. We start with budget constraint one at wage one, and we have our initial tangency at A, OK, with leisure of N1 or little l1. Now our wage goes up. Our wage goes up. Therefore, the budget constraint pivots up. Think of what that means. You can still only have 24 hours of leisure. That's a fixed point. But as you take less leisure, you make more money. So the budget trade now pivots up. Well, that has two effects. The first is the substitution effect. Remember how we get that. We draw an imaginary budget constraint at the new price ratio. The price ratio is just W because I assume the price of goods is 1. The new price ratio, tangent to the old indifference curve, that is point B. So the substitution effect says, take less leisure, OK? The price of leisure has gone up, so holding utility costs, you want to take less leisure. The income effect, however, says, you are now richer so take more leisure. So the income effect goes the opposite way of the substitution effect naturally. You don't need a weird thing for that to happen, like with pizza and cookies. It comes naturally. So for normal goods, the income effect goes the opposite way. Now, in this case, we end up with leisure still going down. We end up with, the wage goes up, leisure goes down, and therefore labor supply goes up. So we end up with our standard intuition, which is, I tell you, if I'm going to pay you more, you're going to work harder or less hard? The standard intuition is I work more hard, OK? But as figure 15-5 shows, it would not be super odd to get a Giffen good effect here, which is, the wage goes up. The substitution effect shifts you to the left, but the income effect shifts you even more to the right, and you actually end up with more leisure. So once again, my intuition, if I say to you the price of pizza went up, what happens to your demand for pizza? You think of a standard-- you say, "Well, I'm going to demand less pizza." If I say to you the wage went up, what happened to how hard you work? It's not clear. Think of a simple example. Think of yourself actually back before you were an MIT student, when you were a kid saving for something. You were saving to buy a bike, and the bike was $150. OK, bike was $200, and you're earning $10 an hour, OK? So you had to work 20 hours to get the bike. Now I gave you a raise to 15 hours-- to $15 an hour or $20 an hour. Would you work harder or less hard? Well, if all you want is the bike, you'd work less hard. You don't have to work 20 hours. You only have to work 10 hours. So in fact, a higher wage caused you to work less hard. That's not that bizarre a case, right? That makes sense. The point is, it's actually quite sensible that you couldn't end up with the labor supply being a Giffen good, with a higher wage causing you to work less. It's not a crazy outcome. Giffen goods and consumer goods are crazy. It's not at all crazy to think that in cases like having a target, a purchase target, a higher wage would cause people to work less. Yeah. AUDIENCE: So does the law of nonsatiation not apply? JONATHAN GRUBER: Absolute applies. Absolutely applies. There's no violation. We haven't violated any of the laws. All we've done is just said income effects-- it didn't apply with Giffen goods too. It's all just saying income effects dominate substitution effects, which we thought was sort of going to be pretty bizarre in the consumption good context, but it's not at all bizarre in the labor supply context. So this is pretty wild. What this says is that basically, you've got a situation where even in the normal world, you can get that paying workers more makes them work less, which is kind of bizarre, OK? Questions about that, about that intuition, or the math, or the graphs? Well, the math we haven't done, but the graphs? We'll do the math on Friday. The graphs or anything? OK. Let's then say, well, does that happen in reality? What does the evidence say? Let's go to the evidence. What does the evidence say? And there may be sort of no question more worked on in economics than the elasticity of labor supply or the shape of the labor supply curve. There is thousands of articles written on this question, OK? And what I want to do here to make the intuition easy, I want to go back to the literature circa probably 40 years ago, when it was sort of the initial burst of interest in this, in like the 1970s. In 1970s, there was a burst of interest in this. And what the literature did was it looked separately at men and married women, because most of women were married, and back then we didn't care about single women, OK? OK, it was a dark time, OK? So the literature looked at men and women, and married women, and asked what was their elasticity of labor supply. Well, let's think for a second about what we'd expect, and to do that, let's think about the substitution effect and the income effect. Let's start with men, the male substitution effect. Let's go substitution effect. Men versus married women, who has a bigger substitution effect and why? That is, when the wage goes up, who has a bigger substitution response to that and why? Men or married women? Think about the world-- think about the Mad Men world or the world, you know, circa 40 years ago. You guys seen enough TV and stuff to know how life was a little bit, OK? So who's going to respond? Who's the bigger-- yeah. AUDIENCE: Are you assuming men were primary providers? JONATHAN GRUBER: Well, they certainly were in the 1970s. AUDIENCE: Oh, OK. In that case, the men. JONATHAN GRUBER: Men have a bigger substitution effect? AUDIENCE: Yeah, they'll work more, probably. JONATHAN GRUBER: OK, that's one option, yeah. AUDIENCE: It'll be married women, because they're only working if they have to. JONATHAN GRUBER: Right. So it's actually married women, because men were already working 40 hours. They can't-- there's no-- So think about a married man in 1975. OK, men didn't raise their kids. Men quite frankly didn't give much of a shit about their kids, OK? Men just worked. That's what men did in 1975, OK? They worked, and they worked their 40 hours, and then went home. OK, maybe they worked less or more than 40 hours, but certainly, the notion of saying, "Well, the wage went up. Maybe I'll take more leisure," never really crossed a man's mind in 1975. Because what were they going to do? They have no one to play golf with. They didn't want to spend time with their kids. What were they going to do? Whereas women had a real substitution possibility, OK? This was an era women were entering the labor force. There were real opportunities for work, but it was also fine to hang out at home. You had-- a lot of your friends were hanging out at home. You could take care of kids. There were a lot of things to do. So women had a much larger substitution effect than men, OK? Because men-- remember, what's the substitution effect? It's about the next best alternative. For men, there was no next best alternative. It was just work. Basically, between 9:00 to 5:00 on a weekday, there was nothing else to do, OK? For women, there was other things to do, which is, you can hang out with friends who weren't working, or you could take care of the kids. Yeah. AUDIENCE: But what about like working overtime? JONATHAN GRUBER: OK, well, let's-- but once again, if I'm a man, you might think that I could then-- but then once again, if I work-- the substitution effect could work that way for overtime. But let's talk about just the decision to work at all, in some sense, or the decision to work sort of your first 40 hours. Overtime is hard, because then you get paid more, et cetera. OK, now let's go to the other side. Let's go to the income effect. So let's not say this is zero. Let's say it's small, because this is big and this is small. Because you can work a little bit overtime or something like that, and some men did care about the kids. I'm obviously being facetious. So it could be, some men were willing to spend time with their kids, et cetera. OK, now let's go to the income effect. For whom is the income effect going to be bigger, men or women? For whom is the income effect going to be bigger? Yeah. AUDIENCE: Maybe men. JONATHAN GRUBER: Because? AUDIENCE: Because they have a goal of like, they need x amount of money to just provide for their families. So if they get this huge raise in wage, then they become wealthier, and they could start doing more leisure in the week. JONATHAN GRUBER: Exactly. There's actually two reasons it's men. One, you're more likely to have your target income. Two is, you can't have an income effect if you don't work. The income effect is proportional to how hard you are working. If you weren't working, then there's no income effect, right? Income effect is essentially-- the income effect for labor is essentially the hours times dH dy. What Manny said was the reason why dH dy might be bigger for men than women, because they have these targets. More relevantly, if women weren't working, they didn't have dH, so this is zero. So the income effect is zero. So for men, this was big, and for women, this was small, OK? Put this together, and what does it suggest about the relative shapes of labor supply for men and women? Someone raise their hand and tell me. What does it suggests what the labor supply curve would look like for men and women in this era? OK, given the intuition we talked about here, what does it suggest the female and male-- the married women labor supply and the male labor supply curve should look like? You guys can get this, come on. Well, let's talk-- what did we talk about? We talked about the substitution effect. If the wage goes up, it leads to more leisure, which means it leads to more labor supply. By the income effect, if the wage goes up, it leads to less labor supply. So for men, with-- for women, with a big substitution effect and a small income effect, this suggests a standard steep upward-- standard upward-sloping supply curve. Think of the income effect being zero. Then we get the standard substitution effect. We know the sign of that. So for women, this suggests an upward-sloping supply curve, just like a substitution effect suggests a downward-sloping demand curve. For men, it's not clear. You could very much get a Giffen effect here, because basically, there's not much option for substitution, but they might work a lot less if they get rich, OK? So that is sort of this-- what I like with this example-- it's hard, but I like that this example sort of illustrates how substitution and income effects can come together to get a bottom line answer. What do we know? What we know is that actually, evidence is that female labor supply was very elastic, that circa this era, female labor supply was in the elasticity of between 0.5 and 1. That if you raised women's wage by 10%, there was a 5% to 10% increase in their labor supply, which is pretty not elastic-elastic, but reasonably elastic, OK? Whereas for men it was pretty much zero. It wasn't negative. It wasn't positive. It was basically zero. Basically, men just worked 40 hours and then went home, OK? So basically, in an era where for women, the labor supply was very elastic and of the standard direction, higher wages lead you to work harder, an upward-sloping supply curve. But for men, it was pretty much a vertical supply curve, maybe even a bit backward bending, maybe even a wrong sign supply curve. But pretty much, you could think of it as zero, OK? Now, what do we think has happened in the 40 years since these two numbers? So elasticity of woman of between 0.5 and 1, and men of zero, what do we think has happened to these two numbers in the 40 years since these studies, and why? What do you think has happened to these elasticity estimates and why? Yeah. AUDIENCE: Are we talking about these together? JONATHAN GRUBER: Let's talk about women. What do you think has happened to the female estimate? AUDIENCE: Probably gotten less elastic. JONATHAN GRUBER: Because? AUDIENCE: More of them are working in a primary role. JONATHAN GRUBER: Right. Well, first of all, this is going to come down, because in fact, it's now more standard just to work, right? In fact, now, for a woman today, in many communities, it's like being a man in 70s, which is if you don't go to work, there's no one to hang out with, OK? So basically, this is going to get smaller. And they're more of a primary winner in the family. This is going to get bigger. So in fact, female labor supply has fallen more to like about an elasticity about 0.2. It's actually fallen over time. Now, for men, the question is, do you get the opposite effect? Actually, men sort of care more about their kids now, and there's more sort of activities going on during the day, but in fact it hasn't. In fact, male labor supply still is pretty inelastic. What's happened is kids are now in childcare. So basically, we've gone from a world where, as wages went up, women went-- men worked. Women either worked or didn't work, depending on the wage, and if they worked, the kids went in childcare. Now men work and women work, and kids are in childcare. And that's basically the change, the evolution of the labor-- roughly speaking, obviously. Still, female labor force participation is only about 70%, OK? Many women still do stay home and raise their kids, and are in and out of the labor force, OK? But by and large, we moved to a world with just overall less elastic labor supply. Yeah. AUDIENCE: Between the average two-income household is richer now, or-- JONATHAN GRUBER: No. The average-- well, OK, we're going to get into this when we talk about income distribution. What this has done is allowed the average two-family household to tread water. So it's, the average two-family household today has the same income as they did in the 1970s. Why? Because workers earn a ton less in real terms than they did, and that's facts about inequality we'll come to, that basically, the average family in America, despite having-- going from the wife not working to the wife working is no better off they were 40 years ago. And that has lots implications we'll talk about, OK? So any other questions about that? So let me end with one final example, an application, OK? Which is to the problem we have in the world of child labor. It's a huge problem around the world, is kids being forced to work. It was a huge problem in the US till the 20th century. It's a huge problem around the world, because A, work can often be dangerous and bad for their health, but B, they can't be going to school and having the opportunity better themselves. If a kid is spending all day working, then that kid is destined to a life of working in the same crappy job, because there's no way to get the skills that allows them to grow and go further. Now, one-- we will talk in the next few lectures-- in a few lectures about international trade. And one criticism of international trade is people say, "Well, if you allow these developing countries to sell more stuff to the developed world, that will-- they'll put the kids to work more." So if we have free trade and Vietnam can suddenly sell a bunch stuff to America, that's more kids they;re going to put to work making that stuff. So one common argument you hear against free trade is it's bad for kids, but in fact, that argument is not necessarily right, because it ignores an important point. Manny? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: No, that's a different issue. The point-- that's right, but the point it ignores is free trade makes families richer. And the families are richer, they may want to buy more education for their kids. So on the one hand, it's true. Free trade makes kids more valuable in the labor force. On the other hand, it makes family richer and they want more education for their kids. So to look at that two Dartmouth professors did a study, who looked at Vietnam, and looked at what happened when Vietnam liberalized trade in rice. So let's go to figure 15-6. Now, we haven't gotten international trade yet, so I'm just going to sort of hand wave through this. You don't need to really understand this graph, except what the bottom line is. OK, what happened was before trade liberalization of Vietnam, before 1989, you could only sell rice made in Vietnam in Vietnam. So what that meant was the supply of rice was s sub v. The demand for rice was d sub v, and the amount of rice sold was q sub v. And kids worked in the rice paddies. When they liberalized trade, suddenly Vietnam could sell to a much larger market. They could sell to the world market, d sub w. That's a bigger market. So they were able to shift up their supply curve and sell more rice. They could sell more rice, because now they're selling to the whole world, not just to Vietnam. You don't need to notice this in the graph so much intuition. If you give someone a bigger market, they're going to make more stuff, OK? Yeah. AUDIENCE: But doesn't that also put them in competition in other countries, whereas if it was just like-- if each country is just selling to themself, then Vietnam would have-- JONATHAN GRUBER: No, they liberalized in the sense that they let it send out. I didn't say they let more in. AUDIENCE: Oh. JONATHAN GRUBER: OK, but we'll come back to international trade, OK? So basically, the point is, there was this demand shock that allowed them to sell more rice. So what effect does that have on the market for child labor? Let's go to the highly complicated last figure and let me walk you through this. Here is the market for child labor, OK? On the x-axis is the amount of child labor. On the y-axis the wage of kids, OK? We start at point one, initial demand and initial supply, wage 1, L1. Now we liberalize trade, and that leads to more demand for child labor, because we want to produce more rice. So that shifts us out to D2 and point two. So we have more child labor. That's bad. But what this ignores is families are now richer, and with the income effect, they will buy their kids education. They'll pull their kids out of working and put them in school. That's represented as a shift to the left of the supply curve. So we move from point two to point three through the income effect. Families are now richer. And indeed, if the income effect is large enough, you could move to point four. You could actually have a reduction in child labor. Why? Because the benefits of more kids working in terms of producing more rice is exceeded by the value of the firms of taking-- of the families of taking the extra money they're making and putting it into education for their kids. And in fact, the studies showed that we did move to a point like point four, OK? We actually found that child labor fell when they liberalized trade, that the intuitive argument, that gee, if they sell more, more kids are going to work, it's wrong. That in fact, when you sell more, yes, more kids-- demand for more kids, but families are so rich, they put their kids in education rather than their fields, OK? And that is a wonderful sort of counterintuitive story of how what-- I'll talk about economies like free trade, how free trade can actually have an unexpected positive effect. We might think it's negative. And there's a question. Come up if you want to talk, but we've got to end now. So thank you for saying a minute extra, and I will see you guys on Wednesday.

MIT_1401_Principles_of_Microeconomics_Fall_2018

14_Oligopoly_II.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: OK, I want to talk today about-- continue our discussion of oligopoly. Last time, we talked about non-cooperative equilbria, but in the start, we said, gee, life would just be better off if everyone would just cooperate. And someone even asked me, "Well, why don't they just cooperate?" So let's talk about that. Let's talk about cartels. What happens when firms try to cooperate-- achieve the cooperative equilibrium in oligopolies? Now, clearly, this is going to be the best outcome. So the fixed examples-- let's go back to our example of last time, American United. Recall, last time, we said that demand was in the form 339 minus Q. Price is 339 minus Q. And that the marginal cost was 147, OK? Now, we talked about the fact that if American was a monopoly in this market, they would simply solve the monopolist's problem. They would set marginal revenue, which is 339 minus 2Q, equal to marginal cost, which is 147, and they would get that the optimal Q would be 96. And then reading back off the demand curve, that would imply an optimal quantity of $243, OK? So that was what we got last time. Now, imagine that American is not a monopoly, but American United cooperate. What if they got together and said, "You know what? Let's behave as if we're one monopolist, flying 96 flights and just splitting them equally. We'll do 48, you do 48. So let's cartelize. Let's achieve the monopoly outcome, and we'll also share 50-50 the fruits of that outcome." So in that case, each firm would fly 48 flights at a price of $243, OK? And total profits in the market-- each firm would then make profits of 48 times-- price minus average cost, which is marginal cost, because it's flat-- times 243 minus 147-- or since I'm not like you guys, I can't do that in my head-- 4,608 per firm, OK? So each firm would achieve profits of-- take the 96 flights, split them in 1/2, and then each achieve profits of 4,608. Now, what we can see is that these profits are much higher than what they got in the non-cooperative equilibrium. Remember, the non-cooperative equilibrium, they were each doing 64 flights at a price of-- they were each doing 64 flights at a price of 211. We saw that last time. So what were their non-cooperative profits? Their non-cooperative profits for each firm was 64 times the 211 they were charging, minus the 147 in marginal costs, or their former profits were 4,096. So their profits used to be 4,096 when they weren't cooperating. They've gone up by 12.5% to 4,608 by cooperating. So simply by getting together, saying, "Don't be an asshole. Let's cooperate. Let's figure how to make the most money." Getting together, they solve the prisoner's dilemma, get to the best outcome, and make a lot more money, OK? So the question is, why don't they always do this in oligopolistic markets? And fundamentally, there's two reasons. Now, normal people-- you'll see when I'm done-- normal people will teach them in a different order than I will. But let me start with the two reasons the other economists would teach them. The first reason why cartels don't form is that they're fundamentally unstable. Cartels are fundamentally unstable as long as firms are self-interested. Each individual firm in a cartel has an incentive to cheat, and that's because they essentially can solve the monopoly problem of poisoning by cheating. Let me explain how that works. It's best to see this through numbers. Let's imagine that we start in this cooperative equilibrium, 48 flights each and profits of 4,608. And now let's imagine, that quietly, American increased its number of flights from 48 to 50. American says, "Well, I know I agreed 48, but on the sly, I'm going to fly two more flights and hope they don't catch me." OK? Well, what's American's profits? Well, their quantity is 50. Q sub A is now 50. What's the price? If they're going to do 50 flights, what's the price? Yeah? I'm sorry? It's 280-- the price is-- let's see if you got that right. I've got to check my notes. No, if the price was 243 when they're a monopoly, now they do two more flights. Remember, they're adding two flights to the total. There used to be 96 flights. Now it goes to 98 flights. So the price falls to 241, OK? So you were thinking about them as alone, but remember, there's still United doing the flights too. They're still doing 48 flights. So there were 96 flights total. Now it goes to 98. So the price falls to 241, OK? So what's their profits? Their profits are now 50 times 241 minus 147, or 4,700. Their profits have gone up. OK, let me back up and do it again because I went fast, OK? They say they're going to do two more flights. We have to respect the demand curve. So there's going to be two more flights. The price has to fall. If the price falls, then the price has to fall to 241. They now make profits of 4,700, OK? Well, if United is caught with their pants down and continues to do 48 flights, what does United make? Well, United's still doing 48 flights, so the profits of United are 48 times 241 minus 147, or 4,512. So American's profits are up and United's profits are down through American cheating. What happened? How by cheating did they drop? What's the intuition of why them cheating drove their profits up and United's down? And in fact, if you add these up, lowered total profits in the market. What's going on? Yeah. AUDIENCE: It's in the quantity. You're lowering the price that you can sell each unit for, but for American, because they're increasing the number of units they're selling, they still make a greater profit, whereas as United has stayed the same, because each unit is sold for less, then they're making less of that. JONATHAN GRUBER: That's almost right. You got most of it, but there's one key wrinkle that's important, which is, a monopolist would have the same argument. The key thing is, what stops the monopolist from raising the price from where he is? Yeah? The poisoning effect, but think about the poisoning effect. Who does the poisoning effect affect? Everyone in the market. Everyone sees a lower price, but only American gets more flights. So they essentially get the benefit. It's like you said, they get the benefit of the extra flights, but only 1/2 the penalty of the poisoning effect. So for them, it is optimal to lower the price and sell more, because they share the negative effect, the negative part with United, but they get all the positive part. Once again, a monopolist, when you try to sell more and lower the price, there's a positive part, which is sell more units, but a negative part, which is the poisoning effect. Well, here, American gets all the positive part and only 1/2 the negative part. So they make money by cheating, OK? Well, of course, United knows this. They saw the price go down. They know American's cheating. So United wants to cheat, and the whole thing breaks apart. So cartels are unstable, because by cheating, you get all the benefit but only part of the cost. And so cheating is incentivized in a cartel, and therefore, cartels will break down. They're not stable, OK? So that's the primary reason economists say we don't see cartels. The other reason people like to bring up is this little thing, they're illegal. But you know, we don't let stuff like that bother the economists. But they are illegal. That's another reason why we don't see cartels. In the late 1800, cartels were quite common. In the late 1800s, big industries like oil and railroad industries came to be dominated by a few large firms, and they tried to become cartels, but it kept breaking down. So in the late 1800s, with oil companies, Standard Oil, and the big railroad companies, they kept trying to have cartels and it kept breaking down. So they come up with this idea. They basically said, "Look, we can't trust each other. So we're going to-- every firm is going to turn over all its decisions to a common trust. And there'll be a trust that's got representatives from every firm on the board. But we will publicly commit to what we're going to produce at what price, and therefore, we can make sure there's no cheating." So essentially, every firm's still involved. They're all on this trust board. But they're making that decision in a way that's at least public to them, not public to the public, but public to them, so they can make sure they're not cheating. So they formed these trusts, and essentially cartelized, and made huge profits. And it worked. It solved the stability problem because cheating could be observed more readily. Now, would it solve this for very long, we don't know, because the government-- the public got pissed, and the government came in and passed what's called antitrust laws. And antitrust laws are laws which do not permit the cartelization of oligopolistic industries in this way. So let's talk about antitrust laws and how they work or how they don't work. I want to do a couple examples. One example is the movie industry. Now, the movie industry, you know, is a classic oligopolistic industry. There's a few players. There's new players. A24 is huge now, what didn't exist 15 years ago. But by and large, there's sort of a few players you've heard of which dominate the industry, OK? And the way the industry works is movie companies make-- they produce the movies and then sell them to movie theaters that show the movies. They show a variety of movies at any one time. But what happened was in the '30s and '40s, the production companies started buying up the movie theaters. And what they do when they bought the movie theaters, they said to the movie theater, "We now own you and you will only show our movies." So they say to the movie theater, "We now own you, movie theater on the corner of Lincoln and Kennedy streets, and we're MGM, and you'll only show MGM movies." And essentially, what that meant was essentially, they were taking over, monopolizing, a given distribution network. And they essentially carved it up. They agreed, OK, you get these theaters. We'll take these theaters. And essentially, that was the way they formed their cartel, was through distribution. And the federal government jumped in and said that that was an antitrust violation. The federal government sued and won. And so that industry, that was broken up. But did that mean-- that didn't mean folks stopped trying to cartelize. It just meant they stopping being so obvious about it. So later moves to cartelize were more hidden. So for example, in the early 2000s, airline industries were in big trouble, because oil prices were going way up due to the Iraq war and other factors. Oil prices were going way up, and the airline industry was in trouble. So in 2004, British Airlines and Virgin Atlantic had secret talks about essentially cartelizing the cross Atlantic market from the East Coast to London. And what they did is they said, "Look, if we sort of obviously set our prices together, people are going to notice. So to do this instead, we're going to add fuel surcharges to the bill. We're going to say, 'Oh, oil is getting more expensive. Your price haven't changed, but there's now a fuel surcharge on your bill.' And that fuel surcharge is going to be something people won't pay attention to because they won't notice that we're rising it together." And these fuel surcharges rose quickly from $10 to $120 per flight, and essentially rose in lockstep. They essentially coordinated, but tried to hide it by making the coordination not over the sticker price, but over this thing that's sort of at the bottom of your ticket, which is the fuel surcharge. So this worked for awhile, but then what happened? Well, what happened was the prisoner's dilemma. Was it lawyers for-- lawyers for Virgin Airlines started worrying they were to get busted. And they said, "Well, if we go to the feds first and bust British airlines, maybe we'll get a better deal." So they were essentially the prisoner that ratted. So Virgin Atlantic was the prisoner that ratted, and the whole thing broke down. There were penalties, more on British Airlines, because Virgin Atlantic ratted on them. But just like the prisoner's dilemma breaks down, it broke down in reality. And so that was something not where the law really worked, but where the cartel was unstable. And the end result was Virgin Atlantic paid no fees, paid no penalty, and British Airlines paid more than $500 million. So British Airlines clearly did not study the prisoner's dilemma, and did not realize that they should have gone first in ratting out Virgin Atlantic. Now, that said, sometimes cartels operate openly in the public and get away with it, OK? Let's talk about probably the biggest open cartel perhaps in America today, the National Football League, OK? The National Football League-- football is the most popular sport in America, the most profitable sport in America. There are 32 teams, and they're essentially 32 businesses whose job it is to produce football wins, OK? Now, these businesses have a huge incentive to collude with their fellow business, however, because they can-- because of television rights. So if the New York Giants and the New York Jets competed over the television rights for their area, they would compete away the profits that could be made by getting a big contract. If they collude and say, "No, you can only-- we will only agree to a contract together," they could get a higher price for that contract, because it's either you give them the contract or you're out. If they competed, TV companies would compete against each other and fight the price down, OK? And in general, actually, this goes more than this. The League sells the rights to televise games as a package. So in fact, the National Football League literally sells, explicitly says, "We have a cartel of 32 teams. We are selling the monopoly right to televise these games." Somebody's got Sunday. Somebody's got Thursday. Somebody's got Monday night, et cetera. But it's a monopoly product they're selling. Now, how did they get away with that? Well, basically, actually 1957, they were busted. That's a long time ago, I think even before I was born, that long ago. OK, they were busted and the court ruled that the NFL was violating antitrust laws, OK? Now, that was 1957. That was 60 years ago. The NFL still makes about $40 billion on its television contract. What happened? Congress just exempted them. Congress said, "Well, you know what?" We know they violate antitrust, but we're going to pass a law which exempts them from antitrust law and let them do it. So it proves basically that Americans like football more than free markets, and basically, we now have a cartelized football industry that-- because Congress basically exempted them from the laws, OK? So those are some examples. Yeah? AUDIENCE: Is this also true for other sports teams? JONATHAN GRUBER: Other sports teams, it is not as-- they are-- it's largely true for other sports leagues. It's largely true. Some of it aren't as explicit as football, but it's largely true for the other sports leagues as well. AUDIENCE: What about on the international level, with things like soccer, that are-- JONATHAN GRUBER: I don't know, actually. I presume it's-- I mean, I don't know they have international antitrust laws. I'm not sure how that works with international leagues. It's a good question. OK, now, another form of cartels-- we talked about cartels and how companies have incentive to put them together. Actually, sometimes, the government can make a cartel. Yeah? AUDIENCE: How well [INAUDIBLE] working? JONATHAN GRUBER: How's what? AUDIENCE: How well is OPEC working? JONATHAN GRUBER: OPEC? So OPEC is, as I mentioned in the first lecture, a series of countries that get together to produce oil. It's not working as well as it was when I was a kid. It worked really well. Because more countries-- A, more countries are cheating, and more countries-- there's more oil being found outside OPEC. But it still works. It's sort of a partially functioning cartel, is I think the way to think about it. OK, so let me actually-- let me go on and talk about-- let's do one more example about a time when a government made a cartel. Here's one more interesting example. So in the early 1980s, before the early 1980s, the US dominated the car production business. Starting in the late '70s, Japan started making huge inroads into car production, and by the early 1980s, we're in recession and car manufacturers in the US were really pissed that Japan was taking so much of our market. And we'll talk in a couple of lectures about international trade and all those sort of issues, but put those issues aside. Right now, you just have this issue that car manufacturers wanted to limit the amount of Japanese cars to come into America. Now, you guys have been reading in the paper about international trade and why economists typically don't like limiting international trade. And Reagan was a standard Republican, the party of free trade. So Reagan said, "Well, we're not going to limit the cars that Japan wants to send in, but we're going to tell Japan, if you guys were willing to agree to a voluntary export restraint, we wouldn't mind." So we said to Japan, they imposed what they called a voluntary export constraint, which basically said, we won't negotiate a deal with you. You will voluntarily agree to reduce the number of cars you send to America, OK? It's not a government policy. This isn't big government. This is negotiations of a private company, voluntary agreement. OK. Japan happily agreed to this, why? Yeah. AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: No. AUDIENCE: Sold cars at higher price. JONATHAN GRUBER: Because? Because you made a cartel. The Japanese companies used to have to compete with each other to sell the cars in America. Now it's like, OK, you guys get together and limit how many cars you send. They're like, great, you've given us an ability to form a cartel, by essentially telling us, get together and figure out how you're going to sell this many cars to America. So essentially, this voluntary export constraint essentially cartelized, and no company could cheat. So if you had a cartel, and a company tried to cheat, they couldn't sell the cars to US. US wouldn't let them come into the US. So essentially, the US provided them a way of enforcing their cartel. What happened? Well, the average price of a Japanese car in America went up by $1,200, OK? US auto profits did go up, but US consumers lost out by way more than producers gained. And on net, the estimates are that US consumers were about three billion-- overall, US was about $3 billion worse because of this policy. Just examples of how different government policies can interact with the cartelization of industries. Yeah. AUDIENCE: Company price matches, isn't that sort of like making a cartel? Because the other company would see it. They're going to price match. They wouldn't want to set a price lower than what the-- JONATHAN GRUBER: Well, it's a great question, and you're pointing out this is not a solid line. And in some sense, the question is, if it's true that-- so for many years, tobacco industry worked this way. There was one large player, Philip Morris. Philip Morris would sort of raise the price that everyone would match. Now, as long as there's no evidence that they agreed to do that, that is not illegal. If there is evidence they agreed to do that, it's illegal. But as long as it's just like, no, this is the way it's going to work, then that's not illegal. So basically, that's sort of an implicit cartel. Now, once again, what's holding it together? Nothing. A company could cheat and try to charge less. But basically, that essentially-- they figured they were working better as a cartel, and essentially, it was hard to-- there was no way to bust it. Yeah. AUDIENCE: I mean, if a company says to consumers like, "If you can bring in a lower price, we'll sell it for that price." JONATHAN GRUBER: That's sort of a different-- I mean, you could imagine you would need every company to do that. That would be a cartel enforcing way if every company had that deal in a market. But if one company had that deal, it doesn't enforce the cartel. Every company needs to have that deal. And so the question is, if every company on their own, OK, we've decided we're going to have that deal, that would essentially be a way of trying to bring-- enforce a cartel, but I think that would be hard to say it was an antitrust violation. Good questions. Other questions? OK, so now, let's ask why do we care about all this. We care about all this because it matters for ultimately what matters to us in this class, which is economic welfare, OK? So now, let's go to a second thing I want to cover, which is comparing the equilbria. We've now covered three types of market structures, perfect competition, monopoly, and oligopoly. Now, let's compare them, and I want to compare them in two ways, quantity sold and profits per firm, profits earned per firm. And we're going to stick with the United, American market, OK? We know, if this is a monopoly, if this is a monopoly, if they can perfectly cartelize, then there'll be 96 flights total. Each will fly 48 flights, and profits per firm will be 4,608. OK, we solved that already. That's the cartel outcome. The non-cooperative outcome, which we call the oligopoly outcome, is they each sell 64 flights. We solved that last time. And as we solved here, they each make profits of 4,096. What's the competitive outcome in this market? What's the competitive outcome in this market? First of all, what's the price? Somebody raise their hand and tell me. If this is a perfectly competitive market, what would the price be? And then what would the quantity be? Yeah. AUDIENCE: 147. JONATHAN GRUBER: Price would be 147, because in a perfectly competitive market, price would be marginal cost. So profits would be? AUDIENCE: Zero. JONATHAN GRUBER: Zero. And quantity would be 339 minus 147, or 192, OK? So here we have, for a given market, a nice table which lets us compare the three different possible outcomes. And what you see is essentially, the more you can monopolize, the higher your profits but the smaller the market, OK? So basically, three lessons here. First of all, generally speaking, the oligopoly outcome is somewhere between the monopoly and perfectly competitive outcome. Where in between, take 14, 12. If 14, 12 was bad, it's how you figure out where in between this outcome comes between these two. It's all about game theory, OK? So that's what's exciting about game theory is, this is a wide range, and game theory is a set of sophisticated tools that let us pin down where in this range companies will end up in a realistic case. 0.2 is, the more you can monopolize, the higher your profits will be. But let us come to welfare. Now, I haven't computed social surplus here. But here's the cheat, I don't really need to. I don't need to because essentially, roughly speaking, social welfare is proportional to the quantity sold. In other words, we know that in a perfectly competitive market, this is-- the welfare-maximizing quantity is 192. We know that there's 192 flights that maximize welfare, because that's the competitive outcome. What we're saying is the more-- Oh, this shouldn't be 64. It should be 128. That's my bad. 128. It's each doing 64. We know that as we monopolize the market, there are fewer and fewer flights. Therefore, we're creating a deadweight loss. Essentially, deadweight loss is proportional-- actually, it's sort of exponentially proportional-- to the gap between the quantity sold and the competitive quantity. Welfare is maximized here by definition. We proved that. So any reduction for that means it increased deadweight loss. So the more you reduce quantity, the more you lower welfare. So essentially, as we go down the column, we lower profits but raise social welfare. Yeah. AUDIENCE: When we're-- I'm not really sure. Are we talking about an oligopoly that acts like a monopoly? JONATHAN GRUBER: Yeah, cartel/monopoly. AUDIENCE: So there's [INAUDIBLE]---- JONATHAN GRUBER: Yes. But otherwise, you do 96, your profits would be twice that. But the bottom line is that essentially, you've got essentially more profits in this market. OK? Other questions about that? So the bottom line is, the more competitive the market, the higher the welfare, but the lower the profits, OK? So that's kind of our bottom line of how we think about this. Questions about that? OK, next I want to cover-- we've only covered the case of two firms. What if there are many firms? After all, most oligopoly markets are not just two firms. We've talked about cars and movie producing studios. There are many firms, OK? Well, the Cournot model is super hard to do when there's more firms, but there's no reason you can't. It literally just becomes three equations and three unknowns or four equations and four unknowns. Literally, as you can see, you could simply see, if you take that model and add more firms, it just expands the state space. It becomes impossible the graph, but you could solve it. Eventually, you've got n equations and n unknowns. The key bottom line result is that as the Cournot-- as the number of firms gets large, the Cournot equilibrium approaches the competitive equilibrium. That is mathematically, if you solve this-- you don't have to solve this-- but the bottom line condition is, the markup that firms earn is equal to minus 1 over n times the elasticity. In sort of a market-- this is sort of in a symmetric Cournot market of the kind we've been working with. The markup is 1 over the number of firms times the elasticity of demand. So think about this for a second. Imagine there is one firm. Then this equation says the market is equal to minus 1 over the elasticity of demand. Where have we seen that before? That's the monopoly condition. That's the monopoly market condition. So when n equals 1, this is an equation we've seen before, the monopoly market condition. When n equals 2, the firms are making 1/2 as much. When n equals 3, a third-- it goes a fact-- factor third, et cetera. What this says is n approaches infinity, we approach a competitive outcome. We'll never get there, but we're asymptoting towards a competitive outcome, which basically says-- you know, it's sort of like my point about contested markets. You get a market that's sort of competitive enough, you're going to shrink the markup as more and more firms enter, OK? So that's sort of a general condition that we could derive, that shows that as more firms are in, then you get a lower markup. Now, I want to make-- there's actually-- But this actually understates the case, for an important strategic reason, which is, more firms lowers the markup in a-- more firm lowers the markup in a Cournot non-cooperative model. But more firms also makes a cooperative model harder. So this is for the non-cooperative model. The non-cooperative model, your profits fall as there more firms. But it also gets harder to cooperate as there are more firms, because there are more people you have to trust, more people you have to keep hold of. So a great example of this, actually, for a long time, mercury, the stuff we use in thermometers and such, only was found in Italy and Spain, in the mines in Italy and Spain. And they had a cartel between the two countries to sell mercury. What happened, other countries discovered mercury, and they couldn't keep the cartel together, and the price of mercury fell a lot. With the question about OPEC, similar thing-- OPEC was much more successful in the 1970s, when essentially, the only source of oil were basically these Arab nations that form OPEC. What happened over time is we discovered more oil around the world, in particular, in Russia and in the US, which has sort of broken the power of this cartel to a large extent. So the reason why a bigger market moves us towards a competitive equilibrium is that it makes it harder to maintain a cartel, OK? Now, let's actually-- the other issue I want to cover here is, I want to talk about what does all this teach us about a key policy issue, which is the issue of mergers? What does everything we've learned here tell us about thinking about mergers? OK, we know about mergers. It happens all the time. Two companies merge. Well, it turns out when companies merge and they're large enough, the federal government regulates that. The federal government gets a vote on whether that merger is going to be allowed to go forward, either the Department of Justice or the Federal Trade Commission, depending on what industry it is. So the federal government has to decide how to evaluate whether two firms merging is a good idea or not, and essentially, what it comes down to is a simple trade-off, economies of scale versus market power. The benefit of two firms merging is economies of scale. If two firms have sort of redundant production processes and they merge, they can be more efficient. There can be positive economies of scale for merging firms. So it's cost efficiencies, basically. Economies of scale deliver cost efficiencies. On the other hand, the more firms merge, the more this n goes down, and the more markets go up, and the worse it is for consumers. So the trade-off is, do you want to reduce-- is reducing n worth it in terms of the economies of scale? Or in other words, does the producer efficiency go up enough to make up for the potential loss to consumers of this less competitive market, OK? People understand that? Now an interesting case of this, which has got very big implications for all of us in America, is hospital mergers. During the decade of the 2000s, there was a rash of hospital mergers, where hospitals said, look, here's a classic case for economies of scale, because hospitals have what's called a peak load problem. They have to have empty beds. Hospitals can't be full all the time, because there might be a car accident, and people need beds. So by definition, it's inefficient for hospitals to be 100% capacity. Hospitals want to have excess capacity. The problem with that, there's two hospitals next to each other, each with excess capacity, that's inefficient. It'd be more efficient to have one hospital, one merged hospital, then they just manage the proper amount of excess capacity. And hospitals made this argument, and we basically approved any hospital merger that they wanted in the 2000s. Well, what happened? What happened is the hospitals lied. They kept both hospitals open, kept all the empty beds, and just raised prices. So essentially, the hospital mergers did not deliver any of the economies of scale they promised, but did deliver a lot of the market power we feared. So a huge cause of the increase in medical spending in the 2000s was these hospital mergers, which essentially took a lot of the competitive pressure out of the medical market and didn't really deliver economies of scale. And this is the hard part of being a regulator. Most of what public policy economists do in the world is regulate. All over the world, there are thousands of economists employed all over the world, hundreds of-- tens of thousands, whose job it is to make regulatory decisions of this nature, and they're really hard. Because we've drawn nice, clean theoretical models here, but we have to know what's epsilon. You know, how much-- what's epsilon, to figure out the effect to consumers. What are the economies of scale? Will they exploit those economies of scale, et cetera? So these are really hard and interesting decisions. Now, let's go on to the last topic I want to cover today, which is price competition. Price competition. Now, the models we've been discussing so far have been what we call quantity competition, that United and American compete on how many flights to send, and then the demand curve tells them what they can charge. But in fact, in many markets, that's not how firms compete. In fact, we even mentioned it. Someone mentioned about best price offers, et cetera. They don't compete on quantity, they compete on price, and that's a different model, named after another French economist. A model of Bertrand competition is a model of price competition, is a model we call Bertrand competition. This model says that basically, two firms compete over what price to set, and then the quantity is determined by the price that results from that competition. So they don't compete over quantity. They compete over price, and the demand curve then tells you the quantity, OK? Now, in this case, what's really striking about Bertrand competition is that unlike the Cournot model, under Bertrand competition, two firms can be enough to get us to the competitive equilibrium. Why? Why do we only potentially need two firms to get to the competitive equilibrium? Yeah. AUDIENCE: I'll do you lower. JONATHAN GRUBER: Why don't you explain a little bit more what you mean? AUDIENCE: One firm [INAUDIBLE] the other one what price are lower, and like-- JONATHAN GRUBER: Exactly. As long as there's profits to be made, it's like our entry/exit decision, right? As long as there's profits to be made, I'm going to come in at a price one penny below you, make one penny less profits, and steal all the business from you. So if there's perfect competition between firms in a Bertrand sense, then you only need two firms to get to the competitive equilibrium in theory, OK? So it's a very different idea. Cournot competition, we need many, many firms to get close to this competitive outcome. With price competition, because firms are always kind of competing on one penny below each other, in a market that's otherwise competitive, you can actually drive the price to competitive price through-- essentially, you can actually drive the price down to marginal cost. It's sort of like I talked about contestable markets and as long as there's profit to be made, someone would enter. Here, as long as there's profit to be made, someone will lower their price, and that'll keep happening until price equals marginal cost. So in Bertrand competition, you actually can get close to or at-- to the competitive outcome with a small number firms. Now, two points to make about this. Your first point is, well, holy shit, how do I say which one of these to use? You've just taught me-- you've just spent a lecture and 1/2 on this fancy model, spent 37 seconds on this model. How do I know which one to use? You didn't write down any math, so I don't know what to do. I'm freaking out. OK, how do I know which one to use? Well, the bottom line is, we're not going to ask you to do much math about Bertrand competition, other than sort of the intuition about competing over price. The more relevant question is, how do you think about the situations where Cournot competition is more likely and Bertrand competition is more likely? So what do you think? In what types of markets do you think Cournot competition would be more likely, and what kinds of markets do you think Bertrand competition would be more likely? Yeah. AUDIENCE: Wouldn't the Bertrand be really efficient in an elastic market? JONATHAN GRUBER: Well, no. Elasticity is the same. So basically, elasticity is going to have a similar effect in both. It's going to basically drive the price down in both, OK? Because the elasticity is higher, it drove the markup. Bertrand, it's going to drive down in both. So it's not actually about elasticity. It's something about production processes. What type of production processes are going to lend themselves to price competition versus quantity competition? Think about it this way, if I offer a price, what do I have to do? AUDIENCE: I think the better the production is dominated by the capital costs, or is it [INAUDIBLE] the variable costs? JONATHAN GRUBER: That's roughly speaking right. Basically, if there's long lags in production, I can't do price competition. So if I say I'm going to compete, people would say, "Great, I want all your product." I'm like, great, you can have it in a year. That doesn't work. So things like auto companies are going to have a hard time with pure price competition, because if Toyota says, "OK, I'm $1 less," someone says, "OK, great. We want a million Toyotas tomorrow," they can't do it. So things which are capital-intensive lagged production processes it's going to be hard to have pure-- real life, of course, there could be some mix of these. But it will tend more towards quantity competition, because you're really going to know what you're going to sell, because that's sort of-- you can't just infinitely supply it. With other things like cereal sales, where you can sort of immediately crank up a million more boxes of cereal in like a day out of your production processes, there would be more likely to be price competition. Things with small production lags, then you'd be more likely to have price competition, because if you lower the price, all of the sudden, you dominate the market. You can meet that demand, OK? So essentially, we can think about price competition as being more likely the smaller the production lag, or maybe the less capital-- it's not really about capital intensity, because you can have a capital-- you can create things quickly. It's more about production lags. Now, we're never going to ask you to tell us which is right, and of course, in reality, it lies somewhere in between. But this just gives you a sense of kind of when one type of competition is more likely than the other. Yeah? AUDIENCE: Does it have anything to do [INAUDIBLE] to protect the cereal in the grocery store? JONATHAN GRUBER: Great segue. You've jumped ahead to the last point I want to make, which is, imagine you're in a Bertrand competition world, like with cereal. That's a pretty awful world if you're a producer, OK? Basically, that's where your markup's tiny, because any time you try to raise the price, you get undercut. What can you do? Well, we've already gotten the answer. What you can do is you can engage in product differentiation. You can engage in product differentiation. OK, so basically, the reason why you're in Bertrand competition is because you're selling the same thing. Once I'm selling something different, I take on the features of a monopolist again. So if I can get consumers to not think of my good as identical to my competitors, then I can price above marginal cost, and people would still buy it. The reason Bertrand competition drives price to marginal cost is because people view the goods as identical. But if they don't view the goods as identical, then I can keep price above marginal cost, OK? And the example-- breakfast cereals is the perfect way to illustrate this. So back around World War II, there were essentially basically like three types of cereal, OK? There was Cheerios, there was cornflakes, and Quaker oats. OK, that's basically what cereal was, not very exciting. Now, but by 1970, there were more than 150 breakfast cereals to choose from, including some which are variations of Cheerios and variation of cornflakes. In fact, you could all say in some sense, all cereal is variations of Cheerios, and cornflakes, and oats. And then-- and moreover now, if you go to a store today, you can actually buy generic versions of brand name cereal. You can buy Oatios or Marshmallow Mateys, which are Lucky Charms, or what's the other one? I love buying these big bags. You guys ever buy these generic Lucky Charms, Marshmallow Mateys. Generic Captain Crunch is like, you know, Ahoy Matey or something. I don't know. They've got these generic things which you can buy, which are really just the same. So essentially, what you do-- what companies want to always do, which are in Bertrand competition, is always try to product differentiate, always try to figure out a way they can create a market where they can price above marginal cost, OK? So for example, let's take General Mills, a company that makes Cheerios, OK? They're making Cheerios, and then all of a sudden, Oatios and stuff started coming along and they weren't making money. What do they do? They created different kinds of Cheerios, like Apple Cinnamon Cheerios. General Mills did not create Apple Cinnamon Cheerios out of the goodness of their heart. General Mills created Apple Cinnamon Cheerios because they were getting killed in the Cheerio market, and so they tried to differentiate by having a new product on which they could charge a higher price, which is Apple Cinnamon Cheerios. Now, how do we feel about this? Well, it's not clear. On the one hand, by introducing Apple Cinnamon Cheerios, General Mills was able to push its price greater than marginal cost. And as price pushed above marginal cost, quantity sold in the market falls. It created deadweight loss. Quantity fell, and that's bad, OK? On the other hand, Apple Cinnamon Cheerios are quite good, OK? So it actually ends up being much like our patent discussion, which is, essentially, by differentiating, they've had two effects. They've lowered consumer surplus and welfare by pricing above marginal cost, but raised it by shifting up the demand curve, by creating a new good that people want. Yeah. AUDIENCE: Isn't like-- doesn't like consumer [INAUDIBLE] not necessarily have to happen, because then different people have different demand curves, and the demand curve for like Apple Cinnamon Cheerios is not-- hasn't been this good. JONATHAN GRUBER: No, the point is-- OK, it's another way of stating my point. Even if the demand curve-- let's say there's a new demand curve for Apple Cinnamon Cheerios. It's way out, OK? That's great, OK? But still, the fact that they're pricing above marginal cost means they'll sell fewer than they would in a competitive market. If they had invented Apple Cinnamon Cheerios and sold it at marginal cost, they'd still be way better off. So the trade-off is, essentially, how far out do we shift demand by creating this new product, versus how much do we restrict sales by pricing it above marginal cost? So essentially-- now, and now what we have is a market with about five firms that dominate it, and about 5,000 brands of cereal, OK? So it's constant product differentiation. And essentially, this is the trade-off with product differentiation, which is we get reduced sale-- we get deadweight loss, because they're not pricing it at marginal cost, but we get new products that people might like. Yeah. AUDIENCE: So if there's a product with like-- JONATHAN GRUBER: Differentiation. AUDIENCE: Yeah, differentiation. Is brand loyalty between things like Adidas and Nike, or like, Apple and Android, where there are various [INAUDIBLE] where you feel strongly towards one, is that good for both of the two things That you're choosing between? JONATHAN GRUBER: Well, actually, that's really interesting. It depends on whether that brand loyalty is based on innovation or blind faith. So this is, once again, this gets into the deep, interesting issues of industrial organization. You talk about game theory, which is, if I can create brand loyalty in a way that makes you slightly better off, but keeps you in my brand forever, then that might be worse. By creating it in a way that makes you much better off, that might be better. So essentially, that's why, for example, you may have noticed you may be getting one or two credit card mailers. Are you guys getting inundated with credit card offers? OK, it's not because they love you guys. It's because if they hook you now, then you might stick with that credit card later. So trying to exploit-- trying to get you, they're trying to give you a good deal now to get you hooked later, so they can charge up closer to monopoly price later on. So essentially, there's a trade-off, which is, if that loyalty is based on real differences in quality, that might be good. If it's not, it might not be, but the welfare gets very murky. It's a good question. Other questions? OK, so these are exciting real world topics. It's more reasons to go on and study more economics. But let's stop now. We will come back and we'll start talking about factor markets on Monday.

MIT_1401_Principles_of_Microeconomics_Fall_2018

23_Market_Failures_I_Externalities.txt

[SQUEAKING] [RUSTLING] [CLICKING] JONATHAN GRUBER: Externalities, so, so far in the class, we once again remember the big picture. We started with the first fundamental theorem overall for economics, which is that the competitive market will maximize total social welfare. Then we said that will not be true under conditions of market failure. Remember, market failure doesn't mean market collapse. It means when there are barriers to the market achieving this first best outcome, OK? One barrier is imperfect competition. One barrier was imperfect information. A third barrier to welfare maximization or a third source of market failure is externalities, OK? That's the third source of market failure, and we're going to talk about that today. What is an externality? Let me be very clear about the definition. An externality occurs whenever one party's actions makes another party better or worse off, OK? Let's do it-- let's make it a person, whether my actions make you better or worse off, but I don't bear the consequences of that. So, when my actions make you better or worse off, but I don't bear the consequences, then there's an externality. So let's talk about that in the context. Let's start with the classic example of externalities, which is a negative production externality, a negative production externality, OK? The classic example is you've got a river. On that river is a steel plant. That steel plant produces steel, OK? But, as a byproduct of that production, it dumps sludge into the river, OK? You guys remember The Lorax? So it's basically The Lorax, OK? So, basically, the steel plant dumps sludge into the river, OK? That sludge floats down the river and kills the fish. Those killed fish mean that fishermen cannot make as much money fishing on the river, OK? So, basically, there's sludge coming out of the factory, and that sludge we're going to make the assumption is directly proportional to the production of steel. So we're going to say, for every unit of steel produced, there's one unit of sludge that emits on the factory just to make this model easy. So, every unit of steel, there's one unit of sludge. That sludge flows down the river and kills the fish. Unfortunately, there are fishermen down this river who are trying to catch fish, and that hurts their livelihood, OK? So this is a classic example of living by an externality, because the steel plant's behavior is imposing a cost on the fishermen. Their sludge is imposing a cost on the fishermen, but the steel plant doesn't bear any consequence of that. They just dump the sludge and forget about it, OK? So that's what we mean by negative. It's a negative externality because my actions are hurting you. The steel plant's actions are hurting the fishermen. It's a production externality because it comes out of the production process, in this case, for steel. So that's what we mean by negative production externality. It means that, when one party's production adversely affects another party, but the party doing the production doesn't bear any consequences of that, then that's a negative production externality. So what effect does that have? Let's go to figure 22-1 and talk graphically about how we think about production externalities, OK? This is the market for steel. In figure 22-1 is the market for steel, the quantity of steel on the x-axis, the price of steel on the y-axis, OK? The market is initially in equilibrium at point A. That is where demand, which is the downward-sloping blue line, equals supply, which is upward-sloping blue line. Now, as we said, in a perfectly competitive market, demand represents the marginal willingness to pay for the good, which is equal to the marginal benefit that consumers get from consuming the good, OK? The marginal benefit of consuming the good is the marginal willingness to pay. And that's what's represented by the demand curve, OK? We're not going to touch that here. We're going to leave that alone, OK? The supply curve is the firm's marginal willingness to supply, which is their marginal cost, OK? So, in the perfectly competitive market where marginal benefit equals marginal cost, we get equilibrium. And that yields the welfare-maximizing outcome. The market succeeds. It does not fail, OK? The difference now is we're now going to drive a wedge between privately perceived benefits and social benefits. So, for the consumption of steel with the demand curve, we're going to see the benefit to individuals is the benefit to society. They're one and the same. That's what we've assumed all course. But, for the supply of steel, we're going to say wait a second. The benefit to society is different than the benefit to the steel producers. Or I'm sorry. The cost to society is different than the cost to the steel producers. The cost to the steel producers, which is their private marginal cost, is the supply curve, but the social marginal cost adds the damage they're doing to the fishermen. That is society encompasses all the actors in society. It encompasses both the steel plant and the fishermen. So the marginal cost to society is the cost of producing the steel plus the marginal damage being done to the fishermen. So social marginal cost equals private marginal cost plus marginal damage. Social marginal cost equals private marginal cost plus the marginal damage. And we see that as the red line. What that means, from a welfare perspective, what we care about is social marginal benefits and costs, not private marginal benefits and costs. So what that means is the social optimum, the welfare-maximizing optimum, is actually at point C. Point C is the welfare-maximizing optimum where the social marginal cost equals the social marginal benefit, OK? And, therefore, we overproduce. What's happening is the steel company, not considering the damage they're doing through production to the fisherman, produces too much. The steel company produces at the point where private marginal cost equals private marginal benefit, which, in our case, equals social marginal benefit, OK? That's the private market decision. But, in fact, it should be producing at the point where social marginal cost equals private marginal benefit, private marginal benefit-- equals social marginal benefit, OK? And the point where social marginal cost equals social marginal benefit is lower production. Why? Because lower production avoids-- reduces the damage being done to the fishermen down the river, OK? So the optimum, from society's perspective, is point C. In other words, there is a market failure. The private market is not delivering the welfare-maximizing outcome. And we can see that creates a deadweight loss. The deadweight loss is the units that are traded that are socially inefficient. Why are they socially inefficient? They're privately efficient. If you see Q2 and Q1, before we introduced externalities, we'd say, well, it's a shame if they don't get produced, right? If we ignore externalities, we'd say, Q2 and Q1, well, they have a benefit higher than their cost. So they should get produced. But, actually, in a world of externalities, their benefit is lower than their cost because their cost incorporates the damage done to the fish. So there's a deadweight loss. Critically, remember, you've got to know how to draw these deadweight loss triangles. Remember, deadweight loss triangles always point to the optimum. The deadweight lost triangle is the area ABC, OK? It's drawn-- it's units that are sold where the social marginal cost exceeds the social marginal benefit. And that's that deadweight loss triangle. So there's an inefficiency arising from the fact the private actors do not account for the social implications of their actions, OK? Questions about that? So we have here a classic example, perhaps the classic example in all of economics, of a market failure. The classic example is a market failure happens when the social implications of your actions are different than the private implications since people maximize their own private well-being. That's what Adam Smith sort of taught us. The notion of the invisible hand is that the market acting in its own interests will deliver the best outcomes for society. We're saying, no, that's not true if the market's own interests has implications for other parties that are not accounted for, OK? Now externalities don't have to just be on the production side. We can also have negative consumption externalities. That would be a case where my literally consuming a good makes you worse off. My consuming a good makes you worse off, OK? So let me start with a simple question. Let's start with a perfectly competitive market. If I consume a good, I raise demand for that good. That raises the price. Is that an externality? In a perfectly competitive market, it's not. Why isn't that an externality? If I consume the-- if I want to consume the good, the price goes up. Everyone pays a higher price. Why is that not an externality? Yeah? STUDENT: Because you also bear the cost. JONATHAN GRUBER: Because you also bear the costs. An externality only occurs when you don't bear the costs of your actions. When you want a car, and, therefore, the price of cars goes up, you pay that higher price. Externalities only occur when you don't bear the consequences. So, in general, consumption externalities don't happen through causing higher prices. Consumption externalities happen more directly when my consumption affects you. So the best example of this would be smoking, OK? When I smoke, it affects you. It affects you in a number of ways. Most directly, if I smoke in this classroom, you get secondhand smoke, and you get ill as a result. But that's not all. It affects you because, if I smoke, and I get sick, and my health care costs go up, then, well, I work at MIT. All my fellow MIT employees bear those costs because we all share health insurance. And, when I retire, all society bears those costs because those costs are paid for by the Medicare program, which is financed by taxation. So my health care costs are an externality. Secondhand smoke is an externality. What are some other externalities from smoking? What are other externalities that can occur from smoking? Yeah? STUDENT: Environmental damage from the production. JONATHAN GRUBER: Well, that would be a production externality, OK? We're going to leave that alone for now. I'm just talking about from consuming cigarettes. From consuming cigarettes, what else-- what other damage comes? So I make you-- I may make you sick through secondhand smoke. I might raise health care costs. Well, I might raise health care costs, OK? What else is another externality? What else does smoking do? Well, it turns out there are 100,000. This is a number which I triple check because I still can't believe it. 100,000 people every single year die in fires caused by smokers, not in the US, worldwide, which is a crazy number. But, if you think about how tightly packed slums are in developing countries, one person falling asleep with their cigarette burning can kill thousands of people, OK? That is an externality because my action to smoke has killed you, OK? And I'm clearly not going to compensate you for that. So that's another externality, OK? What about the fact that smokers are less productive at work. They have to take more smoke breaks. They might get sick more often. Is that an externality or not? The fact that smokers are less productive at work, is that an externality or not? And why or why not? Yeah? STUDENT: Not necessarily because, if they're less productive, they're going to do less work and get paid less. JONATHAN GRUBER: Exactly, it's not an externality if they're paid less. This is the key thing, which is it's only an externality if you don't bear the consequences. If smokers are less productive at work, and they get paid less as a result by exactly the same amount they're less productive, there's no externality. But, if their wage doesn't fully adjust, and, therefore, their lower productivity affects everybody else in the firm or the firm's profits, that is an externality, OK? So this is the deep aspect of externalities. You have to think about whether people are compensating for it, OK? You have to think about that. OK, most importantly, the fact I kill myself by smoking is not an externality, OK? Smokers die seven years earlier on average. Roughly speaking, every cigarette you smoke lowers your life by seven minutes. It's pretty linear, OK? But you know what? If I sit by myself on a rock in the middle of nowhere and smoke until I die, no problem because, in that case, the social implications are the private implications. I've made my privately optimal decision, and there's no effect on anybody else. So it's also socially optimal. There's only an externality if we have one these mechanisms, like if I'm smoking, I'm in the woods, and I start a fire, and the fire company has to come. That's an externality. But, as long as I just sit by myself, and I don't bother anybody-- I just smoke until I die-- there's no externalities, OK? So externalities come through the effects on others. So let's think about the externalities of smoking. Let's think about a negative consumption externality, OK? Here we have the market for cigarettes. On the x-axis, we have the number-- I'm sorry, figure 22-2. On the x-axis, we have the number of cigarettes, quantity of cigarettes consumed. On the y-axis, we have the price of cigarettes per pack. We have an initial equilibrium at point A, which is where the private marginal benefit equals the private marginal cost. Here we're going to assume there's no externalities from producing tobacco. Let's assume there's no sludge produced, whatever. That's a separate issue, OK? There may be production externalities too. We covered those. You already know how to think about those. But here let's assume there aren't any. Let's assume the social marginal cost equals the private marginal cost, no production externalities. But there is a consumption externality. Every cigarette I smoke is bad for society. What that means is the social marginal benefit is below the private marginal benefit. The social marginal benefit is the private marginal benefit minus the marginal damage I'm doing. MD is the Marginal Damage, the marginal damage I'm doing. That's estimated to be about-- absent secondhand smoke, the damage of smoking is about $0.50 a pack. The secondhand smoke part is really hard, and the estimates are anywhere from $0.01 to $2 per pack. So that's hard to know how big that is, OK? But, certainly, we have this negative consumption externality, which is that, basically, every pack of cigarettes I smoke is worth at least $0.50 less to society than it's worth to me because I have these external effects on society and perhaps a lot more than that. As a result, I should smoke-- I choose to smoke at point A, but the social optimum is point C. So, once again, I've created a deadweight loss. Once again, I'm over consuming. There's overconsumption here. Just like there was overproduction of steel. There's overconsumption and a deadweight loss because there are units that are privately optimal to consume, but not socially optimal. Now, externalities, this is kind of a fun topic because it is interesting. I talk-- so let's talk for a second about secondhand smoke and whether that's actually an externality. So almost all the damage of secondhand smoke is not done by smoking in a crowd. It's done to family members. Almost all the damage of secondhand smoke is done to family members. Mostly, it's that you make your family members sick by smoking. Is that an externality? When or when is it not an-- under what conditions might it not be an externality? Yeah? STUDENT: Well, I guess it wouldn't be an externality if like let's say you die. And then the consequences of them getting sick doesn't affect you at all. Or, well, I guess it-- JONATHAN GRUBER: No, no, then it-- no, then it would be. OK, so what-- under what conditions-- under what conditions would it not be an externality? When-- yeah? STUDENT: If your-- like, if your family gets sick, and you are impacted by that. JONATHAN GRUBER: Yeah, if I care about my family, in particular, if I maximize family utility, then it's not an externality. If I maximize not my own utility, but my family's utility, then I will essentially internalize, internalize, the externality. Just like a lower wage means that I bear the consequences for being a less productive worker, if I care about my whole family's happiness, and I make my kids sick by smoking, then my smoking decision will actually reflect the total consequences for my family. I will smoke only if it's optimal for my family for me to do so. It doesn't mean I won't smoke. It just means I must-- I'll have to enjoy it enough that it's worth making my kids sick, OK? It doesn't mean that's an incorrect decision because, after all, the odds are you don't make your kids that sick, and, you know, you might like smoking a lot, OK? It's just that you will internalize the externality because you only smoke to the extent that it is optimal for the whole family. So it's not necessarily an externality, OK? So we'd like, actually, to test whether it's an externality. There's actually a clever test of this, which is there's a test of whether people maximize family utility, which is, if people maximize family utility, what would be the implications of giving a father $1 versus giving a mother $1? If both the father and the mother are maximizing a family utility function, the same family utility function, then should it matter whether I give $1 to a father or $1 to a mother? No, it shouldn't because that's $1 to the family. We're maximizing family utility subject to a family budget constraint. It shouldn't matter who gets the dollar. So one test of family utility maximization is does it matter who gets the money. And it turns out it does a lot, OK? So there was a great test of this. In the UK, they used to have a tax system where, essentially, there was a credit they gave for every kid. And the way the credit worked is there would be a check sent home, OK? Then they changed it. So, instead, they said, well, instead of sending a check home, we're just going to add it into pay, into wages. So, instead of getting a check sent home, we're just going to raise your wages. Well, it had no effect on family budgets. They literally just changed it. The difference was, back in-- this was in the '70s. Men worked and women didn't. So, when the check came home, women controlled it. But, when it was in wages, men controlled it. So, if there's family utility maximization, it shouldn't matter. But it turned out, as soon as they changed the way they paid it, spending on kids went down, and spending on drugs and alcohol went up because, basically, guys don't care about kids as much as women do. Sorry, guys. It's just-- I don't know what the evolutionary biology is. At least, in the '70s, they didn't. So that's a rejection of family utility maximization. Who had the dollars actually mattered. It's kind of a neat study for how you think about these theories. So it suggests that, secondhand smoke, probably, people don't perfectly maximize family utility. So there probably are some externalities, OK? So that's a negative consumption externality. There can also, of course, be positive externalities. So let's talk about a positive consumption externality. Let's talk about my neighbor. I don't get along with my neighbor, OK? And, partly, it's because my neighbor has a habit of starting big projects and leaving them half done. And, about 25 years ago, 20 years ago, he started a big project to landscape his yard, created these huge mounds of dirt that I look at directly from my kitchen, and then stopped. So, for 20 years, I've had to stare at huge mounds of dirt, OK? Now let's think about my neighbor's decision to go ahead and get rid of those piles of dirt. Let's say that that would cost $1,000. So the cost is $1,000, OK? And let's say the benefit to my neighbor from doing so is clearly less than $1,000, or he would have done it. Let's say it's $800. So that's why he leaves those piles of dirt because the cost is $1,000 to remove them, and it's only worth $800 to him. But what he's not accounting for is, if he removed them, there'd be a positive benefit to me of another $300. So, actually, the total social benefit of removing the dirt piles is higher than the social cost. So, from a social perspective, he should do it, but he doesn't because, privately, it's not optimal to do so. So that is a positive consumption externality. Yeah? STUDENT: Does this mean you'd be willing to pay him like $200 to do it? JONATHAN GRUBER: Well, this leads to a very deep question, which is can't all externalities simply be internalized. Let's take this example. Why can't I just go over and offer to pay him, OK? Well, and, in fact, with any example, we can do that. Why can't the fishermen just go pay the steel plant, OK? Why can't you pay me not to smoke in class, OK? Indeed, with any of these externalities, there's a question why can't they all be internalized. And, indeed, there's a school of thought, which suggests that externalities aren't really a problem. They can all just be internalized. But, of course, that's totally wrong, OK? Let's start with a hard example. Let's talk about the biggest environmental externality, which is global warming, OK? With global warming, every single time you drive, you are bringing people of Bangladesh that much closer to being under water. How could you possibly negotiate that? How could you possibly negotiate where the people of Bangladesh would come and say, well, drive a little bit less so I don't go under water? OK, that's not happening, OK? But, even with these simple cases, think about this. Why can't I just go to my neighbor and offer him $200? There's three problems. First problem is there's the fact that I don't really know what his costs are and what his benefits are and that I might-- I don't want to offer more than I have to to get him to do it, OK? The second problem is he doesn't know how I feel. So there's an information asymmetry, which makes negotiations hard. There's a third problem too, which is it'd just be deeply weird to do that, right? I mean, it's just not how society works. You think about a classic case of an externality you've probably all run into, which is your neighbor playing their music too loud, OK? If your neighbor plays their music too loud, that is an externality on you, OK? Now, in principle, you could go to your neighbor and say, well, look, I'm studying for a test. This test will raise my grade by 10 points. A higher grade in this class will raise my earnings by $1,000. So I'm willing to pay you $83 to stop playing music because I've calculated my lifetime earning effect of your playing the music. OK, even at MIT, that would be sort of deeply weird to do. So what do you do? You either shut up about it, or you go yell at them. OK, but yelling induces-- it's not necessarily an efficient way to resolve it because maybe they really want to play the music. The efficient thing would be, if it's worth more than $83 for them to play the music, they should get to play it and just pay you $83. But, in fact, that doesn't work. So, in fact, private solutions to these problems simply do not work, OK? It's just hard to figure out how you can really get people privately to internalize these externalities because negotiation is difficult and because it's just socially awkward. There is a famous apocryphal story told of a famous economist who was on a flight and wanted to get work done and couldn't because the person next to him wouldn't stop talking. So they actually offered them $10 to shut up. I don't believe that actually happened, but, you know, it makes for a good story, OK? So that's a positive consumption externality. Finally, we have positive production externalities. The classic example of a positive production externality is R&D by private firms, OK? When a firm does research and development, they don't just create learning for themselves. They create learning that might benefit other firms as well, OK? And, indeed, the best economic estimates suggest that the social returns to $1 of R&D are 2 and 1/2 times the private returns, that every dollar of R&D a firm does benefits society by 2 and 1/2 times how much it benefits the firm. And, as a result, firms under invest in R&D, OK? As a result, firms under invest in R&D, OK? And they are-- and that's leading them to-- that's leading to too little R&D being done in society, and that affects all of us because that affects growth. That's what my new book Jumpstarting America is all about. It's about why we need the government to come in and invest more in R&D because firms under invest, OK? Essentially, firms don't account for the spillovers that their investments have on others. So a great example used in the book is the example of when two drug companies were racing to invent statins. You guys are too young to know about statins, but statins are basically a cholesterol-lowering drug that's a miracle. It saves hundreds of thousands of lives every year. Best guess is about 200,000 lives a year are saved by people lowering their cholesterol through being on statins. Statins were being invented in the early 1980s by two rival drug companies, Merck in the US and Sankyo in Japan. And they were racing to develop these statins. And then Sankyo suddenly stopped. And Merck found out through the grapevine it was because some dogs got sick in drug-- in the animal trials. And so Merck went to Sankyo and said, hey, we heard some dogs are sick. What's going on? Sankyo said we're not going to tell you. You're our competitor. So Merck offered to pay them money. They offered to partner. Sankyo said no way. This is private R&D, and we don't want to share it with you. So Merck stopped too. Five years later, some academics got permission to run trials on statins. It turns out that what happened to the dogs had nothing to do with the drug. They were fine. It's totally safe. And statins were invented and save 200,000 lives a year, but five years after they should have. Literally, one million people died because there was-- they could not benefit from the spillovers of R&D knowledge, OK? This is an example of what we mean by under-investment in R&D, OK? So we have externalities can be negative or positive. They can be production side or consumption side. Questions about that? Yeah? STUDENT: So, regardless if they're negative or positive, externalities still create deadweight loss. JONATHAN GRUBER: Yes, regardless if they're negative or positive, externalities still create deadweight loss because, if they're positive-- I should-- Jason, just sent me a note. Next year, we should have in the handout a positive graph. Well, I'll just do it here. I can-- I'm capable of drawing a graph. So let's think about R&D. OK, here's the quantity of R&D, OK? Here's the cost of R&D, the price of R&D. So basically-- and here's the-- so here's going to be the demand for R&D, OK? Here's going to be the demand, which is private marginal benefit. And here's the supply, which is the private marginal cost, OK? And let's just say that there's no externalities from actually-- this is a consumption externality. This is a production externality. So, basically, the point is that, when a firm does R&D, they do it until the benefits to the firm equal the costs to the firm. So they do an optimum-- they do an amount of R&D of, you know, Q1 at a price of P1. But what they're missing is that, in fact, what they're missing is that the costs to them are actually well below, well below, what it truly costs because they are benefiting others by doing it, OK? So the supply curve, the true social supply curve, is down here. The social marginal cost is the private marginal cost minus the social benefits that we get from doing that R&D. So, as a result, they should be doing Q2 R&D, but they're not. They're doing too little R&D. And that's making a deadweight loss. The deadweight loss, remember, is with reference to the optimal point. This is the deadweight loss. This is deadweight loss from under producing R&D. The difference between-- no, I'm sorry. I got that wrong. These triangles are always confusing. In the drawing, I got that wrong. OK, it's the difference being the social marginal cost, between the social marginal cost, and the private marginal benefit. So, basically, let me think about this for one second. This is always a little bit hard to do. So, basically, there are units-- there are units they under produce. So, essentially, what their-- they should be producing this many units, yeah, between the-- it's where the private marginal-- yeah, it's this. I had it right. That's the deadweight loss, OK? Yeah, so, basically, what you have is you're going to have underproduction of R&D, just like you had overproduction of steel, OK? Yeah? STUDENT: Why would the social marginal cost go down when the social marginal benefit goes up? JONATHAN GRUBER: Because the marginal benefit is what's the marginal benefit of another dollar of R&D. That's basically-- that's, essentially, what's the knowledge created, OK? You could view this either way, but the idea here is I'm producing R&D. I'm not consuming R&D. This is the benefit of consuming R&D. This is sort of the benefit of society of consuming that R&D. So you think of it as lowering the cost of producing the R&D is sort of the way we think about it, OK? But the main thing is not-- yeah? STUDENT: So is the marginal change between the social curve and the private curve, is that linear? Or, as it grows-- JONATHAN GRUBER: That's a great question. I'm always making it-- I'm making it constant. I'm assuming marginal damage or marginal benefits are constant. STUDENT: So, mathematically, whichever triangle you choose is the same? JONATHAN GRUBER: Yeah, exactly, because I'm making it constant. And, in fact, you could imagine it could be growing or shrinking, OK? Now, with this in mind and realizing that private sector can't solve this for the reasons we talked about, let's talk about the role-- let's talk about the role of government, government solutions. So, once again, remember the basic logic of this class, which is that-- the basic logic of the class, which is that if-- the basic logic of the class is that the market knows best unless there's a market failure. If there's a market failure, the private market will deliver a deadweight loss. Now we have to ask can the government actually make it better. Remember, we talked about monopoly regulation. The government may or may not make it better, OK? Information asymmetries, it may or may not make it better. Same thing with externalities, the government may not make it better. So let's talk about how the government, in theory, could make it better, OK? Well, there's two ways the government could make it better. One way is by regulation. So go back to figure 22-1, OK? The government could literally regulate and could say, look, we know the optimal level of steel to be produced is Q2. We're just going to tell you to produce Q2. That's it. Problem solved, OK? We just say, hey, steel plant, we know the optimum is Q2. You produce Q2. Problem solved. The problem with that is that requires the government to know quite a lot. The government needs to know both the supply and the demand curves to figure out where Q2 is and what they should regulate. Let's say all the government knows is the damage being done, and let's say the damage is linear, or they can approximate it as linear. Then there's a much easier solution, which is a corrective tax. What if the government came in and said, look, I don't where demand and supply curves are? I don't really know. It's really hard to figure it out. All I know is that, for every unit of steel you produce, which is a unit of sludge, you're killing $100 worth of fish. That's what I know. That I can study environmentally, OK? What if I simply tax the steel plant by $100 for every unit of steel they produced? That is, if I imposed a tax on the steel plant-- STUDENT: Don't you mean sludge? JONATHAN GRUBER: One unit of steel is one unit of sludge in this example. So I'm going to tax every unit of steel they produce, which is the same as producing one unit of sludge, OK? What if I impose that tax? Well, let's look at figure 22-3. What does that do to the firm's decision, OK? Well, before the government came in, the firm was producing at point A where their private marginal costs equaled the private marginal benefit, which is the social marginal benefit. Now the government comes in and levies a tax. It levies a tax at exactly MD, the marginal damage. What does it do? It shifts their private marginal cost curve to the social marginal cost curve. It has caused the firm to internalize the externality because now the firm is paying an amount exactly equal to the damage they're doing to society. So a corrective tax can cause the firm to internalize the externality. Corrective tax caused the firm to internalize the externality, OK? Essentially, a corrective tax by the government can get us to the right answer because it gets firms to do the right thing, OK? It gets firms to pay attention to the social costs, not just the-- not just the private costs. Similarly, we could do same thing with a positive externality. What could the government do with a positive externality? Yeah? STUDENT: Subsidize production. JONATHAN GRUBER: It could subsidize by-- so imagine I knew exactly how much social benefit there was per unit of R&D. If I subsidized the firm doing R&D, then I would lower their costs, right? If I offered them a subsidy of this amount, their cost curve would shift down. I would lower the cost. So R&D would get to the right point. So a corrective tax of the amount of damage gets firms to internalize the externality. A corrective subsidy of that amount gets firms to internalize the externality or individuals to externalize the internality. And we can get to the optimal outcome by the government imposing a corrective tax or subsidy of the right amount. Now we could of course also get there with regulation. It's just a lot harder. Questions about that? OK, this is our first example of good taxes, OK? Taxes have been bad throughout this course. The role of taxes has been distortionary to the economy. We haven't talked about it a lot. We'll talk about it more in a couple of lectures. They've been distortionary to the economy. This is saying, no, a tax can actually play a positive role because a tax can correct a market failure. Now, as always, if the tax is set incorrectly, it could make things worse. OK, if you set a tax that was five times the marginal damage, it would make things worse. But, if you set it correctly, it can make things better. We're offering the potential for government intervention to make things better here. Questions about that? OK, so what do we have? We have a situation where the private market is not delivering the optimal outcome. The private market is not delivering the optimal outcome where it seems hard to think of private solutions, but where a government solution, either through regulation or easier corrective taxation, can get us to the optimal outcome. So now let's ask how does this actually work in practice. And let's talk about two examples. Let's talk about environmental externalities and health externalities. Start with environmental externalities. And, of course, the most important is global warming, OK? Currently, the amount of carbon dioxide in the atmosphere is at its highest level in 400,000 years. Basically, every year becomes the hottest year on record almost linearly. Almost monotonically, every year is the hottest year on record. We're heating up. Scientists predict that it's possible-- the central prediction is that temperatures will rise by more than 2 degrees Fahrenheit by the end of the-- I'm sorry, by more than 2 degrees Celsius by the end of the century, but it could be more than that. There's actually, currently, the best estimate is that there's as much as a 10% chance that temperatures go up by 10 degrees by the end of the century, which would end human life, basically, in most of the world, OK? There is a non-trivial chance we're all gone by 2100, not my problem, largely not your problem, certainly your kids' problem, OK? OK, by and large, we are basically-- we are basically-- we have a-- we know for sure there's going to be negative implications. Basically, we are essentially-- unless there's a radical new technology invented, Bangladesh is gone. It's over for Bangladesh. Cape Cod is gone. Much of Florida is gone. That's already happening, OK? At this point, the question is can we actually stop the entire East and West coasts and much of the South from disappearing as well and many other countries in the world from disappearing, OK? That's the sort of decision we have to make now, OK? So, basically, this is a classic negative externality because, that negative situation, you were not thinking about that when you filled up your car last time. You're not thinking about the fact that the fossil fuels you're emitting are contributing to that, OK? It's a classic negative externality. So what can the government do? Well, the natural solution would be corrective taxation. The natural solution would be to have a carbon tax, to literally say this is the amount-- we actually have a pretty good sense from engineering models what the cost of carbon is, what the marginal cost of carbon is, OK? And we could literally impose a tax on the use of carbon of that amount. I think it would amount to something-- I don't know the numbers these days. It's something like between $0.25 and $0.50 a gallon of gas. So it's a lot, but it's not-- we've seen gas prices in the last year go up and down by that much. OK, that's not an outrageous amount, OK? In Europe, they already have gasoline taxes well above that level, OK? So corrective taxation, in principle, could be the answer. We could literally just use engineering models to compute the costs, social costs of carbon. We could put a tax on it. And then at least we would stop global warming going forward. You know, Bangladesh may be gone, but we can maybe save a lot of the rest of the world, OK? So that's in theory. In practice, in 1994, Bill Clinton proposed a $0.03 gas tax and lost Congress, OK? In practice, people don't like gas taxes. It's very hard politically in the US and other places. And that is why the world has turned to a different approach, which is quantity regulation, which is say, look, in practice, we should have a global carbon tax. In theory, we should have a global carbon tax. In practice, that's hard. That's why we have negotiations. That's why we try to have a global negotiation to try to get a global cap on carbon emissions, actually have a quantity regulation, to actually have a quantity regulation. We started this. The first true global negotiation was in Kyoto, Japan in December 1997. I was fortunate enough to be there for that negotiation. I was in the Clinton administration at the time. And we got to go over and do that negotiation. It was actually pretty neat because they decided I was going at the last minute, and the only plane left to go on was Air Force Two. So I got to fly over with the vice president on Air Force Two, which was pretty cool, super cool. They have really nice seats and stuff. And so I sat down, and the phone next to me rang. And I was like-- I answered it. I was like hello. They're like, hey, John. I'm like oh my god. It was them calling from Japan, but like getting a personal phone call on a plane was super cool. So, anyway, so I went over to Kyoto. I learned how these negotiations work, which is, over five days, I slept four hours. In Japan, they sell-- they sold, at the time, this coffee in cans. So you just chug these cans of coffee all the time and stay awake. And, basically, everyone is so tired by the end they just agree just to kind of get it done. And that's sort of way that negotiations work. So we agreed to the Kyoto global warming treaty, which would have lowered emissions worldwide, but the US did not sign on. The US refused to sign on. There's been continuing negotiations. Most recently, we know about the Paris round of negotiations, which the US did sign on to, but the current administration has pulled us back out of. So we have a problem, which is that, basically, we're heading to this environmental catastrophe, and the world can't agree on actions to take. And it's not really a choice. I mean, we have to do something, or our grandkids will all be under water, or our great-grandkids will all be under water. We have to do something. The optimistic case that we'll do something comes from the example of what's called chlorofluorocarbons. When I was a kid, many, many products were made with what's called chlorofluorocarbons. They were in refrigerators. They were in aerosol sprays, et cetera. Scientists realized they were actually damaging the ozone layer, which protects us from ultraviolet rays from the sun. And people were like, yeah, whatever, much like they are with global warming now, yeah, whatever. But then a fucking hole opened up in the ozone layer. Like, literally, it was like, oh my god, there's a hole in the ozone layer. And 180 countries got together almost overnight and banned chlorofluorocarbons. Like, literally, almost overnight, they were gone. It was amazing, international cooperation, terrific international cooperation to take an environmental catastrophe on and deal with it. The problem is global warming doesn't quite work that way, OK? By the time we say, oh my god, Bangladesh is under water, it's too late. So the question is sort of how do we get politicians and the public interested in taking something on when we don't have the symbol, like a hole in the ozone layer, to actually represent the damage that's being done. And that is the challenge facing something like global warming, but we have to take it on, OK? So that's environmental externalities. The other big type of externalities are health externalities, are health externalities. I talked about smoking, but, indeed, there are huge externalities levied by a bunch of activities that we do that impact our health. So, for example, drinking, drunk driving causes 13,000 deaths per year, 13,000 deaths per year, OK, to put a sort of blunt face on it, four 9/11s every year from drunk driving plus 400,000 injuries every year from people driving drunk. Consuming gasoline, global warming is a huge externality. Perhaps one of the biggest externalities facing us, the biggest social externality, is obesity, OK? Obesity causes a lot of illnesses that cost a lot of money. Projections are that children born in the year 2000, so about your kid, about your generation, one third of them will get diabetes before they die based on current weight projections. Now, as you notice, looking around this room, that's not a problem of the elite East Coast people, OK? It's not a problem of-- it's a problem of the less educated. It's a problem concentrated more in the South, but, nonetheless, if you look at a number of southern states, the obesity rate is above 35%. Literally, more than one in three people in the state are obese, OK? This is a huge problem, and it's going to cause huge social-- it's going to have huge social consequences for our country. The question is what do we do about these. What do we do about things like smoking and drinking and obesity? And there's essentially-- there's, essentially, four answers. The first is information. Can we just inform people about the damages? And, indeed, this has been shown to work with smoking. OK, we knew smoking was bad for you in about 1954, OK? But we only really got through to people starting really in the 1970s and '80s, but it's had an enormous effect. Smoking has fallen incredibly in the US through that information. But here's the interesting thing, OK? Smoking rates-- smoking used to be 50% in the entire-- every adult, 50% of all adults smoked. It didn't matter race, gender, class, whatever. Now smoking is essentially down to zero among the well-educated and still about 20% against the less educated. So information works, but it works in a very inequitable way. OK, so information is one solution. The second solution is taxation. And, indeed, this has been shown to work for cigarettes once again. Smoking is price sensitive. The elasticity of smoking with respect to the price is about minus 0.4. About every 10% you raise the price of cigarettes, there's about 4% less smoking. It works. In particular, youth smoking is very price sensitive. Youths are very price sensitive because youths have less money. So they're very price sensitive. So it actually works. But that's sort of the easy case. Taxing cigarettes is easy because every cigarette is bad for you. Taxing alcohol is trickier because, after all, most of the damage is done by a tiny share of drinkers. Most of us will consume alcohol responsibly most of our lives and not cause any external damage, OK? But most of the damage is done by a tiny share of drinking. So taxes is trickier. If I proposed a big rise in alcohol taxes, people would say wait a second. I'm responsible drinker. Why are you taxing-- why are you taxing me? So that's a little bit trickier. Not to mention obesity, taxing food is maybe the trickiest of all, OK? So taxes are trickier. You could maybe try-- an alternative thing you could do is penalties. So, instead of taxing alcohol, we could just steepen the penalties for drunk driving. You know, if you killed a few drunk drivers, there would be less drunk driving, OK? But the problem is that's a pretty extreme penalty. And what if you got it wrong? You'd feel kind of bad about killing someone because the breathalyzer didn't work. There was a series of articles, actually, in The New York Times about how terrible breathalyzers are and how inaccurate they are, OK? So problem with penalties is we can't enforce them perfectly, OK? So that's the third solution. The final solution and the one that's really most discussed right now is illegality. What if we just made these things illegal? And this comes to the discussion of marijuana. Should marijuana be legal, OK? Well, illegality is an extreme form of lowering externalities. Now, obviously, when pot is illegal, people still smoke pot. But it's still true, when you make it legal, it's consumed at much higher levels. OK, legality does matter. So, for example, people have done studies. Yes, people under 21 drink, but, literally, if you look at people the day after their 21st birthday, they drink much more than the day before their 21st. Not on the 21st birthday, that's the party. We ignore that. But, the day after their 21st birthday and thereafter, they're drinking at much higher levels than before. Legality matters, OK? It's also true, the day after a 21st birthday, people are much more likely to die in a drunk driving accident than the day before their 21st birthday, OK? So legality matters. So the point is we have a whole series of tools to think about this. And the question is how should we combine and use them. And the answer is take 14.41, and I'll teach you all about it. But we don't any more time here. This just raises the issues to think about with externalities. It's an important topic to think about it. I realize that's a lot to cover in one lecture, but I just wanted to sort of give you a taste for how economists think about and analyze this kind of market failure.

MIT_1401_Principles_of_Microeconomics_Fall_2018

22_Government_Redistribution_and_Taxation.txt

[SQUEAKING] JONATHAN GRUBER: So today, we're going to continue our discussion of equity and efficiency. We started last time talking about the equity-efficiency trade-off. Well, first we talked about why we think redistribution might be necessary, and the striking facts on inequality and poverty in the US. Then we talked about the trade-off, the equity-efficiency trade-off, and the fact that when you try to deal with a problem like redistribution through taxation or transfers, you're going to have a leaky bucket. That there's going to be inefficiencies associated with both taxing individuals to raise the money, and transferring to individuals to spend the money, are both going to cause deadweight loss. So what I want to to today is take that abstract framework from last time, put it in some real terms. By talking first about how taxation works in the US, and the kind of issues we face in taxing people, and then turning to redistribution programs. And concluding with a very positive story of what I call a patch to the leaky bucket, and how we can redistribute in a very efficient way. So let's start by talking about taxation in the US. And when I talk about taxation, I want to talk about two topics primarily. The first topic is, who bears taxes? Who bears taxes? Now, this is a topic that we call-- in public finance, the field I specialize in-- the topic of tax incidence. Now, you might think this is a silly question. If you pay tax, you bear the tax. Why is this an interesting question? And the answer is, it's an interesting question because it's not true that the person that bears the tax pays the tax. That in fact, when you impose taxes, they have complicated effects on multiple parties because of the operation of the market. So as a result, the party that actually sends the check to the government may not actually bear all that tax. And so that's the interesting insight we'll start with today. So to think about that, let's go to figure 22-1, and let's think about the market for gasoline. You've got demand and supply. You've got an initial 100 billion gallons of gas being sold at a price of $1.50. Those were the days, back when gas was $1.50. Now, imagine the government comes in and says, we're going to levy a $0.50 tax per gallon on the suppliers of gasoline. So the gas stations, or the gas companies, or whatever. For every gallon they sell, they will send us $0.50. You literally cut a check to the government, $0.50 for every gallon you sell. What does that do-- and you might stop there and say, OK, well, then the incidence of that tax is-- that tax is borne by producers. That's great. I don't have to worry about it. I'm buying gas. It's those guys selling gas who have to bear it. But you'd be wrong. And the reason you'd be wrong is shown in the right-hand-side diagram, which is to think through the market effect of such intervention. What does intervention do? Well, it has no effect on the demand for gas. The fundamental demand curve. The underlying utility of the marginal gallon of gas hasn't changed. But it has changed the supply curve, because essentially, we've introduced a new marginal cost. The marginal cost of gas has gone up by $0.50 a gallon. Every gallon of gas you produce and sell now bears an extra $0.50 cost. So that's a shift upward of the supply curve. Remember, the supply curve is the marginal cost curve. So if I increase the marginal cost of every gallon by $0.50, that's literally just a parallel shift upwards in the supply curve. Supply curve shifts from S1 to S2. That's the new supply curve. Well, at that new supply curve, you cannot continue to charge the old price. If you can charge the old price of $1.50, people would still want to buy 100 billion gallons of gas, but companies would only want to sell 80 billion gallons. Why? Because the marginal cost has gone up. And we have an upward-sloping supply curve, so at a higher marginal cost, they're going to want to sell fewer gallons. So you have a disequilibrium. What happens is that the gas company adjusts by sliding up the supply curve to the new equilibrium point D, where they sell 90 billion gallons at a new price of $1.80. This is all just standard supply and demand stuff, we've seen that before. What's interesting here is to note that what this means is that part of the tax is borne by consumers. Part of the tax is borne by consumers. What I mean by that? What I mean is consumers used to pay $1.50 a gallon, and now they pay $1.80 a gallon. So the tax has increased what they pay by $0.30 a gallon. Now, they're not sending that check to the government, but it's the same thing. It's the same effect. The point is this tax has increased the amount that consumers pay for gas by $0.30 a gallon. So the incidence on consumers is $0.30. The incidents on producers is a little more complicated. They now get $1.80 per gallon. That's great. But for every gallon they sell, they have to send a $0.50 check to the government. So their net price per gallon is $1.30. So we can see that by taking the new price at point D and subtracting off the $0.50 tax they have to pay, which says that instead of $1.80, they get $1.30. We call that the tax wedge. The tax wedge is the difference between the tax including the price and the tax after paying the price. So it's just a wedge between what people get before they pay the tax and what they get after paying the tax. That's the tax wedge. We have a $0.50 tax wedge here. And so what is the burden of the tax on producers? Well, they used to get $1.50 a gallon. Now they get $1.80 but pay $0.50 to the government. So the burden on the producers is $0.20. So even though the producers are sending the check to the government-- the gas stations, the gas companies sending the check to the government, they actually pay less than half the tax. Now, nominally, they pay the $0.50, but we don't care about that. We just care about where they end up after the market is adjusted. After the market is adjusted, they are $0.20 per gallon worse off than they were before. And consumers are $0.30 per gallon worse off than they were before. This is the fundamental insight of tax incidence, that you can't just look at the paper version of who pays the tax. You have to consider who really bears the consequences of the tax. And that's why it gets interesting. Because the party that bears the consequences is not necessarily the party that actually sends the check to the government. That's the fundamental insight of tax incidence. Questions about how that all works. OK. So bottom line is all you do is you say, look, think about how the tax affects the market. In this case, it's an upward shift in supply curve. Shift the supply curve up. Find the new equilibrium. That gives you the new equilibrium price. And then the burden on the party not paying the tax is the difference between the new price and the old price. The burden on the party paying the tax is the new price minus the tax they have to pay. OK? Now, what's interesting about this is that within this framework-- so the first bold claim I'm going to make is, actually, it doesn't matter who sends the check to the government. The first bold claim-- let me tell you about it. The first bold claim I'm going to make is that the amount set on the check to the government is not the amount you actually bear of the tax. The amount you bear of the tax comes out of this analysis. Here's the second bold claim I'm going to make. It doesn't matter who sends the check to the government. If you impose the same $0.50 tax on consumers of gas in the same market, you get the same outcome. How's that possible? Well, let's look at figure 22-2. Now we have a different tax. This tax is now every time you buy a gallon of gas, you pay $0.50. Imagine the outcry. Imagine if the government had a tax which made gas companies pay $0.50 of a gallon they sold, and they said, you know what, we're just going to switch that and make people pay the $0.50. Could you imagine the headlines? The outcry? Government screwing the little guy to favor the oil companies. But in fact, those headlines would all be wrong. Because in fact, it doesn't matter. To see why, let's look at the diagram 22-2. Now imagine we have a $0.50 tax on consumers. Now the supply curve doesn't shift because marginal cost hasn't changed. But the demand curve shifts down. Because for every gallon you buy, you have to send $0.50 to the government. Well, the demand curve represents your willingness to pay. You are now willing to pay $0.50 less per gallon. If you're willing to pay before at point A, the 100 billionth gallon sold, they're willing to pay exactly $1.50. Well, if they only pay $1.50 and now you send $0.50 to the government, well, now they're only willing to pay $1 for that gas. So what that means is consumers now, with the shifting demand curve, if the price stayed at $1.50, they would only want 80 billion gallons at point C. They would only want 80 billion gallons at a price of $1.50. Why? Because the price to them isn't $1.50 anymore. They pay the $1.50, plus they pay the $0.20. They'd have to pay another $0.50 check to the government. So they only want-- so to them, they don't want 80 billion gallons. So you have to move to a new equilibrium. Once again, we know about these adjustments. We talked about that. You're going to slide down the supply curve, and you're going to end up at a new point D. D is the new equilibrium where the new demand curve, reflecting the much lower willingness to pay, intersects the supply curve. That point, interestingly, is at 90 billion gallons. Flip the page back to 22-1. That's the same quantity we had before. So we get the same effect on the amount of gas sold. What's the difference is the market price. In 22-1, the market price rose to $1.80. Now, the market price falls to $1.30. But the burdens are the same. Think about the burden on the consumer. The consumer used to pay $1.50. Now they pay $1.30. So they saved $0.20. But they have to send a $0.50 check to the government. So the burden on the consumer is $0.30. They save $0.20 on the price and send a $0.50 check to the government. What's the burden on producers? Well, the producers used to get $1.50. Now they get $1.30. So what's the burden on producers? $0.20. Same as before. The point is for any given total tax wedge, for any given total amount of tax, who bears it does not depend on who sends the check into the government. That's irrelevant. The side of the market's irrelevant. All that matters is the underlying demand and supply curves and the size of the tax wedge. As long as you have an underlying set of supply and demand curves and a given size tax wedge, you don't care who actually pays. Because the market will adjust to offset that. And so this is an incredible insight of tax incidence, which is that something-- I'd imagine most of you walk in this room today, and I quickly said to you, does it matter if the gas company pays $0.50, you pay $0.50, you'd say, yeah, it matters. But it turns out it doesn't. Turns out it doesn't. As long-- given a given tax wedge and given a set of supply and demand curves, it doesn't matter who actually pays. And that's the fundamental-- it's another fundamental side of tax incidence. And that's why, when you read articles in the paper about this tax is bad because it's on people, this tax is good because it's on corporations, that's not the way to think about it. The way to think about it is what's the total tax wedge, and what does the market look like? Yeah? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: Excellent point. All of this, of course, is within our standard 14.01 framework. Once you depart from that-- by the way, it's also true even if there's monopoly and stuff. That's still true. So this does not depend on perfectly competitive markets. This is also true in non-perfectly competitive markets. So monopoly, oligopoly will still feature these two rules. However, once you depart from perfectly rational consumers, then things can change. So for example, there's a lot of interesting research on what's called tax salience. And as I said, since I won't have time for a lecture on behavioral economics this semester, I'm going to throw some behavioral economics nuggets at you. But if you find this interesting, take 14.13. It's really exciting. And here's a kind of insight, which is, what if people pay attention to taxes when it's in the price, but not when it's rung up at the register? So they ran a cool experiment in supermarkets in California. They randomly, in some cases, increased the price on the shelf by the sales tax. In other case, they kept the price fixed, and the sales taxes rang up at the counter, like we normally do. And they found that affected demand. That affected how much people wanted the good, even though the net price was the same, because it was more salient when it was built into the price than when it's like at the end, when you just ring it up, you don't pay attention to it. Yeah. AUDIENCE: [INAUDIBLE] research as to whether when the consumer knows that they're being taxed rather than the producer, if that changes how the demand function works [INAUDIBLE]. JONATHAN GRUBER: Well, that's exactly what I said. That's exactly what-- there's two questions. One is how this affects the demand curve. That's what I was saying. That basically, the demand curve-- it's not quite your question. Your question's about sort of a moral, like, do I feel differently if one's being taxed another-- or you're talking about the salience point? AUDIENCE: Maybe like a little bit of both, just how people react when they know for a fact that they're paying the tax-- JONATHAN GRUBER: Yeah. I think we don't know for sure the moral aspect. But the salience aspect, we do know for sure, which is, how the tax is presented affect people's demand, even though it shouldn't. Now, there's another question which I thought you were going to ask, which is, how does the fact that you can see a tax affect how people feel about taxes? So here's a super cool study my colleague Amy Finkelstein did. You guys know E-ZPass, the thing on the highway where you-- OK. So when states moved-- when I was a kid, you just wait in line and pay cash to get through the tolls. Now you just drive through with your E-ZPass. So what's happened is it made the tax less salient. It used to be you'd feel pain every time you go through the toll. Now it's just some bill you get at the end of month. What she found is when states switched to E-ZPass, they rapidly raised their taxes on roads. They raised their tolls, because people in mind as much, so they could raise them more. So there's an interesting politics aspect to this as well. So this matters. Having said that, I'm going to ignore it. But this stuff does matter, and that's why behavioral economics is fascinating. OK? Now, there's a third point I want to make about taxes. So I talked about who bears taxes. I talked about the side of the market. Side of the market is irrelevant. And then the third point I want to make about taxes is it's all about the elasticities. Which is, who bears a tax? For a given tax wedge, who bears the tax is all determined by the elasticities of supply and demand curve. And in particular, inelastic parties get stuck with taxes, while elastic parties avoid taxes. So inelastic agent, inelastic firms, inelastic consumers, they get stuck with taxes, while elastically supplied firms, or elastically demanded goods, they avoid taxes. So to see that, let's look at figure 22-3. Let's go back to our tax on the suppliers of gasoline. Once again, I hope you know by now it wouldn't matter if demanders of gasoline. But it's a little easier to see with suppliers of gas, so let's go back to our tax suppliers of gasoline. So we have a supply curve that's been shifted up by $0.50. Let's consider two markets, one with perfectly inelastic demand, one with perfectly elastic demand. In the market, it was perfectly inelastic demand. We used to sell 100 billion gallons at a price of $1.50. Now we levy a $0.50 tax on suppliers. Where do we end up? We end up still selling 100 billion gallons of gas. You have to, because inelastic. Well, if you're going to still sell 100 billion gallons of gas, then the suppliers can't bear any of the tax. They have to be able to fully pass the tax onto price. That is, they have to be able to charge $2. Think of the logic. Suppliers have to be on their supply curve. The supply curve has just shifted up by $0.50. Therefore, if you're going to sell the same quantity, the price must go up by $0.50. So in that case, who bears the tax? Who bears this tax on gas? Someone raise their hand and tell me. Yeah. AUDIENCE: Consumers. JONATHAN GRUBER: Consumers bear all of it. Consumers bear all the tax, suppliers bear none. The supplier, you sent-- you just sent a $0.50 check to the government, but you're getting $0.50 higher price, so you don't care. Insert 50 Cent joke here. In the gas? I don't know you do a 50 Cent joke. Whatever. OK. So you don't care. Now let's flip the case. Let's imagine demand for gas is perfectly elastic. Now, the supply curve has shifted up by $0.50. Supply curve was shifted up by $0.50. Now what happens is consumers say, look, I don't care how you feel, Mr. Supplier, I'm not paying more than $1.50 for gas. I have perfectly elastic demand. Supplier has no choice, then, but to eat the whole tax. Because if they try to pass any of it onto the consumer, the consumer will bolt. So the new equilibrium is that-- where the price stays the same of $1.50, and the quantity falls. In this case, the producer bears the whole tax. The consumer pays no more than it did before. The gas company gets the same price it did before, but has to send a $0.50 check to the government. So the entire burden is borne by the producer. What is going on? Why would inelastic demand-- what's the intuition here? So with inelastic demand, consumers end up getting stuck with the tax. And with elastic demand, consumers avoid it. Yeah. AUDIENCE: With elastic demand, consumers are willing to pay any price, so if the producers want to put all of the tax on them, they can, and consumers can't do anything about it. But if you [INAUDIBLE] elastic demand, then they can go anywhere else to get gas at $1.50. So if you raise your price, then they're just not going to buy it. JONATHAN GRUBER: Right. Exactly. I like to think of this-- that's a very good explanation. I like to think in terms of almost like negotiating power. It's not the right way to think about it, but I think it's useful intuition. That when you've got inelastic demand, you've got no negotiating power. You just want insulin. You're going to pay anything for insulin. So the supplier knows that, so it's going to make you pay the whole tax. It doesn't matter if you send the check to the government or he sends the check to the government, he's going to pass the whole cost to you. But if you think about fast food, if you think about a tax on hamburgers, if you're going to charge a penny more for hamburgers, I'm going to buy a hot dog or a slice of pizza. Then the supplier knows they're screwed. They have no leverage. You have all the leverage. You've all the negotiating power. So they can't raise the price on you. So the bottom line is, inelastic factors get stuck with taxes and elastic factors avoid taxes. And that's sort of the other lesson we get here as we think about taxation. So at the end of the day, all that matters-- if you think about any tax being imposed in the US, it's never as simple as just a gas tax, often a complicated giant set of taxes. You think about simple-- a single tax, all you need to know is the elasticities, supply and demand, and the size of the tax, and you're done. You can figure out who bears the tax. And the Congressional Budget Office, for example, does these exercises all the time, and has estimates of how given tax changes will affect the income distribution as a result. Now-- and actually, there's a pretty cool study of this. Let me ask the question. We have hospital taxes. We have-- hospital. We have hotel taxes. I've got too much health care on the brain. We've got hotel taxes in many major cities. There's some instance of a hotel tax. You can imagine a diagram, we've got supply demand. How does the incidence of the hotel tax change when Airbnb comes in? Yeah. AUDIENCE: The consumer is suddenly much more able to switch around, and so a lot more of the tax, or maybe all of it, is borne by the hotel. JONATHAN GRUBER: Exactly. What you see-- and how would you test that? How would you test that? Yeah. AUDIENCE: Comparing prices in areas with different taxation rates? JONATHAN GRUBER: Exactly. What you find is when Airbnb comes in, hotel taxes are passed less onto price. Hotel owners are more bearing the taxes compared to consumers, because consumers are more elastic. So that's an example of how this can matter. OK. So now, let me say one other thing about taxes. The last thing I want to say about taxes is what to tax. And this is a fundamental debate that goes back to the 17th century, which is essentially, should we tax people based on what they produce? That is, their income. Or should we tax people based on what they consume? Their consumption. The philosopher Thomas Hobbs in, like, 17-whatever, talked about, why should we tax people based on the fruits of their labor? Let's tax them based on what they take out of society-- that is, what they consume. And this is a debate economists have had for centuries. Should we tax consumption or income? This debate, which shows in the real world, because in Europe, they rely much more on consumption taxes than we do. Yeah. AUDIENCE: Wait, would it matter? JONATHAN GRUBER: I'm going to tell you why it matters. So in Europe, they rely much more on consumption taxes than we do. So in the US, most of our taxes come from income tax, and in Europe, a lot of their tax revenue comes from what's called the value added tax. And if you've traveled in Europe, you'll know about the value added tax, which is a form of consumption taxation. So this is a debate that plays at international-- in the US, we have sales taxes, but they're quite small as a share of revenues compared to European nations. Now, why do we care? Well, we care because the following equation. y, your income, can either be spent on stuff or saved. Your income can either be spent on stuff or saved. And as I talked about a few lectures ago, savings is a major engine of growth in the economy. More savings means a broader pool of capital. Means a broader pool of capital, which means a lower interest rate, which means firms can invest more. So we like promoting savings in the long run. In the short run, in a recession, we may feel differently. But in the long run, we like having more savings. So in the long run, we like having more savings. If you tax income, you tax both my consumption and my savings. If you tax my consumption, then you don't tax my savings. That is, you relatively favor savings over consumption. You promote people to save rather than consuming. So that's why many economists-- I think probably if you did a poll of economists and said, should be tax income or consumption, the majority would say consumption. And the reason they would give is because we need more savings in society, and we need to promote that. Yeah. AUDIENCE: Wouldn't it be equivalent to taxing income and they're giving a tax break for saving? JONATHAN GRUBER: That would be identically equivalent. If you give a 100% tax credit for all of your savings, that would be identical. Now, in the US, we have some partial credits like that, but far from 100%. So the question is, what's the counterargument? Well, the counterargument is all about fairness. Which is that it turns out that the rich save and nobody else does. Basically almost all the savings in society is done by the top probably 10% of individuals in society. And the bottom 50% of individuals in society have basically no savings. So essentially, what this means is for most Americans, for the typical American, this debate is irrelevant, because they have no savings. You tax their income, you tax their consumption, it's the same thing. For rich Americans, it'd be a much better deal to tax consumption, because they have savings that then wouldn't be taxed. So the problem with the consumption tax base, it'll mean a massive redistribution from the poor to the rich, because the rich folks do the savings. Now, there would be a simple answer to-- there would be an answer to this. We could answer this, because ultimately, the rich folks die. It doesn't matter how rich they are, they die eventually. And if at that point, we taxed all their savings, we could then equalize things. So in other words, if we took all the money that was left-- let's say we had this consumption tax, but we counted its consumption the money you left behind when you died. Then it would solve the problem, because the rich would-- during their life, they'd pay less tax, but they'd pay it all at the end. So that's why we have a critical debate in this country over the estate tax, or the so-called death tax. This notion of whether you should be taxed on their estates becomes very important for thinking about this. If we had an estate tax which was literally at the same rate as all other taxes, then we could move to taxing consumption. It would be the same. In fact, we'd essentially have a consumption tax at that point. But in fact, we have an estate tax that is paid only by the top 0.04% of people who die each year. And it's paid on only a fraction of their assets. So we don't have that. So that's why the fairness debate comes in. So I just want to point out, just another-- these are all just topics I'm hitting to whet your appetite. These are interesting things we need to think about when we think about how to set up our tax systems, is basically things like, do we tax income or consumption? OK? Questions about that? OK. That's taxes. Now I want to turn to-- so that's one side of the equation, which is taxes. We know they cause inefficiencies. We know they cause deadweight losses. And we know that there's some question about who bears them. But the bottom line is taxes are putting some leak in the bucket, and that's a problem. The other side of the leak in the bucket is transfers. Which is, as we saw in the diagram last time, if you essentially condition my getting money on my working, I'll work less. If you say to me, hey, John, you're currently making $5,000 a year. I'm going to give you $10,000 no matter what you do, but any money you earn will come off that $10,000, I'll say great, I'll just quit. So the problem with transfers is they act like taxes. If you take the transfers and phase them out, take them away from people, they have a similar effect. In fact, we call transfers an implicit tax. [INAUDIBLE] transfer system from last time, which is the amount you got was the max of 0, was the max of 0, or 10,000 minus your income. This is essentially 100% tax rate on everybody's incomes below $10,000. Because for every dollar you earn, we take it away, because you get $10,000 no matter what. So this is a tax, basically. But in fact, this tax is unavoidable if we want to target money to poor people. If we just gave everyone $10,000, it wouldn't be a tax. We just gave everyone $10,000. Now, it would impact how hard people work. Why? Why would it impact how people work? Yeah. AUDIENCE: [INAUDIBLE] logarithmic [INAUDIBLE].. JONATHAN GRUBER: No, no, I'm going to labor supply theory. What's the name of the effect for why that would affect [INAUDIBLE] if we gave everyone $10,000? The income effect. So it wouldn't affect how our people work. But it wouldn't really distort people's labors. We think of distortions that arise in the substitution effect. People making different decisions because of tax rates. The reason we get a problem, the reason we get deadweight loss is because I'm ta-- not that I'm giving you $10,000, it's that I'm taking away as you get richer. That's the problem. So then the question is, what can we do about this? What can we do about the fact that we're essentially imposing implicit taxes on poor people by giving them money? So there are a couple answers. The first is categorical transfers. That is, instead of giving the money to everybody, just give it to deserving populations. So for example, one of them-- the largest single cash transfer program, pure cash transfer program in the US, is to-- that's literally giving cash is what's called the SSI program, the Supplemental Security Income program. It's about $80 billion a year. And what this does is give cash grants to low income families that have disabled children. So it's not just that you're low income, but that you have a disabled child in your house. Likewise, we have something called the TANF program, which is what is traditionally called-- I've used the term "welfare" in this course to mean well-being. Traditionally in America, when you say welfare with a sneer on your face, you're referring to TANF. TANF is cash grants to low-income single-parent households. Essentially, if you're low income and you're a single mom, typically, you get a grant from the government. Now, the question people have always asked is, why do we impose these conditions? Why do we make people disabled? Why do we make them be single moms? Why don't just give them the money? Why not have this? Why have these other conditions? And the answer is because-- that basically, we think this is a way of reducing the distortion that arises from transfers. Think of it this way. Imagine all of us were born with, on our forehead, an unremovable tattoo that said "hardworking and lazy." No, let's not do that. Different thing. Let's do "low skill and high skill." So I know your underlying ability. Then I would simply say, look, if you're high skill, you don't get any money. I don't care if you're poor. If you're poor, it just means you're lazy, because I know you're high skill. If you're low skill, I'm going to give you money. And that would not distort behavior at all, because people couldn't change what's on their head, so they would just continue to work as hard as they always worked. This is the idea of trying to find signals like that. You're trying to find ways of identifying the people who need the help. And by doing so-- who need the help, but need the help in a way that's not changed by their behavior. We think that a kid being disabled, they didn't choose to be disabled. Or mom being single mom, she didn't choose to be a single mom. That starts to get a little more interesting. So essentially, the idea of categorical welfare is to basically say, can we find things about people that they didn't choose on which we can base giving them money? So blindness is a good example, et cetera. Now, the question that then raises, well, can people choose these things? Well, you might say, well, of course, a kid can't choose to be disabled, but people can choose to be single moms. In fact, the evidence is the opposite. Let me explain what I mean. If you give people money tied to being a single mom, it doesn't cause them to be a single mom. The evidence is very clear on that. So there have been hundreds of studies showing that paying women money conditional on being a single mom doesn't cause them to get divorced or have kids out of wedlock. That's been clear. But if you give family money based on kids being disabled, more disabled kids show up. Why? Because disability is a tough thing to assess. Most disabilities are musculoskeletal or mental, and those are hard things-- you can tell if a kid's missing a limb, but it's hard to evaluate truly musculoskeletal or mental disabilities. And interestingly, despite your intuition what it might've been a few minutes ago, the place where in subsets we see the most people changing their behavior to qualify-- at least changing behavior meaning not becoming disabled, but claiming they're disabled-- is here and not in single motherhood. So categorical transfers help, but not as much as you might think, because the categories are hard to measure. And that's why another tool we've used to try to get at this is in-kind transfers. In-kind transfers. OK. Part of the problem-- yeah, Manny. AUDIENCE: For the TANF, does it account-- when it says for single parents, does it account for people that live in the household? So like-- JONATHAN GRUBER: It's a great question. That's a tricky thing to have to deal with, which is how do you deal with co-residing people? And that's something they have a complicated set of rules about. AUDIENCE: [INAUDIBLE] the stuff you didn't choose thing, because they seem to try to simplify the cons [INAUDIBLE] choose, whereas I think in real life, it's probably related to the circumstances of your birth-- JONATHAN GRUBER: Absolutely. AUDIENCE: --and other stuff like that. And so is it possible that these indicators are making it worse for people who are in situations where they couldn't have avoided being in poverty, but they happen to not be a single mom or not be disabled? JONATHAN GRUBER: Awesome question. And so basically, here's the trade-off. Okay, give me a better way to teach it. Here's the trade-off. The trade-off is, the more you target-- let's say I said I'm going to replace our entire welfare system in America with just transfers to the blind. And let's assume people don't blind themselves to get money. You laugh, but there is a city in Florida where they found that over a certain period of time, the vast majority of US claim for lost limbs came from one city in Florida. And it turned out that people were actually cutting off their limbs to qualify for government money. They actually called it Nub City. So you laugh. This stuff does happen. But by and large, we assume it doesn't. OK, so let's say I want to replace the entire US welfare system with one just for people who are blind. On the one hand, that'd be great, because you wouldn't cause any distortion. People shouldn't work any less hard because you can't change whether you're blind. On the other hand, you'd leave all of these people out in the cold who need help. So that's the trade-off, which is the broader you spread it-- the more you move toward a universal system, the more potential distortion you cause in terms of people changing their work behavior, but the broader set of people you can help. And that's why there's a big move now for what's called universal basic income, UBI. It's a big movement now around the world to say, look, forget these things, SSI, TANF. They're all messy to measure. They're stigmatizing. People can change them anyway. Let's just give people money. And that's sort of one motivation for that. Now, but the other way to get at this that people come at is, look, the problem all here is a simple one, which is everyone loves money. Rich or poor, we all love money. But what if what we gave people was not money, but things that poor people need and rich people don't? Like for example, mediocre public housing. Not a mansion, but an apartment. A rich person isn't going to pretend they're poor to get a crappy apartment or a mediocre apartment. But a poor person who'd be homeless otherwise will happily take it. What about medical care? Rich people have private health insurance. They don't need government medical care. Poor people need government medical care. What about food stamps? Rich people can afford food. Poor people can't. So the idea is basically, by giving in-kind transfers, like medical care, or housing, or food, we get people to what we call self-reveal that they're poor. Here the problem is people might pretend they're poor to get the money. They might quit. Even if they can afford to work, they might quit. Here we say, well, look, you're not going to quit your job to get a mediocre-- you guys with your-- make it 100 grand a year at MIT, I'm going to quit your jobs to get some mediocre public sector apartment. But if I said I'll give you 100 grand whether you work or not, you might quit your job. But say you live in a mediocre apartment, you won't. So the idea, by giving people in-kind benefits, is to get them to self-reveal whether they're actually poor or not. Now, that's not the main reason why giving in-kind transfers. This is the economist reason. The main reasons is because politicians are what we call paternalistic. And this comes all the way back. You guys remember when we discussed food stamps, and we did budget constraints, and I said, well, we never want to give people food stamps. We want to give them money. And they say, but what if the [INAUDIBLE] labeled cocaine? Then we would want to give them food stamps, because we don't want them to spend the money on cocaine. Well, that's how politicians feel about poor people. They feel that poor people, if you give them money, will just waste it. And that's why the vast majority of transfers we do in America are in-kind. If you add up all the money we give people in cash versus the money we give in-kind, particularly medical care, the in-kind dollars vastly outweigh the cash dollars. And that's because politicians are paternalistic. They're afraid-- no. The economics reason is because there's all this model of self-revelation, but that's not why politicians do it. They do it because they're worried people will waste the money if you give them cash. And I think we showed with the food stamps and the evidence, that's not really the right way to think about it. OK? Questions about that. OK. Now, this is all kind of negative and nebulous, so let me conclude with a great example, which is an example of a public policy which actually doesn't stop the leak in the bucket, it actually puts a patch in the bucket. And that policy is called the earned income tax credit. The EITC. This is what's known as a conditional transfer. What that means is it's a program where you get money, but the money you get is a function of how much you make up to a point. So it's actually-- the other thing it's been called is a wage subsidy. So here's how the EITC works. Let's go to figure 22-4. I'll show you how the EITC works. Here's how the EITC works. For every dollar-- on the x-axis is how much earned income you have. On the y-axis is the check you're going to get from the government. What this says is on every dollar you earn, until you earn $13,870-- these numbers are a bit out of date. It's a little more now, but roughly gives you the idea. Until you hit $13,870-- that is roughly the poverty line. It's a little bit more than the poverty line. For every dollar you earn, the government gives you $0.40 more. So if you're someone who starts with $0 and you earn $1, you take home $1.40. It's a negative tax. It's a subsidy. So every dollar, you take home $1.40. Until you reach earnings of $13,870. At that point, you've achieved a check of $5,548. That's 40% of 13,870. At that point, the government says this is the biggest check we're sending you, and we're to keep that flat until you're at an 18,110. And then we'll start to take it away. We're going to start to take it away so that by the time-- and what we're going to do is we're going to take that 5,548 check down by $0.21 cents for every dollar you earn. It's going to flip from a negative tax to a positive tax at a 21% rate. So instead of a negative tax at a 40% rate, we switch to a positive tax at a 21% rate, so that by the time you've earned $44,454, you've zeroed out your EITC. So for most of you, the EITC will be relevant. For most of your families, the EITC is irrelevant. So it's a targeted transfer program. Conditional transfer. It's conditional because basically, it's conditional on working. But it's targeted in the sense that it phases out such that middle class and upper class people don't get it. It's targeted to lower-income people through this phase-out. The problem is this makes the EITC effect complicated. So let's consider. Let's consider the three segments of this graph. Let's start with the first segment. Let's say you were earning zero, and I said, for every dollar you earn, I will give you $0.40 more. What effect does that have on your labor supply, starting from zero? Yeah. AUDIENCE: [INAUDIBLE] more because having a lot of leisure can cost a lot more. JONATHAN GRUBER: The substitution effect will make you work more, because essentially, we've raised the price of leisure by 40%. What about the income effect? AUDIENCE: [INAUDIBLE] JONATHAN GRUBER: There is no income effect. So the point is, for someone at $0, this unambiguously increased their labor supply. So if you take guys who aren't working, this definitely will-- I mean, it might be a zero effect, but it's non-negative effect on labor supply. Likewise, with low-income, working low hours, the income effect will be small. Remember, the income effect's proportional of how much you're actually earning. So for these people on the left-hand segment, upward-sloping, it should cause them to work more, because the substation effect will be big and the income effect will be small. So if you mark the people on this left-hand segment, you should mark they'll work more. Now what about people in the flat segment? Someone else raise their hand and tell me. What's this going to do to the labor supply of people on the flat segment? They're going to work more or less? Yeah. AUDIENCE: I think they'll work the same, because it's not like any change in their earning. JONATHAN GRUBER: Well, their wage hasn't changed, but something has changed. AUDIENCE: If they're on the flat part, they'll work less because they only have to work until they make $13,870 to get [INAUDIBLE].. JONATHAN GRUBER: Well, that's not quite-- that's an extreme model. More generally, it's because of income effects. More generally, the point is I've taken someone and made them $5,000 richer. So compared to a world without the EITC-- that's why I show-- compared to a world without the EITC, I will now work less. There's no substitution effect, only an income effect. So now I'll work less. So the first part [INAUDIBLE] substitution effects. The second part is just an income effect. Now what about the third segment? We're assuming leisure is normal. Do I work more or less? Do I work more or less, assuming leisure is normal? Yeah. AUDIENCE: Less? JONATHAN GRUBER: Less because? AUDIENCE: Substitution effect. JONATHAN GRUBER: Substitution effect and income effect, because your income is falling as-- because the point is, as you work more, your income is falling. So here, both the subsection and income effect combine to make you want to work less. So if you look at this graph, this doesn't look very promising. On the left, we've got a bunch of guys working more. In the middle, we got guys working less. And on the other side, we got guys working way less, because you have a substitution and income effect. So the question then is, what effect does the EITC have? And the answer is it turns out it has an enormously positive effect on labor supply. That it gets a lot of guys who were working zero to stop working zero. But it doesn't seem to lower the hours among those who are working more than zero hours. Now, why is that? Unclear. It could be because there's different elasticities along the income distribution. Probably, it's because of tax salience, wherever I wrote that. Wherever the hell I wrote that. Tax salience, which is that people understand, gee, if I go to work, I get a big check. They don't understand, by the way, the marginal hour I work is being taxed at $0.21 more. So probably, it's tax salience, I don't know. But the bottom line is the studies show convincingly that interest in the EITC massively increased labor supply and distribute to the poorest people in society. This is a reverse leaky bucket. Literally, we managed to take money, give it to poor people, and make the pie bigger. So this is an enormous victory for thinking about how we can-- for getting around this problem of how we can transfer to people. Now, it didn't solve all our problems because, [INAUDIBLE] up here, it doesn't help people who don't work. So there's still-- this isn't the only solution we need, because some people literally can't work, so they can't benefit from a program like this. But for workers, this is an enormously successful program that is really one of the great government success stories in terms of both transfer to poor people and increasing the size of the pie. All right? I'm going to stop there, and so we'll come back-- I'll see you guys on Wednesday.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Exploration_3_I_2024_I_Lecture_13.txt

All right. It should be up in a second. But you go ahead and get started on your refresh your understanding. All right. When you turn to somebody near you and see if you got the same answers for this. This question asks you to think back to what we were learning about last time in terms of posteriors over what the parameters might look like for a multi-armed bandit. So check in with someone nearby you and see whether you got the same idea. [BACKGROUND CHATTER] OK, we're going to go ahead and come back together and go through the answers for these. All right. So the first one of these are true because in this case, for a beta 1, 2, where we're weighed more towards an arm that more frequently gets something like a 0 instead of a 1, then we're more likely to sample these three parameters. The second one is also true because if you have a flat uniform over all of the different arm parameters, you're more likely to keep distribution. And the third is false, because when you have a 1, 1 prior, that's a uniform somewhere between 0 and 1, so the true arm parameter could be a 0 or it could be a 1 or anything in between. And then the second one asks you to think about using Thompson sampling to sample arms. And so the first one is true. So given these priors, you could sample either of those values for the underlying parameter for your Bernoulli variable. The second one is false. So let's assume that the real parameter here is 0.4 and 0.6. What this question is asking you to reflect about is that Thompson sampling is not guaranteed to give you an upper confidence bound. So it may instead just select a parameter that is consistent with your prior. And for these particular sample betas, it will happen to choose the true optimal arm for this round. Awesome. So I want to just-- let's see if I can make all the AV work. Want to briefly show you this nice example. Let's see if I can make this go away. All right. So I wanted to show you this nice example of, somewhere where you might want exploration. So we've talked about exploration so far in terms of cases like, you're an online advertiser and you'd like to figure out which ads work for people. It comes up in health care. I want to show you an example of an application which we thought about in collaboration with Chelsea Finn and a bunch of wonderful grad students recently. So this is the assignment. This is an assignment that's used in Stanford where students actually encode the game. So in this case, compared to the settings we're at where we assume you have the environment and then you're learning an agent to act in that environment, here, students are actually creating the code to make the Breakout assignment, so to make the game environment. And this, generally, is often really engaging and fun for students, particularly when they're learning to program. Many people like computer games. So this is a really great opportunity for people to learn and can be really engaging. And a lot of different people use these type of assignments. So it's not just at Stanford, but many, many other places, including code.org and others use this assignment to try to teach students about programming. Here's the problem. Even though it teaches lots of different introductory computer science concepts, there's a challenge, which is, if you want people to learn from writing this assignment, you need to be able to provide them with feedback. And providing them with feedback involves grading the assignments. So in this case, we normally have a rubric of different things that the program is expected to do correctly. Like, is the paddle drawn correctly? Is the ball drawn correctly? When you bounce, does that respect the desired transition dynamics? Things like that. And so normally, just like when you guys get feedback from Gradescope, someone has to go through and play the game to do this. And so that is really expensive because that means people have to figure out, when the ball bounces here, does it actually do the right thing? And then you have to do that for each of the different rubric items. So there, for example, it jittered. It didn't do the right thing. So what you can think of here is that essentially someone is manually designing a mental policy, a grader is designing a mental policy in their head for how to play this game in order to uncover whether the game dynamics are correct. And so the way we normally do that right now is, each individual grader figures out how to do that. And then they play this. So this means that it would take probably around eight minutes per submission. So you can't just do a unit test in the normal way because you actually are trying to figure out how the game behaves in different scenarios where it might take multiple actions to even get to that scenario. So if you think about doing eight minutes per submission, if you have 300 submissions in a course and there's actually many, many more people than that have played this game on code.org, or tried to code this, that's an enormous amount of grading time. That's an enormous amount of human resource time. And that means that some of the people that offer this challenge to students don't grade it at all-- just too expensive. And so that means students get the opportunity of trying to do this exciting assignment, but they don't get any feedback back, which can really hinder their learning process. So there's a lot of things that make this hard. It's a stochastic setting. There's not kind of simple heuristics. And there are multiple errors. So my student, Chris Pietsch-- another professor here-- and I started think about this problem a few years ago of saying, couldn't we design like a reinforcement learning agent to play this game? And what we want is that this reinforcement learning agent can explore the parts of the domain so that we can try to uncover how they're doing and whether the game is coded correctly. So we did this work and then we did an initial approach to this. And then Evan, who is the key author of this, extended this to try to think about rubric items. So the idea is that instead of having humans graded, what we're going to do is we're going to replace humans by a machine learning agent. And in particular, what Evan did is he built on our, and I and Chris's initial work and said, let's actually phrase this as, think about how we can use meta reinforcement learning and exploration. The reason this is an exploration problem is because you want to learn an RL policy here so that in a new environment you can quickly use behaviors to grade the assignment. And so that's where efficient exploration is coming in. So you don't want this to have to take 20 minutes to try to grade it. You want to, as quickly as possible, whether for an agent or a human, figure out what strategy you should use to play the game in order to correctly grade whether this is a good environment. And so, Evan had a really nice NURBS paper building on our NURBS paper. These are both machine learning contributions of how to-- there's a series of papers, there's a first paper on how we could do this at all. There's a second paper by Evan who is looking at trying to do explicit exploration, really fast exploration, and then we joined forces to think about how we could do fast exploration in this setting. And then more recently, we published a paper showing that this could actually significantly reduce grading time and actually improve accuracy when you combine this with humans. So, I just give this as an example to illustrate another exciting exploration case where if you can design agents that can learn quickly, and can quickly explore an environment, it can end up being really helpful. And we'll come back to DREAM and this idea of meta exploration later in the course, later today. So today will be our final lecture on fast and efficient reinforcement learning. And then next week, we're going to start talking about Monte Carlo tree search, which was one of the key ideas behind AlphaGo. I hope that homework 3 is going well. Feel free to reach out to us with any questions. And feel free to come to our office hours. All right. So just to remind ourselves about where we are, we've been thinking about different frameworks for evaluating the correctness of algorithms and how efficient they are at learning and making decisions. And so far, we have focused mostly on bandits, which is this much simpler version of reinforcement learning where the decisions we make don't affect the next state. So we saw how to do that for both standard bandits and Bayesian bandits. And today we're going to start to lift all those ideas up to Markov decision processes. So we did that by design because a lot of the ideas around optimism under uncertainty or posterior sampling or Thompson sampling can be lifted up to the tabular Markov decision process case. And then all of these ideas also then can be extrapolated up with some care to the function approximation setting. So that's where we're going to go today. The main approaches for trying to act efficiently in Markov decision processes and we're going to start by focusing on the tabular setting, will again be optimism under uncertainty, and probability matching or Thompson sampling. And we're going to see ideas of how to do that in this setting. OK. So here is one of-- it's not the oldest algorithm to do probably efficient exploration in tabular Markov decision processes, but it's one of the quintessential ones. And I think it illustrates a lot of the really nice ideas. So this is a lot. Let's just step through it. So the idea in this case is that we're going to be making decisions in a tabular Markov decision process. We're going to be taking actions with respect to some specific Q-function that I'm going to define in a second. We'll observe the reward in the state. We're going to update a whole bunch of things, update that special Q tilde and repeat. The key thing that we're going to be trying to do is similar to what we saw for the upper confidence bound algorithms. We're going to think about how do we construct an upper confidence bound on the Q-function. So that's going to be the key-- we're going to be doing-- this is an upper confidence bound algorithm. So this is going to, again, use the idea of optimism under uncertainty. And we're going to think about how do we bring this to MDPs. So the key idea in this case is what we would like to do is we'd like to construct an optimistic upper bound on the Q-function. This is a model-based approach, which means the way we're going to do that is we're going to try to construct optimistic estimates of the reward function, and optimistic estimates of the dynamics model. It shouldn't be immediately obvious what it means to be optimistic with respect to the dynamics model, and we'll go through that in a minute. In practice, what we're going to do is the following. The reward is the easiest to start with. So in the reward case, we're going to maintain counts of how many times we've taken an action in a particular state. We're also going to maintain counts of how many times we've started in a state, taken an action, and went to a particular next state. And we've seen these ideas before for tabular Markov decision processes. We've used them for certainty equivalent planning back in the first couple of weeks of class. So the reward model is perhaps closest to what we've seen for the bandit before. For the reward model, what we're going to do is we're going to compute the empirical average over in this state and this action, what's our average reward we've seen so far. And then, we're going to think of there being an upper confidence bound to that. What we're also going to do in this case is we're going to maintain an empirical estimate of the dynamics model. Now, when we do this, we're going to do the normal Bellman equation, except for we're going to include a bonus. So this part should look familiar to what we've seen for Hoeffding, which is when we are going to compute a Bellman backup, instead of uncertainty equivalence, we would just use the empirical estimate of the reward function and the empirical estimate of the dynamics model. Instead of doing that, we're going to include this bonus term. This is just bonus term. And there's a few different ways to do model-based interval estimation. I'm picking one here that just uses a bonus term, but I'll talk about some other ones. There's a number of variants. So what this is saying is when I do my Bellman backup of what is the expected discounted sum of rewards from starting state S and taking action A, this is going to try to approximate Q star. I'm going to plug in my empirical estimate of the reward. I'm going to use my empirical estimate of the dynamics model, and then I'm going to add in a bonus. And if I have not taken that state and action very much, that bonus is going to be really large because those counts of the number of states and actions is going to be really small. So this will be large if the counts are small. The key difference compared to what we've seen with bandits before is this is a Bellman backup. So we will then repeat this many, many times. So you do this for all states and actions, and then you back up and you do this many times. So intuitively, what's happening here is this is like pretending the expected discounted sum of rewards you'd get. If you start in a particular state and take a particular action is much higher if you have not visited that state and action very much. So that's where this optimism comes in. You end up adding in this bonus term here. And this bonus term will be really large. So beta is defined up here. This bonus term, this is 1 over 1 minus gamma. So if you imagine that all rewards are scaled between 0 and 1, and gamma is really small, you can think of that as being like H times that special term divided by the square root of the number of times you've been in that state in action. So what that means is when you do these repeated Bellman backups, it will drive your policy to visit parts of the state and action which you have not visited much. OK? Because those are the parts where you're going to have these really large overestimates, probably overestimates-- optimistic estimates, I should say. You have these optimistic estimates of how good the value could be in those states. And the reason this is important is because it might be-- so this is going to work when you're doing a series of episodes, or you're working in the same MDP for a long time, what will happen is that your MDE will explore in your MDP, and it will drive you to cover the state and action space, if you think that it might possibly have good rewards in those places. So this will drive exploration. And by doing these repeated backups here, you're propagating your optimism under uncertainty backwards so that you develop this policy to drive you that way. So this is one of the quintessential algorithms for doing tabular optimism under uncertainty based planning. It is also a PAC algorithm. So we talked about PAC last time. PAC, it means it's Probably Approximately Correct. I'll just write that out again, just to remind ourselves. Probably approximately correct. But now, we're going to talk about, in particular, Markov decision processes. So we talked about how an algorithm is probably approximately correct if most of the time it makes a decision that is close to optimal and only makes mistakes on a polynomial number of times. So we talked about that last time. We saw that you don't have to guarantee this. You could make mistakes forever. Like, if you're acting randomly, that's not-- you would continue to make this forever. MBIE is a PAC algorithm. So what it says is that let's let script a t denote MBIE-EB's policy at time step t, and s t denote the state at time t. With high probability, the value of the action the algorithm takes is at least the value of the optimal action for that state minus epsilon. And it's true on all but a finite number of steps with high probability. So this is the number of steps. And the important thing here is this is a polynomial in the size of the state space, the action space, 1 over epsilon, and 1 over 1 minus gamma. Now, I always encourage my research students to plug in for bounds because theoretical bounds are beautiful, but it's nice to know whether or not they are at all related to practice. So, for example, in this case, you might imagine, let's say, we have s equals 10 and a equals 10. And you've said epsilon is equal to 0.1 and gamma is equal to 0.9. All right. So let's just work out what that would be. That would be roughly 10 to the 3 times 10 to the 9, or 10 to the 12. So that's a lot. So what that would say is that we are sure by using this algorithm that we will only make mistakes on this 10 state MDP, 10 to the 12 time steps. Now, I don't know about you, but I would hope that a 10 state GridWorld MDP, that we could learn to act substantially faster than that. So I use it to highlight that these bounds, while this might officially say this is a PAC algorithm, they can be pretty conservative in how many mistakes you might make. Now, in practice, often this optimism under uncertainty algorithm can work very well. It doesn't say you will make this number of mistakes, it just is an upper bound on it. But it's good to plug these things in just to see how tight or not you think it is relative to real performance. All right. So this is a PAC algorithm and the paper goes through an interesting proof of it to show the different components. But one of the key ideas is something called the Simulation Lemma. And that I'm going to go through at least briefly because the Simulation Lemma is one of the many core ideas when we think about doing efficient exploration. And the key idea for the Simulation Lemma is the idea that we can relate the accuracy of our models to the accuracy of our learned Q-function. OK, so that's the key idea. It's going to say, if we have bounded-- yeah, I guess I'll just leave it there. So if we just ensure that we have good predictive models, we can relate our error and our predictive models back to our value function. So let's do that at least, sketch. So this is going to be for do for tabular settings. So we're going to assume that we're back in a finite set of states, finite set of actions. So this is one proof of the Simulation Lemma. We're going to assume that we have pi as a fixed policy. And we are going to assume that we have a max norm on the reward. So we're going to assume-- if you remember back, let me do R1 minus R2-- we're going to assume that we have two different MDPs. So MDP 1, and MDP 2. And these might have slightly different reward functions and slightly different dynamics models. So remember that if we have the infinity norm, you can express this as this is going to be the place where the two reward functions differ the most over our finite state space. So if one of them gives the rewards of 1, 2, 7, 3 and the other one gives the rewards of 2, 6, 1, 7 we would figure out for which state do the two rewards differ the most. Let's assume that's upper bounded by alpha. We're also going to assume that we have an upper bound on the dynamics model. So we're going to assume that T of s prime given s a minus T 2 of S prime given s a is also bounded. And I'm going to assume that's bounded by beta. So that means from the point of view of your predictive models, the two MDPs differ. You can bound the amount. And we're going to show if that's true. Then we also are going to only differ in their estimated Q-functions for a particular policy by a bounded amount. So what we have is we want to compare what is the Q value of under model 1 for state s, a and then following policy pi versus Q pi 2, s, a. And the reason this is going to be important is because in general, what we're going to have is the R1 and R2 are going to correspond to our uncertainty. So if you think back to the Hoeffding inequality, as I told you about, we talked about how our empirical estimate could differ from the true estimate by a bounded amount. So that should make you think about this part. R1 could be our empirical estimate of the reward and R2 could be the true unknown one. And Hoeffding can give you an upper bound on what that alpha is. And similarly, we can get a bound on the dynamics model error as well. So the idea will be to say, if you end up plugging in, say, like an empirical estimate of the reward model and the dynamics model, how far away could your estimate of the Q-function be from if you actually knew the true reward model and the true dynamics model? So that's why we're doing this. OK, so this is the difference. Let's just write down what that will look like. So this is going to be R1 of s, a plus gamma, sum over s prime, T 1, s prime given s, a, V 1 pi of S prime, minus R2 s, a plus gamma sum over s prime T2, s prime. I've just used the definition of the Q-function there to write out what is the difference between the two Q values. All right. We're going to upper bound this as follows. We're going to just use the triangle inequality first. So we're just going to say, this is less than or equal to R1 minus R2. So I use the triangle inequality plus gamma times the difference in the second terms. That was in parentheses. OK, I'll just use my triangle inequality. I split the two terms. Remember this? We've already said is going to be less than or equal to alpha. Because we've upper bounded our R. So then, we have to think about the second term. Then we're going to do something that we often do in reinforcement learning, which is we add and subtract 0, or we add 0. And we're going to do that by trying to relate between-- right now, we have the dynamics model of one thing and the value-- the dynamics model of model 1 and the value function of model 1. And so now, we want to have some in between terms so we can directly think about the difference in the value functions under one particular dynamics model and the difference of the dynamics model separately. OK, what we're going to do in this case is we're going to say, this is less than or equal to alpha plus gamma, and then we're just going to add and subtract some terms. So sum over s prime. And I'm just going to use shorthand here so that I can fit everything. So I'm just going to introduce add and subtract 0. Careful with my-- make sure that's clear. OK, so I just introduced, I added a new term that, this intersection term, and I added and subtracted it. And the reason that's helpful is now I can just think of terms where they only differ in the dynamics model or terms that they only differ in the value function. So this is going to be less than or equal to alpha plus gamma times sum over s prime T1 of s prime times the absolute value of V1 pi. prime S V2, [INAUDIBLE] plus gamma times sum over s prime, T1 of s prime minus T2 of s prime. All right. So what have I done there? I've just rearranged the terms. I just moved these two here, and then I move those two there. And I'm starting to apply my absolute values a lot, just repeatedly do the triangle inequality. All right, so what is this? This part looks a lot like this thing. So that's going to be like a recursive term. So we can turn this part into the following. This is going to be-- this part here will be less than or equal to alpha plus gamma. Let's call this difference delta. I'm going to call that delta. Then if I do that, I have sum over s prime does T1 of s prime. So this could just be like a max difference in your value functions. And then the second term I have, I'm going to use the fact that my value function is upper bounded. So here, our max divided by 1 minus gamma. So if you get the maximum reward at every single time step, and your discount factor is gamma, this is an upper bound on Vmax. So that allows me to take this term out, and then that just leaves me with my difference in my dynamics model. So I have this. And that was this term, here comes that, OK? All right. So that's what I have in this case now. And so, this has to hold for all states and actions. So here, the delta that I've defined. Delta is kind of the worst case error over any of these. So over any states, what's the maximum difference between the value functions? So that also has to hold on the Q side. So we get this, we get delta is less than or equal to alpha plus gamma, delta plus gamma Vmax beta. And I'm going to subtract that, so now I have 1 minus gamma times delta is less than or equal to alpha plus gamma Vmax beta, or delta is less than or equal to 1 over gamma alpha plus gamma Vmax beta. OK, what are we just shown? We've said, the worst case error in the value function for the same policy between one model and the other model is upper bounded by 1 over 1 minus gamma times the error in your reward model, plus gamma times your maximum value, times the error in your dynamics model. So that is one version, at least, of the Simulation Lemma. It comes up in lots of different other areas too. People use it for a lot more advanced, complicated settings. But the critical idea here is to say, if you can bound your error in the dynamics model or the error in your value function, and the error in your reward function, that also means that your Q-functions can't be too different. So that's the main important point of that. And so, this idea, in general, is a-- excuse me-- a helpful one because it means as we explore and we learn these predictive models better, we can be sure that our value function is also simultaneously getting better and better over time and getting more accurate. And in the proofs of PAC algorithms that's often used to say, you can't infinitely learn in particular states and actions and not end up with a value function that is getting more and more accurate. All right. So now, I'll pause there in case anybody has any questions before we move on to Bayesian Markov decision processes. Yeah? So here, we define the difference in the value function as delta, but why we could also represent the difference in Q-function as [INAUDIBLE]? Yeah, because this is an upper bound to this term and this has to hold for all of the states and actions we could ever be at. And that also has to hold for this, for any state. So you could make A here be pi of s. Yeah. Should there be a factor of the number of states because we're summing over all the states, or is that-- because just from the error bound, I thought that's just for one state. For example, the dynamics, just for one state, it's less than or equal to beta. Then the proof, it's summing over a bunch of states. Good question. So what happens here is this is just assuming that given you have a bound on this, this tells you, this is for every single s prime, you have a bound. What will end up coming-- normally, where the number of states will come in is when you start to think about how much data you need to achieve this. And you want this to hold for every single state action pair. And normally, that's where you will end up getting a dependence on [INAUDIBLE] data in order to get sufficiently small confidence intervals. And with your union bound, you need to make sure all of these bounds hold. In terms of this part, the state space doesn't appear. But this is just for the Simulation Lemma. It doesn't then tell you all the way to how many samples you need to achieve this. OK, that's another part of the proof. You could probably imagine already, given what you know about Hoeffding, that you could imagine having some way to compute how many samples you need to get alpha sufficiently small in the dynamics model, kind of just a similar idea. And you can do this for other parametric models too, like Gaussians, et cetera. Anyone else have any questions on this part before we go on to Bayesian? [COUGHS] All right, let's step onto Bayesian. So in all of these cases, we weren't using any notion of priors, or any of the things that we saw last time. So now, we're going to think about how we can lift some of the ideas we saw from last time to think about this for Markov decision processes. So just as a refresher, remember in the other way we can think about this, a common way to think about trying to do efficient exploration, is to imagine that we have some prior knowledge over how good we think different states and actions might be or how we think the dynamics might work. And then what we're going to do is try to use that information to figure out how to act. And we saw Thompson sampling as being one method that was an efficient way to try to make decisions when we have these priors and these posteriors. And now, we're going to think about lifting these ideas to the sequential case. So what we saw before is that we'd have these priors over the model parameters-- like in this case, the reward models-- and if they were conjugate, [COUGHS] excuse me, then after we would actually observe a reward, we had this nice closed form expression for the betas. So we could think of these as just being the number of successes and the number of failures. And I talked about but didn't actually illustrate that you can do this for other sorts of things like Gaussians, et cetera. All right, so you might think this should work clearly for the reward part of a Markov decision process. Can we do this in general? So this is what we did just, again, to remind ourselves that Thompson sampling for multi-armed bandits involves maintaining this prior. We would sample from it, meaning we would get a particular set of values for our coin flips, like what you saw before. We would then act optimally with respect to those, observe reward, and update our posterior. So now, what we're going to do is a very similar thing, but we're going to maintain priors over Markov decision process models. So we could have a reward model in this-- right now, we're going to again start with the tabular case. So we're going to start with the tabular case. There's a finite set of states and actions. So in this case, you could imagine maintaining a different reward model for every single state and action and being able to sample from it. So you could sample like a parameter for every single one of those. And we're going to see how we can use that to actually do something very similar to Thompson sampling for the sequential process case. OK. So the idea now is that we're going to maintain a prior over all of the dynamics models and all of the reward models. We will sample from that. Now, if you remember what I just showed you, in the case of bandits, once we did the sampling of the parameters, it was really easy to figure out a decision. Because like in the case of the Bernoulli bandits, as soon as you know that this coin flip is going to give you one with higher probability than this other one, it tells you how to act. For a Markov decision process, it's more complicated because as soon as you see the dynamics and reward model, you don't know how to act yet. So you actually have to solve a planning problem. So it's like you sample a Markov decision process, once you're given that Markov decision process, then you have to do planning, like value iteration, or something like that to actually get your Q star. Once you get your Q star, then you can select the optimal action given that computed Q star. So computationally, it can involve a lot more work than what we saw in the bandit case. Then the next question you might have is, how do we do this sampling? So, this is the PSRL algorithm. It was invented by Ian Osband, Dan Russo, and Ben Van Roy, who's here at Stanford. And these guys were here at Stanford when they invented it. The idea is as follows-- and I'll talk about sampling the dynamics model in a second-- there's going to be a series of episodes. At the very start of an episode, given your prior, your current posterior, you are going to, for every single state action pair, sample a dynamics model and sample a reward model. Given that, you're going to compute Q star for your sampled MDP. Once you have that sampled MDP, you're going to act according to that policy for the entire episode. So this computes a Q star for the entire episode. You're then [INAUDIBLE] for t equals 1 to h, you're going to assume your episodes are finite. You're going to act according to your Q star. You observe your reward in your next state, you're going to repeat. At the end of the whole episode, you're going to take all of the data that you just got, and you're going to update your posterior. So for the reward model, it can be very similar to what we saw last time, where you just update your counts. For the dynamics model, it may not be clear what you would do in that case. So in this case, what we would often probably choose to do-- all right, let's write up here. So what we would often do is reduce a Dirichlet model. A Dirichlet model is a conjugate prior for multinomial. OK, multinomials is what we can use for our normal dynamics model here because on multinomial allows us to express what is the probability of going to any of the next states, given this current state and action. So in general, we would have one multinomial for each state and action pair. We are now in the Bayesian setting. And so now, so we would have one per s, a pair. And this specifies our probability distribution over all the next states. That specifies p over s prime given s and a for all s prime. And it has to sum up to 1. That's the multinomial part. The part where we're being Bayesian is we're assuming we don't know what all these parameters are. And so, we have a prior over them. And the Dirichlet is a conjugate prior, which means that if we start with a Dirichlet over multinomial parameters, we observe something. So let's say we're interested in understanding what happens when we're in state 1 and we take action 1, and we observe that we go to s 3 seven times. And we observe, we go to s 7, three times. Well, I'll do two times. What that means is that at the end of that episode, we would use that data to change our Dirichlet distribution over those multinomial parameters. Very similar to what we saw for the beta distribution. And I don't expect-- I mean, it's an interesting thing to do, but I don't expect you to do that in this class. Some of you might want to for part of your projects. But the key idea here is that it is conjugate. So it means that the posteriors you get are in the same family as your priors. And so, you can use this to sample multinomials. Essentially, it's just sampling dynamics models. So we do in this case. And we do this over and over again. And the really key things to notice here compared to what we were seeing before is that we have to sample this entire MDP and we have to compute its optimal value before we act. So we do all of this computation before the start of an episode. And you might-- yeah? I'm a bit confused about this sampling the MDP, but the sampling dynamics and reward models part. Oh, this is just explaining that. OK. This is like a comment. You sample the MDP. What this means is just for each of the state and actions. Yeah. Good clarification question. To sample an MDP, what I mean is that you're going to define the MDP. So that means we can completely specify an MDP, given a known state and action space, and a discount factor by specifying a dynamics model for every single state and action pair, and specifying the rewards model. And then we'll have to compute the optimal action. Now, one thing that you might wonder about is, is it important or necessary that we do all of this once per episode? So when we talked about Bayesian bandits, after every single observation, we updated our posterior. So we would try buddy taping the toes, and we'd see that that helps someone recover, and then we would update our prior. This is a bit different. We are only doing this every h steps. Now, you might think maybe that's computational. You might think that that's being done for another reason. But it's something that's an interesting thing to think about. Let me see if I have this on the next slide. Yeah. So let's do a check your understanding and then I'll give you talk a little bit more about why this is done in PSRL. So this asks you to think a little bit about doing strategic exploration and MDPs and in Thompson sampling in the algorithm I just showed. [AUDIO OUT] All right. I want you compare your answers with someone near you. [AUDIO OUT] OK, thank you-- yeah. So now it should be back on. What I was saying is that in Maria Dimakopoulou's work, she was thinking about concurrent reinforcement learning, which is something we've also thought about. And for this much more realistic setting, the idea is whether you might need to coordinate exploration and how frequently you update. Now, one of the challenges of this setting, even before you get into concurrent reinforcement learning, is that if you update your prior a lot within a task, like within a single episode, you're essentially sampling different MDPs within the same episode. The reason that can be bad is that now is going to totally change your behavior, and there may be some cases where you essentially thrash. Let me give an example. So one of the canonical hard Markov decision processes that people talk about is a chain. It's really just an illustrative one. And there are lots of different slight variants of chains. And the idea is that you might have something-- I've shown stuff that's similar to this before where on one side or on the other side-- oh, it's not reconnecting just in the back there, you need to reconnect. That you would have high reward on one or the other sides. What you could imagine in this case, if you were thinking about it being like a Bayesian bandit, is that some of the times, it might pick a Markov decision process where this is the good, the best state, and some of the time, it might pick a Markov decision process-- oh it's still not showing on there. Thanks. Hopefully that will come up. And some of the time it might pick one that is here. So if you start off acting, let's say that you first sampled an MDP where this is the best state, you do your planning and then your agent is going to start going this way. OK? Let's say you observe that there's some zero reward here and your Thompson sampling updates. And now, it says, hey, this is the best state because you just have some prior over the model parameters. And so your agent turns around and it's like, oh, I shouldn't go this way, I should go this way. And then as you're doing that, it's getting more rewards and it's updating its posterior. And so then, it samples again and it's like, oh, this is good. And so, it can lead to this kind of thrashing behavior because it's sampling a new Markov decision process each time. And so your agent can end up toggling back and forth between its ideas over which MDP it's in. So it's for this reason that often you will want to essentially commit to the Markov decision process you're in for the whole time. You don't always have to do this, but that's one of the reasons why this can be helpful. This commitment is also in the 2013 NURBS paper that-- well, not-- the algorithm that we saw earlier, right? They both commit? Yes. So far, sorry-- This is just exactly the same as the PSRL algorithm. I'm about to tell you about the seed sampling. But yes, this is just in the 2013. So yeah, exactly. This is the in PSRL itself. It commits. And this is one of the reasons for that. And so, in Maria's work, she discusses some of the important benefits of it. And then, she thinks about how would you actually maybe try to couple and coordinate exploration if you have many agents that are going through the same environment at once. And it's in some ways, it relates to this idea, too. You might want everybody to commit to exploring different parts of the space. Because if you have many agents in the same domain, you might want to say, you're going to think that the best reward is here. You're going to think the best reward is here. Go explore, and then we'll unify in our posterior afterwards. So she has a nice demonstration of that. And then she extended it to the deep learning case shortly afterwards. But maybe if I can play that. [AUDIO OUT] [VIDEO PLAYBACK] OK, so eventually, it happens. But then you can get two concurrent UCRL where in this case, you can start to-- if you don't do something smart, again, this can end up being not very effective. And let me just see if I can skip ahead to the last part. Seed sampling. OK, good. [VIDEO PLAYBACK] [MUSIC PLAYING] [END PLAYBACK] OK, so seed sampling in her case is what they're doing when they essentially do concurrent reinforcement learning. And you might have even missed it because that part is really fast. OK, so I'll move it to this, just so I can talk over it at the same time. So this is seed sampling, which is what their idea was. And this just talks again about doing strategic, coordinated sampling. So you can see in this case, we're leveraging the fact that you've got concurrent agents that are exploring the environment, they're committing to it, but they're committing to it in a way that they coordinate that. So that you don't get all of the agents. So here, by 324, all of the agents have shared information about where the cheese is, and everyone's solved. All right. So that just illustrates why you both need this committing to a particular exploration strategy. And then if you're in the case where you also have concurrent agents-- which is very realistic-- that having this additional coordination is really helpful. Now, I think one of the interesting things to note there is that this is a nice place where there's some-- is it connecting? Hopefully it'll connect in a second. There's an interesting disconnect between theory and experiment. It's still not? Maybe there's a problem with a connector. We'll try to get that fixed for next week. There's a disconnect between theory and practice because theoretically, you don't need to do this exploration. So we have a paper from 2015 showing that if you don't do coordinated exploration, it's still totally sufficient. You can still get basically almost a linear speed up. Oh good, finally came. Yay! All right. So that covers how you can do Bayesian exploration and optimism under uncertainty in the tabular Markov decision process case. But of course, what we'd like to be able to do is to do this for much more large state spaces and realistic problems. So this is very much an ongoing area. Again, you'll see this similarity to the types of ideas we've seen before. Very popular ideas are optimism under uncertainty and Thompson sampling. They're not the only ones, but they're probably the dominant strategies people try to use. For-- I may have just not caught this, but specifically, what is actually different between the two algorithms? What is the difference in this sampling, between 2013 and 2018? Yes. So two things. One is that the PSRL does not think about concurrency. So they just assume there's a single MDP. You have a single agent in it. The other case assumes you have m agents all in the same MDP. So like the mice trying to find the cheese, there's not just one mouse, there's a whole bunch. And the idea was seed sampling is also to think about how do you choose which MDP they each think they're in to distribute the exploration. OK. Yeah. And the other case, you don't have to do any coordination because there's only one agent. OK, good. So in terms of generalization, we're going to think about this. The reason why this starts to get more tricky is a couple of things. One is that for optimism under uncertainty, it means we have to have a notion of uncertainty. And it just gets much harder to represent uncertainty when we have deep neural networks. Similarly for Thompson sampling, as we start to move up to really complicated domains, we need posteriors over really complicated settings and that's also computationally challenging and hard to approximate. So let's first start with contextual bandits. And some of you guys will probably be doing some of this for your project. So instead of having our multi-armed bandit, now we're halfway between a Markov decision process and a bandit. So we're going to assume we have states, but the action we take doesn't influence the next state. And so now, if we think about rewards, we'll have a reward per action in state. And just like what we've often done before, if we have a really large state and action space, we're going to assume that we use some parametric representation to model the relationship between state and action and output rewards. Perhaps not surprisingly, there is an enormous benefit of doing this. So if you think about a setting where this is the number of arms you have, if you did something like upper confidence bounds-- and this is a regret. Regret is on the y-axis. So if you did something like upper confidence bounds and you have 1,000 arms and 4,000 pulls-- sorry, you have 1,000 arms and then you're pulling these over time. So this is, I think, just regret after a fixed number of time steps. Unsurprisingly, if you have a lot more arms to pull, you'll have a lot more regret. Because in upper confidence bounds, in the things we've seen so far, you don't share any information across the arms. If, on the other hand, use something like linear UCB, which assumes that your arms are represented by a set of features-- so showing someone like a Trump campaign today and a different Trump campaign tomorrow might have the same effect. Because they're going to have a shared set of features about Trump, at least would be one thing that would overlap. You can leverage that structure. And so, what you can see in this case is that if you leverage say, a parametric linear representation in this case, even as you scale up the actual number of arms, if your parameter space is still the same, then your regret doesn't scale badly. So for example, this is k, but you know your theta in this case. The-- Your theta in this case might just be low dimensional. So we might have a theta, which is in R d, so we have a d-dimensional representation. And this just shows that this can be really helpful. In general, you want to leverage structure. So one common thing to do is to model the reward as a linear function. Of course, this could be built on top of a deep neural network, or on top of a large language model or something like that. You can often just use some really complicated representation of the state and action space. And then, say, for the last layer, my actual reward is going to be a function of these complicated features, dot product with some theta parameter. And one common thing is to assume that it's just a linear function plus some noise. And the nice thing about this is that if your features are interpretable, then your reward function is also very interpretable because you can just think of relatively, how much do each of those features contribute to your reward? All right. So one thing to think about in this case is in these settings-- well, I'll go a little fast through this part because I want to make sure we get to the MDP part two. But when you have this, even if you have a linear set of models, you can use them to represent more complicated functions. Because, let's say-- technology is getting-- So let's say this is your reward model for three different actions. This is a 1, this is a 2, and this is a 3. And this is what your reward is. And this is your state and space. So let's imagine that you had a linear representation. Then, you could represent policies that are just joint linear because if you were taking the max here, this is what the value would be of your policy. Because it would say, a 1 dominates for this part of the state space. A 3 dominates for this part of the state space, and a 2 dominates for this part of the state space. So linear ones-- I guess the main point here is that even if you have a linear reward model, it doesn't mean your policy has to be linear. Your policy will be disjoint linear. It can be made up of these sorts of functions. So it's fairly flexible. OK, how would we work in these cases? Well, in this case, what it means to have uncertainty is we need to have uncertainty over this linear vector. So, we want to capture uncertainty over theta through some sort of uncertainty set. And there's been a lot of beautiful work to try to quantify the types of uncertainties we have through things like the elliptical potential lemon, things like that, which give us, basically, just sort of an uncertainty set over vectors. And you can do this in a computationally tractable way. And what this means is it gives us a principled way to get an upper confidence bound on the reward function, given that we have uncertainty over linear model. And this was shown to be very useful for news article recommendations about 14 years ago. And you can also look at chapter 19. So these are really useful. This is one way to represent a contextual bandit setting when you want to handle generalization. We'll now talk briefly about how you might do this for Markov decision processes. OK, so if we think back to the MBIE-EB algorithm for finite state and actions, we have to modify a few things. So if we think about this, we were keeping track of counts. And we were doing this-- we were building a model separately for every state and action. So this count-based term here that we're using as a bonus, we've already seen how we might be able to do Q-functions with deep neural networks. But the big problem here is the count-based bonus. We have an infinite number of states. If you think about Atari or something like that, you certainly don't want to count. You're mostly only going to see one Atari screen once ever. And so, these sort of count-based bonuses aren't very realistic. And so we're going to need ways, essentially. But why do we have that count-based bonuses? We have the count-based bonuses to try to quantify our uncertainty over how well do we know the reward model for this particular state in action, and how well do we know the dynamics. And so, one of the ideas when deep RL came around was to think about, could we lift this idea and try to quantify our uncertainty in the deep RL setting? So we're going to need to move beyond having these very simple counts to think about something that's a higher level representation of that. Now, if we could get that-- and I haven't told you how we can get it yet-- you could imagine that a lot of the algorithms we've seen before could be extended fairly easily. So in particular, if you think about something like function approximation with Q-learning, we could imagine just adding some sort of bonus term in here. So instead of having our empirical reward plus gamma times our target, like our observed next state in action with some parameter weight, we could just plug in some bonus. That's kind of what MBIE-EB is already doing. It's just that our bonus before was determined by our counts. And now, we need some other way to lift that so we can do that for much more general settings. But once we have that, we can imagine plugging it in here. So there's a lot of different approaches that have been developed to try to think about something of density, or quantification of how many visits we have or how much certainty we have over different parts of the state and action space. So one of the things that Marc Bellemare and others did, which was pretty successful, is they tried to build pseudo counts over parts of the state and action space. So you could imagine maybe even some particular rooms in a video game many, many times. And so, you try to essentially reduce your uncertainty over those. There's all sorts of important details here around whether you-- normally, in MBIE-EB, every round you update all of those counts. In reality, if you think back to deep Q-learning, we maintained a buffer of state action rewards next states. Now, you would need to include those bonus terms in there too. And if those bonus terms are changing, how much do you update your buffer? Just to give you a sense of some of the different wrinkles one has to think about. But the high level important thing is that this matters a lot. So in Montezuma's Revenge, which was early on considered one of the hardest Atari games-- probably still is-- if you did a standard DQN for 50 million frames, which is a lot, it never got past the second room. With epsilon greedy exploration, it was not strategic. It just got very bad performance. But what Marc Bellemare and others showed is that by incorporating a notion of count-based lifted to the generalization case, you could do far, far, far better. So that's just to highlight that there are ways to lift up this notion of optimism uncertainty for this type of setting. There is similarly ways to lift Thompson sampling. So, we've done some work there where we think about particular representations and parameters. Ian Osband, who introduced PSRL, then tried to lift it up to the deep Q-learning case. They did it where they were just bootstrapping samples as an approximation. That is a pretty coarse approximation of uncertainty. Something else that often worked pretty well-- surprisingly well, given how simple it is-- is essentially to do something just at the last layer. So the last layer do something like Bayesian linear regression to try to get an uncertainty estimate, and then sample from that. So this is a pretty simple thing one could try. There's a lot of work to do this. Let's go back to thinking of other really recent approaches which try to think about doing this not just for one task, but many tasks where you need to do generalization. So early in this lecture, I introduced the DREAM algorithm to you, which we later used to actually go grading of the Breakout assignment. The notion in DREAM is that you have many different tasks and you're going to learn how to explore in them efficiently. So that was one example where we're now really thinking about how do we develop efficient exploration strategies by leveraging structure over the tasks, where an agent is going to do a series of tasks. Similarly, in some of our recent work, we introduced decision pre-trained transformers. This was, again, a meta-learning case. The idea is that your agent is going to do a series of bandit problems, or a series of RL problems, and we want to learn how to optimally explore in those settings. So I'll just show you briefly how it works. The idea in this setting is we're going to use a pre-trained transformer. One of the interesting things is you map reinforcement learning to supervised learning, similar to behavior cloning. But instead of relying on the data you collected in the past, if you can compute what would have been the right action to take there, you can train it to predict that optimal action. It turns out that when you do that, we can exactly map that back to doing the equivalent of Thompson sampling. So in all the settings for which Thompson sampling has theoretical guarantees, this decision pre-trained transformer can inherit those guarantees, which is pretty cool. The nice thing too is that, empirically, it can allow you to take advantage of structure that is present in your domain that you didn't have to code. So let me just give you an example of that. So what I showed you earlier in this lecture is that if you have a domain where you have some linear structure, if you give that linear structure to your algorithm, then you can do quite well. So that's the green line here. So this is the amount of data you have over time. And this is your cumulative regret. Lower is better. So, most historical algorithms have assumed you give that structure to your bandit. You write down, there are these 300 features that you need to pay attention to news articles and people to figure out what the reward will be. If you give it that structure and that structure is right, you often do pretty well. You could not leverage that structure, and you would get something like this. So this is a Thompson sampling algorithm, which just assumes that it doesn't have that linear structure. What are the cool things that we found by this approach is that in this setting, if you really have a linear structure in your domain, and you're doing many tasks and all of them have this linear structure, what are decision pre-trained transformer will learn is that even though you're not telling it, it will realize it can more compactly encode that structure. And so, when you deploy it on a new task, you will get behavior almost as if you gave it the unknown structure. So I think this is really interesting because often one of the brittle aspects of machine learning is that we originally wrote down these sort of representations. And of course, one of the really amazing things for deep learning is that we're trying to not write down specific representations as much, and get much closer to the input raw data. And this is illustrating that in terms of sequential decision making and meta exploration for multiple tasks, we can do something similar here, where we can inductively learn that that's a more compact way to represent the domains, and get this much more efficient exploration in new tasks. All right. So just to conclude, we're wrapping up our notion of data efficient reinforcement learning today. You should understand this tension between exploration and exploitation in reinforcement learning. I haven't used these words. They're not great words. So I don't use these. I haven't used a lot in there. But exploration, meaning you're taking time to learn about the domain and exploitation, meaning that you're leveraging that information to make good decisions in the context of reinforcement learning. You should be able to define and compare different notions of good, whether empirical, convergence, regret, and PAC. You should know for the algorithms we've talked about, do they have-- for example, is greedy sublinear regret? Which it's not. You should understand the proof sketch I did of why upper confidence bound is sublinear in regret. All right. And then next week, we're going to talk about AlphaGo and how do we think about doing SMART adaptive tree search in really large games. See you then.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Emma_Brunskill_Dan_Webber_I_2024_I_Lecture_15.txt

Hey, everybody, we're going to go ahead and get started. And we'll start with a refresher understanding, thinking back to DPO and RLHF. All right, why don't you turn to someone near you and see if you got the same answer, particularly for the third and fourth one? But-- All right, so let's come back together. The first one is true. The DPO model does assume that we have a particular model of how people are responding to preferences, in particular the Bradley-Terry model. The second one is also true. Even though we've been thinking a lot about when we actually have preferences, we can also use this in cases where someone just directly provided you reward labels. And so RLHF is a paradigm. It's totally compatible with the idea of just getting rewards from some way. But normally, when we think about that human feedback, it's normally from preferences. The third one is an interesting one. Does somebody want to argue why they think that is not a good way to learn about the reward model for board games? With multiple optimal points, you know what is-- Yeah, you feel there's multiple options. Yeah, I think that could be one. I was thinking of something even simpler. Does anybody else want to add why you might not want to do this for board games? It might be really hard to compare this in a game like chess, where the reward is at the end. And self-play might just be that. Yeah. So it might as well-- and also because we normally know what the reward model actually is in games. And so if we know that at the very end, we can say this is a plus 1 or this is a minus 1, there is no reason, necessarily, to assume we want to look at two-game states and ask a human to try to judge which of those two is better. We know the ground truth reward. And so it's probably better just to directly use those. And it may be that those pairwise rankings for intermediate game states might not be very reliable, too. And then the last one is also true. DPO and RLHF can both be used in extremely large-- with extremely large policy networks. All right, so where are we? Last time, we talked a lot about Monte Carlo tree search, and we talked about AlphaZero. And what we'll do today is we'll talk briefly to finish up that part, just quite briefly. I want to clarify a couple things that I mentioned last time. So the rest of the day, we're going to have a guest lecture, almost the rest of the day. We're going to have a guest lecture by Dan Webber, which is a way to introduce the last part of our course, which is to think about where the rewards come from in terms of how we make judgments about which rewards we might want to prefer or not. And we'll talk about that today. And we'll talk about that after the quiz as well. But before we do that, I just want to do a little bit more on Monte Carlo tree search. So let's see if we can get take 2 for the video first. So I think that sort of captures-- it's a nice-- we don't normally get documentaries made about the work that happens in artificial intelligence, at least not yet. But I think that it's a pretty powerful examine of why people were so excited about this result and the implications it had when you can exceed the best performance in the world at something by computers. And we've seen examples of this in the past. For those of you who heard about it, there was an IBM Watson for Jeopardy case a number of years ago. And I remember being in the audience when that was happening. Many, many people watched it in different watching parties at the time, and I was in one of those watching parties. And it was a similar moment in AI, thinking about what are the levels that we're going to be able to achieve in AI and what are the implications of that for of human expertise and human excellence. So Monte Carlo tree search, of course-- so, of course, they did win the game. Deepmind did win against Lee Sedol. Let's now just go back and think a bit about what Monte Carlo tree search and AlphaZero are doing. So this is another refresher understanding. And I'm also doing two of these today, just to give you an example. These are the types of questions you also might see on the quiz. So we'll do another one of these. And then I'm going to clarify a couple points about AlphaZero from last time. Why don't you find somebody near you and compare your answers? OK, good. I'm hearing a lot of discussion about this, which is good. So the first one is true. The first one is it does approximates a forward search tree. The second one is false, and I know this is a little bit subtle. So Monte Carlo tree search tries to approximate the forward search tree. But as you might remember, the forward search tree can scale exponentially with the number of states and the number of actions because you're expanding by both of those at each level. And so what Monte Carlo tree search does is it uses its dynamics model to sample a next state. So you don't have to enumerate all the possible next states. So it uses sampling to help with the state branching factor, but it doesn't tell you what to do about the action factor. One thing you can do is you can do upper confidence trees. And then that tells you how to use a bandit to figure out which action to select next. In AlphaZero, we see that even that is likely not to be sufficient when you have an enormous branching factor. And so you may need some sort of additional weight, like the probability, like a policy to select among those actions. The second one is-- the third one is also false. So this was true in the original AlphaGo. And I think in the Lee Sedol 1, 2, that you saw in the video, they did have two networks. But in AlphaZero, they just have a single network, and it outputs both a policy and a value. So it just has two output heads. The third thing is true. So amazingly, even if you spend 40 days and you've got many TPUs, et cetera to learn a policy output and a value output in the network, they still do it at test time. Like if you're playing Lee Sedol, they still do additional guided Monte Carlo tree search at that point, and it makes a big difference. So I think it was something like going from an Elo score of 3,000 to 4,500 or 5,000. I'm going to get the numbers wrong, but it was a huge gain by doing a little bit more of extra local computation. And the third thing-- the fourth thing is also-- or I guess the next thing is also true, which is self-play does form-- provides a form of implicit curriculum learning, because the agent is always essentially working with an opponent that's very similar to its level. In fact, itself. So it's exactly at its level. And that means that the density of reward it gets is going to be much higher than it would get if it was playing an opponent that was much higher or much lower. The other thing that I wanted to clarify is-- so when we talked before, we talked about selecting a move in a single game. And we talked about how it both maintains this Q-- maintains a Q s, a for a certain node, as well as this upper bound here, which is going to be proportional to this policy function that gets from the neural network divided by 1 plus the number of samples. So I mentioned in class-- I just wanted to make sure to clarify this. I mentioned in class that I thought that all of the S's here are just the nodes. But S is a little bit weird of a notation because you could imagine it could either be the state space or the node, like that part of the tree search. I looked back on it. This is actually the node. So I was correcting what I said last time, but I just wanted to make sure that was clear. So they're thinking of each of these points as being like a particular s, a. But in theory-- and again, I'm not a Go expert. So I'm not sure how often this happens. You could end up at the same game state lower down in the tree, and you would maintain totally different statistics for down there. So you're not sharing across those. And there's just simplicity to do that in terms of-- there's simplicity-- that can help in terms of the architectures that you need to derive and just simplify some of the storage for this. So it is-- these are per node. And then the other thing that I just wanted to clarify also is that when you get to the root later and you're making a decision over which of these actions to take, I mentioned that what they do at the final end is that they're going to do something proportional to an s, a at the root divided by 1 over tau. So this is going to be the policy at the root. I just wanted to make sure to be clear about what that does. So this is sort of prioritizing parts of actions that you've taken more, that you've explored more in your neural network. So let's just see a little bit about what this would look like. So if tau is equal to 1, what that would mean is that your probability of taking action A from the root would be equal to the number of times you've taken action A in the root divided by the sum of the times you've taken. Well, just really the total number of rollouts you've done from the root. So that would be strictly proportional. In that case, if you have a tau less than 1, that means that you are going to upway some of these things. So then you would have-- so let's say if tau is equal to 0.5, then you would have an A root squared divided by the sum of N a, root squared. I'll put a prime there. So what that would mean is that if you make tau closer and closer to 0, then this is basically going to do a winners-takes-all approach, and you'll basically select whichever action you took most from the root. And then as you go to more towards 1, it's sort of equally spread across all the times you've taken each of the actions. And as you might imagine, that's going to have different implications for how exploratory you are. Now, note that none of these things are doing it based on what the value is at the root. All of these are just based on essentially how much of the time you've explored different parts of the tree. So I just wanted to make sure to clarify those in those cases. So I may have any other questions about alpha 0. And I do just want to say that I-- as I mentioned before, there's a number of different other derivatives that have happened since this. So there's mu 0, which doesn't even need to know the rules of the game. And there are also a lot of sophisticated approaches which have to do with hidden information, games like poker. So in this case, there's full information. You know exactly where all the white stones are. You know exactly where all the black stones are. There's no hidden information that either player has, and there's only two players. But there's been a lot of work on thinking about cases, like poker and others, where there's some cards that one agent doesn't see from the other. And so then, how do you play optimally in those games as well? So do you have any questions before we move on to our guest lecture? Yeah. With these other models, specifically user that doesn't make the rules the game, do we just observe that even though it doesn't know the rules, it learns just as well, or does it do better without knowing the rules? What's the consequences of doing that? Yeah, that's a great question. I'd have to go back to the paper and remember the exact results. It certainly can do just as well. So it can quickly-- you still have to give it some feedback. It has to know whether or not it won, but it doesn't have to know all the individual rules. And so you can-- I just don't remember how much additional data you needed in that case. As you guys might remember from last time, we saw that there is a really substantial impact of architecture. So depending on the architectures you're using and we're using a convolutional neural net or some other different types of networks, those also make a massive difference to the amount of data you need and the quality of the result. So I think that's something to keep in mind when we think of removing information, like what the rules are. You could imagine that if you do that but then you have some other innovations in terms of the architecture, it might be that you need only the same amount of data or even less than what we're needed here. So generally, they don't do full ablations over all the combinatorics of the ways these systems just specified. It's a good question. But that work certainly suggested-- and they've also extended this to other games and things like chess and others just to show that you can use very similar techniques to conquer those games as well. All right. With that, let's switch over to Dan. So I'm really delighted to have Dan talking today. He is-- I guess I'll keep this here until he comes up. He is a postdoc fellow here at Stanford. He'll introduce his own background a little bit more, but he has a lot of expertise and thinking about different frameworks for how do we think about rewards and what are the implications of the different ways. We're going to define those in terms of the subsequent type of systems we might develop. Please, please hold your applause until you see how it actually goes. Oh, yeah. All right, I'm going to need a sec to get this hooked up. While I do that, I should note, I am going to ask you, at various points maybe, to talk to some of the folks next to you. So if you're not in a good position to do that, that might be a time to move to get yourself in such a position. OK, is that good? Is that too loud? Just right? Love it. OK. And we go to 250. Yes? 250. Yep, perfect or-- or we can at any rate. Great. OK. So yeah, I'm Dan. Dan Webber. Here today to talk to you about value alignment. But before we do that, maybe it is just worth saying a little bit about, who is this guy? Why should we listen to him or care at all about what he has to say? He's not the professor. So as I mentioned, I am a postdoc here at Stanford in HAI and EIS. That's the Institute for Human Centered Artificial Intelligence and the Center for Ethics and Society. If you've taken a lot of CS classes at Stanford, you've probably seen somebody who has my job at some point or other. Yeah, big part of my job is embedding ethics into computer science courses like this one. Before I came here to Stanford, I got my PhD in Philosophy at the University of Pittsburgh, where I wrote my dissertation on moral theory, which basically means just trying really hard, maybe too hard to think systematically about value, which is what brings me to you today. So before that, even I got my bachelor's in Computer Science at Amherst, I did a couple of years in software development after that. So I'm not completely new to CS. I know this world a little bit. I did take an introductory course on AI. That was 10 years ago. I think the field has changed immensely since then. I don't even think we covered reinforcement learning at all. So you all are going to know the reinforcement learning way better than I do. I'm not here to be an expert about that. What I am hoping to do is give you a bit of a window into how to think about value and how it might be more complicated than you think. So we're not going to solve any deep problems about value in the next hour. We're not going to be able to go very in depth on a lot of this stuff. If you're interested in that, I recommend courses in the philosophy department. But just try to give you a quick, maybe, lay of the land, sense of the range of possibilities when we're talking about value and value alignment. So, OK, it might help to start with an example of value alignment or maybe, more accurately, an example of value misalignment. One of the classic examples in this literature is paperclip AI. But this example from Nick Bostrom in 2016, maybe you're all used to this in reinforcement learning, but it tells you something about the state of this literature that a classic example could be from 2016. So Bostrom describes an AI, designed to manage production in a factory, which is given the final goal of maximizing the manufacture of paperclips. Does anyone have an idea maybe of how this example continues? Maybe you've seen it before. Anyone knows this one? No? OK. Well, in Bostrom's example, at least this AI proceeds by first converting the Earth and then increasingly large chunks of the observable universe into paperclips. OK. Now, Bostrom is thinking in particular about superintelligent AI. That's what his book is about. So he's got the destruction of the entire universe in view. But even a less powerful AI system might pursue a simple goal like this in surprising ways. Does anybody maybe have a more mundane example of what could go wrong if an AI system were, say, in charge of a paperclip factory, given no further instruction than to maximize the production of paperclips? Yeah? [INAUDIBLE] people for a lot of shifts, like through the night and then fired workers who complain and hire new ones. Yeah. Good, good, right. Yeah, we could maximize production if only we trapped people inside the building and made them work around the clock, right? Excellent. Any other-- yeah? [INAUDIBLE] about the quality of the paperclips, right? So they could all be really bad. Good. Yes, exactly. It might be that the easiest way to maximize the number of paperclips I produce is to produce really terrible paperclips. That's not really what I was looking for probably. Great, thank you. Anyone else? Yeah? I mean, if the price of electricity changes at different times of day, it could be like trying to make paperclips but just economically and efficiently. Yeah. Good, right. So it's maximized the number of paperclips, but there's no sense of other goals that you might also want to pursue here, like efficiency or minimizing the amount of electricity you use or anything like that. Great, yeah. Or you could imagine-- I mean, it depends what levers the eye has to pull, but you could imagine it recycling the factory's plumbing for raw materials or locking out humans who could interrupt its process. Something like that. So great. So, in general, we might say the problem of value alignment is this problem of, how do we design AI agents that will do what we really want them to do? What we really want is usually a lot more nuanced than what we say we want, right? Humans work with a lot of background assumptions, and these assumptions can be hard to formalize, easy to take for granted. If I told you as the manager of the factory to maximize the production of paperclips, you would realize that you should do that consistent with existing labor laws, or that you should make paperclips that actually work, or that you should be on the lookout for keeping your costs down, things like that. But because these can be hard to formalize, they're easy for us to forget about. It's hard to solve this problem just by giving better instructions to AI agents. And here, I mean, if anybody wants to give it a try, what would be the better-- how would you solve this problem maybe just by trying to give a better instruction to the AI? Anybody have what they think might be an improvement on just to maximize paperclip production? Yeah? [INAUDIBLE] Too much? Good, yeah. So yeah, specifying, A, that you want paper clips of a certain quality and giving a sample of what that looks like. Good, that would help-- that could help address this problem potentially of, can you maximize production just by making worse paperclips? Right? It might not go far enough to, say-- and, by the way, you shouldn't work the factory workers around the clock, but great start. Yeah? Maximize the long-run profits of the paperclip factory. Good, good. So yeah, giving a broader goal-- right, I want to maximize the production of paperclips, but that's something I want probably because I want to maximize the profit that the factory generates. Good. Is that going to be enough to avoid all of the problems that we've seen come up? I mean-- yeah? I mean, it's most of them, right? We need high-quality paperclips. You can't turn the universe into paperclips. The profit will be zero. You can't be using too much electricity or doing things that are economically inefficient because it won't be profitable. I mean, the labor laws are probably-- I think that you'd be violating. Yeah, right. I mean, if there's enough people willing to work in this factory, maybe we're able to keep a lid on how poorly we treat people. We could get away with maximizing profit while still-- but good, OK. So that's getting us some of the way there, but still there's a worry about essentially treating people well. OK. I mean, we could keep doing this all day. But hopefully, this is a little bit of an illustration. Even trying to think of better instructions, you might just realize, oh, there's another thing I forgot, there's another thing I forgot. I mean, you can compare this maybe to the difficulty in manually specifying reward functions. I mean, in some sense, this is the same problem. OK, I think I know what the thing is that that I want. OK, it turns out to be much more complicated than that, much harder to specify, especially when you're thinking about making a system that's going to take instructions from users maybe, who are not experts in reinforcement learning. Folks in this room are going to be relatively good at foreseeing these kinds of problems with giving incomplete instructions. If you're designing a system that's supposed to take instructions from non-expert users, they might not be so good at foreseeing these issues. OK. Maybe any-- I should-- I say, any questions now? And in general, going forward-- I mean, if anybody has any questions at any time, don't hesitate to raise your hand. OK. So we have this problem. How do we design AI agents that will do what we really want? But that's a little underspecified, right? I mean, there are lots of things that we might mean by a phrase like "what we really want." So here's one of them. You might think value alignment is the problem of designing AI agents that do what we really intend for them to do. The problem with Paperclip AI might be that it failed to derive the user's true intention, which is to, let's say, maximize production subject to certain constraints, maximize production without overworking the workers while making sufficiently good paperclips and while keeping costs down and so on and so on and so on. Deriving that nuanced, complicated intention from the underspecified instruction maximize production. If that's how we think about value alignment, then, of course, the solution is going to be to design AI systems that can successfully do this translation, take under-specified instructions, figure out what the user's actual intention is that they're trying to express, and then act on that instead. How is this from a technical perspective? Here's Iason Gabriel, a researcher in Philosophy and Ethics of AI. So what he says about it, he says, "This is a significant challenge." And he means from a technical perspective. "To really grasp the intention behind instructions, AI may require a complete model of human language and interaction, including an understanding of the culture, institutions, and practices that allow people to understand the implied meaning of terms." That's what he said in 2020. How do folks in this room feel about how this quote has aged maybe in the last four years? Does this seem like a significant technical challenge? Does it seem less significant maybe than it might have seemed four years ago for any reasons? Seeing shaking your head. Why not? Well, you're probably trying to imply-- trying to allude to GPT. But I don't think that's enough, because GPT might omit certain aspects of the world model that might still cause loopholes like that. So I don't think the problem has really been solved. Good, yeah. So, yes, I am not a subtle man. I was indeed thinking, yeah, require a complete model of human language and interaction. Hmm, that maybe sounds like a model that a lot of folks have been hard at work developing. But, yes, I agree with you. Yeah, so you might think-- yeah, could you use something like an LLM to affect this translation as part of the system? But, yeah, how complete do we think those models really are if I give this under-- if I say-- If I give the ChatGPT the user wants to maximize production in the paperclip factory, what do you think they really intend? Is it going to catch all of the nuances that are typically communicated when one human is talking to another? Yeah, I agree. There's reason to doubt that, but see what the future holds. But that's the technical challenge. This is a philosophical challenge here as well, which is that you might think our intentions don't always track what it is that we really want. So classic cases of this might be cases of incomplete information or imperfect rationality. We've sort of already broached this one. I mean, suppose that I intend for the AI to maximize paperclip production, again, subject to these constraints, because what I want is to maximize return on my investment in the factory. If the AI knows that I would get a better return by producing something else or by selling the factory, has it given me what I really want if it does what I intend, which is for it to maximize paperclip production? Well, in one sense, yes. But in another sense, no. You might think that other sense is the more important one. It's not giving me the thing that I really wanted because that thing is coming apart from my plan about how to get it. OK. So you might think the solution here is that-- what you really want is an AI agent that does what the user prefers, what they actually prefer, even if this isn't what they intend. On this interpretation of the problem, Paperclip AI is misaligned because I prefer that it not destroy the world, or I prefer that it not lock all the users in the factory. Users, all the workers. OK. Now, the problem here is that, if you want to align to what the user actually prefers, there's going to have to be some way for the agent to know what the user prefers when that differs from the intentions that the user expresses. How are you going to go about doing that? Solution to this might be to work with the user's revealed preferences. Preferences that are learned from observing the user's behavior or feedback. Obviously, you've learned some techniques for how to do this kind of thing, but not every technique is going to be like this. You're going to have to do something, like inverse reinforcement learning or reinforcement learning from human feedback that allows the agent to train on observation of the user to try to determine what they prefer based on how they've behaved or what they've told it its preferences are. Of course, you're going to run into this problem that from a finite number of observations of the user's behavior or preferences, there are, at least in theory, infinitely many preference/functions that could represent inferring that could be a challenge. And it might be especially hard to infer preferences about unexpected situations, like emergencies where you don't have any direct-- you're unlikely to have directly observed the user's preferences about unusual emergency situations because they arise so rarely. But you might think it's precisely an unusual or emergency situations, where it's so important for an AI agent to be aligned to our values. So those are some of the technical challenges. But here, again, we have a philosophical problem, which is that, just as my intentions can diverge from my preferences, it seems like my preferences can diverge from what's actually good for me, or so some people might think. So, for instance, a lot of people prefer to smoke, but you might think it's not really good for them to do that. Or I might prefer to maximize profit on my paperclip factory at all costs, but maybe it would be better for me to be less focused on money and spend more time with my family, right? So the thought here is that your preferences might actually, in some cases, come apart from what's really in your best interests, objectively speaking. And that this is something that you might try to align an AI agent to instead. We want to do what's actually in the user's interests, even when that's not what the user themself prefers to do, right? If you think this, you're going to think paperclip AI is misaligned because it's objectively bad for me, for the world, to be destroyed. Or objectively, bad for me, for these things, to-- the pipes in my factory to be ripped out or what have you. Here is a sort of combined technical and philosophical problem, though, which is that-- unlike the intended meaning of my instructions or my revealed preferences, what's objectively good for me is not something that can be determined empirically. This is a philosophical question, not a scientific one. So it's not just a matter of building the right model of human language or observing the user enough. To figure out what's actually in my best interest is not entirely an empirical endeavor. You've got to actually do some substantive moral philosophy to solve this. Now, the bad news for solving this problem is that there's a lot of disagreement about what is objectively good for a person. I say philosophers disagree about this, but I think non-philosophers also disagree about this as well. Is it just a person's own pleasure or happiness that's good for them, or is it the satisfaction of that person's desires or preferences that could be different from pleasure or happiness? I might have preferences that will be satisfied only after I'm dead or something. I'll never derive any pleasure from their satisfaction, although they could still be satisfied. Or do we want to say that things like health or safety, knowledge, human relationships, these things are objectively good for us, even if we don't enjoy them, don't prefer them? These are all sort of live options in the theory of value. And depending on how you answer this question, you're going to be looking at a different kind of value, even if you already know that what you want to do is aligned to what's in the user's best interest. The good news, though, is that behind this disagreement, there is quite a lot of agreement, I would say. These things like health, safety, liberty, knowledge, dignity, happiness, almost everyone agrees that these things are at least usually good for the person who has them. Even if you think that really, ultimately, all that matters, all that's good for a person is their own happiness, well, these things typically make the person who has them happy. So you might think you don't really need to resolve this underlying philosophical dispute to have a good sense of what's in the user's best interest. I mean, these are things that, for the most part, are in a person's best interest, no matter what theory you endorse behind it. OK, any questions about any of that so far? OK, one complication about aligning to the user's best interest is that one thing that we normally take to be good for a person is autonomy, which is the ability to choose for yourself how to live your life. Even if you don't always make the best choice, it might be good for you to have this kind of control over your own life. We want to avoid paternalism. We want to avoid choosing what we think is best for someone, rather than letting them choose for themselves. So even in a case where you're aligning to the user's own best interest, you might still need to take their intentions or their preferences into account. It might be that part of what's best for them is to have their own intentions fulfilled, to have their own preferences honored. OK. So this has all been pretty abstract. I want to move into slightly more concrete case study. But first, maybe just to recap what we've covered so far, value alignment is this problem of designing AI agents to do what we really want them to do. But this can mean a lot of things. It could mean doing what we really intend them to do, what we really prefer that they do, what it would be actually in our best interest for them to do. And all of these things can come apart. They're not necessarily the same thing, and they might impose certain technical or philosophical constraints on your approach. OK, let's talk about how this works or what kind of difference this could make in practice. Think a little bit about LLM chatbots. So everyone who talks to ChatGPT is talking to the same chatbot. OK, there's different-- there's GPT 3.5. There's GPT 4. Ignore that. I mean, fundamentally, it's the same chatbot for everyone. But plenty of chatbot providers are now offering a wide range of different chatbots with different personas. Some of these designed by users themselves. So these examples are all from character.ai, which promises personalized AI for every moment of your day. So here, this comes out maybe a little small. But you can talk to the creative writing helper. You can talk to the are-you-feeling-OK bot. You can talk to the dating coach. These are some of the relatively normal ones. You can talk to depressed roommate. You can talk to Torybot. I am Torybot. I believe in the free market. You can chat with AOC. You can chat with Donald Trump. You can chat with Feminist Faye. I am a feminist that hates Donald Trump. OK, lots of variety, lots of options here. You could imagine yet stranger and stranger personas that you might build into a chatbot or that your users might. None of this-- all of these are designed by users. None of these are coming top down from the provider of the LLMs. OK, so think about this a little bit. Imagine you're building an LLM chatbot to serve as a source of news for users. I mean, maybe this is going to strike you already as crazy, but I think there are a lot of people out there who already treat Google as their primary source of news, a lot of people who are replacing Google and other search engines with LLMs. So I think there's demand for this. Imagine you were wanting to fill it. You can think a little bit about these questions. How would you make-- in what ways would you make the chatbot personalizable if you were interested in aligning to the user's preferences? In what ways might you make it personalizable if you wanted to align to the user's best interests? And think a little bit about the pros and cons of this. So I think take a minute to think about this and then maybe chat with somebody near you, compare notes, see what you're thinking. And we'll come back in a couple of minutes for a larger discussion. [SIDE CONVERSATION] All right, I've been hearing a lot of good conversations that I'm not eager to cut short, but maybe there are conversations that we can now bring back to the whole room. So anybody have any thoughts from their discussions that maybe they want to share? I know some of you have thoughts because I was hearing a lot of good ones out there. So don't be shy. You probably have better thoughts than I do. If you don't say anything, then I'm just going to tell you what I think. And then you're going to be stuck with that. Yeah? I guess for the first point, it would be-- I think it's pretty simple. You'd probably be-- you'd use a preference optimization approach. And you'd offer 10 different questions of, hey, do you prefer this answer or that answer? And then you would optimize the news that's being fed to that user accordingly. Yeah, good. Yeah, like you said, fairly simple. If I want to align to the user's preferences, I'm going to figure out what it is that the user prefers. I'm going to give them news that fits that profile, right? Is that what everybody was thinking about this first question? Anybody have something they want to add to that? Yeah, great. OK, I think-- yeah, I think that's exactly right. OK, what about if you were trying to align to the user's best interests, their own good, objectively considered? Yeah? I have a thought on that one, which is that you don't-- it's pretty hard to know what someone's best interest is, as well as avoiding the tenet that was on the previous slide of, don't be paternalistic. So really, the only way you could have any hope of doing this would be optimizing for best interests of an entire population. So if it doesn't apply, if the policy of best interest doesn't apply to everyone, then I would argue that you can't actually do it for an individual user. So that's the way you would personalize it if it's [INAUDIBLE] the line to a user's best interest is you wouldn't ask that question to begin with. You would just have it set already for the entire population. OK, yeah. I mean, good. I think-- well, without I need to constantly resist the temptation to just turn every one of these lectures into a philosophy class. So I love that answer. I'm curious about why it might be less difficult to determine what would be in the objective best interest of a large group than maybe of one person. But this is a question we'll come back to maybe later. Anybody else have thoughts about this second one thing? Something different come out of your discussions? Yeah? I think just take the movie Her as an example. When you got to know the person very well, the person opened it up, a lot of data, a lot of the information, then you will be able to maybe prioritize on how you make the suggestion. And also depend on the person using the tool. For example, some tools are better on delving into the news, trying to understand the sources. Some of them are better at-- you just want to take the most important thing and then just have to spend time on an email on random news and all that. So I'm talking about two components at least. One is, you know the person better, the other person. The other thing is, you know the behavior and how they would use the tools. And just referring-- I mean, the tools should be refrained from extending too much and just grabbing too much attention of the user. Yeah, good. Thank you. Yeah, and I think that there is something to this that it might be that just from observing someone's preferences for long enough. Getting that much data about them, you might be able to get a little bit of insight, maybe, into what's in their best interests, even when that diverges from what they want in the moment. So yeah, great. I see-- yeah? We have similar-- we similarly had an idea about maintaining some sort of state for the user's best interest. Maybe you could have some sort of structure that would represent different aspects of the best interest and which could be personalizable to the user. And with every interaction, it would reprompt the LLM and then change this if appropriate. And every time you are trying to get an output for the user, you could put this as part of the context and write a prompt accordingly alongside whatever the user is asking in order to fit that goal better. Good. Well, that sounds to me a little bit more like maybe aligning to the user's preferences. Maybe I misunderstood. This sounds trying to figure out what it is that the users want to get, what they're looking to get out of the bot, and then determining what to return based on that. Maybe I misunderstood. I don't think that's necessarily true. I think you could write a prompt that would-- like the internal prompt for keeping up the state of the user's best interest could be written, and the fields could be provided such that it would try to meta-- you could ask it to meta-reason about what the user's interests likely are. Oh, I see. I see. OK, good. Yeah, great. Yeah, any anybody else? I mean, maybe there's, in some sense, a more basic question behind this, which would be something like, what is maybe in a news-seeking agent's best interest? What kind of news would it be best to provide somebody? Yeah? Probably news that shows a variety of perspectives that you check that it's actually correct as well, I think. It's in the user's best interests that they're properly informed as opposed to maybe only seeing news that puts them in a good mood or aligned with their existing opinions. Yeah. Good, right. Yeah, you might think-- yeah, in contrast to the approach we discussed earlier of, we're going to query the user about their preferences, every time that we give them news, we're going to say, did you like that? Was that what you were looking for? Yes/no. We're going to adjust and give you the news you want based on that. Yeah, you might think it's actually better for people to be exposed to high-quality news, unbiased news, to be exposed to a variety of opinions and arguments, rather than-- What's the worry about aligning too heavily to the user's preferences is that you might be putting them in a kind of echo chamber, where they're getting all of their news from talking to Donald Trump bot or talking to Feminist bot, and they're not getting other perspectives. Yeah, good. Does anybody else have a different answer to that question maybe? What would be in the user's best interest to receive as news or how you would approach that from a design perspective? OK. Well, good. I think that's definitely right. I mean, in terms of pros and cons. Does anybody have get into this? If you were designing the news chatbot, which of these approaches would be better? What would be the pros of one, cons of another? Yeah? For me, I think optimizing for best interests is almost like paternalistic because you are assuming that you know the interests of the user. You have a good approximation. You might really not know at all. So it's like that user had some tragedy or whatever in their life recently. And then some sort of recent news event has a lot of mass death. Like, maybe they don't want to be exposed to that, even though maybe it's a very important event. You should know about this, but you don't have complete state of the user's psychic state, how they actually feel. So maybe just using the preferences that are already-- that you actually have just from using the app, what did they click on, might be better. Good. Yeah, I think that's great. So there are-- even if we can say things maybe at a very general level about what is in a person's interest, what is actually good for a person in general, that leaves a lot of room for variation from person to person, especially if you think that quite a lot of what's good for a person is built out of subjective interests of theirs or their desires or what makes them happy, what makes them unhappy human. That's not a thing you might have full access to. So there is this problem that if you're trying to align to what's really good for the user, your only real way to do that is by aligning to what it is that you think is good for the user, right? And you might be good at figuring that out. You might not be. And absolutely, that's where you run this risk of paternalism. So an advantage of just aligning the user's preferences, giving them what they said that they want means that you avoid that risk. You avoid trying to position yourself as saying, no, I know what's really good for you when maybe you're not in a position to determine that. Yeah, anyone else on this point? OK. [INAUDIBLE]? Yeah. I just thought that the counterargument is that running the risk of being paternalistic, you actually give convenience. But then giving them low-quality choices, you actually waste them a lot of time. So apps evolve. Yeah, I think that's right. And right. I mean, to the earlier point, it's-- yeah, there might be some aspects of the user's best interest that are easier to determine than others. We might be reasonably confident that it would be in any user's best interest to be given high-quality sources of news, to be exposed to a variety of opinions that might be-- you might want to align in part to that general human interest, while still allowing some room to align to the user's preferences. So these are not necessarily mutually exclusive goals in alignment. I mean, it might be-- in some ways, it might be worth focusing more on the user's preferences. In some ways, in some cases, contexts you might want to focus more on what do we think is actually good for the user, because what they prefer might be junk information or convenient bias-confirming information, things like that. OK, great. Well, there's one thing I think that has been not completely absent, maybe, from our discussion, but I hope noticeably absent from my lecture and from my slides. So far, is there may be a big piece of the puzzle that we're missing, something that you would have thought? This would be a-- this is what we're going to talk about with value alignment. And why haven't we gotten there yet? Anybody at all? We've talked about aligning to the user's intentions. We've talked about aligning to the user's preferences, to the user's best interests. Yeah? A good way to measure alignment? A way to measure alignment. Yes, that has been absent. I just think-- yeah? Maybe aligning to a society's overall interests, rather than just a person's? Yeah. So you are correct. But I was thinking-- yeah, I mean, that there are people other than the user. Where's my text? There's my text. Yeah, there are other people whose interests are important, maybe, to take into account than just the person who is giving instructions to the agent. So you might think there's really another possible interpretation of what we're after with value alignment, which is that an AI agent is value aligned, if it does what's morally right, right? I mean, the main problem with paperclip AI isn't that it does what's bad for me. It does what's bad for everyone if it destroys the world. Or it does what's bad for the factory workers if it makes them work around the clock making paperclips and so on. So earlier, we were focusing on, what do we mean by what we really want? What does really want mean? This would be to focus a little bit more on the we. What is it that we really want? Because, of course, what the user intends, prefers, even what's in their individual interests might be bad for others. We probably don't want to say that paperclip AI is value aligned if it maximizes production by exploiting the workers in the factory, even if I, as the user, have no qualms about exploiting the workers, right? OK. That said, it wasn't a waste of time to start by focusing on the user, right? Even if we want to align to morality or to the interests of more people than the user, we also do want to align to what the user wants when what the user wants is morally acceptable. So it still matters how we understand what it is that the user really wants, even if we need to place that in a larger moral or societal context. But, of course, here, too, we have a philosophical problem. I mean, which things are really morally right? There's a lot of disagreement on this one, too, not unlike the question of what is objectively good for a person. Is it right to lie to spare someone's feelings? Is it right to pirate copyrighted material? Is it right to buy luxuries when you could donate to charity instead? Is it right to kill one person to save five or a thousand or a million? These are at least some of them. I hope you think difficult moral questions. Certainly, they are moral questions that people disagree about. Again, philosophers and non-philosophers alike. So how do we align to what's morally right in the face of this disagreement? This is you might think, where my field of study comes in. You might turn to moral theory, which is basically just a systematic attempt to answer questions like these. So a moral theory, you might have heard of, it's called consequentialism. It says that an act is right if whatever produces the greatest net good of any act available. Or you might have heard of utilitarianism, which is a kind of consequentialism. It says that you should produce the greatest total happiness that you can across all people. If you have a theory like this, this can be used to answer some of these difficult questions people disagree about. Is it right to lie to spare someone's feelings? Well, if you're the consequentialist, you'll say it might be. If you can get away with it, if no one discovers that it's a lie, and it makes somebody feel better, that might produce more good than not telling the lie. So there's an idea here, which is that we could align AI to morality, to what's morally right. If we align agents to the correct or best moral theory, there's going to be a philosophical problem with this. Does anybody think they know what it's going to be? Has a similar form to all of the philosophical problems we've encountered so far. Well, there's a lot of disagreement about what the correct moral theory is. So there's disagreement not only at the order of ground-level moral facts about whether you should tell a lie to spare someone's feelings, but also about the best theory for systematizing this kind of stuff. We already saw a consequentialism. But there's a whole host of others, and just to put a few of these on the table, just to give you a sense of the range that we're looking at. You could be a prioritarian, where you would think that, really, what you want to do is not to maximize the total good but to produce the greatest weighted sum of good, where the interests of those who are worse off is given more weight. Or you could take this to extreme, a maximin, or minimax view, where what's morally right is to make things as good as possible for the person who's left the worst off by what you've done or to minimize the negative consequences for the person who suffers the most. So in cases where you have to think about a quantifiable good-- but if I have four people who I can-- how would I do this-- assign goods to, I have the option to say-- I have options to distribute goods, say these ways among different people. If I'm the consequentialist, I'm going to say, well, I want the one that produces the most total good. That's this first option. If I'm a prioritarian, well, I'm going to need some kind of way of weighting this. Say that the way to give more weight to the people who's worst off is that we weight the good to you on a log scale or something, right? Then in-- well, in base 10, at least this is going to be the prioritarian's choice. We want to-- by giving more priority to those who have it worse with the-- sorry, I'm not explaining this very well. I'm trying to move too quickly. If I was taking the log of each of these as the prioritarian, I'd say here, we have-- this is coming out to 6. This is coming out 7. That's better. If I want to make things as good as possible for the person who ends up worst off, I might choose this last option, even though in this option, we're getting the least total good, right? The person who ends up worst off is doing better than the person who ends up worst off in the other options. So all these options and more are available to you in moral theory. You might take a satisficing version of any of these views, instead of trying to maximize the total good. You might think an act is right if it just produces a sufficiently great sum of good or weighted sum of good. We haven't even yet touched deontological views, which hold that, even acts with the best consequences can be wrong if they violate certain moral rules or rights. Often, these rules will be rules like, don't murder anyone, don't steal, don't lie, keep your promises. Right, you might think that an act can't be right if it involves stealing from someone, even if it produces a lot of good. This is something that a view like consequentialism might not capture. Although you might think that these rules or rights are themselves justified by their good consequences, it would be best if we accepted rules like this and followed them. OK, returning to this problem of paternalism that we encountered earlier, there is another problem here. So one is there's just, what is the best moral theory? Who knows that's-- I've been working on that for a decade and haven't gotten much closer to it. But even if we knew what the best moral theory was, it might be bad to design AI agents to act on moral values that their users don't share. This could be because we want to avoid a kind of paternalism where we say, no, these are the correct moral values. It could be for more practical reasons. Just the users won't trust AI agents if they disagree with them about moral matters. OK. So there's some difficulty trying to align to the best or the correct moral theory. But also, like with the objective good, where there's a lot of disagreement here, there's also quite a lot of agreement about what is the morally right thing to do. In simple cases, we all agree you shouldn't kill people. You shouldn't lie to them. You shouldn't steal from them. So another idea for aligning to morality would just be aligning AI agents to what we might call common sense or consensus morality. Common sense, moral ideas, that most people agree on. Instead of trying to make AI morally perfect, we should just aim to have it make moral decisions like a normal person would. Right? This view probably ends up being pretty deontological and satisficing, right? Most of us think you follow certain moral rules, you respect other people's rights, then you're not morally required to do the best you can. It's fine to do less to prioritize yourself in some cases, things like that. Now, one advantage of aligning to something like common-sense morality, rather than to a particular moral theory, is that moral theories often have surprising implications. I know we're just about out of time, so I'll skip to the chase on these. I mean, you can think about the consequentialist requirement to maximize net good. I mean, suppose you had an AI agent that was a surgeon. His five patients dying, each of which needs a different organ transplant to save their life. Well, if you're thinking about just maximizing the net good subject to no constraints, maybe what you think is, well, that nurse walking by in the hall has all of the organs that I need. Maybe if I just harvest the organs from the nurse, put them in the five people, save five lives, the cost of one five is greater than one. We just maximize the net good. That's probably not what you wanted your surgeon AI to do. Think about cases where you might want to break a deontological rule against lying as well. AI agents aligned to a particular moral theory might discover some of these surprising implications before we do. And they might discover them in practice, rather than in the philosophy seminar room, which is where we prefer for them to come up. So by contrast, aligning to common-sense morality, you might end up with an agent that behaves more predictably, making moral decisions like a regular human. It might be unpredictable in some edge cases, where common sense arguably runs out. Would an AI aligned to common-sense morality kill one person to save a million? I don't know. That's what we-- We got into moral theory to try to answer hard questions like this. If we've just taught the agents to think about morality like we do, it might be as unsure as we are about what to do in a case like this. I need to let you go. So I'll just leave you with the thought, how bad would that be? How bad would it be if AI was as unsure about morally hard cases as we are? OK. We've covered that. I will let you go to enjoy your Wednesdays. If you are interested in talking more about any of this ethics in general, feel free to reach out. Set up a meeting. We can talk more. Any questions now before we depart? Or I can stick around for a few minutes, if folks want to talk to me offline. OK, great. Well, take care.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Exploration_2_I_2024_I_Lecture_12.txt

Hey, everybody. Welcome back. We're going to start talking more about the state of efficient reinforcement learning today. But before we do that, we're going to start with a Check Your Understanding. So this asks you to think back about what we were learning from multi-armed bandits. I would probably do one and six first because they're warm-ups, and then the rest of these. Just to clarify, in terms of notation, I'm using f of delta here to be a function of delta. Because I was slightly loose on exactly what the dependence is on delta, in terms of whether it's like delta over t or what we're going to choose for that function. So I just wanted to be agnostic to that there and put it as a log of a function of delta. As usual, feel free to look back at your notes from last week, if you want to refresh your brain on the notation. All right. One more minute to write down your initial answers and then I'll ask you to turn to a neighbor and compare. All right. Why don't you compare your answers to someone that's nearby you? [SIDE CONVERSATION] All right, great. Let's come back together. So I think most people converged on the same answer for the first one, which is, yes, algorithms that minimize regret do also maximize reward. Ooh, hold on. Pen is not working. Let's see if I can grab a different one. So the first one is true, if I can get this up here. OK. So the first one is true. If you minimize regret, you also maximize reward. For the second one, is that one true? Do you want to say why it's true? Are you saying it's false? I'm saying the second one is true. Hold on. Let's see if I can get my thing to power up. My pen isn't working. So the second one is-- let's double check that I-- I'll keep it back onto here, in terms of the answers. I moved things around a little bit, last minute, which is always dangerous, but I wanted to include a couple additional ones. OK, let's see if we can make this do the right thing. OK. So the second one should be true. This is basically the UCB algorithm, which is this is the empirical estimate of the performance of each arm. So in the case where you just have a finite set of arms, which we can also think of as a finite set of actions, we just look at what their average reward was. Nt of a was how many times have we pulled action a after t time steps. And log of f of delta was just the term that we had to try to express the dependence on delta. Delta was used to look at confidence intervals that we were using for the upper confidence bound. So this is true. The third one is also true. So, in general, with our confidence intervals, you will be selecting all arms an infinite number of times. But it might be really slow later on. Let's say you have a really big gap between arms. Then that log term-- and you'll have a t dependence in there, in general. That will continue to grow a little bit. So you'll sample another arm again, which helps with the fact that you might have been really unlucky and gotten a really weird estimate of the arm performance, so far. OK. This one was a little bit subtle. And I realize it could be not quite clear here whether I was asking you to think about the T over delta part or the 1 over square root Nt of a. I wanted you to focus on the first thing. So what this is saying here is that, instead of shrinking our confidence intervals by a rate of 1 over square root Nt, we're shrinking them at a rate of n to the minus fourth. Let me just write that. [ERASING WHITEBOARD] That's pretty squeaky. OK. All right. So let me just give-- so we're shrinking it. So will that mean that our confidence intervals are wider or narrower for the same number of counts? So let's say versus Nt of a to the minus one half. So, for example, if Nt of a is equal to 100, you've pulled this arm 100 times. Which of these two is going to be bigger, the one on the left or the one on the right? The one on the right. That's right. So instead of it being-- oh, the other way around. So this is going to be-- so if you have 100 to the minus 1/4 versus 100 to the minus 1/2, this is going to be 1/10. And this is going to be approximately 1 over 3. There's several different inverses here. I know. What this means, basically, is that we're growing-- we're shrinking our confidence intervals slower. So another thing you might see often, as a bonus term-- DeepMind often uses this, in particular, for some of their algorithms, is "to the minus 1." So that's a faster rate. So then that one would be 1 over 100. You can think of these as trading off different amounts of, essentially, how much exploration you're going to get. Because this is saying how quickly are you collapsing your confidence interval as you have more data. Now, as somebody pointed-- was asking me about-- when I was going around, they were like, well, we didn't just randomly pick this. We picked this because of the uncertainty bounds that we derived from Hoeffding. So Hoeffding said, if you have an empirical estimate of your mean, how far away could that be from the true mean? Well, under pretty mild conditions about your variable being bounded, we could get this 1 over squared N rate. So someone was asking me, very reasonably, well, why would you pick, say, this or something else? You might pick this because you just don't want to explore as much. So even though this holds for our theory, it's often somewhat conservative in practice. So you might just pick something like a faster exploration rate because, empirically, you want to just explore less. And you could think of that as being related back to what we saw with PPO, that a lot of their theoretical derivation said, this is what your step size should be, but it was way too conservative for most realistic applications. So they just changed it, and they introduced the clipping thing. On the other hand, there might be cases for which you might not be sure you could get this sort of rate. Or you might have other reasons to think you might need more exploration. So, for example, maybe things are non-stationary. And you think, my customer preferences are actually changing over time. And so I want to explore more, over time, than I would if I assumed that I was stationary. And we'll talk more about stationarity in just a second. So given all of that, this means that this expression would actually have wider confidence intervals and probably a higher upper confidence bound than our original algorithm, which means that we would still expect that, over time, it will learn-- [SNEEZES] --pull the optimal arm-- bless you-- more than any other arms. But it probably won't have as tight regret bounds because we may be exploring too much. Now, this was an interesting one. Will this, if we add this particular bonus term, make the algorithm optimistic, with respect to the empirical rewards? Somebody want to say if it's going to make-- if we add on a bonus term, will it make it optimistic, with respect to the empirical rewards? Just the empirical rewards, so compared to your empirical mean. I'm not trying to make it a trick question. Yes. Yes, exactly. So if you just add 20 to your empirical estimate, it will be optimistic, with respect to your empirical estimate. But is it guaranteed to be optimistic, with respect to your true mean? So imagine if I had said B was 0.001. It would still make it be optimistic, with respect to your empirical rewards. But would it necessarily be optimistic, with respect to your true mean? In general, no, right? So if you think back to our bandits, which just had binary rewards, let's say you have a coin that actually has a 0.5 probability of getting a heads, which we'll call a 1. If you flip it once and you get a tails, your empirical estimate will be 0. If you add a bonus of 0.01, your empirical estimate will be 0.01. The true value of the mean is still 0.5. So one of the key ideas from using Hoeffding and explicit upper confidence bounds is that, in general, it's not easy to figure out a simple bonus term you can add in order to make things optimistic. And so that's why you might, in general, want to be using these Hoeffding or other explicitly derived confidence intervals. OK, great. And then the last one is true. Does anybody have any questions about these? Yeah? Can you explain number three again? I'm not sure whether dependence on-- It's inside of here. So if you go back to the slides from last time-- let me see if I just have them up. Yeah, we create these upper confidence bounds. So, in general, we define these upper confidence bounds. We talked about how we need the bounds to hold over all-time steps, t. And so, in fact, this was not a perfect expression that we're going to have some sort of t dependence inside of the log. And in general, we'll have something like t or t squared inside of the log. And so that will introduce a dependence on the time-- either the time step, so far, or your total time horizon inside of your upper confidence bound. Any other questions about this part? OK. All right. I'll make sure that the solutions are aligned. So last time we talked about Bayesians. Sorry. We talked about bandits, which were this single-state version of Markov decision processes. Your actions didn't make any difference to the next state because you're always in a single state. We talked about how people often use the word "arms" as an equivalent for actions. And, there, we were trying to be really explicit about uncertainty over the rewards. And we talked about algorithm, upper confidence bounds for trying to be optimistic, with respect to that uncertainty. Well, today what we're going to focus on mostly is Bayesian bandits. And we'll get there in a few minutes. Before we do that, I think it's nice to think about-- I think it's exciting to think about all the application areas where these come up. And I wanted to go through this example, which I think I mentioned briefly in lecture one, just again, to think about all the complexities which come up when we want to try to use these in practice and where bandit algorithms, in particular, might be used. So this is a really beautiful paper by Hamsa Bastani, which was in Nature a few years ago. And they were trying to tackle a really important problem, which was, at the time, as everything was shutting down with COVID, all these countries had to decide on a quarantine protocol and who to test. So, before, a number of countries basically almost entirely shut down travel. But particularly in the beginning, people were letting-- And even then, often, there might be exceptions. So as people come into a border crossing, countries had to decide who to test. Now, they couldn't necessarily test everybody because resources are finite. They're also having testing facilities they're using for testing all of their own individuals. And tests are expensive. In addition, when someone is tested, they are going to ask them to quarantine. Depending on where you were in the world, that might actually have been funded by the government. So you have to go to a quarantine hotel, which also costs the government money. So there are a lot of reasons, in this case, that your resources are limited and you don't necessarily want to test everyone. Also, in general, it may not be necessary to test everybody, if you're trying to minimize the probability of letting in people that have COVID, in terms of limiting spread. So if there's someone that's from somewhere where there's no COVID, then you don't necessarily need to test them. So this is the setting. What happens is that when people were coming into Greece, they would submit a form in advance, like when you go to the airport or before you go, et cetera. And then what they would do is they had this approach called Eva, which tried to use the prior testing results to figure out who to actually test when they came. So what would happen is that, then, when somebody comes, like the next day, either they would say, we're not going to test you at all-- and then you would leave the premises. Or, for a subset of people, based on the form, based on where they were coming and based on prior results they had, they would decide to test someone. Then, after you got that test, you would send it to a lab. And it normally would take 24 to 48 hours. I don't remember exactly what kind of test they were using there. Maybe it was some sort of rapid PCR. I don't remember. Those would go to a central database. And then they would use those results. All these people would quarantine for 24 to 96 hours or so, during this time period. They would get the results back. If you're clear, you can go and proceed. Otherwise, you need to continue to quarantine. And then they're going to use this information to go back to Eva and update their algorithm. So this is really cool because this is an opportunity to try to be very careful about resources, but really do so in a way that still preserves the safety of the individuals in the country, as much as possible, and the public health. So I like that, in the Bastani paper, they describe this as a non-stationary contextual batch bandit problem with delayed feedback and constraints. OK. So that's quite a mouthful. But I think it's really nice to think about-- as we go from this simple setting of just thinking there are k arms, we can think about all the practical things that we might have to deal with in this setting. So, here, in some ways, the k is very small. It's only two. Either you're going to test someone, or you're not going to test them. So it's a very small action space, which is nice. In this case, compared to what we've seen, so far-- but we'll see this case later. We're going to have context. Context, you can think of as just being like states. So people will have a feature vector that describes what country they're coming from, a bunch of other details about them, and that gives you a state that we're going to use to decide whether or not to test someone. So that's why it's contextual. It's non-stationary because COVID was constantly evolving and, often, a lot of the information we were getting was lagged. So if you're in Greece, you might be able to see information from Sweden and from China and from the US. But all of that information is often, likely, probably at a population level. Those people may or may not be the same people that are traveling to Greece. Probably, in general, they're different. And because of the lag, it may or may not be informative. And in fact, in their paper, they argue a lot of that information was not as informative as this kind of real-time information. It's batched. What I mean by that is that-- and we'll see this more today. You don't get to make a decision after every test or not test. You don't see the result immediately. So what happens here is they say, 200 people fly in on a plane. You have to decide, for every single one of them, whether or not you're going to test them. And then you wait two days. [LAUGHS] So it's this delayed feedback, and you have to make a decision for everybody before you get to observe that feedback. And so that makes it quite tricky. And we'll talk more about why that might be tricky for some of the upper confidence bound algorithms we've seen so far. I think this batching is really important for many, many application areas. So if you think back to our guest lecture and you think about direct preference optimization, this is another area where, in general, you're going to be able to get a batch of data, label it all and then continue. So in some of the work that my lab is doing and some other people's work, when we're thinking about doing adaptive data collection for preference optimization, we, again, need it to be able to handle this much more realistic batch setting, compared to getting information after each decision. So the delayed feedback is this 24 to 48 hours. And the final thing is constraints. So there are lots of constraints in this setting, which also generally changed the setting from a lot of the ones we've thought about, so far. So one is that you might have resource constraints. You might say, at most, we can handle, let's say, 100. I forget exactly what it was in the paper. 100 tests a day, so you're going to have constraints on that. The second is, politically, you might have constraints, too. It might be tricky for Greece, if they decide that they're not going to let in anyone from Sweden. So there might be different quotas, or there might be other reasons to say, we have to think about some broader types of risks and benefits in these cases. So that's also challenging. One way you can think about implementing this is this could essentially change your policy class that is reasonable. So instead of your policy class saying you can make any decision for any individual, you may now have a population-level constraint as well. This is something that my lab has thought about some with our partner, Sharad Goel, who's at the Harvard Kennedy School. And there, we've thought about cases where you might have resource constraints and fairness constraints that mean that you can't just make decisions for people individually. But you need to think about overall trade-offs, in terms of your policy quality, that happen at the population level. The reason that's important is because it often introduces a lot of challenges computationally, when you can't just think of each individual separately. All right, so we won't be able to cover all of the ways that they handle this algorithmically. But I encourage you to read the paper, if you're interested in this space. And I think it's a really beautiful example of using reinforcement learning, particularly multi-armed bandits, to tackle this problem. One of the things that they had to do-- so this was a real system. They really deployed it in Greece. I think, when I talked to Hamsa, she said it came together in a month. It was a really amazing effort. And then one of the interesting things they also had to do here is to understand how much of an impact it made. Because they weren't going to do a randomized controlled trial in COVID to understand this. So another interesting thing that this paper looks at is using offline methods, like the batch methods you've been seeing in the past, to try to estimate the counterfactual of how much impact this had. So I think it's a really nice example of a lot of the different ideas that we've been seeing in this class. All right. So that's one of the many, many ways that bandits are useful. Clinical trials is another one. A/B testing, ad placement, there's many, many others as well. But I think this is a really nice example in public health. OK. So now let's continue. We're going to talk about, specifically, some of the algorithms that could be relevant to this and, in particular, Thompson sampling, which is particularly relevant to this kind of batch setting. All right. I'm going to do, very quickly, just notation. Remember, regret is the opportunity loss per one step. Total regret is the total opportunity loss. We're using Q to denote the expected reward for a particular arm. I'm blanking on who suggested this last time. Forgive me. But someone came up to me and said, hey, couldn't we have used just a smarter, optimistic initialization? Do we have to actually have these upper confidence bounds? And I think that's a very reasonable suggestion. And that was a great follow-up to some of the stuff we're going to talk about today. So one simple thing you can imagine you could do, instead of worrying about these upper confidence bounds which you have to update all the time, is you just optimize-- you just initialize your q hat to some high value. And then you just update that estimate over time. And when you do that, you know that, eventually, you're going to converge to the right thing. Asymptotically, with the law of large numbers is-- you're not changing that initialized value. That initialized value may or may not have been right. It'll be an upper bound, and it'll just converge to it. So this is an interesting thing you can do. The challenge with that is that, in general, you don't know how high to make it. And so if you make it really high-- let me just be clear here, what I mean by "really high." Often, this might be much, much larger than the actual range of possible rewards. So maybe your arm rewards can be between 0 and 1. And you initialize this to 70. So sometimes the initialization might be far higher than what is actually practical. It does encourage a lot of exploration early on, which might be really valuable. But in general, unless you get the value exactly right, which you generally can't know-- because that's why you're trying to learn, in the first place. Then you can still lock on to a suboptimal action. And what do I mean by "lock on," is that you converge to a suboptimal action and then you never try anything again, which means you'd get linear regret. The other thing that's bad is that, if you initialize Q too high, then you're also just not going to benefit from-- you're going to be making bad decisions for much longer than you actually need to do. Even though, in theory, this could be a good thing to do or-- sorry. In principle, you might imagine, this is a good thing to do. In reality, it's very hard to set. Now, it's also an interesting question of how you might do this with function approximation. I know you didn't implement deep Q-learning. But if you think back to deep Q-learning, where we used a neural network to represent the Q function, do you guys think it is easy to initialize that, so the values are optimistic? Let's see. At least [MUTED] shaking his head. Why not? You're right. It's just [INAUDIBLE] for the network to output specific value. Yeah, it's hard. Right? Maybe you could train it on fake data. But then you'd have to know how big the Q is. Yeah, in general, this is really-- if it was a table, it's at least easy to write down, like, 90 for all of those things. And that's what you initialize. In a deep neural network, it's really unclear, how you initialize those parameters, so that, for all the states that you would reach, you would have even a good shot of it being optimistic. So I think that's another challenge here, is-- and that's a challenge for a lot of the optimism algorithms we'll see, in general, is can we do it with function approximation. Now, there's a lot of work on thinking about how to do things with function approximation. And we'll get into that soon. So if you do carefully choose the initialization value, you can get good performance under a new way of measuring what good performance actually means. OK. So let's go back to regret. So in regret, we just try to think about how do we quantify the performance as we make lots of decisions. So T, here, is the number of decisions we make. And we're just trying to think, in this case, about how many decisions we make over time. Let me see if the pen is finally charged. Not today. So we could either be making lots of little mistakes or infrequent, large ones. And what you might imagine that we want to do is to think about a different form of loss. And so we're in, particular, another form of performance. That is going to be PAC. So let's draw what that'll look like. So I think I drew this last time, but I'll draw it again. So make this times step t. And this is Q of at, Q of the actual arm that you pulled. And this is q of a star, So let's imagine that you have an algorithm that is pulling arms, like the following, all right, which means that-- then maybe sometimes it's pulling the right arm, hopefully. So in this case, sometimes the algorithm is doing something that's just a little bit suboptimal, and sometimes it is making a really big mistake. So what we can do here is we can quantify how big our mistakes are. And you might have a situation where you say optimal performance is really hard. It's really hard to learn what everyone's perfect ad preferences are or things like that. Maybe I'm going to relax my criteria. I'm not going to require optimal performance, but I want pretty good performance. I want epsilon optimal. So what we do, in this case, is we count every time we make a bad decision. Meaning, something that is worse than epsilon optimal. And otherwise, we think of all of those as basically being in an equivalence class of optimal. So that's going to be what we think about when we think about PAC. OK. So I'll define what that is. So a PAC algorithm-- and raise your hands if you've seen this in machine learning. If you've taken machine learning, you might have seen PAC. OK. Yeah, so at least one or two people have. So, often, in machine learning, particularly if it's a machine learning class that includes some theory, they'll talk about PAC and probably approximately correct algorithms. And that's where this idea comes from. So it came from the machine learning community, and then reinforcement learning borrowed it. So the idea in a PAC algorithm is that on each time step, a PAC algorithm is going to choose an action whose value is epsilon optimal. Meaning, the value of the action that's taken is at least the value of the optimal action minus epsilon. So that means that we're in this region, with high probability on all but a polynomial number of time steps. So essentially, it's saying that the majority of the time, your algorithm is making good decisions. Good, here, being defined as epsilon optimal. But sometimes we'll make bad decisions. But we're going to say, with high probability, the total number of bad decisions we make is not too many. What we mean by "not too many" here is something that's polynomial, in your problem parameters. So that generally means the number of actions you have, epsilon, delta, et cetera. As you might expect, if epsilon is smaller, generally, the number of samples you need will go up. Normally, something like 1 over epsilon squared. So if you care about being more optimal, you're going to need more data. Or, in other words, your algorithm might make bad decisions for longer. If delta is smaller, meaning that you want this to hold with higher probability, you'll also need more data. And if there are a lot of actions to learn about, in general, you need more data. So it gives us some notion of the complexity of the problem to learning. So this is a different type of-- a lot of algorithms, you can get both PAC guarantees for and regret guarantees. But it is just a different notion of optimality. Most of the PAC algorithms for reinforcement learning are based on either optimism, like what we've seen from last lecture, or Thompson sampling, which we're going to see later in this lecture. And there do exist PAC algorithms that just initialize everything to a really high value. I don't know of any practical algorithms that do that, ones that people use in practice. But there is theory and papers about that case, so it is possible to do. All right. And we'll see more stuff about PAC shortly. Let me just give an example. So remember back to our fake trying to learn how to treat broken toes example from last time, where we had surgery and taping, buddy taping the toes together. Again, this is not medical advice. Imagine that this is-- epsilon is 0.05. So in this case, before we thought about this is what the optimal sequence of actions you should take-- but, of course, you don't know that because you don't have data. If you had this sequence of actions, [INAUDIBLE] and optimistic algorithm, this would be the regret you would get in each case. But under the PAC case-- let's see if I can type this here. Under the PAC case, this would be epsilon. So the important thing to notice here is that, because the reward of a2 is within the epsilon bound of a1, which is the optimal action, this action would also be considered optimal. So from the perspective of the PAC algorithm definition, this would be not denoted as a mistake. The only mistakes would be when the algorithm takes a3. So when we talk about this PAC definition here of counting up the number of time steps, we don't make a really good decision. The only decisions that would count for that, in this setting, is the a3 decisions. In contrast to that, when we talk about regret, anything that's suboptimal counts. So you get penalized for all the a2 decisions. I thought we were allowed to make mistakes for a polynomial number of steps. Yes, you are allowed, and it still will be PAC. That's exactly right. But I'm just pointing out here that the-- so the only actions you're taking that count towards that polynomial is a3 here. It's not a2. Whereas, a3 and a2 count towards your regret. OK? Yeah, So does screening become easier if I gradually reduce epsilon? Good question. So normally, in these cases, you fix epsilon in advance. And it defines the number of samples you're going to need for each of the actions or states in actions in the MDP case. So it's like an exploration term, and you keep track of counts. There are algorithms-- with me and my former PhD student, [MUTED], part of the work that we did there was to talk about what if you want to have guarantees over many epsilons at once. I'm thinking more along the lines of epsilon greedy algorithms we did in [INAUDIBLE] where, because we gradually reduce epsilon [INAUDIBLE] to the [INAUDIBLE]. If that's the same, they can be-- do the same thing there. Yeah, it's a little bit subtle. It's a great question. So in general, the bounds will depend, something like 1 over epsilon squared. So if your epsilon is going to 0, that will say that you have to do an infinite amount of exploration. If you're interested, I have this. I mean, one of our papers thinks about trying to have simultaneous bounds over lots of epsilons. But in general, the basic version of this, you commit to an epsilon in advance. Great questions. All right. So going back to where we are and reminding ourselves, in terms of algorithms-- this relates to your epsilon greedy. Constant e-greedy, decaying e-greedy and optimistic initialization all have the problem, in general, of having sublinear-- of having bad performance. It's, in theory, possible to have sublinear regret. But you often need to have stronger knowledge than is known. Optimistic initialization also can have the PAC guarantees that we just talked about. And I guess I'll just say, too, you can convert these results into-- so epsilon greedy is not a PAC algorithm. But you can think about different types of other exploration strategies and whether or not they're PAC. And we'll get back into those soon. OK. Let's jump into Bayesian bandits. They're a pretty elegant idea. So, so far, we've made almost no assumptions about our reward distribution. So we've maybe said they're bounded. It can be between 0 and 1. And that's basically all we needed for Hoeffding. We need them to be bounded. We needed them to be-- they're independent and identically distributed. But we haven't made any other assumptions. So we haven't said it's Gaussian, or it's a Bernoulli or something else we might know. And when we're being Bayesian about this, we're actually going to leverage knowledge we have about the structure of the way the rewards are generated. And what I mean by that is, normally, some particular statistical model, so it's a Gaussian model. Or it's a Bernoulli model, things like that. And the reason that that might be helpful is that, often, if we're doing these in a domain like public health or others, people might know lots of information about-- [SNEEZES] --what the reward structure is. Bless you. And could we leverage that to get better algorithms and better performance? OK. So before we do this, it's probably helpful to do just a quick refresher on Bayesian inference. Some of you guys might have done a lot of this. Some of you might have done very little. We'll go through just a quick reminder about this because this is going to be used a lot for what we're going to see today. So the idea is that we're going to start with a prior over the unknown parameters. In our particular case, that's going to be the unknown distribution over the rewards for each arm. So it's like if we have a coin flip. Or, if we think about the toes example, what's the probability that someone's going to heel, if they're given surgery? We don't know what that parameter, theta is. And so we're going to have a prior over what that theta could be. Once we're given some observations about that parameter-- for example, if we observe, when you do surgery, that someone was healed, that is going to change your uncertainty over the unknown parameters. So let's do a particular example. So if the reward of arm I is the probability distribution depends on a parameter of phi i, we have initial prior over that parameter. Pull on arm, we observe a reward. Then we can use Bayes'-- that should be "Bayes'--" Bayes' rule to update that. And I think it's really helpful to visualize how the priors change over time, so we'll see that in an example shortly, just so you can see what that might look like. So what we're going to have here is Bayes' rule. All right. This is our prior probability over the parameter governing the reward distribution for this arm. This is the likelihood of observing a particular reward given a specific parameter value. And this is the probability of seeing that reward in general. And when we do that, this is Bayes' rule, and then we use it to update what our new probability is over the parameter that generates that reward. So in the case of surgery, it would be before we had some distribution over how successful we think surgery is on average, we give surgery to someone, we update it, we observe that they are healed. And then that changes what we think about the underlying parameters. So, bring that out here. This is the prior probability. This is the probability of reward given a particular parameter. This is the probability of getting the reward in general. And we can rewrite this by using the joint distribution of the reward and the parameter and then marginalize out the parameter. All right. So this is beautiful. Oh, yeah? Can we go back to the previous slide. I'm just kind of confused on the setup. If I imagine that phi as a primer for a Bernoulli variable and using background knowledge, I have some prior as to what it should be, what does it mean to have a distribution over that? Yeah, it's a great-- in general, it may not be obvious that we can compute this. So for example, we're going to see in some cases, this is analytic. You can analytically update this, which is super elegant. What I mean in that case is as a simple-- so let's say-- we'll see an example shortly. But phi i could be as the probability recovery-- I'll do this for surgery-- for surgery. So this would be, say, 90% of the time, someone's recovered. And let's say-- or something like that. And this could be a particular prior. So I could say, I think my probability that your recovered mostly from the surgery is 0.9. So I'm pretty confident that the surgery is going to be highly effective on average. But I think that there's some probability that the surgery is not so effective. And then I would say, well, I think that maybe it was 10% probability the surgery isn't as effective. But on average, people are going to recover at rate 0.4 with the surgery. And we'll see some specific examples of this. This is not the priors we're going to use, but this just illustrates how you can have distributions over distributions which can get confusing pretty quickly. But on the other hand, it's also super elegant and a place where we can put in prior knowledge, just like clinicians and others may have information where they can actually directly specify these priors. All right. And so there's many questions you might have in this case of, like, where do these priors come from? And even if we have these priors, how do we do this calculation? So in general, this is complicated. So you can see here, you've got to have a functional form for this. This, in our case, was like flipping a coin. And so if your coin has a bias of 0.9, what's the probability you'd get reward 1? It would be 0.9. So you have to have a probability distribution here, probability distribution here. You have to marginalize one out over here. And when you do all of that, you get your new posterior, which is after you observe something, now, what is your new distribution? So you might imagine that now I update this, maybe I see that the surgery was successful. And I'm, like, oh, maybe I can update this to be 0.95 and 0.05. So in general, this is going to be computationally tricky to do exactly without additional structure. There's lots of ways to approximate it, but the really cool thing is that in some cases, you can do this analytically. So this is idea of these conjugate priors. So this is beautiful idea of the exponential families. And if you have a representation of your prior that is conjugate with-- this is often called your likelihood function. Then, after you do all of this updating, this new thing is in the same statistical family as what this was in before. And we'll see some specific examples of this in a second. So the high level, really beautiful idea in this case is that it's analytic. When you do all of this, let's say this was initially a Gaussian, this is still going to be a Gaussian if you use conjugate priors. So let's see how to do this for Bernoulli. So for Bernoulli, there is a conjugate prior, which is really cool. And the conjugate prior is called a beta distribution. And it's going to have a really nice, beautiful interpretation that we'll see in just a second. So the equation looks terrible. The equation says the probability of a particular theta-- remember, this is the bias of your coin-- given some alpha and beta-- these are just two other parameters-- is theta to the alpha minus 1, 1 minus theta of the beta minus 1 times the gamma function of alpha plus beta divided by gamma of alpha, gamma of beta. So this looks fairly terrible, but it is conjugate, which means that after we observe something, our new posterior is also going to be the same. But it turns out that it has a really simple explanation, a really simple intuition, which is, imagine you start with your prior being a beta alpha beta. And then you observe a reward that's either 0 or 1 because your variable is just 0 or 1. It's a Bernoulli. Then your new beta, your posterior, is just r plus alpha, comma, 1 minus r plus alpha. What does this mean? If you observed a 1, then you increment your first parameter. It's like you increase the number of successes. If you observe a 0, you increase this number, the second number, like you increase the number of failures. So you can think of what the beta is doing is essentially just keeping track of how many heads did you get, how many tails, or how many ones did you get, and how many zeros? It's just keeping track of those, and it can use those to explicitly update what the probability is of your theta. So it's really beautiful because you don't-- computationally, that's really easy to keep track of. You're just going to add one depending on what you see. And what you can think of this here is being as, how confident are you in advance of how many pseudocounts did you see of success versus failure? So, for example, if I'm really confident that the surgery is going to be successful, maybe I'm like, yeah, I'm so confident. It's as if I've seen 100 successful surgeries and only two failures. But if I'm really uncertain, what I would do is I'd say, well, I'm going to treat like one successful, one failure. I really don't know. And we'll see what this looks like in just a sec. Excuse me. So now when we have this, this is basically giving us a distribution over the reward parameters, and we can use this to actually make decisions. All right. So there's a couple of different ways to do this. And one of the ways to do this is by getting a confidence interval, similar to what we've seen before. But the other thing is called probability matching or Thompson sampling. And let's go through Thompson sampling now and see an example. All right. So in probability matching, we're going to assume we're in the Bayesian bandit case. And what probability matching does is says, OK, the way we might want to explore is by sampling actions according to the probability that they're optimal, given everything I've seen so far. So what it says is given some history, which is like the past things I've tried and whether I've gotten ones or zeros for them, I want to select a new action based on the probability that its true mean is higher than the mean of all the other arms. And I'm not going to tell you yet that that's formally a good thing to do in terms of regret, but you might imagine, that's a reasonable thing to do, sort of says, oh, well, if I think that arm is likely to be the optimal one with 60% probability, I'll try that with 60% probability. And then if I think there's another arm that might be optimal, I'll try that with 30% probability. Now, in general, it's not clear how you would compute this. It seems kind of an interesting idea. It's not clear you can compute it, but it turns out there's a really simple algorithm to compute this. So this is called Thompson sampling. And I think it was first invented by Thompson maybe in 1919. Maybe 1919, maybe 1920. Around 1919, Thompson sampling. So it was around forever. I mean, it's been around for like 100 years. But at least from the machine learning perspective, I think it was forgotten about for the first, I don't know, 90 of those? It really came back into prominence about 2010, 2011, when some people discovered that it actually had some really nice empirical properties, unlike Hoeffding, which has been used for a long time. How does Thompson sampling work? We're going to have a prior over each arm. Then for each iteration, what we're going to do is we're going to sample a reward distribution from the posterior. We'll see an example of exactly what I mean by that. We compute the action value function, given that sample. We take the arm that is maximum, given those Q's. We observe a reward, and we update our posterior. And then we're going to do this many, many times. And again, this will all seem much more concrete when we do an example. And it's going to turn out that this exactly implements probability matching. So let's come back to this in a second, and let's first do a specific example, because I think that will make it a lot more concrete. All right. So let's go back to our broken toes example. What we're going to do, so we're going to place a prior over each arms parameter, and I'm going to choose beta 1, 1, What does a beta 1, 1 look like? That looks like the following. I'm sorry, my pen isn't working today, otherwise that would have been helpful. But I'll draw it up here. This is 0, this is 1, this is theta. We know that for a Bernoulli variable, the value for a theta has to be somewhere between 0 and 1 because you can either always get 1 or always get 0 or somewhere in between. What a beta 1, 1, looks like-- so this is going to be the probability of theta. This is my prior. What a beta 1, 1 looks like is this, which is a uniform distribution. What it says is I have no idea what theta is. It could be 0, it could be 1, it could be 0.5, it could be 0.7, it could be 0.9. It just says someone is totally agnostic. This is called often like an uninformative prior, saying, I have no idea what my probability is for surgery, et cetera. But this is what that looks like. So this is our prior. And now, what we're going to do is we're actually going to sample a Bernoulli parameter, given the prior of each arm for the three arms. So what does that mean? That means I'm going to sample something for surgery. I'm going to sample something for buddy taping, and I'm going to sample something for nothing, for do nothing. All of them have this particular prior for now. So for the first one, it's like I'm just sampling from a uniform distribution between 0 and 1. So it could be anything between those 0 and 1. Let me just check which number I'm-- should I use the numbers I'm going to use for the next ones? So let's say, for example, that I happen to sample 0.3. That's a totally reasonable thing that I could sample, given this uniform distribution between 0 and 1. Then for buddy taping, let's say I sample 0.5. Again, a totally reasonable thing I could sample, given this distribution. And for do nothing, I'm going to sample 0.6. So this is just the distributions that I have over my prior over the parameters, and this is a particular set of parameters I could sample. Given that, I now-- what Thompson sampling says I should do is I should select the action that is maximal given the parameters I've sampled. So under these three parameters, if you want to maximize the probability that someone will recover from surgery-- or sorry, recover from their-- recover in terms of their broken toe, should I do surgery, buddy taping, or nothing? Which one has the highest chance? In this case, nothing. In this case, nothing, right. So as soon as you-- in our case, it's pretty simple once you see the theta, because the theta is exactly equal to the expected reward. So what this would say is in this case, you should do nothing. So this is going to be-- So this will say do nothing. All right. We're going to observe the patient's outcome. Now, in this case, we're going to assume that doing nothing is actually not so effective. And so we're going to observe a zero. And now, what we're going to do is we're going to update the posterior over doing nothing, given that observation. Now, the other two haven't-- the other two arms, their prior hasn't changed because we haven't gotten any observations about surgery or buddy taping. The only thing we've got an observation about is doing nothing. So what I said before-- so we have alpha. This is our alpha beta parameter. So this is our prior. In particular, it was beta 1, 1 before. And when I pull this arm and I get a reward of 0, what I said we would do here is the first one you can think of as being the number of successes, and the second one is the number of failures. So what this becomes is it becomes beta 1, 2. And that is going to look different. It's going to look like this. This is a beta 1, 2. And this is a beta 1, 1. Does somebody want to tell me intuitively why it makes sense that this looks like this for doing nothing? Does this put-- where does this put weight in terms of parameters? Since we received a 0, we should think it's more likely at 0. Yeah, we should think of the parameter value is likely lower. And so we've shifted our probability mass. And so we're, like, OK, for the things that we don't know, we're still totally agnostic about whether they're effective or not. For the thing that we just tried, do nothing, we got a 0. So it is more likely that our actual theta is lower because a lower theta in general, will generate more 0's. And so we've changed our distribution. Let's see what that looks like here. So this is our new posterior. We're using again, remember, this is conjugate. So this is our new theta, and we haven't changed it for the other two. Now, here's the next important thing. What we're going to do now is we are-- so this is what that beta looks like, just for beta 1, 2. Now, we're going to do our next step at Thompson sampling. So what we have to do now is we now have to resample. So we are going to resample where we have two distributions. These two ones are a beta 1, 1. And this one is a beta 1, 2. We're going to throw away all of our old parameters from last time. They were just samples. We now have an updated distribution for one of the arms, and we have the old one for the other two. So in this case, we might resample, and we would get this. It's more likely now that we would sample a theta 3, which is lower because our beta puts more weight on the lower part. So this is what this looks like here. So under this, we're going to pull arm one because it has the highest expected success. And that one is going to give us a beta 2, 1, because remember, again, that we increment the first one in terms of the number of successes and number of failures. So as you might expect, it's exactly symmetric to this one. And we have something that looks like this. I'm not being perfectly precise of the intersection. So now, again, we throw away all the old parameters that we've sampled so far. So we're going to throw away the 0.7, the 0.5, and the 0.3. And we're going to resample. So this time, let's imagine we sample 0.71, 0.65, and 0.1. And we again observe that the outcome from surgery is successful. This is what a beta 3, 1 looks like. So it stops looking like a straight line. It starts having a curve. So I really like these graphs because I feel like it gives one a much better intuitive sense of how as you get information that translates to your posterior over what you think the theta likely is. So as you see more successes, it will tend to go weighted to one way. As you see more failures, they're weighted to the other way. And as you might expect-- so we're not seeing that right here. But if you can see cases where it starts to concentrate in the middle or somewhere in between, just depends on what actual observations you're getting. So this is how Thompson sampling works. And I think-- so let's say we did this, then now we have this something that's even more peaked. We get another one, and you can see it just continues to curve. OK, yeah? Could you use Thompson sampling with random variables that just have many more parameters as opposed to just using Bernoulli's? Yes, absolutely. Yeah. And one of the examples we'll see later today, they're using it for advertising, and they have a large number of features. Yeah. You can extend all of these to the function approximation case. Good question. So I think one of the things-- I mean, obviously this is a small example. What we saw in this particular example I just did is that we quickly started to converge to a1 in this case. Now, notice so far, we've actually never pulled a2. We had some probability of pulling a2 because if we had sampled a really high value for a2, then we would have pulled it. But we haven't done that yet. And a1, which actually does have generally, a higher probability in this case of having good outcomes, is starting to be pulled. So it's quite different than the optimism methods because in optimism, we had to at least pull each arm once, so we could even start to initialize our confidence bounds. That's not the case here. We already have a prior over what the values are, all of these, and we can immediately start using that to make decisions. All right. So what is Thompson sampling doing when we're doing these polls, and what results do we have in this case? So what it is doing in this case is it's-- well, actually, let me just step back because I wanted to get to the example. So I went through that part a little bit fast. Let me just go to how the matching is working. So let's just go back to here for a second. So what we can see in this case is what Thompson sampling is actually doing, is that each time point, it is trying to select actions according to this probability. And it'll often end up being optimistic in the face of uncertainty because we're doing an argmax with respect to our empirical estimates. But it won't-- in general, as you might imagine, uncertain actions have a higher probability of being the max. So if you are really sure that your parameter is at 0.5 and you have another parameter you have very little information about, you have a beta 1, 1, then you're more likely to accidentally sample a much higher value for that parameter. So the elegant thing here is that you can think of this as the following. This is really useful also for the theory. So in posterior matching, that's this first line. That's sampling things according to this. What Thompson sampling does is it doesn't do that explicitly. It just samples a reward for each of the-- like a reward parameter for each of the different arms. And then it picks the one that's argmax. And so it's really elegant that that is in fact, the same thing as doing probability matching. That gives us the fact that that ends up working in terms of these. So the key idea in this case is that as you're computing these with respect to the data that you have so far, in fact, the probability that Thompson sampling picks an arm is exactly equal to this true probability, given all the data you've seen so far. I'll do a pointer. If we have time at the end, maybe I'll to go through the proof briefly for the Bayesian regret case. But there's also a really nice explanation of this inside of Tor Lattimore and Csaba Szepesv'ari's book. There's quite a lot there. This is a quite mathematical version of it, but it gives you some really nice background. Let's go back to here. Let's first just talk about, how do we evaluate performance? So what we saw in frequentist regret, like what we saw last time, is that we're assuming a particular unknown set of parameters. Our arms are actually 0.9, 0.7, 0.6. We just don't know what they are. And then our regret is always evaluated with respect to the optimal arm, given that fixed set of parameters. Bayesian regret assumes there's this prior over the parameters. And so when we talk about regret, we're actually taking an expectation with respect to that prior. So it's still-- this looks like exactly the same as the frequentist regret, but now we have this outer expectation over theta. One of the key ideas of this, in terms of how one might prove things in this case, is if we think back to how we proved some ideas around regret, we didn't do the full proof. I just try to give some sketches. One of the key ideas in the proof for frequentist regret in upper confidence bounds is that we try to construct these upper confidence bounds UT that we thought would be higher than the true value of the arm with high probability. And we use that in order to figure out how many times we would pull suboptimal arms. We leverage this fact. So it turns out that you can do Bayesian regret bounds under a pretty similar decomposition. You can think about computing an upper confidence bound and the likelihood that it'll hold. We might come back to that later today, but I want to first get into extending these up to higher level settings as well. Before we do that, I just want to highlight that if you try to get standard bounds, like what we saw last time, for standard Thompson sampling-- and what I mean by that is the type of Thompson sampling I just showed you. To my last check, they don't actually match the best bound for upper confidence bound and frequentist algorithms. However, often empirically, they can be really effective algorithms. And I'll just highlight here that in general, you can't compare directly between Bayesian regret bounds and frequentist because one of them is with respect to this prior over parameters. So let's look at that for a particular domain and why Thompson sampling might be particularly helpful for a lot of real world cases. So this is a really nice paper by Olivier Chapelle and Lihong Li, which sort of re-initiated a huge amount of interest in Thompson sampling a little over a decade ago. So I think they were both at Yahoo at the time, if I remember, right? They were thinking about a contextual bandit case, so they were thinking about making news article recommendations, et cetera. And so there you would have a context, like you'd have a bunch of features about an individual. And also, often you would have a bunch of features about the arms, so explaining maybe news articles, and all the features, or ads and stuff like that. But we're still going to see in the context of sampled iid at each step. So if I give a particular ad at this time point, it doesn't impact what's going to happen to [MUTED] So it's still a bandit. There's no sequential dependencies there. Arms are articles. Reward is binary. Either you click on it or you don't. In this case, you can model it using logistic regression because you have this binary output. So what are we seeing here? So this is CTR, which means it's a clickthrough rate. It's normalized because they're not going to tell us exactly what they get on their real world data. The important thing to look at here is the x-axis, which is delay. So in many cases, just like what we saw for a public health setting, there will be some form of delay even for online customer cases. So Amazon will show you something, and they don't find out for a little bit of whether or not you're clicking on it or whether you bought the thing. And so what they're comparing their algorithms with here is the following. So TS is Thompson sampling. OTS is optimistic Thompson sampling. You can try to add in a little bit of optimism in these. UCB is upper confidence bound. EG I think, is epsilon greedy. And exploit is you just do whatever the mean looks like so far. These are all hyperparameters. As often is the case, the hyperparameters matter, so it's useful to look at these. I think the really interesting thing to look at in this case is to look across the time. So if you-- this is the shortest delay, and this is the longest delay. And you can see for the blue algorithm, it varies very little in terms of its performance, even if things are delayed a lot. But if you look at, say, UCB, its performance tends to drop a lot in terms of as you have longer delay. And so that is one of the reasons why you might want to do Thompson sampling in these cases. So let's think more about that and do a check our understanding. So let's think about an online news website with lots of people logging in every second. Often, someone will come online before you've seen the outcome of the previous person. It asks you to select all of the things that you think are true as we think about Thompson sampling versus upper confidence bounds. All right. Why don't you compare your answer to someone nearby? So let's come back together. So as we were just discussing, we pointed out that Thompson sampling could cause much worse performance-- this one is true-- than optimism if the prior is very misleading. So this is true. Because if for example, maybe surgery is really effective and someone starts off and thinks surgery isn't effective at all, and so you put a lot of probability mass, you could have a really sharp prior on it over here, then it could take a long time essentially for your data to overwhelm your prior. So this one can be a problem. The first one is also true. So if you think back to the algorithms that we saw last time for optimism, there is no randomness in there, unless you have a tie. So if your upper confidence bound for arm one is higher than the upper confidence bound for arm two and arm three, you're just going to take arm one. And that's fine, but if you have a delay, that means you can't update those upper confidence bounds. So if the next customer comes, you're, like, oh. Or the next patient comes, and you're, like, I still think surgery is best. I still think surgery is best. And you're not going to try anything different. Whereas Thompson sampling just has this prior or posterior. And so if I have someone come, I can just sample from all of my priors. And then if another person comes, I'll again sample from my priors. And so because of this distribution over parameters, unless it's collapsed to a delta function, in which case you know what the right thing is to do anyway, you'll get natural exploration. So that's one of the really big benefits of Thompson sampling, is that even if you don't get new data, you naturally will try out different things, and that can be really helpful. It is true that optimism algorithms generally are better than Thompson sampling in terms of their regret bounds. That may or may not translate to empirical benefits. But they don't actually necessarily have strong regret bounds for this setting. So this is false. And that's because all the bounds we've been talking about so far, don't think about that batch setting. They're being derived for the case where you get information, you update your confidence bounds, you continue. So this highlights some of the particular benefits and the potential weaknesses of Thompson sampling. If your prior is reasonable and you've got this delay or batch setting, it can be very helpful. If your prior is really bad, it can take a long time to get past that. So before we end today, I think an interesting question to consider is whether or not Thompson sampling is optimal. Now, we can get nice regret bounds for this case. I know I didn't have a chance to go through that particular proof today, but it's not optimal in general. So it would be really cool if we could get something that was basically perfect. You might imagine that if you have a prior and you have a known horizon that you could compute a decision policy that would maximize your expected rewards, given that prior and the horizon. So I haven't, at least in this class, taught you all the tools you need to do that. But at a high level, you could think of it as a Markov decision process over parameters, which is kind of wild. So if any of you guys have taken Mykel Kochenderfer's class-- actually, who's taken Mykel's class, anybody here? So you can think of like a POMDP. Your state is your parameters, your actions are pulling things, and then your belief state is your new probability of your parameters. So it's really elegant. In theory, you can compute something that will exactly maximize your expected reward by doing POMDP planning. The problem, also for those of you who've taken Mykel's class, is that often, POMDP planning is really intractable. So it's often not clear that we could do this in a computationally reasonable way. In general, one of the challenges here is that if you wanted to do this, it would have a decision policy that's a function of the history, which means all the prior actions you've taken and all of the rewards you've observed. And that's going to increase exponentially with the number of decisions you made. So there's this idea of an index policy. And an index policy says we don't want to have to think about this exponential history or state. An index policy is, one, a decision policy that computes a real valued index for each arm, and it plays the arm with the highest index, using statistics only from that arm and the horizon. So that means I don't have to pay attention to this combinatorial exponential thing. I can just say for this particular arm, maybe what were my rewards that I've observed so far, and then I can use that information to make decisions. So, for example, a greedy algorithm which just relies on your empirical average of the performance for each arm, is an index policy. So it's an upper confidence bound algorithm because it just relies on the upper confidence bound for the rewards you've seen for each arm. So there are a lot of index policies. Surprisingly, there is an index policy that is optimal. So Gittins proved that there exists an optimal policy for maximizing the expected discounted reward in a Bayesian multi-armed bandit, that you can compute, that only depends on these statistics separately for each arms. So that's really cool. It means that it is possible in some settings to actually exactly optimize your expected sum of discounted rewards for these type of Bayesian bandits. Thompson sampling will not do this in general. So Thompson sampling is generally not equal to what the Gittins index would be, but it can still be a very good thing to do. All right. So just to summarize some of the things that are useful to understand from this part of the section. And next time, we're going to start talking about these ideas for sequential decision processes, like Markov decision process. You should be able to define regret and PAC. You should be able to prove or know why the UCB bandit algorithm has sublinear regret, like up to the proof sketch we did in class. You should be able to give an example of why e-greedy, and greedy and pessimism can result in linear regret. I don't think you need to be able to do this for Gaussian rewards. But you should be able to do Thompson's sampling for the case that we've just talked about, at least in pseudocode land. So if someone said, you observe another count, what would your beta parameter be? And then also that you should be able to understand the UCB bandit algorithm as we've covered in class. So we've been building up all of these things to think about now how we can do exploration and data efficient learning for sequential processes. So next time, we'll think about how to do this in a standard decision process as well as thinking about, what do we do when we're in really large state spaces or really large action spaces, and how do we lift this all up for function approximation? I'll see you on Wednesday.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Policy_Search_2_I_2024_I_Lecture_6.txt

Hi, everybody. Welcome back. We're going to be talking more about policy gradient methods today. And we're going to start off with a quick Refresh Your Understanding. All right. Let's go ahead and go through these. So everybody said the last thing was false, which is correct. It is not guaranteed to converge. They're not guaranteed to converge to a global optima. They're just guaranteed to converge to a local optima of the policy gradient space. The first one is true. There are different ways to write this down, but in general, what we're doing is we're going to be trying to find, take steps in the policy parameterization space. We're parameterizing our policies by theta, so that we're going to be trying to move in the direction of the log of the policy parameters times their value, the return you get from them. The second one is false. There's a bit of disagreement over this. So because you can see from this first derivative, we are going to look at the direction of the derivative with respect to theta of the log of the policy parameters. But it's weighted by the return, or weighted by the Q-function. So whether we push it up or not will depend whether or not we're getting high rewards when we go in that direction. So this one is false. And this one is also true. But in general, what we're trying to do is we're trying to find parts of the policy space, such that when we follow that policy, we visit states and actions, which have higher estimated Q-function, or higher estimated rewards. Do you have a question? OK. Great. All right. So last time, we started talking about policy search, which was this idea of saying we're going to be directly trying to search in the policy parameterized by some theta. This could be a Gaussian policy class. This could be softmax, or this could be, as it will often be, a deep neural network. And what we're going to talk about today is we're going to get to finish off that part, and then talk about more advanced policy gradient methods. And in particular, today we're going to cover at least the majority of PPO. So this should be enough for you to be making significant progress on Homework 2. So in particular, what we're going to be covering is, we've talked last time a lot about likelihood ratio or score function policy gradients. We're going to talk more about the notion of a baseline and why introducing that is not going to incur any bias in the estimate of our gradient. We'll talk about alternative targets, and then we're going to talk about PPO. And again, just to remind ourselves, PPO is what they used in ChatGPT and a huge number of other application areas, as well. So it's a really, really useful technique. All right. So let's just remind ourselves, we talked about how we can take a derivative with respect to the value of a particular policy. So this was the policy parameters. And we showed that it could look like this. And this was an unbiased estimate of the gradient, but it could be very noisy, in part because it looks something like our Monte Carlo estimates that we saw before because we're looking at these returns. And so we talked about a couple of different fixes, or we started to talk about fixes to make it tractable. So one was to leverage the temporal structure, meaning that your reward on time step three can't depend on your actions after time step 3. And so we could use that to reduce the variance of our estimator. And now, the next thing we're going to talk about-- we start talking about this last time-- is baseline. So as we talk about this, I think it's just useful to keep in mind throughout this that we're always trying to converge as quickly as possible. So we want these estimates of the gradient to be as low variance as possible so we can try to be taking better steps in our policy space. All right. So let's look at the baseline. We started talking about this before and we said, well, when we are thinking about how to move our policy inside of the policy space, we want to think about, not just how much reward we're getting, but really, maybe how much reward we're getting relative to other things we could be doing. So we want to know how much better this policy is compared to other stuff. And I said, you could introduce this baseline, b of st, which was only a function of the state. So note, not a function of a, of theta or a. Now, there's been other work, including from my lab, thinking about whether we can introduce baselines that may be a function of something beyond the state. But for today, we're just going to assume that it's only a function of the state. And what we're going to prove now is that, for any choice of the baseline, as long as it's only a function of the state, this gradient estimator is unbiased, which means that we could introduce this here, and we're not changing, on average, what the gradient estimator is. And a near optimal choice is going to be the expected return. So now, we're going to, again, just be trying to think about how much better is taking the actions under this current policy compared to other things we could do. So let's see. We're going to step through why adding a baseline does not incur any bias in our estimate of the gradient. So what we're going to do this is we're going to think about how our gradient comes together. OK. So remember, tau here is our trajectories, and then this is our gradient. So the goal is to show this is equal to 0. Why is that? Because if we think about what this term was, this first term was an estimate, unbiased estimate of the gradient. And now we've subtracted off this term times this term, and we want to show that, in expectation, subtracting off that term is 0, which means that we didn't introduce any bias. OK, so let's just step through how that works. So the goal is to show this as 0, and we're just going to step through this. So our expectation is over tau. Tau is our trajectory, so let's just write it out. You can write it out as the states we've seen up to a time step t plus the states that we see from that time step onwards because that's like we're just writing out our full trajectory. So we can just think of our trajectory here as being s0 to t and a0 to t minus 1. And that's just the full trajectory. So that's tau. And so we're just going to decompose this expectation. So we break this up. And after we've done that, we can notice that we can pull one of the terms out. I'm going to pull this out because this is not a function of this future expectation. And I could pull that out there because this is just a function of the current state, and it doesn't depend on the future state, so the future actions that I take. All right. So that's what I did. And then next, I'm going to notice that, well, this term here is only a function of the state and the action. It's not, again, a function of all the future actions and future states. So we can just rewrite that as expectation over the action t. All right. So what are we going to do next? The next thing we're going to do is, I'm just going to write this out more fully. So I'll repeat this. So we've got our baseline here, st, and we're going to write out what this expectation is. This is an expectation over the actions. What actions are we taking? We're exactly taking the actions according to our policy. So I've just rewritten what that expectation is. The expectation we're taking over actions is exactly the probability we take each action according to our current policy. OK, once we have that, we can play the likelihood ratio, or we can think of what is this derivative. This derivative is equal to bst sum over a. The derivative of log is just going to be the derivative of the things inside divide it by this one I should have added. Let me put a theta in here to make it clear that all of this is a function of my current theta. So I just took the derivative with respect to the log. But when we see that, we realize we can cross those out. So we can cross off this. We can cross off this because that's the same. So now what do we have? We have the expectation over states. b of st, sum over a, derivative. Remember, we're taking the derivative with respect to theta pi of at st theta. So this is what this looks like so far. And now what I'm going to do is I'm going to switch the derivative and the sum. OK? So if I'm going to say this is b of st, derivative of theta, sum over a. OK, why was that important? Because now, and let me just-- but we know here that the sum over all actions we could take at this time step has to sum to 1, because that's true for any policy. So this must equal 1, which means we have b of st of the derivative of theta of 1, which is equal to 0, because that's a constant. Let me just show it here because it's more neat. So there's two main insights for how we did this proof. The first was we thought about our expectation over all trajectories, and we broke it up to the part of the trajectory that happened before, the state of interest, the st, and the part that happened afterwards. After we did that, we showed that we could rewrite that expectation just in terms of at, because we didn't care about all the future stuff. This only depends on at and st. And then we take the derivative, and then we can see that we can switch these, and then that just becomes 1. And it's a constant that doesn't depend on theta. So the derivative with respect to theta is 0. And so that is why introducing a baseline that only depends on the state does not introduce any bias because an expectation, its value is 0. So that allows us to derive what we often call a sort of a vanilla policy gradient method, which incorporates both the temporal structure and a baseline. And so the idea in this case is that we're going to collect a set of-- we're going to take our current policy, which is parameterized by theta. We're going to roll it out a number of times, and then we go for each time step t in each trajectory. We're going to compute the return, just like our Monte Carlo estimate. And then we can compute the advantage estimate, which is we take that current return from that state till the end of the episode minus our baseline. And someone asked me about this last time. Generally, we're going to change refitting the baseline each time. So we can re estimate the baseline. Again, it doesn't matter what we pick for the baseline. It will always be unbiased, but there will be better or worse choices. So you can imagine, if the baseline is 0, it will never make any difference. The goal is to, hopefully, have a baseline that's pretty informative and has a value close to the value of your policy. And so then we'll update the policy using our policy gradient estimate, which is a sum of these terms. So we're going to use all of these terms, where we've got that derivative with respect to the log of the policy parameters times our advantage, and then repeat. So this is a vanilla policy gradient algorithm. Yeah. There is no discount factor in the return. Is that intentional, or what? Yeah, good question. So right now, the question is, there's no discount factor. There's no discount factor right now because we're assuming we're in the fully episodic case, so we don't have to have a discount factor. You could certainly include one if you want to. Yeah, so for right now here, we don't have a discount factor. Now, one thing that you might think about when you're starting to look at this is to say, well, a lot of this feels like the Monte Carlo estimation that we did earlier in the class. We've been using these G estimators, this return, to estimate what is the performance of the policy from this time step to the end of the episode. But as you might imagine in this case, that generally is a pretty noisy estimate. So then the question is going to be, could we maybe do better. So there's two places we could imagine. I guess there's two things here that we could imagine plugging in other choices for. There is what is the return of the policy from a particular action till the end of the episode, and what is my general estimate of the performance in that state. So one thing you can imagine here is, if we think back to Q functions and value functions, maybe we could plug those in, instead of using the return and using a generic baseline. So you could plug in-- instead of saying, what is the return from this state and action to the end of the episode, you could imagine plugging in the Q value for the current policy from the state and action to the end of the episode. And we can either make gamma equal to 1 or not. We're going to generally assume no for now. Assume episodic. So you can set gamma equal to 1. And the state value function could be a good baseline. And just to remember here, on this slide, you can think of G as kind of being like a Q function, and V as being a value function. So this would be an alternative we could do. OK, so let's think about how we generally could reduce variance. So what we've seen so far is we're mostly using Monte Carlo like returns. Now, let's see if we can do something better. So one thing we can do now is we're going to try to plug in and use things like state action values. And this is where the idea of actor critic methods come in, which are also really popular in reinforcement learning. So the idea here is that we could reduce the variance of this estimate of the value function at a single st from a single rollout by bootstrapping or doing function approximation at that point. So you could think back to deep Q-learning or something like that as a way for us to approximate what the value might be, or just general sort of deep learning for the value function. So when we do this, we end up with what is called actor critic methods. The idea is that the actor is the policy. So the actor is the policy often parameterized by theta. And the value function, or the state action value function, is the critic. And it's representing a V or a Q function. So that's why they're called actor critic. Actor is our policy parameterization. Critic is our state action value. And the great thing is that we can use both of those inside of a policy gradient algorithm. So you are constantly updating an estimate of the state action value, as well as having an explicit policy parameterization. And you use them together to, hopefully, increase the rate at which we learn to get a good policy. Now, in this case, normally what we're doing here is we're gathering data using the policy. And then we're using that data to fit a critic. And the reason we call it a critic is because the critic is sort of trying to estimate the performance of an explicit representation of the performance of the policy. So the actor makes decisions, and the critic says that's how good it was. So that's why it's called actor critic. A3C is a pretty popular actor critic method. There's quite a lot of others. So many of the reinforcement learning algorithms will end up being essentially actor critic algorithms. And so it'll be useful to have both representations. So if you think of it, you'd have a sort of a deep neural network to represent your policy, and you'd have a deep neural network, a separate one-- you could share parameters, but you don't have to-- to represent your value function. All right. Once we do that, we can think of rewriting our policy gradient formulas. So this was what we had before. We could approximate this now as saying, well, what if we just plugged in instead of that return g, which is that sum over the rewards, we plugged in a Q function, and we plugged in a parameterized Q function with a parameter w. So these were our weights. So now, just to highlight here, we're going to have these two sets of parameters w and theta, theta for the policy, w for the value function. And if we let the baseline be an estimate of the V, then we can just directly write down a state action advantage function, where we look at the difference between the Q and the V. So now V is serving as our baseline. And I'll just highlight here that using the advantage function was one of the first things that-- I got best paper maybe in 2016. Right after deep Q-learning had started coming out when people thought about these different adaptations, one of the things that was proposed is to think about trying to maximize with respect to advantages. But here, we're going to be using this within a policy gradient approach. Now, one of the things you might wonder here is, OK, well, we've got these extremes. On the one hand, you could have this Monte Carlo return of what is the value of the state in action that you get from starting that state in action and rolling out to the end of the episode. And the other is you could plug in a Q function. Now, there might be some sort of blending between these two. So these are known as n step estimators. So a critic, in general, doesn't have to pick sort of a temporal difference. Temporal difference in the way that we've seen it so far is normally-- so I'll just write down here. We've seen it as TD0, which means we have the immediate reward plus gamma times V of s prime. So in TD0, you take you think of your immediate reward, and then you plug in or bootstrap immediately when you say ns on the next state. And I'm going to plug-in my value for that next state. So if you think back to our tree representation, it's like you see one actual observed return, and then you plug-in your estimate. But in general, you could trade off between taking one step or taking two steps, or three steps, and then plugging in your estimate of the return. So in particular, here's a number of different types of estimators you could look at. So let's call this r hat 1, which is get your immediate reward. This is what we've seen before, plus gamma. Then you bootstrap. A second one would be you take your next 2. Again, your gamma can be 1 or not, and then you plug in it. And sort of r hat infinity would be your normal Monte Carlo return, which is you don't do any bootstrapping. You just sum up all your rewards. And you can think of each of these as being estimates of your Q function. And then you could just get advantage estimators where you subtract off the V of your current state in each of those settings. So those are all things you could do. They're called n step estimators, where n is the number of time steps until you bootstrap. And one of the important things to think about is where you might want to trade off between these ones. So we'll do just a Check Your Understanding now. If we think about introducing these type of blended estimators, how does bias and variance trade off? So why don't we go ahead and do that now. Can you select like more than one? You should be able to. Does that work? OK, I'll give you one more minute to think about it and put it in your answer. And then there's a lot of variability in what people are saying. And so why don't you talk to your neighbor in a second? All right. Turn to a neighbor. Compare what you got. [INAUDIBLE] What do you think about it? [INTERPOSING VOICES] Yeah? you are subtracting the V function. But if you think back to Monte Carlo and TD methods, did they have the same bias? [INTERPOSING VOICES] Well, this is the question. Do these both have the same bias? So if you ignore subtracting a hundred [INAUDIBLE] the first part of the estimate, do you think the first one has higher bias or the second one? Yeah, I don't know why it has high bias. If you think back, the first one should be like the temporal difference method, and the second one should be Monte Carlo. And remember, I guess, it should be clear that these Vs are all estimates, so they're not converged. So then the top one has higher bias. Exactly. The second one has lower bias because of the actual value. Exactly, yeah. And then what do you think that means for the variance? I guess higher variance also for the top one because, I guess, because it's an estimate. Close. Yeah, so for the first one, you're totally right that it's got higher bias because you're immediately bootstrapping. But in general, it will have lower variance. OK. Is that just generally a trade off? Yeah, yeah. Normally, the Monte Carlo methods, because you're summing them up, they're normally totally unbiased. But they have lots of terms. So you could think of there's lots of stochasticity. OK. Yeah, and then the other way is normally aggressive. I see. OK, that makes sense. Cool. [INTERPOSING VOICES] All right. I'm going to ask if people all turn this-- yeah, I'm going to ask, did it may change their mind after talking to someone? OK, at least a few. Yep, all right. So I think one of the things, too, I just wanted to clarify here is that I find it easiest to-- I think it's often easiest to think about this without subtracting the V, because the V is the same in both of these. So you can just focus to understand which of them has high variance or high bias. You can just focus on the first things. And when you look at just the first things, it should remind you of Monte Carlo methods versus temporal different methods. So the first one has low variance and high bias. Does somebody want to say why? So the first one, which looks kind of like a TD0 update, so this one has low variance. So a1, low variance, high bias. Does anyone want to share why that is? Which is it? Sorry, it wasn't on top. It wasn't, no. Just to make sure, I'm going to explain what each of them has. But yes. Do you want to say somewhat? Yeah. And remind me your name. I'm not entirely sure, but I think that intuition, at least when I was thinking about it, was that using the actual values of, for example, rt plus 1 is more accurate, versus if you bootstrap very early, it's more of an estimate. It's not as accurate. Yeah, that's exactly right. So it's the same as in temporal difference methods. In general, it's a little misleading sometimes to look at these. Maybe I should put hats over all these next year. But all of these Vs are just estimates. They're given finite amounts of data and however many backups we've done, et cetera. V is an approximation. So this is an estimate. So this isn't true. And if it's not true, it's probably biased. So in general, V will not be an unbiased estimator. And so this means that we're only using 1.r from the true policy, and then we're immediately bootstrapping. So in general, this is going to be high bias or higher bias, but it's generally going to be pretty low variance. And the way to think about this is that each of the rewards, in general, are going to be from a stochastic process because you're taking a series of steps. And then at each one, you're going to sample a reward from there. So you only have one really random thing here, and then one thing that is fixed. It might be wrong, but it's fixed. In contrast, this one, which looks like a Monte Carlo estimate, is going to have high variance because it's got all of these different rewards that are all being sampled from a stochastic process. So if you think of it this way, imagine it's something where your robot can walk anywhere over the room. And under your policy, your policy can go in all of these different directions. But when you actually just execute one trajectory, you're just going to get one of those. And so its variance generally is enormous. And this might be true even if, on average, you kind of have a trajectory like this or something. So in general, this one is going to be really high variance, but it's generally going to be low or 0 bias. Why is it low or 0 bias? Because this actually is a return from the policy that you're executing. So in expectation, this really is equal to the value of that policy. So it's generally low or 0 bias, but it can be really high variance. And so that should maybe give some intuition over-- and we were just discussing this. In general, it's going to be, unfortunately, be a trade off between low variance and low bias. And so often you're going to want things in terms of n. So if that's the n step, and you want to minimize your mean squared error, often you're going to end up wanting to do something where it's a couple steps, and then bootstrap to get a nice trade off between bias and variance. And I think we'll probably talk a little bit more about that next week. So I'll just highlight this here that this has low bias and high variance on this one. So this one has low variance and high bias. The other one is the opposite. All right, cool. So just to think about this, so when we are thinking about these targets, we can go between these different ones. And then, oops, sorry. Somehow, these got copied. OK, so these are all things that you can plug in. You can make different choices over whether you plug in these n step methods or others. And then you can use this all as part of your actor critic policy gradient method. So now what we're going to do, and I'll just make it a slide to delete those slides. So now what we're going to go into is more advanced policy gradient methods. OK, so those are the basic ones, and they're kind of the backbone between all of the algorithms that we do now. But there's been a lot of interest in these types of methods and how do we scale them up, and how do we make them better. And we'll talk about what we mean by better here. So actually, we'll probably talk about some of this next week because I wanted to make sure that we got through the algorithm today so that you guys can have all the knowledge you need to be starting to do the implementation. And then we'll do more on the theory next week. So policy gradients, so far, we've been talking about them being great, and we know that they're used in some really important application areas. Why do we have to go beyond the methods that we just saw? Well, there's a couple of different limitations to them. One is that the sample efficiency is generally poor. And so in general, you have to do many rollouts with the same policy in order to get a good estimate of the gradient. Because you want a good estimate of the gradient, because otherwise, when you take a step, you might get somewhere that's not as good or, you don't get to the place you want to as quickly. And the other, and we're going to see an example of this shortly, is that the distance in the parameter space doesn't necessarily equal the distance in the policy space. So this is a little bit weird of an idea. But the idea is that, if you have a policy parameterization, whether it's like a deep neural network or others, there's some parameters in there. And when you change them, when you change your theta, you're going to get a different policy out. But whether that is really smooth, if I change theta, let's say theta is a scalar. So maybe theta is like 0.7. If I change it to 0.75, does that smoothly change how much more I take a particular action, or might it be really discontinuous? Might it suddenly say you were taking this action with 20% probability, and I changed it a little bit, and now I'm like taking that action with 90% probability. The main idea here, and we'll see an example of this in a second, is this part may not be smooth, which means that even really small changes in your theta, in your policy parameterization, might actually lead to really big differences in how you're making decisions. So you could imagine your robot picks up things one way, and then you change your theta a little bit, and suddenly, it drives off a cliff. OK, not quite that extreme, but it's not clear that, as we smoothly take gradient steps in theta, if that's going to smoothly change our policy parameterization or policy decisions. So let's look at those both. All right. Sample efficiency. So what we've been seeing so far is the idea is that we take our policy, we roll it out one or more times, and then we take a gradient step, and we take one gradient step, and then we roll out our policy again. And in general, it would be really nice to be able to take multiple gradient steps. But so far, we have not seen that. And it's called on policy expectation. So this is similar to SARSA and other methods we've seen before, where you're learning about a policy and its value by actually executing it. So the problem is when we think about-- let me just go back to here. When we think about doing these gradients, we've assumed that we've gotten trajectories from the policy and the theta that we're at right now. And then we use that data to take a step. So now we're at some point. Here's our theta. We're at some point, and we estimate the gradient from that point. So we estimate it from trajectories that are generated under theta, and then we take a step. Now, the problem is, now we might be here. And what we would like to do is to take another step before getting more data. But the problem is, we don't have-- let's call this theta prime. What we have is we have data from theta. We don't have any data from theta prime. So a priori, it shouldn't be clear that we could take more than one step. You can take one step because we have data about theta. We estimate the gradient at that point. Now, we would like to just be able to continue to take gradient steps before we actually go out in the real world and gather more data. But it shouldn't be obvious how to do that yet. And so when we talk about policy gradient right now, we've been talking about on policy methods, where we just try to estimate the gradient for the policy we just executed. So similar to SARSA. Now, so what we would like to be able to do is-- so that's what we've been doing so far. We collect sample trajectories from the policy, then we form a sample estimate. It's pretty stable. We get our gradient, we take a step, we rinse and repeat. Another thing we could do, thinking about Q-learning or others is, well, what if we could use that old data to estimate the gradient at some other theta, some other policy. Could we do that? This is known as off policy estimates. This generally can start to be pretty unstable. We're going to think about different ways we could even do that. But we really would like that. We would like to be able to use our old data, take multiple gradient steps before we actually have to gather data. And you can imagine that might end up allowing us to be much more data efficient so that the total amount of times we have to gather more data is much less. So we're going to think about a way today, and we'll talk more about this certainly over the next few weeks, as well, of how do we use our old data to, essentially, move faster in our parameter space. Here's the second big challenge. So in general, we're going to be doing stochastic gradient ascent with some step size. We've repeatedly thought of there being some sort of learning rate or step size. One of the challenges-- and this was important for deep Q learning. We thought about it even for TD learning and stuff. What was the step size? How much do we update our estimate every time we get new data? Turns out it's much harder here. Now, we saw before that, under some pretty loose, loose requirements on the learning rate, we could, at least in tabular cases, guarantee to converge, et cetera. Policy gradient methods are a little bit different. Here, the step size really matters. And if we take a step size that is really quite bad, we can collapse in our performance. Does anybody have an idea of why? Why does that happen? We why can we suddenly collapse? So I [INAUDIBLE] might mix the optimal target, and then go for the win rate. Yeah, that is great. Great. So let's look at an example. So remember, the way that we're getting our data is from our policy. So let's say we're trying to get to this point. You have a big learning rate. So we took big steps. You might now get to part of the space, which is really bad, really, really bad policies. By really bad policies, I mean they have really bad value functions. If you have really bad value functions and trajectories, which are visiting states and actions, which all have really bad reward, it's really hard to estimate a good gradient of where to go. In general, you might be in a really long plateau place. So this could be a really long. And so the gradient here might be really hard to estimate of how do I get back to that local optima. In general, it's not going to be impossible, unless it's completely flat. But it might be really close to completely flat. And so that's a big problem, is that you don't necessarily know how large your step size should be. On the other hand, if you use a really small step sizes, that's bad, too, because each time, your step size is basically determining how much you change your policy before you get new data. And so you would like to take as big a step sizes as possible that don't overstep and that allow you to quickly get to the optimal local optima. Now, things like of atom style optimizers, et cetera, help, but it won't necessarily solve the problem. And one of the challenges here is, we're only getting information about the states and actions that are visited under our policy. And you just might get to regions where there's very little information to estimate those gradients. Here's another challenge. And this relates to that we're taking steps in the theta space, not directly in terms of the actions we take when we update our policy. So let's think about a parameterization, which is a pretty simple parameterization. So this is a logistic function. And this is 1 over 1 plus e to the minus w. So in this case, this is the parameterization. It's kind of like a softmax. You just have some probability of going to one action, and the rest of the probability goes to the other, and you've parameterized it with theta. So if theta is equal to 0, it's 50/50. You'll either take a1 or a2. If theta is equal to 2, you suddenly take a2 is much less. And if theta is equal to 4, you basically never take a2. And that's just because of how our relationship goes from theta to pi of a. In this parameterization, it's pretty extreme. So as we make sort of small, relatively small changes to theta, I didn't make theta a million or anything. I've basically shifted, even with 0, to 2, to 4, I've radically shifted how much of my probability mass goes on to a1. And that's just to illustrate this issue of smoothness that, as I make what might be considered relatively small changes in my theta space, that might make my policy near deterministic. And we know that if our policy is deterministic, we can't learn anything about other actions. So let me just make this a little smaller so you can see the question. So the challenge in this case is that step size can matter a lot in terms of efficiency. We don't necessarily know what the right step size is, and it may be hard for us to know how small changes in our parameter space relate to changes in the action distributions we actually follow. And so what we'd really like to do here is to actually come up with an update rule that doesn't over-change the policy too quickly, but still allows us to make rapid progress. So we'd like to move as far as we can in a way that we think is, really, ideally, is going to just directly increase the value of our policy. And I guess I'll say, well, we'll see a bit more of that. And also, ideally, we would like this all to be monotonic. We would like it so that if we think back to the policy improvement algorithms that we've seen before, policy improvement algorithms for the tabular case where we knew how the world worked had this great property that every time we updated our policy, we got a better policy or we were done. Now, the world is much more complicated now. We've got these sort of continuous parameterizations. We're not guaranteed to get to the optimal policy, but it would be really cool if we could still guarantee that we're going to just get sort of monotonic improvement, unless we get to a local optima. And the things that I've shown you so far don't necessarily have that property because they'll still converge to a local optima, but you might overstep, like we see here. So you might go over. You might be having monotonic improvement and then crash, and then you have to go back and forth. So we have not guaranteed monotonic improvement so far. But that would be really nice and that could be important in a lot of real world domains, like you'd imagine. If you were using this for health care applications, you would really like to have monotonic improvement, and not suddenly performance crash. All right. So let's think about how we might be able to get here. And you can think of a lot of this lecture is motivating the things that you're going to be doing in Homework 2, including the theory. So in general, what we'd like to have is we'd like to have an update step that uses all the data that we just got as efficiently as possible, and that takes steps that sort of respect this distance in the policy, like the decision space, as opposed to just smoothness in the parameter space. And in order to do that, we need to understand how does the performance of two policies relate. So we have data from one policy, and we're considering trying to move to a new policy. And we'd really like to know, OK, given the data that I have from policy one, what does it tell me about how good policy two might be? Because ideally, policy one's data would allow us to tell us which policy two I should move to next. So this is what you're proving in Homework 2. You're going to prove the performance difference lemma. And the performance difference lemma, it allows us to relate the performance of one policy to the performance of another policy, given data from one of the policies. Let me just state this out. So what does this say? This is the value of one policy, policy one, policy pi prime. I'm just using j here, but this is value V. You can think of this as just V. So what is that equal to? That is equal to the expectation over trajectories that are sampled using pi prime. And again, if this is finite, can use gamma equal to 1. We sum over the distribution of trajectories you could get if you followed policy pi 1 times the advantage under policy pi. So this part here is just equal to the difference between if you took an action minus. But note here, because this expectation here is over pi prime, the way we're selecting these actions, I'll just write it out a little bit more. Imagine we have deterministic policies. So it's like we're thinking about the Q value, if we first take an action according to policy pi prime and then follow policy pi for the rest of time, versus what we would have gotten if we just followed policy pi the whole time. So you can think of it as sort of breaking down the difference in the value between two policies into a series of small differences of, well, how much gain would I have gotten at this state if I had taken pi prime's action instead of the one I actually took. OK, what about here? And then you of want to sum up all those additions. Every day, I'm happier because I went to Stanford instead of Harvard. And I just add up all of those, and that tells me over the course of my whole career how much happier. I will have been, hypothetically. OK, so this here is over trajectories. Now we're going to make a transformation and move it into state action distributions. Because what this is going to be here-- so now, this was over trajectories. We're going to rewrite this just in terms of state action distributions. What we're going to say is, all right, as we think about adding up all these advantages, what I'm going to do is I'm going to cluster together all the advantages that have to do with the same state. So I think of there as being a distribution over states I might reach and actions I might take under policy pi prime. So if I just follow this policy, I'm going to visit some states, and I'm just going to think about what is the advantage in each of those states weighed by how frequently I visit them. So we're sort of transforming things from thinking of it as being trajectories and thinking about weighting over time steps, to weighting over a finite set or a space of states and actions. And so we'll have a distribution. It might be like, I visit state 1 half the time. I visit state 7 only 1 in 10,000 times. So this allows us to reweight the advantages. What does this distribution look like? This looks like the following. Essentially, you just think of what is my-- so here, we're allowing us to have discount factors because we're looking for the infinite case. But you can adjust this. What this is just saying is that, well, my weighted distribution for state s is equal to, well, how likely was I to be in state s on time step 1 under that policy. Well, how likely was I to be in state s in time step 2 under that policy? What if I was in time step 3? And so you just sum up all of those. And as you could imagine, in the infinite horizon case, those could easily go to infinity, particularly if you have states you can visit a lot. And so the discount factor here makes sure this becomes a distribution. So we want this still to be normalized. And then similarly, this is also with respect to taking the actions under pi prime. So you'll be proving this in the homework, but we'll see how this can be helpful. So why are we making you do this? So the nice thing is it's going to define the performance of pi prime in terms of advantages from pi. OK, so that seems good because we're like, well, we have an existing policy. But the problem is it still requires trajectories sampled from pi prime. So you could think of pi as potentially being the policy we have right now, and maybe we can estimate the Q function for it. We can estimate its advantages. And now, we want to figure out how good would be this new policy we might take. But the problem is we don't have any trajectories for the new policy. We only have data from the old policy. So we really want to get to something where we can estimate how good is pi prime using data only from pi. That's our goal. So our goal, estimate j pi prime only from data from pi. So you can think of pi prime as the new policy. So what we really want to do is take a step so that our new policy is the best one we could get to in terms of its improvement over the previous policy. And we want to be able to do that by only using an estimate from our old data. And it shouldn't be clear yet how we could do that. This is looking promising, but we still seem like we need data. So this is still data from the new policy. So let's look at it from a different angle. So this thing d, d pi of s is a distribution over states. It's a discounted future state distribution. And we're going to use that to rewrite the relative performance of policy performance identity. So why is this relative? It's because it's with respect to the performance of our current policy. So that's why we have a subtraction there. We're going to rewrite that there. I want to see if I have it. OK, yeah. So I'm going to step through this. So what we can do at this point is we can rewrite it as follows. So right now, remember, this is in the time or the trajectory notation. So I'm going to, again, rewrite this so that I'm going to move it into the state action representation. So instead of thinking about as trajectories, I'm going to think about it as what's the distribution over states and actions that I'm visiting. So I'm going to write it as follows. 1 over 1 minus gamma sub over-- I should write this as the expectation. Expectation over s prime according to pi [INAUDIBLE] and then under-- I should write it this way. OK, what have I done there? So this should look pretty similar to the previous slide. What I've said is, OK, this is the discounted future state distribution. I saw on the previous slide that I can rewrite this expression as 1 over 1 minus gamma, e times s times this d times a. So that's what I'm doing now. So I've rewritten it in terms of this state action distribution. And this is where I'm in this problematic case that I've got the wrong-- let me make sure I put a quote there. So this is just to be really clear. This is over pi prime. So this is all with respect to the new policy. So now I'm going to note the following. OK, so what does this look like? This looks like it's 1 over 1 minus gamma, the expectation over s prime sampled according to d pi prime. And then what is this expectation? So this is going to be sum over a pi of a, given s pi prime. That's horrible notation. Let's try this again. That's what this means. I'm taking an expectation for each of the states. I'm taking an expectation with respect to here. Let me just try to make this neat. There we go. OK, so I'm saying, imagine I sample states from my d pi prime distribution. How do I do this expectation over a sampled from p prime? Well, I just sum over all the actions, look at the probability of me taking that action for that state under p prime times my advantage. So now, the key thing we're going to do is we're going to try to change this so that we are using more-- we're going to try to get to a point where we only need data from pi. So the first thing we're going to do here is we're just going to rewrite this, and I'm going to multiply and divide by the same thing. So I'm going to say this is pi prime of a given s, divided by pi of a given s, times pi of a given s. OK, I have not done anything, except for I've multiplied and divided by the same thing. Why did I do that? Well, the good thing is I know how to get samples from this. I have samples from this. This is from my old data. This is from the actual policy that I took before. OK, so what this says I can write this as, I've got 1 over 1 minus gamma e of s according to d pi prime. And then I have this expectation over a sampled according to pi, not pi prime, of my reweighted advantages. And what do I reweigh them by? I reweigh them exactly by the probability I take that action under the new policy versus the old policy. And that's OK because I'm going to assume that I have access to the policy parameterization of the new policy. I'm just trying to figure out how good it is. I don't have samples from it, but if you tell me, hey, this is the action you took in that state, I can say, OK, well, that's how likely I would have taken that under my new policy. So I can do that reweighting. And this is an instance of something called importance sampling. And we're going to see a lot more about that soon. So this is the first step I can do. So this is great. Right? Because now, I don't need to have samples from actions taken by my new policy. I can just reweight the data I already have. So that's super cool. But there's a problem. This is still pi prime. So I still don't have any data over states from my pi prime. All right. So this is that just written out more neatly. We'll see a lot more on that in the future. But we still have this big problem because we don't have any states from pi prime. We have data from pi. So what are we going to do about that? Well, we're just going to ignore it. Always an option. So we're just going to ignore that and proceed. And this is what this is going to happen in the rest of the class for the rest of this lecture. We're just going to pretend that those states are the same. Now, as you might imagine, that is going to slightly induce some error in my estimate. When might that be bad? Well, it might be really bad if the two policies would actually visit totally different parts of the state space. But if they visit things that are really close, maybe it's not going to be that bad. OK? So let's imagine that you have policy 1, and it goes, since this is your robot, and it goes to most of this part of the space, and then you change your policy. And maybe it also goes to that part of the space, and it goes a little over here, too. But there's quite a bit of overlap. The places where it would be bad is something like if your policy goes like this, your new policy. And then there's no overlap in your state space. So it's going to turn out that if pi and pi prime are close, and we're going to define what we mean by close, then this is actually not a bad approximation. It's not perfect. It's not too bad. So in the paper, they prove that we can bound how bad this approximation is. So in general, we're going to define this to be L, math script L, of pi with respect to pi prime. If this was perfect, this thing would be 0 because this would exactly equal this difference. So this minus this would be-- these two things would exactly be 0. But what it turns out is that how far off this approximation is depends-- bless you-- on the KL divergence in the policies. And I'll define what KL divergence is in a second. So in particular, it depends on the KL divergence with respect to-- let me just undo that so I can just do this part. With respect to states if you were sampling them according to d pi. Now, d pi is good because d pi is the actual discounted states that we visit under our current policy. That means we actually have data about it. So I always like to work with my-- in my lab, we always try to instantiate our theoretical bounds because I feel like it's super informative to be like, is this 10 to the 10, or is this 0.5. And one of the big questions often that comes up when we try to do this is that sometimes you can't instantiate your bounds at all because it will depend on constants you don't know. So it's a beautiful theorem, but you can't even check how big it is. The nice thing about this is it's checkable. At least this part is because you can look at your actual trajectories from your current policy. And if you have a new policy, pi prime, you can see and evaluate what your KL divergence will be. So this is actually evaluatable. So we'll see. What is C? So C is a constant. Any information about its value? No. We're not going to need it for now. But in the paper, you can read about exactly what that is. Yes, good question, though. OK, let's see what KL divergence is. So what this says, just at a high level, is that this approximation is not totally insane if the policies are close. In fact, this is going to be a pretty good approximation. So as we're going to see, this is tight if the policies are identical, which is exactly what you'd expect this to be tight. So if your two policies are identical, their difference should be 0. And this bound would tell you it's 0. All right. What is KL divergence? Some of you guys might have seen this before, but in case some people might be new. So what KL divergence allows us to do is to compare two probability distributions. So in our case, what this is going to be is over actions we will take. So pi of a given s, versus pi of a given s. So both of these are probability distributions that sum to 1 for a particular state. And so in our case, what we would be summing over here, x would be a. So we'd be summing over all of these. If you have the same probability distribution, the KL divergence is 0. Otherwise, it's strictly positive. It's good to know it's not symmetric because we've made a choice here in the ordering. These are good properties to know about. It comes up all the time in reinforcement learning. So in our case, we can look at, for a particular state s, what is the KL divergence and what the policies would do for that particular state. So that's what we've got there. So it says, essentially, how different are the actions you would take. Now, why is this good? Well, we've been spending some time saying, hey, we really don't want to think just about how close we are in theta space. We really want to get to thinking about how different are our actual policies, how different are the actual decisions we make in particular states. And the nice thing is, what this bound says is that the difference between two policies is not just about how close you are in some parameter space. It's really about how different are the decisions you make in all the states you'd reach under your current state distribution. And so if you'd make all the same decisions in the states you're already reaching, your policy value is going to be really similar. If you would make very different decisions, then your policy value might be really different, because you'd go off and explore really different parts of the state space. So this is really elegant. And now, we have something where we can just use our old data. So we have our old data. We can use it to estimate what the performance improvement would be if we try to get to a new policy. And so what you might imagine in this case is you could use it to search over or decide which new policy to try. This allows you to compute it for lots of different pi prime. It doesn't just have to be the pi prime for one gradient step. It says, in general, even outside of policy gradient methods, you can evaluate the value of changing your policy to pi prime with respect to your current performance by this expression. And this will be more or less tight depending on how different your policy is at making decisions in states that you would currently reach. And we'll talk more about this. We haven't talked about this yet, so we will talk about-- This also relates to some really nice literature from the last 20 years of thinking about how do we do monotonic policy improvement in policy gradient methods and policy search methods. It also relates to the notion of a trust region, which is this idea of, when you're changing your policy, how far can you go and still sort of trust the performance of it and trust you can get improvement. So there's a bunch of different nice papers related to this. OK, let's talk about the algorithm. Proximal policy optimization is going to be inspired by all the things that we just talked about. So what we want to do is, it wants to be able to take multiple gradient steps, and it wants to be able to do this in a way so that we don't overstep. We try to focus on policy parameterization in terms of the actual decisions that we make. So there are two different variants. One is it solves an unconstrained optimization problem where it uses this approximation. So that's the approximation we had on the previous slides. I'll just write down, from prior slides, where we use data from the current policy, and we add up these weighted advantage functions. So what it says is, well, the thing you want to do is you want to pick the policy that maximizes our estimated difference, subject to a constraint on the KL divergence. Because it's realizing that L approximation is going to get worse and worse as the KL divergence gets really large. So it's directly incorporating this bound. So it's thinking, OK, I want to think about what this is, but then I also have to consider the fact that my estimate might be off by as much as this square root of KL. So you really want to improve with respect to something that considers both of those. So this is one version of policy. This is not the way most people do PPO. We'll see the other really common one. But it's a nice baseline to know about. And here, when we think about what that KL is, as you might have noticed, KL is defined for a single state. So for a single state, we can say what is the distribution over actions I take in one policy versus another. But of course, we have lots of states. And so what we can do here is we can take an expectation over the KL, over all the states we visit. And that was part of the theoretical bound, too. Another really important thing you can see here is this waiting between trying to optimize this policy improvement, while respecting this KL penalty. And you can change this over each iteration to approximately satisfy the KL divergence constraint. So this does not guarantee that you will. This does not guarantee that you're going to get monotonic improvement, but it's trying to get towards that. So let's see how that works. So this is the algorithm. What it does is, you can compute the advantages using any advantage estimation algorithm. You compute your policy update, and you can do K steps with that. So the nice thing is that you can use your old data here, and you can take multiple gradient steps. After you do this, you can also check if your KL divergence for your new resulting policy is large. If it is, then you may increase the penalty. If it's small, you can decrease the penalty. And that just allows us to trade off between how much we pay attention to this KL divergence constraint versus not. And as I noted here, you might violate the KL constraints, but most of them, they don't, empirically. This is one reasonable thing to do based on everything we've seen. Now we're going to see something else, which is inspired by that is a much more common thing to do, which is, well, let me just highlight here. Multiple gradient steps is really good. So one of the benefits is that we're not just taking a single gradient step. We're taking multiple. So just to really highlight that. All right. What is the other thing we want to do here? We haven't talked about natural gradients. But for any of you that are familiar with these, they're another way to try to think about taking gradient steps. We're not going to talk about that for now. So the other thing we could do is equipped objective. So what we're going to look at in this case is, remember how we talked about we had this kind of ratio between this is what we're using to weight our advantage function, was the difference between how likely you were to take that action under our old policy versus our new. And we're using it to weight our advantage function. What the clipping does is it says, well, I don't want this to get too high or too low. OK? This could become really high or really low when my policies are really different. If my policy is really-- you can imagine that, if my policy puts really low probability on something that the current policy puts high probability on, this ratio here is going to go towards-- this r is going to go to about 0. And if this puts very high, let's say this is 1, and this puts very low probability on that, this could be extremely large. This could be like 10 to the 6. It could be very, very large. And both of those are being used to weight the advantage function. Right? So your advantage function could be getting shrunk towards 0, or it could be getting blown up by a factor of, say, 10 to the 6 if you have a big difference in the actions you would take under one policy versus another. And in general, we don't like things where we're thinking of policy gradients, where we might have terms that are exploding or vanishing. And that's part of the point of the KL divergence constraint is to say, you want your policies to stay close. So the clipping is sort of inspired by this general idea, but says, well, maybe something similar we can is we're just going to clip. We're just going to say you can't have weighted advantage terms that are going towards infinity, or minus infinity, or 0. And so if this ratio is too extreme, I'm just going to clip it. I'm going to not allow it to be less than 1 minus epsilon, or greater than 1 plus epsilon. And epsilon is just a hyperparameter. And essentially, it's sort of meaning that your policy might change further than that, but that's not going to benefit your loss function. So this, again, is going to constrain your policy class to stay within this region, for which it's making similar decisions. So we're still really focusing on what actions are we actually taking. Are we taking similar actions in these states, as we would be normally, regardless of how much my theta is changing? And then you just do your policy update by taking an argmax over this. So this is your clipped objective function. All right. So let's see how this works. Let's think about what it's doing. So we'll do a quick Check Your Understanding. So this shows you what L clip does. This is L clip, as well. And what I'd like you to think about is, what does this look like, depending on the advantage function. So L. Let me just write it down. L clip. OK, so this is r. So on the x-axis is r. And then on the y-axis is L clip. And what this is asking you to think about, this is from their paper, is to think about what does clipping do in terms of your loss. And so I'd like you to think about, in this case, which of these two, if either, match within the advantage function as positive or the advantage function as negative. So a here is the advantage. Let me just make that clear. A is equal to the advantage. So just think of this for a single term. Consider for one term. So just this part. So just for one single rta, what is happening here? And just to be clear here, what we're doing in this case is we're taking the minimum between the normal thing we do, which is this reweighted advantage function times a clip of the r times the advantage function. And feel free to flip back and forth or play with numbers, just to get some intuition of what is this doing to our loss function. Or I shouldn't say loss. Our objective function, in this case, because we're trying to take the argmax of it. So we're thinking of this as sort of an approximation of how much is our policy going to improve when we change our theta. So we're going to want to take a max of this over with respect to a new policy theta. And we want to think about, this is sort of bounding what that new performance benefit could be, and how does that vary with respect to the advantage. Nobody thinks it depends on the value of e, which is correct. So this does not depend on the value of e. Why don't you turn to your neighbor and see if you got the same thing? [INTERPOSING VOICES] Cool. It looks like talking converged most people, which is great. So the first one is correct. So this is a greater than 0. This is a less than 0. Does someone want to explain why? Most of you that voted got it right. Well, it is quite simple because the simply let a coefficient to the value. So it has a positive slope, and it is positive and negative slope. Then it's negative. Yeah, so if we just focus on this for a being equal to greater than 0, what will happen is, as that ratio r gets higher, and higher, and higher, you'll just linearly go up because it's just something between 0 and 1 that's getting larger and larger. r can never be negative. So it's just useful to see in this case. So as it's getting larger and larger, it's just going to increase your L clip value. But at some point, you're going to run up against this part. And so at this part, you're going to clip it, and you can't get higher anymore. Remember, in general, we're always trying to maximize L clip. In this case, when the advantage is negative, you're trying to reduce the amount of probability mass you put on that action, because you don't want to take that anymore. I got a negative advantage, so I need to stop doing that. So essentially, you want to be sweeping, changing your policy in the opposite direction. You'd really like to be able to push r all away to 0 and say, I never want to do that action again. It gave me a negative advantage. But you can't do that because that might change, radically change your policy. And so once you get to 1 minus epsilon, you cap it, and you can't further shrink it to 0. Great. OK. And another way to see this as from the paper is, you can think about these different types of constraints and different clipping. And essentially, again, it's sort of making the objective pessimistic as you get really far from theta. Now, just in the last couple of minutes, I want to make sure to show some plots. So this is just the same algorithm, but we're doing clipping. I will just note here that next week, so next time, we'll discuss-- next time, we're going to discuss the choice of a further. Just like what we saw earlier today, you can do n step estimators, et cetera. And you can do what's called generalized advantage estimation. You don't need to know that for this. We won't cover that for today, but we'll cover it more next week. So just to note, there's some additional choices here. But let's just look at what the performance is. So at this point, TRPO and some other algorithms were out there. They have the PPO clipping in purple. This is a number of different MuJoCo domains, similar to MuJoCo domains you're going to be working with. And what you can see here is that, in general, so TRPO has this trust region idea, and it's similar in some ways to what they're doing in PPO. But trust region policy optimization is quite a bit more complicated. And what you can see here is that this light purple, which is PPO, this is the number of steps, is generally doing just much better than these other ones. So they're not claiming they're going to get to a better optima. They're just saying, we're going to be able to get there much faster with much less data. And in some of these cases, this is really an enormous performance improvement for the same amount of data. So this is one of the reasons why it became extremely popular, is it is a pretty simple algorithm to implement, and it has extremely good performance in many cases. Now, so you can go to the original paper, proximal policy optimization algorithms, or go to the blog post from 2017. I do think one thing that's good to know, and really, throughout much of our recent history, is to also understand what are the implementation details. So this is optional. You don't have to read it, but there was a nice paper that followed up from this work because, again, PPO has been hugely influential from some colleagues at MIT that said, well, really, which of these things are most important. Because in general, in these algorithms, there will be these things, but then there's also some hyperparameters or architecture choices, et cetera. And so knowing how these choices are made often do make a big difference in reality. And so that's always good to know. It's whether or not is it an algorithmic improvement, or are there additional things that we're not treating as part of the algorithm but actually are really important for practical performance. So you should know everything now that you need to for making good progress on Homework 2. And we'll continue to discuss this next week. Thanks

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Introduction_to_Reinforcement_Learning_I_2024_I_Lecture_1.txt

Hi, everyone. We're going to go ahead and get started. I'm Emma Brunskill. I'm delighted to welcome you to Reinforcement Learning, CS234. This is a brief overview of the class and what we're going to be covering today. And I just want to start that probably everyone's heard of reinforcement learning these days. That wasn't true about 10, 15 years ago. But you can describe what is happening in reinforcement learning by a pretty simple statement, which is the idea of an automated agent learning through experience to make good decisions. Now that's a pretty simple statement to say. It sort of encapsulates a lot of what me and my lab and many, many others have been trying to work on for the last 10 to 15 years. But it's sort of deceptively simple because it involves a lot of different really challenging and important things. So the first is that any sort of general agenda to try to achieve general artificial intelligence has to include the ability to make decisions. There's been absolutely enormous progress in what we would call sort of perceptual machine learning, things like being able to perceive faces or cats or identify cars. And we call that often perceptual machine learning because it focuses on trying to, say, identify something. But, of course, in reality, what we're all trying to do all the time is also to make decisions based on our perception and based on our information we're receiving. And so it's critical if we think about what it means to be intelligent to consider how to make decisions, and not just any decisions, but what it means to have good decisions. This sort of question over how can we learn to make decisions, particularly faced by uncertainty and limited data, has been a central question that people have been thinking about at least since the 1950s, particularly pioneered by the ideas of Richard Bellman. And we'll hear a lot more about Bellman's equation, which many of you might have seen before, later even in this lecture or next lecture. So there's one sort of argument for studying reinforcement learning, which is it's an essential part of intelligence. It has to be part of a general agenda of artificial intelligence. And so we should study it to try to understand what it means to be intelligent. And that certainly for me is one of the really big motivations. So I think there's just a lot of fundamental questions about what is the data needed to learn to make good decisions. But there's another really good motivation to study reinforcement learning, which is that it's practical and it allows us to solve problems we'd like to solve. So in particular, over the last roughly decade, there started to be a lot of really impressive successes of using reinforcement learning to tackle problems or to get unprecedented performance in a lot of really important domains. So the first one is the board game Go. So who here plays Go? OK, a few people. Maybe not. You can talk to the people that raise their hands. So it's an incredibly popular board game. It's also an incredibly hard board game. It's far harder than chess, and it was considered a really long outstanding question of artificial intelligence. But roughly like, I guess about eight years ago now, eight to nine years ago, there was a team at DeepMind, which was still a fairly small organization at that point, that thought that they could make significant headway at teaching AI agents to be able to play Go. And the idea in this case is that we're going to combine between the ideas of reinforcement learning and Monte Carlo Tree Search, which is something we're going to hear about later in this class, to create a system that played Go better than any humans in the world. And so there's even a movie now about sort of one of the seminal games in that sort of endeavor and how humans felt about that and how the creators of the AI systems felt about that. But this feat was achieved far earlier than what people expected. And one of the key reasons for that was using reinforcement learning. Another really interesting place that we've seen progress of using reinforcement learning to tackle incredible challenges is in the idea of fusion science. Fusion is a potential approach for trying to tackle the huge energy issues that we have in trying to transition to more sustainable options for that. And one of the challenges here-- and I'm not a fusion expert-- is to manipulate and sort of control things within a vessel. And so what the reinforcement learning question in this case is, how do you command the controllers, the coil controllers, in order to manipulate this into different types of shapes? And so this was a Nature paper from two years ago where they showed you could use deep reinforcement learning techniques to accomplish this in a way that was far more flexible than had previously been imagined. One of my favorite examples of the applications of reinforcement learning comes from a pretty recent important case, which is COVID testing. So this was a system that was deployed in Greece. They had limited resources, and they were trying to understand who you should test in order to help control the epidemic, because as many of you may know, there's a lot of sort of free movement within Europe and there was a lot of transitions. And they were trying to think about how to leverage their resources in a data-driven way because, of course, the epidemic was changing. And so this is a beautiful paper by a Stanford graduate, Hamsa Bastani, and her colleague. She's a professor over at Penn now that used reinforcement learning to really quickly do this. And it was deployed. So Greece used this for their testing at the border. But perhaps the most famous example recently is ChatGPT. So I think that as many of you might know, natural language processing has had incredible successes over the last decade. And there was a lot of work trying to use transformers to make really, really capable natural language systems. But up till around, I guess, like a year and a half ago, most of that work was not known to the broader public. So even though we were getting these amazing advances in natural language processing, it wasn't at the state yet where everybody was using it. And so the key idea of ChatGPT was to use reinforcement learning to create vastly more capable systems. And I like to talk about ChatGPT not because it's perhaps the most well-known success for reinforcement learning, but also because it exhibits a lot of the different technical challenges and questions that we're going to be covering in this class. So let's just walk through sort of how at a very high level sort of this figure from ChatGPT of how the ChatGPT system works in terms of training. So the first thing it does is it does what we would probably call behavior cloning or imitation learning. We'll be covering that in this class. And we'll be talking more about it even in this lecture. So what did it do? So, again, just to remind, I suspect everybody in this class probably uses ChatGPT probably multiple times a day or Claude or Gemini or one of the other large language model systems. But just in case you have not, the idea in this case is that you might have some sort of prompt or task you want your language system to do, like explain reinforcement learning to a six-year-old, and then someone gives a response, like we give treats and punishments to teach, et cetera. And you can think you can try this out with ChatGPT and see how well you think it explains it. And then what that was treated as is sort of a direct supervised learning problem. So just trying to take that input and then to produce that output. And we will call that also imitation learning or behavior cloning in this class. And we'll talk about why. So that was the first step. And this is sort of what people have been doing in natural language processing. And the systems were good, but they weren't that good. So the next idea was to try to explicitly think about utility or rewards, like how good were these particular labels or these particular outputs. So here we're going to actually build a model. We're going to build a model of a reward which relates to model-based reinforcement learning. And the way we're going to do this is-- or the way they did this is we collect preference data. We ask people to compare or rank across different forms of outputs. And then we use that to learn a preference model. And we're going to cover that in this class. That's going to be one of the differences to this class compared to a couple of years ago that I think preference-based reward signals are really important and very powerful. And so we're going to be covering that in this class this term. So in this case, they would learn a reward. And again, don't worry if you haven't-- if you're not familiar with what rewards are and stuff. We'll go through all of that. I just want to give you a high level sort of sense of how ChatGPT is related to some of the things we're going to cover in the class. So they learned a reward signal. And then they did reinforcement learning using that learned reward signal. So now they're going to do reinforcement learning. And this is called RLHF because it is reinforcement learning from human feedback. And I'll just note here that this was not the first time this idea was introduced. It had been introduced maybe about four to five years before this for sort of simulated robotics tasks. But ChatGPT demonstrated that this really made a huge difference in performance. And so I think it's a really nice example of the types of ideas that we're going to be covering, as well as sort of the incredible successes that are possible. Now, even before ChatGPT came along, there was starting to be a huge interest in reinforcement learning. So some of you-- we have an optional textbook for the class. It's by Sutton and Barto. Richard Sutton is from Canada and is one of the sort of founders of the field. And when I started in reinforcement learning and I would go give my talks in conferences, it used to be like me and Rich and 30 other people. Nobody cared about reinforcement learning. I mean, it's not for, you know. A few of us did because we thought it was really amazing. But as you can see, sort of through the 2000s, which is when I was getting my training around here, there just weren't that many papers, and the community wasn't nearly as large. But this nice paper by Peter Henderson-- the y-axis is papers-- shows that there's been this sort of enormous increase in interest. And I think a lot of this was really due to the fact that kind of around here there was some amazing successes on the Atari video games, where people showed that you could learn directly from pixel input to make decisions. And then there started to be the successes in AlphaGo, and then there became more and more successes. So it is an incredible time for reinforcement learning. This curve is continued to go up. However, I think it's also important to notice that there is also a number of skeptics. So there was a pretty famous talk by Yann LeCun in 2016 at one of the major machine learning conferences, NeurIPS. So Yann LeCun, for those of you who don't know him, is one of the sort of seminal figures in neural network research. He has won the Turing Award. He is an amazing, amazing researcher. So he gave a keynote at NeurIPS. I believe it was a keynote. He certainly gave a very famous talk there, where he was talking about the role of different types of machine learning questions and subareas in terms of making progress on machine learning. And he very famously talked about machine learning as a cake. And so he said that the main cake is really unsupervised learning. And that's really going to be the body, the most important aspect of machine learning. Things like representation learning from unlabeled data, that's really going to be the core, and that's where we're going to have huge amounts of data and we're going to make a lot of progress. And then supervised learning was the icing. So that's still pretty important. It's like very important part of the cake, at least in my opinion. But it doesn't have as much. We don't have as much supervised learning, and it's sort of [INAUDIBLE]. And then he argued that reinforcement learning was just the cherry. Now cherries are important, but not nearly as much, perhaps, as the rest of the cake. And he went on and talked about some places where he thought that RL still might have a role. But it was considered a really important talk because what he was sort of demonstrating is that reinforcement learning was having a part to play in machine learning, but maybe only a very minor part. Now I think it'd be interesting to talk to him today. I haven't talked to him recently, so I don't know what his current opinion is, but I think it's a really important thing to think about like, where are all of these different techniques important, and where will we be able to make the most progress in terms of advancing AI? And so with that, we're going to try and do our first poll, which is about why you guys want to take this class. So we'll look through these. You'll have to bear with us a little bit with-- we had a few technical difficulties that we're working with CTL on, but it should work out. So if you go to either the first link in Ed or you go to this HTTP, you can-- if you have any issues like, do you want to be registered? If it's hanging, just skip the registration, refresh, that should all sort it out, and then just add in your SUN ID as your screen, and just take a second and write down a bit about why you want to take this class. And it could be anything. It could be that you're really curious about something. It could be because you're doing an internship and they told you had to take something about reinforcement learning. Any of the things are fine. Just take a minute or two. Thanks for all the great reasons. I will talk about some of those when I talk about also what we're going to cover today and try to address why. I think a lot of the things people are bringing up are things that we're going to be touching upon. So I think if we want to think about-- I think it's really important to start thinking about what is reinforcement learning about, because if we understand what it's about, then we know what types of questions we're interested in this space. And we also understand what sort of applications it might be helpful for. So, of course, your creativity is unlimited, so you can see what you might come up with other ideas that people may not have thought of for applied RL. But the four things that people typically think about when they think about reinforcement learning as a discipline and as the sort of what reinforcement learning involves is optimization, delayed consequences, exploration, and generalization. So the first is optimization. And the optimization aspect is really just saying that we're thinking about the best way to make decisions, which means that we explicitly have to have some notion of utility. An example of this would be something like finding the minimum distance route between two cities given a network of roads. This means you can directly compare different solutions, because if one solution has a smaller distance than the other, it is strictly preferred. So there are many, many important optimization questions and reinforcement learning because it is concerned with making good decisions, cares about us being able to rank or decide across those different ones. The second one is delayed consequences, the idea being that the decisions that we make now can affect things far later. So maybe saving for retirement now has some immediate cost, but it leads to some significant benefit later. Or maybe there's something you can do early in a video game that later has a lot of benefit. There are two reasons why delayed consequences is challenging. One is for the reason of planning. Many of you might have actually-- raise your hand if you've taken AI at Stanford. OK, so about half of you. So you probably saw planning in AI. And planning is the idea that even when we understand how the world works, it might be really complicated to try to decide what the optimal thing is to do. So you can think of this like chess. All the rules are known. It's still really complicated to think about what's the right thing to do. So when the decisions you make involve reasoning not just about the immediate outcomes but the longer term ramifications, the sort of planning problems are even harder. But the other reason this is really hard is when we're learning, meaning that we don't know how the world works and we're trying to understand how through direct experience. So when we're learning, temporal credit assignment is hard, meaning that if you take some action now and later on you receive a good outcome or a bad outcome, how do you figure out which of your outcomes caused that good or bad later result? So this happens all the time to us as humans. How do you know why you got into Stanford? Well, I don't know. Was it because you colored you won a coloring contest when you were 6, because you scored well on the SAT, because you went to a good high school or you wrote a really good essay? It's really hard to understand this. In some cases, it may be impossible. But when we're getting to make repeated decisions, it's really important that we can start to use the prior experience to figure out which decisions were important or led to good outcomes so that we can repeat them. So that's one of the reasons why this is hard. Exploration is one of my favorite things in terms of reinforcement learning. And the idea of this is that the agent can only learn about the world through direct experience. So it's like trying to learn to ride a bike by trying and failing and trying again and through that direct experiencing, learning the right way to ride a bike. And the key idea about this is that information is censored in that you only get to learn about what you try. So, for example, right now, you don't know how much worse your life would be if you were MIT. I went to MIT for grad school. MIT is also a great place. But you generally can't ever understand what that counterfactual life would have been like. It's one of the central challenges. It's also a huge challenge in causal inference, which is another big interest of mine and something my lab works on. So there's this general challenge that you only get to learn about the actual things that you do as an agent, as a human, as an agent, et cetera. And so the question is, how do you use that experience to figure out how to make good decisions? So as a concrete example of this, you can imagine you're a company, and you give some promotion to all your customers. You can't know what it would have been like if you didn't give the promotion to those customers. And even if you can give it to one customer and not another, they are not the same people. So I can't rewind and say, Dilip, who is our head TA, this time, I'm not going to give you the promotion. Let's see how that world would have worked out. That's one of the central challenges. So we'll talk a lot about exploration later because it's one of the key things that is different compared to many prior approaches. And generalization has to do with this question of really wanting to solve really big, challenging problems. So we'll talk a lot about what decision policies are. But in general, you can just think of them as a mapping from experience to a decision. And you might think in those cases, you could just preprogram it. So if your robot goes down the hallway, if it hits the end of the hallway, turn left. But let's think about a video game, which we can think of as just sort of generally having some input image. So let's imagine that it's something like 300 by 400. And let's say we have at least 256 different colors. So now we have set of images that we could see that is at least 256 to the 300 cross 400. So those are at least the space of images. That's probably an underestimate. And now we get to think about what we would do in each of those different scenarios. So the combinatorics are completely mind-blowing, and we can't write these down in a table. So this is why we would need something like a deep neural network or something else in order for us to try to make decisions in these realistic settings which are extremely large in terms of the type of scenarios, the number of scenarios we want to make decisions on. So you've probably seen all of these ideas, or at least most of them in other classes for other types of AI or machine learning. So I think it's useful just to contrast what is reinforcement learning doing compared to these other ones. So the first is AI planning. So in AI planning, generally, we're doing some form of optimization, trying to minimize a distance or something like that. We are often trying to handle delayed consequences. And those are the two main things. So we might also have to do generalization if the size of the space is really large. OK. So that is how-- so RL, in general, will involve all of these. So this is how those would compare. If we think about something like supervised learning, supervised learning does involve learning. So we learn from data, whether something is a cat or not. And we have to do generalization. So we have those two things. And again, this is going to be compared to reinforcement learning, which has all of those. In contrast to supervised learning where you get the correct labels, in unsupervised learning, we don't get any labels. But we're still learning from experience, and we're still trying to do generalization. Now the next thing-- and this has become a really popular thing-- is to think about whether we can map reinforcement learning to imitation learning. We talked about this really briefly about ChatGPT, and we'll talk about a lot more in the course. So in imitation learning or behavioral cloning or reducing reinforcement learning to supervised learning, we generally assume that we get access to expert trajectories. So this could be like someone saying what they would do in response to those prompts. It could be someone driving a car, and then you want to mimic their behavior or some other similar example. So these ideas is that we get input demonstrations of good policies. And that allows us to reduce reinforcement learning back to supervised learning. So we're sort of taking this, and we're reducing it back to here. I think, in general, the idea of reductions is incredibly powerful. For those of you that have taken CS theory classes, that's what we do all the time. We reduce things to set or other things like that. And, in general, I think in computer science, it's one of the strengths of it that they think of how can we reduce one problem to another and then inherit all the progress that's been made on that problem. So in this way, reinforcement learning is similar to other aspects of computer science in that we will try often to reduce reinforcement learning to other problems. This is particularly done in the theoretical aspects of reinforcement learning. Yeah. Yeah. So just before-- Whenever you ask-- just because I'm going to try and learn names, could you say your name, please? Yeah, my name's [AUDIO OUT]. So just to be clear, imitation learning then isn't like a separate technique. It's just an application of supervised learning to the specific reinforcement learning context? It's a good question. So I think some of you-- I mean, there's a lot of techniques that think about when you're doing imitation learning specifically for kind of decision data. You can just think of it just reducing it back. If you want to do imitation learning where you might recover like the reward function-- we'll talk more about that soon and others-- then you may need to use other types of techniques as well. But the most straightforward aspect of it is just to say, I've got demonstrations. I'm going to ignore sort of like this delayed consequences aspect and exploration, and I'm just going to reduce it back. Yeah. And name first, please. Wait, what do you mean by input demonstrations of good policy? What does that mean? Great question. So let me give you an example. So people have thought a lot about this. Maybe one of the first examples of this, or one of the first really public examples of this, was for driving. Like at least what you could do is I could drive a car. It could record everything that I do in terms of controlling the steering wheel. And then we could-- if I'm a good driver, they could say, that's a good demonstration. So instead of the car trying to learn from itself how to steer the wheel in order to, say, successfully drive, you could have humans drive it, and it could try to figure out at each point how should I steer the wheel in order to have good behavior. So the idea is that you actually have access already to good demonstrations of what is a good policy. Yeah. Name first, please. What do you exactly mean by optimization here [INAUDIBLE]? Optimization and what? What do you mean by optimization and imitation? Ah, OK. Good question. So what I mean is that when we do imitation learning from good trajectories, we are assuming that we want to do well. So we want to actually get a good policy. So imitation learning, normally, we're not normally trying to imitate bad performance, but you could think of this as sort of reinforcement learning but without the exploration part, because it's not trying to pick its own data. Why don't supervised learning and unsupervised have that thing? Have the optimization? Yeah. Yeah. So I think because we normally don't have the notion of utility in those. So you might say this is a cat or it's not a cat. It's not like a good picture of a cat or not. Whereas in decisions, we often have a real valued scalar value of like it was like a 0.7 good decision. Yeah, name first, please. [INAUDIBLE] also have the loss function, which we intend to optimize. Yes. So we do often-- we always have loss functions. And that's a great-- but in those cases, there's not normally a utility that goes. So if you could get-- you could maybe have some smooth notion there of how well do you match like a stochastic policy, a stochastic output there. But for many of those, it would be more-- if it's like, did you say it was a cat or not, you would have a binary 0-1 loss. Yeah. Hi. [AUDIO OUT] So does that mean, if you have the data for imitation learning, it's like almost always better than reinforcement learning? Because avoiding the fasteners, you can directly learn what is good. Say that again. So if you have the data for imitation learning, if you have someone actually driving the car, does that mean that you will probably learn a better policy than reinforcement learning or that it's almost always better? Great question. So we'll get into that. So the question was-- if you can hear that-- is if you have good demonstration, say, of driving behavior that you're using imitation learning, can that be better than reinforcement learning? It will depend on your reinforcement learning algorithm. In general, reinforcement learning should always be able to equal or exceed the performance of imitation learning. Yeah. So can you explain the difference between IL and RLHF? Yes, great question. So in imitation learning, what we would have-- and this is what-- this was the first part. You would say people give me-- given a prompt, I look on the internet, and I assume that those were good. So in the internet, I see if someone said like how to explain reinforcement learning to a six-year-old, this is what they said back. And so I just train on those. What RLHF said is that, well, the internet is a big place. Probably not all of it is good answers. So now let's actually ask people which of these two responses they prefer. And now we're going to try to do reinforcement learning on that to actually get to a better policy. Yeah. Something I'd like to ask-- so AlphaGo actually discovers some Go strategies that are not invented by humans that we have never experienced before. So does it mean that if we apply imitation learning too much, it might actually hinder the model's capabilities to explore like what is actually good instead of what humans have thought of, which is probably wrong? Absolutely. And actually, I think this is on the next slide. Let's go back. Good. OK, perfect. So this turns as to where are some of the places that you might hope that reinforcement learning would be better than these other strategies. So one of them is where you don't have examples of desired behavior. So this is exactly like the example that was just brought up. If you want to go beyond human performance, you cannot rely on human performance just to do imitation learning because you're not going to be able to get better than it. So there are a lot of application areas, I think, particularly in areas like health care or education and others where we think we can go beyond human performance. And so in those cases, reinforcement learning because it's trying to optimize performance could go beyond. It could be a particularly useful technique. Another is where you don't have any existing data for a task. So there might be something where you think of it as a decision-making problem, but you don't have prior data. And you need to learn from scratch, and you want to directly optimize. So that's another place where reinforcement learning can be very powerful. Another category is interesting because in some ways, it's also kind of a reduction technique. And this is the place where we have an enormous search or optimization problem with delayed outcomes. So there's been a number of examples of the work of doing this from DeepMind, which have been really extremely elegant. So what I put up here is AlphaTensor. If you haven't heard of it, it's a faster way to do matrix multiplication, which is kind of mind-blowing. So what they did is they said, all right, there's standard ways to do matrix multiplication. This comes up all the time. Could we learn an algorithm that would be better at matrix multiplication? Not me as like a scientist try to write down an algorithm. Have an agent learn one. And they showed yes. And the way they did that was with reinforcement learning. And they've done this in other cases, too, like learning faster, sorting algorithms. So I think this is a pretty incredible frontier. The idea is saying, could we have AI actually be inventing new algorithms? And one of the ways that they framed it here-- and you can think of AlphaGo as similar-- is that it was a really, really, really large search problem. And the challenge with really, really large search problems is that even there, we may not have great techniques for solving them. And so it's sort of a reduction. You can think of people taking a planning problem and trying to reduce it to a reinforcement learning problem to make it more tractable. So that's pretty wild. Most of the time we think of sort of RL been reduced in the other direction or involving planning or above that. But here, in some ways, you can think of these as like either adversarial planning problems or Expectimax problems that are being reduced back to learning as a way to just more efficiently go through the search space. So those are two of the areas that I think are particularly promising in terms of why reinforcement learning is still a really practical and really important area to think about. I think I saw a question back, but maybe-- Yeah, so for-- Oh, what was your name? [AUDIO OUT] For AlphaTensor, is that like it's fast but within some error of the correct matrix product? It's faster but with some error. Some error? Or do you actually get the correct value? No, you get the correct value, which is wild. Yeah. Yeah. So no, it's just better. Yeah. And one of the really clever things they had to think of in this case was how do you know that the answer is correct. How could you provably verify that? So it's incredibly elegant. All right. Now we're going to go quickly through some course logistics before starting to dive into some content. And feel free to interrupt me throughout this or anything else if you have other questions. So in terms of the content, we're going to start off by talking about Markov decision processes and planning. And then we're going to talk about model-free policy evaluation and model-free control. Don't worry if you don't know what I mean by model. I'll specify it. Then we're going to jump into policy search. Policy search is things like proximal policy optimization and reinforce and other approaches. Some of you guys might have already seen related ideas, say, in robotics if you've taken them. And then I'm highlighting here that this is one of the important differences compared to prior years. So we're going to do a deep dive into offline reinforcement learning, offline here meaning that we have a fixed amount of data. And we want to learn from it to get a good decision policy. And during this, we're going to talk about reinforcement learning from human feedback and direct preference optimization. So that's going to be a new third part of the course that we haven't done assignments on before. So I think that'll be pretty exciting. And we'll also talk about exploration and do advanced topics. So the high-level learning goals of the class is that by the end of the class, you should be able to define the key features of reinforcement learning. You should be able to given an application, specify how you would write that down as a reinforcement learning problem, as well as whether or not you think it would be good to use RL for it, that you can implement in code common RL algorithms, and that you understand the theoretical and empirical approaches for evaluating the quality of an RL algorithm. So as you could probably imagine from those papers going up, there's going to be continued progress in this field, and there's going to be a huge number of different RL algorithms. And so one of the key things that I hope to talk about is sort of how do you evaluate and compare them, which might vary depending on the application area you care about. So the way that the course is structured is that we'll have live lectures. We'll have three homeworks. We'll have a midterm. We'll have a multiple choice quiz. We'll do a final project. And then we'll have what I call check or refresh your understanding exercises, which will be going through the Poll Everywhere. And we'll have problem sessions which are optional. Problem sessions are a great chance to think more about the conceptual and the theoretical aspects of the class. And they'll be held starting next week. So one of the main application areas I think about a lot is education. I think education is one of the greatest tools we have to try to address poverty and inequality. And so I'm really interested in evidence to think about how do we educate effectively. So with respect to that, I wanted to share this paper that came out, I guess, almost a decade ago now, where they did a study to look at how people who are taking massive open online courses, how they spent their time and how that related to their learning outcomes. And what they found is that if you do more activities, there seem to be a six times larger learning benefit compared to watching videos or reading. And you might think this is just based on time, but it wasn't. In fact, it seemed like students spent less time per activity than reading a page. And I bring this up because sometimes I have people who come talk to me right before the midterm. And they say, I rewatched your lectures like three times. What else can I do? And while I'm flattered that they want to watch the lectures three times, I really highly recommend you don't do that, that, instead, you spend time doing problems or going through problems from the sessions, going through the homework, going through the check your understandings. It's far more effective and efficient, in general. So in general, engage practice, particularly forced recall, where you have to sort of think about things without checking the answers, is shown to be very effective for learning. And so to achieve the class learning goals, I encourage you to spend as much time as you can or the time you have available for the course on those type of sort of directly engaging activities rather than more passive ones like reading or watching. Yeah. [INAUDIBLE] Name first. [AUDIO OUT] Do you have a time frame for when the problem sessions will be held? Great question. We will announce those by the end of tomorrow. For those ones that we know it's like impossible to coordinate schedules-- so if you can't make it, we encourage you to come in person. But if you can't make it, we also release all the materials and the videos afterwards. OK. I will highlight-- I guess just also on this too-- and I saw several people asking about this. Well, we'll just go back to this part because [INAUDIBLE] cover. So several people mentioned that they were excited about having some more theoretical aspects. This class does involve theory. It is perhaps-- there's probably more theory, I think, probably than the normal machine learning and AI classes, probably a little bit more, and not as much as like an advanced seminar on theory. So normally, most problem sets will have one-theory question. And if you're not familiar with some of the sort of theoretical techniques, totally fine. You can come to problem sessions. You don't have to have any prior background in doing proofs to be able to succeed. Another thing people asked about were Monte Carlo Tree Search. Several people brought up reinforcement learning from human feedback. We will be talking about that. Some people asked about multi-agents. We're going to be thinking about Monte Carlo Tree Search and other ways to have multiple agents that are making decisions. And a number of people said they wanted to get up to speed on sort of the latest ideas and reinforcement learning so they could read papers or do things in their applications. And I think this is all very relevant to that. The final thing is just we have five wonderful TAs who will be supporting. The main ways to get information about the class is to go to the website or go to Ed. We'll be releasing our office hours by the end of tomorrow. And we'll start them for the rest of the week. And all of you guys are completely capable of succeeding in the course, and we're here to help. Yeah. [AUDIO OUT] Yeah. Please. Go back to the course topic slide. Do some of those topics include model-based approaches as well? Yeah. So the first part-- great question. So when we first start talking about-- here, we'll talk about models at the beginning and particularly when we're defining Markov decision processes. And then we will likely be talking again about that more when we get into the offline approach. There's a lot of very interesting questions about when we're picking different-- there's a lot-- we'll get into the fact that there's a lot of different representations you can use for reinforcement learning. And there's a lot of questions over which to use when or when you combine them. And in particular, where do errors propagate in the different types of representations in terms of leading to error in the final decisions you make? But model-based reinforcement learning can certainly be a really powerful tool. Any other questions on the logistics? All right. So let's start to dive into the material. All right. We're going to start with a refresher exercise. So raise your hand if you've seen reinforcement learning at least a little bit in the past. So most people, not all. If you haven't, if everything I am about to say doesn't make sense, don't worry. We're going to cover it. But I like to kind of get a gauge in case people are like, I've seen all of this before for the very beginning of the course. So this is going to be a refresher exercise. We're going to do it on Ed. I'll put the link up again, or you can go to Ed. It'll be the second link. So here's the question. We're going to think about how would we formulate a particular problem as a reinforcement learning problem or as a Markov decision process. So one of the first application areas to use reinforcement learning for education used in roughly the following way. Not exactly. The idea was that you would have a student that didn't know a set of topics. Let's here just consider addition, which we'll assume is an easier topic for people to learn, and subtraction, which we're going to assume is harder. Imagine that maybe the student doesn't know either of these things. And what the AI tutor agent can do is they can provide practice problems. They can provide subtraction practice problems, or they can provide addition practice problems. And what happens is the AI agent gets a reward of plus 1 if the agent-- if the student gets the problem right. And they get a minus 1, if the student gets the problem wrong. And so what I'd like you to think about here is to model it as a decision process. What would like the state space be, the action space, the reward model? If you've taken classes with Markov decision processes before and you don't remember, it's totally fine to look up and refresh your memory. This is not a test. I'd like you to write down sort of what would a dynamics model represent in this case. And then in particular, what would a policy to optimize the expected discounted sum of rewards do in this case for how I've set up this scenario? So I'd like you to write down your answers, enter them into Ed, and then we're going to do some small group discussion in about 5 minutes. And if you're not familiar with these particular words like state space, et cetera, it's still fine just to think about, given what I've told you about the reward for an agent, what might happen in this case? So [INAUDIBLE] only gives the first question. Ah, OK. Sorry. You might have to switch to the Ed. OK. All right. Try to enter in something. It's OK if you're not sure. And then turn to someone near you and compare what you did. [INTERPOSING VOICES] All right. We're going to come back. Hopefully, I heard a lot of really fruitful discussions. So let's see. I know at least one group I talked to had a great idea for what the state space could be. Do you guys want to share what your state space was? And maybe tell your name as well. Sure. You said the state space can be just like a set of word pairs of like two natural numbers or any kind of numbers of like how good the student is at addition than how good the student is at subtraction. Yeah. So you could imagine something which is at addition and subtraction. So you could imagine something like this where you just have a vector pair where it's like maybe they're 0.9 close to mastery for addition and like 0.4 close to mastery for subtraction. This is not the only way you could write down. There's lots of choices for the state space, but that would certainly be one reasonable one. Those are challenging in some ways because you can't directly observe them, but it's a pretty natural way to write it down. And in fact, there are commercial systems that essentially do that where they have like-- for those of you familiar with hidden Markov models, it's basically a hidden Markov model over whether someone has mastered something or not. Don't we have a different type of state space that they wrote down? Yeah. [INAUDIBLE] talked about-- we basically wanted to [INAUDIBLE]. Oh, could you say your name first, please? The knowledge that the student has and also maybe the questions that have already been asked to capture the environment, the current environment that we're at. So I guess this is a better representation of capturing the knowledge the student has. We were thinking of also just-- like the history of questions and students' answers, whether they got it right or not. I guess that's harder to represent on [INAUDIBLE]. No, that's beautiful. So exactly what [AUDIO OUT] [INAUDIBLE]. So that was the other one I was hoping people might come up with, which is the idea of this just being a history, like a history of all the previous questions you've given or all the questions the robot has given the person and what they've responded. So you could imagine it's like a observation question reward dot, dot, dot. And in fact, those two representations here, the history and how the student-- how good the student, can be, depending on your representation, be exactly isomorphic. So sometimes this can be a sufficient statistic to capture that history. And as [AUDIO OUT] was pointing out, one of the challenges with histories is that they grow unboundedly. So if you want to have your neural network be predicting something, you might be able to use something like an LSTM, or you might want to summarize the state. So those are both great ideas for what the states could be. There's not a right answer. Both of them would be great. But there's also other ones. The actions I heard many people share what the actions are. Someone want to tell me what they are in there? I know you guys mentioned what the action space was. Sure. Just whether you pose in addition or subtraction. Exactly. So these are just like what the agent can actually do, the teaching agent, addition question or subtraction. And the reward model is plus 1 if the student gets it right. I saw some questions about what a dynamics model is. And inside of the responses, people are putting on the form. What I mean by a dynamics model here-- and we'll talk a lot more about this-- is what happens to the state of the student after a question is given. So in this case-- and I talked to some people about this who had a great understanding of this already. The idea would be sort of, how does either that history change after you give a question to the student, or how does this the sort of internal knowledge of the student change? So the hope would be as long as this sort of curriculum is vaguely reasonable, that after you give the student an addition question, they now know more about addition, or they're more likely to have mastered addition. So that would be sort of this idea of there being a dynamics process that where you start in one state, you get an action, and now you transition to a new state afterwards. And we'll talk a lot more about that. Now, what is the challenge with this particular representation? Yeah. And can you say your name first, please? Depending on your implementation, there's a risk that the agent just gives really easy problems. Yeah. In fact, [AUDIO OUT] exactly right. And in fact, that's exactly what we think will happen. So we think that an agent that is maximizing its reward should only give easy questions. So in this paper, which I took the inspiration from for this example, it was very close to this, where they tried to pick not correctness, but how long it took people to do problems. And so if the student took less time to do problems-- which isn't necessarily bad in itself. It might indicate some notion of fluency-- the agent got more reward. But, of course, what that means is that you should just give really easy questions that will take the student no time to do because then the agent can get lots and lots and lots of reward. And this is probably not what the intent-- like the designers of this system to try to help students learn things intended. They probably actually wanted the students to learn both addition and subtraction. But I bring this up because this is an example of what is often called reward hacking, where the reward that we specify does not necessarily provide the behavior that we really hope to achieve. And we will talk a lot more about this. In this case, it's a fairly simple example where we can see it fairly quickly, but there are a lot of cases where it's a lot more subtle to understand whether or not the system really will do what you hope it will do. And we'll talk about that more throughout the course. All right, great. So we're going to now just start to talk about sort of sequential decision-making more broadly. And some of this will be review for some of you, but I think it's useful to go through and refresh our memories. So the idea in sequential decision-making under uncertainty is that we're going to have an agent that is taking decisions or actions. So I'm going to use actions and decisions interchangeably, which are going to interact in the world. And then they're going to get back some sort of observation and reward signal. So in the first example I just gave you, it's like the agent provides a problem to the student. And then they see whether the student gets that correctly or incorrect. And then they also use that information to get a reward. So it's giving reward and feedback. And the goal in this case is for the agent to select actions to maximize the total expected future reward, meaning both the immediate reward they get now, as well as the rewards they're going to get over time. And this generally is often going to involve balancing long-term and short-term rewards. So there are lots and lots of examples. I'll just go through a couple of them just to give you a sense. So one is something like web advertising. In this case, Amazon, for example, might choose like a web ad to show you or a product to suggest to you. They might observe things like view time and whether or not you click on the ad, whether or not you make a purchase. And the goal in this case could probably be for them to optimize either click time or view time or revenue. In the context of something like robotics, the control space or the decision space might be something like how to move a joint. And then the feedback that the robot might get back might be something like a camera image of a kitchen. And perhaps they just get a plus 1 if there are no more dishes on the counter. Now, just a quick question, could this potentially be a reward-hacked specification? I see some smiles. What could happen? Yeah. [INAUDIBLE] Oh, sorry. Robot could just push everything off the counter. Which I will say with-- it's tempting, right? Like, I'm just going to make it all go away. But in fact, this does not solve the problem. And now you just have broken dishes and food on the floor. So that would not be a good thing to do. So yeah, this would be probably not a great reward to put. You probably want a reward more like that the dishes are inside of the dishwasher and finally clean. So not just that they were put in there, but actually that you ran the dishwasher. So this would be a second example of a setting. Another would be something like blood pressure control, where you could imagine that the agent gives recommendations like exercise or medication. The feedback is things like blood pressure. And then you would define some reward like maybe plus 1 if you're in a healthy range, else some sort of sloping penalty for being outside of the healthy range. All right. So all of these are nice examples of the numerous ways where we often try to make sequences of decisions under uncertainty. In general, we're going to assume that we have a finite series of time steps. So we're not going to be thinking about continuous time in this class. Lots of interesting things there. We're not going to cover it. What we're going to assume is that the agent is making a series of decisions. So we're going to think of there being a series of time steps like 1 minute, 2 minutes, 3 minute, 4 minute. The agent will take an action. The world will update given that action and emit an observation and a reward. And then the agent receives that, updates, and then makes another decision. We just close this loop. It's a feedback cycle. In this case, as we sort of just talked about at a high level, we can think of there being histories, which is sequences of past actions, rewards, and observations up to the present time point. So the history, ht, would consist of all the previous actions of the agent, the observations it receives, and the reward it's got. In general, this is something you could use to make decisions. You could just keep track of everything you've experienced so far and then condition on that to try to make your next decision. But we often are going to assume that there's some sort of sufficient statistic that we can use to summarize the history. It will be much more practical in many cases. Yeah. Oh, sorry. Just [INAUDIBLE] observation is basically like the history, like a previous history of the [INAUDIBLE] And what's your name? [AUDIO OUT] So the observation in this case would be something like the immediate information you get back after the last action. So in the case of the student, it would have been whether they get the last problem correct or not. So just like a single time step. And then the history would be everything like up to this time point. Good question. So in particular, often to make things tractable and because often, in reality, it's not a terrible assumption, we're going to normally make the Markov assumption. And the idea is that we're going to try to come up with some sort of informative information state that is a sufficient statistic of the history. So we don't have to keep around all of the prior history of everything the agent's ever done or seen or gotten a reward for. And what we say is a state, St, is Markov if and only if the probability of going to the next state given the current state in action is the same as if you conditioned on the whole entire history. So another way to say this, which I think is kind of a nice evocative idea-- this is not from me, this is from others-- is that the future is independent of the past, given the present. That means if you have a rich representation of your current state, you don't have to think about the previous history. And, of course, in general, this will be true if you make S equal to ht. But in general, we're going to be thinking often of sort of projecting down to a much smaller state space. So for example, you might say, well, I could think about someone's blood pressure from all of time, but maybe it's sufficient just to think of their blood pressure over the last like two hours in order to make my next decision. Yeah. Uh-huh. Is there a difference between state and observation in this case? Great question. Yes, in general. So I'll give you a particular example. Atari, which is these video games that DeepMind learned an agent to play, what their state in that case was the last four frames. So not just the last frame, the last four frames. Does anybody have any idea why you might want four frames instead of one? Yeah. Maybe like-- so you can see if there's momentum to an object already moving. Exactly. It gives you velocity and acceleration. Yeah. So there are a number of cases where you might think that there are parts of the state that really depend on temporal differences. And then in those cases, you're going to want more than just the immediate state. Great. Great questions. All right. So why is this popular? It's used all the time. It's simple. It can often be satisfied. As we were just discussing if you use some history as part of the state. Generally, there are many cases where you can just use the most recent state. Not always, but many cases. And it has huge implications for computational complexity data required in the resulting performance. What I mean by the resulting performance is is that in many of these cases, just like in a lot of statistics and machine learning, there will be trade-offs between bias and variance. And so there'll be a trade-off between using states that are really small and easy for us to work with but aren't really able to capture the complexity of the world and the applications we care about so that it might be fast to learn with those sort of representations. But ultimately, performance is poor. So there will often be trade-offs with how we actually-- the expressive power of our representations versus how long it takes us to learn. Right. So one of the big questions when we talk about sequential decision-making processes is, is the state Markov? And is the world partially observable? So partial-- oh, yeah. My question is that, doesn't the Markov assumption make this reward attribution problem somehow harder? All right. Good question. Well, I don't know. I guess you could imagine it might make it easier or harder. There's still the question of you might only get periodic rewards. And you still would have to figure out which decisions caused you to get to a state where you got those rewards. [INAUDIBLE] Yeah. So let me think of it-- so you might have a case where the reward might be a function of your current state. Yeah. Let me think if I can't think of a good example. OK, so let's say maybe you want to run a marathon. And you get a plus 100 if you make it. Boston Marathon is a competitive marathon to get into, so you get a plus 100 if you can qualify for Boston. And you do a lot of different things in your training regime. You eat healthy, and you sleep, and you train. And you get zero reward for any of that. And then on the day of your race, you see if you qualify for Boston. So your state only-- like your reward for getting into Boston only depends on that current state. But you don't know which of those decisions. Was it that you ate well? Was it that you slept? Was it that you trained every week for 17 weeks caused you to get to the state in which you qualified for Boston? And so that's independent of the Markov assumption in that case, because you still have the question of what series of decisions allowed you to get to a state that achieved high reward. Great question. So another thing is where the world is partially observable. We will mostly not be talking about this in this class. Mykel Kochenderfer has a great class where he talks about this a lot, but this does relate to the case we talked about with students. So for students, one way you could think about that is that there's some latent state that you can't directly access, which is whether or not they know addition or they know subtraction. But you get noisy observations when they do problems where they get it right or get it wrong. And the reason it's noisy is because all of us make mistakes on addition sometimes, whereas I have complete faith that everyone here actually knows how to do addition. And sometimes you might guess right even if you don't know it. So the idea is that it's latent. You don't directly get to observe it. This comes up in a lot of robotics problems, too, so I'll just give a quick example here. If you have a robot that uses a laser rangefinder to figure out these little arrows or lasers to figure out its environment-- so it could have 180 degrees of laser rangefinders. And what it's getting back is just the distance in all of these different angles to where it hits a wall. So as you can imagine, many rooms would look identical. So any room that has like kind of the same dimensions would look identical to that robot. And it wouldn't be able to tell is it on the third floor or the second floor. So that would be a partially observable case where it can't uniquely identify its state based on its observations. So we won't talk too much about that, but it's important to know about. Another thing is whether the dynamics are deterministic or stochastic. So there are many cases where things are close to deterministic. Like, if I put down a piece on a Go board, it goes there. But there are other things that we often treat as stochastic. Like, when I flip a coin, I don't know whether it's going to be heads or tails. So that will be an important decision. And then the final thing is whether the actions influence only immediate reward or reward in next state. So as an example of this, you might imagine if you were making a policy for what ad to show to people. And you just imagine for each person coming onto the web, you just show them-- onto your website, you show an ad, and then they go away. And they either buy something, or they don't. A bandit would be a case where you just have-- bless you. You have a series of customers coming in. And so whether or not I show a particular ad and he clicks on it or does not impact whether or not Ellen comes along and likes an ad. So that's a case where it impacts your immediate reward, but not the next state. We can talk more about that. All right. Let's think about a particular sort of running example. We'll think of a Mars rover. So Mars rover is a Markov decision process. Imagine that Mars is really small. We only have seven places in Mars. So in this case, we would have the state is the location of the Rover, which is one of seven discrete locations. We could have actions called try left and try right, meaning that our Rover is not perfect. So sometimes it tries to go a direction, and it doesn't succeed. And let's imagine that we have rewards, which is that there's some interesting field sites. And so if you spend time over here, you get a plus 1. And you have spent over here, you get a plus 10. And else, you get zero reward. So this would be a particular case where we could think of there being these states and these actions and rewards. So when we think of a Markov decision process, we think of there being a dynamics and a reward model. So in particular, the dynamics model is going to tell us how the state evolves as we make decisions. We will not always have direct access to this, but the idea is that in the world, there is some dynamics process and things are changing as we make decisions. So in particular, we generally want to allow for stochastic systems, meaning that given we're currently in a state and we take a particular action, what is the distribution over next states that we might reach? So for example, I'm that Mars rover, and I'm going to try to go to the right. It might be that I can go to the right with 50% probability, but I'm not a very accurate Rover. And so 50% of the time I go to the left, or maybe I stay in the same location. So this dynamics model just specifies what actually the distribution of outcomes that can happen in the world when I make a decision. The reward model predicts the immediate reward, which is if I'm in this state and I take this action, what is my expected reward? I want to highlight here that there are different conventions. You could have the reward be a function only of the current state-- excuse me. It could be a function of the state and the action you take, or it could be a function of the state, the action you take, and the next state you reach. You'll see all of these conventions in reinforcement learning papers. Probably the most common one is this. But we'll try just to be specific whenever we're using it so that it's clear. And you can always ask me or ask any of the TAs if it's not clear. Bless you. So let's think about sort of what a stochastic Mars rover model would be. So I've written down a particular choice for the reward. And let's imagine that part of the dynamics model is the following, which is if I start in state S1 and I try to go to the right, then I have some probability of going to S2, else I have some probability of staying here. What I want to be clear about here-- and this relates to the question before about models-- is that this is like the agent's idea of how the world works. It doesn't have to be how the world actually works. So what I told you in the previous slides is that imagine in this world, in reality, this gives you plus 1, and this gives you plus 10 in terms of the reward. That's how the world actually works. But the agent might have the wrong model of how the world works because it only learns about the world through its experiences, or it just might have a bad model. So this is an example of sort of like a model-based Markov system where the agent would have a particular representation of the dynamics model and a particular assumption over how the rewards work. In these settings, we have a policy. A decision policy is just going to be a mapping from states to actions. It's like an if then table. If it's deterministic, we just have a single action that we would take in a particular state. Like, maybe we always show this one ad to a particular customer. Or we could have a stochastic policy where we randomize this. So this would be something like, oh, when this customer shows up, I show a vacation ad, or I show a board game ad with 90% probability versus 10%. Both types of policies are really common, and it can depend in part what sort of domain you're in and whether you're trying to learn from that experience. OK, so let's see what that would look like in this case. So for Mars rover, you could say that no matter where it is, it always just tries to go right. So that would just be one example of a policy you could have. And it just requires you to specify for every single state what is the action you would take or what is the distribution over actions you would take. So in this sort of setting, we're normally interested in two main-- oh, yeah. Yeah, question. So it's like making decisions based on the state that it's in. And it learn to switch from different types of policies. So not just different actions for some of the state, but also switch to checking the past state, the future state. In the same way like in deep learning, it tries a bunch of different functions. Can it do that, or can it not do that? Great. And remind me your name. [AUDIO OUT] Yeah. So great question. It will in general. In general, when we're learning, it will change its policy a lot over time. So it might start with a particular policy. And then over time, it will explore lots of different policies in trying to search for something that's good. That's a great question, and that relates to what I was just putting here, which is two of the central questions we're going to talk a lot about, particularly at the beginning, is evaluation and control. Evaluation says someone gives you a fixed policy. And you want to know how good it is. Like, maybe your boss says, hey, I think this is the right way to advertise to customers, and we're going to make a lot of money. And you go out, and you just deploy that particular decision policy. And you see how much money you make. So that would be evaluation. Control is you actually want to find the best policy. And so in general, to actually find the best policy, we're going to have to do a lot of trial and error. And we want to do that in a strategic, efficient way so we can quickly learn what that good policy is. So in general, we're going to be talking about things. I just want to highlight we're going to sort of build up in complexity in terms of the type of problems we're talking about. So we're going to be thinking about both like planning and control and sort of thinking about how complicated these spaces are. So we're going to think about evaluation and control because evaluation is often a subpart of doing control. If you know how good a policy is, you may be able to improve it. And then we're going to talk about tabular function approximation methods, because we're going to want to be able to solve really large problems. And then we're going to talk about both planning and learning. In planning, we're going to assume someone gives us that dynamics model and that reward model and the state and action space. And we're just going to try to find a really good policy. And in learning, we're going to actually have to control the decisions we make to give us information that allows us to identify an optimal policy. All right. So what we're going to start with is sort of the simplest of the settings, which we're going to assume that we have a finite set of states and actions. We're given models of the world, meaning someone writes down for us what those look like. And we want to evaluate the performance of the best decision policy and then compute the optimal policy. And we can think of this really as AI planning. OK. So to think about how this works, we're going to start with Markov processes and then build up to MDPs. And this is relevant because it turns out you can think of evaluation as basically being a Markov reward process. OK, so how does a Markov chain work? And just raise your hand if you've seen Markov chains before. Awesome. OK, so most people have, which is great. So this is a memoryless random process. There's no rewards yet. There's a finite set of states in this case. And we have a dynamics model. And if it's just a finite set of states, we can just write this down as a matrix. Just says, what's the probability of going to the next state given the previous state? And so you could just have this say in our-- this would be a Markov chain transition matrix for our Mars rover case. And if you wanted to get an episode, you would just sample. So let's say you always touch down in state S4. You just sample episodes from that particular chain. Yeah. [INAUDIBLE] rows and columns down to 1? All of the-- and what's your name? Yeah, so all of the rows have to sum to 1. OK. Then is it coincidence that columns sum to 1? Yeah. OK. Yeah. I was thinking just now that I should have changed that question because-- and we'll see also why that's important later. OK, In a Markov reward process, it's a Markov chain plus rewards. So same as before. But now we have a reward function that tells us how good each of those states are. And we're also going to have a discount factor. And I'll talk about that in a second. We still have no actions. And we can express R as a vector. So in this, we could imagine our Markov reward process where we have a plus 1 and S1, 10 in S7. So plus 1 [INAUDIBLE] and 0 in all other states. In this case, this is where we start to see the ideas that are going to be really useful for decision processes, which is we can start to think about how good particular trajectories are. So we're going to have a horizon, and you're going to see this in your homework 2, which is the number of time steps in each episode. It could be infinite, or it could be finite. It's like basically how many time steps do you get to make decisions. And the return, which we're going to call Gt, is just going to be the discounted sum of rewards from the time step, current time step, till the end of the horizon. And a value function in this case is just going to be the expected return. In general, this is not going to be the same as the actual return unless you just have a deterministic process, because the idea is that you're going to have stochasticity in the trajectories you reach. And because of that, you're going to get different rewards. Right. So you might wonder-- if you haven't seen it before, why do we have this discount factor thing? So we're sort of weighing earlier rewards more than later rewards. Well, one is that it's just mathematically really convenient. It's going to help us not sum to infinity, particularly if we have infinite number of time steps we can make decisions. And it turns out humans often act as if there is a discount factor. Like, often, we sort of implicitly weigh future rewards less than immediate rewards. And this is true for organizations too. And if the episode lengths are always finite, you can always-- bless you-- use gamma equal 1, meaning you don't have to make a large discount. But when you have infinite horizons, it's generally important to make this less than 1 so your rewards don't blow up. Part of that is because it's really hard to compare infinities, so it's hard to say that this policy that has infinite reward is better than this other policy that has infinite reward, whereas you can keep everything bounded if you have a gamma less than 1. All right. Next time, we will start to talk about how we actually can compute the value of these types of Markov reward processes and then start to connect it to decision processes. I'll see you on Wednesday. Thanks.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Value_Alignment_I_2024_I_Lecture_16.txt

All right, welcome back. Welcome to the last lecture for CS234. What we'll do today is we'll do a review and a wrap up. And we're also going to discuss the quiz a little bit. But before we get started, I just wanted to remind us where we are. So last time, we did the quiz. Today, we have a review of the course and looking forward. So what we're going to do today is we're going to do a combination of the quiz recap. And then looking forward to reviewing some of the things we've done in the class, as well as looking forward. So we're going to jump into the quiz. The quiz, we'll have back to you guys within about a day. And we're just going to step through some of it, because I think it's a nice summary of some of the different aspects. So we'll go to the quiz. I'm going to start on question three. So the quiz, as everybody knows, was comprehensive. It covered the entire course. We're not finished grading it yet. But we noticed that there were some problems that people had more challenges with than others. One quick clarification is that when we said a justification for your choice, we expected you to put something different than the choice itself. So we wanted you to actually provide an explanation or rationale for why you picked what you did. So let's just step through the quiz. And inside of the solutions, we'll also do that. You might have noticed that the second question was identical to the midterm. So it was a good chance to refresh if you hadn't remembered that from the midterm. The third one was slightly tricky. So I want to just make sure to go through it. It's a nice way to review PPO. So the third question really asks you to think about proximal policy optimization, which was something that you implemented. And one thing that might have been slightly confusing, or a good thing to refresh, is that we really emphasized in class that PPO allowed us to use data and make multiple gradient steps. And when it made multiple gradient steps, those would be off policy. But the very first step that PPO makes is always on policy. So this is true. Because if you've just gotten the data, and then you're doing a policy gradient step on that, that part is considered on policy. After that, you are trying to take a further step. So if you had sort of a one dimensional policy, your first step is going to be on policy. And then any further steps you take are now going to be off policy, using data that you collected from the previous round. So I know that was often a good thing to make sure to refresh. And the second part was, we do not have any guarantees on B. And the third part is true. And we want to emphasize here that we're only doing importance sampling over the actions. What PPO does and what some of the other algorithms it was inspired by do is that they don't try to directly handle the state distribution mismatch. And instead, they try to create a new policy that's close enough that they hope that the fact that you're going to be visiting different states under a new policy, it's only going to be slightly observed. And the last is that you can use lots of different types of advantage estimators. And so D is not true. You could use generalized advantage estimation, but you could also use other methods as well. And throughout this, if anybody has any questions, feel free to ask me. So the fourth question was given by our guest lecturer. For those of you that had a chance to attend or to watch it later, he talked a lot-- Dan talked a lot about thinking about the alignment problem and thinking about what things are important for that. The first part is not true. It's generally hard to think about. There's different ways to think about autonomy, but that was not what we were focused on. The second one is true. So one of the things Dan talked about a lot in his lecture was the fact that, often, when we think about preferences and alignment, we often are focusing on people's individual preferences. Like someone says, they like option one instead of option two. But that focuses, really, on the utility to a single individual as opposed to the implications for the broader society. And so the second one is true because, as Dan brought up, moral theories give us a way to think about more broad benefits to society and to collections of individuals instead of to individuals. He also talked a lot about how autonomy is often a core principle when we think about the value of different decisions we can make. And so the idea of an AI agent to allowing people to have some autonomy would say that an AI agent that thinks about someone's suboptimal decision-- so it might be that somebody really wants to do something that we know is not very good for them. That an agent that is aligned and allows that person autonomy would still support that because-- in the interest of upweighting the degree of autonomy. And the last one is also true, because you could think of this as a form of paternalism. So if the agent decides, well, it's not really a good idea for you to smoke. And so I'm not going to tell you where you can buy cigarettes. That may or may not be true. Of course, we know that smoking is causally associated with lung cancer. So you could imagine, in that case, it's not in their best interest. But that would be considered a form of paternalism. And so that would undermine user autonomy. Yeah, I don't know if I agree with even the explanation because it seems to me that best interest and suboptimal decisions are definitionally-- well, best interests entail like optimal decisions. And you're saying we should let the user make a suboptimal decisions. I don't see how those are-- it is in fact in the best interest of the user to make decisions, then those decisions are no longer suboptimal. We should be a contradiction, it seems to be. I don't see how that follows. It's an interesting question. So I think what it says here is that there is different notions of what the objective is. And there are different notions of what is considered optimal or not optimal. So there might be some cases where, for general population, or even for humans in general, there is one part of your reward function that says, this particular decision, like smoking cigarettes, is not considered to be optimal because of long-term health outcomes. However, you might have another part of your reward function that talks about the importance of user autonomy. And so if you value user autonomy higher than, say, someone's health, perhaps, in this particular instance-- for that particular constraint, then in that case, you might say, well, if I'm supporting that person-- so if the user's best interest-- if the best interest of the user is to more value their ability to have autonomy, then for them to make this particular health decision, then you would give them the information about where cigarettes are. Yeah. So just the first clause, since it is in the user's best interest, I thought it's really hard to generalize about what a single user's best interest was. And so that was not a true statement by the start. Because maybe some people aren't best making their own decisions about things. And so I wasn't sure how confidently you could say that. Yeah, it's an interesting question. So what Dan argued is that it is generally a principle that everyone needs some amount of autonomy. And so if you go with that argument, then you would say, if we believe that it is important for all of us to have some autonomy, then under that, that should also allow us the freedom to make bad decisions some of the time. And in that case, an LLM that is supporting us also needs to be able to respect those bad decisions. And you could disagree with this. You could disagree with that as a premise, in terms of the type of theories that promote that everyone should have autonomy. And we give different people in society, different amounts of autonomy. Children generally have less than adults. But if you assume that that's the case-- as long as you assume that it's always important for every individual to have some amount of autonomy, that would include allowing them to make bad decisions sometimes. If our justification is like, that's not something you can assume. How would that be? I don't know how that would [INAUDIBLE]. I didn't grade this particular question. But you can definitely see what they say in terms of that. Yeah, we will look at everyone's justification in terms of that. Good questions. The next one that I wanted to go through was Monte Carlo tree search. So this is another one that people-- there was a little bit of differences over in terms of whether this was something people had some questions on. So the first one is true. So Monte Carlo tree search, the MCTS, the M and the Monte Carlo tree search stands for Monte, not for Markov. And the way that we described it in class, you can use it in both. So what we do in Monte Carlo tree search is we sample from the dynamics model to get to a next state. And as long as we can sample from that, whether that's a Markov model, or if you required all of the history so far in the tree to make that dynamics, that would be OK. So there's not something inherent in Monte Carlo tree search that means you always have to have a Markov system. The second is true. So the way that Monte Carlo tree search uses sampling is it samples the next state. And what it does there is it means that instead of having to enumerate all possible states, you could just sample a subset of them and still get an accurate estimation of the expectation. And the fourth is false. So this is not true because in this case, like in a lot of settings, including AlphaGo, the reward model is known. So we're not trying to learn the reward model. But upper confidence bounds are still useful because they allow us to prioritize among the actions. Because ultimately, we want to be thinking about taking maxes. And so in these cases, upper confidence bounds, like upper confidence bound trees, are using UCB to actively think of how we're expanding out the tree. And then the fourth one is also true, because that's exactly what AlphaZero does, is they use self-play to improve the network, to predict values and action probabilities. Yeah? The fourth on is false? It's false. It's false? It's false. Yeah. Yeah? I'm a bit confused about the false wording. It says Monte Carlo tree search is used in AlphaZero. But doesn't AlphaZero use a different variant of a tree search that is not Monte Carlo? I am a bit confused. AlphaZero uses Monte Carlo tree search. It uses Monte Carlo tree search with self-play to train a network that predicts values and action probabilities. That's part of what it does. Well, wasn't there a different thing that had the confidence for-- Upper confidence trees? Yeah. Yeah. Well, it uses a particular form of upper confidence trees as well. So upper confidence trees is a type of Monte Carlo tree search. It's like Monte Carlo tree search is a superset of UCT. Let's go through these two, because all of these are true. And so I think this is a useful one to go through as well. So ChatGPT did learn rewards from humans, preparing preferences over prompt-output pairs. And then use PPO to train a better prompt. So in fact, we did a ranking where they did do this. In general, for long horizon problems and really large action spaces, that would be somewhere where forward search would be really expensive to do. So using something like AlphaZero, which essentially builds a subset of the tree search, it can be really helpful. In PAC, probably approximately correct methods. We are guaranteed to learn an epsilon optimal policy. But the epsilon might be non-zero, which means we're not guaranteed to learn an optimal policy. So if you want to get a-- if you're OK with your kitchen being slightly messy, which I am, then it would be OK to use a PAC RL algorithm. That would make a finite number of mistakes. But most of the time, would keep your kitchen pretty neat. But maybe not perfectly neat. And then offline RL may be particularly beneficial for health care and other high-stake settings where online exploration might be risky or very expensive. So in this case, all of these were true. And then the next one I was going to talk about is 9, where we go through some of the theoretical properties. So in this case, what we have is that optimism-- in this case, the first one is not true, because we don't have any guarantees in general for the REINFORCE algorithm. The second one is true for the reasons we just said. A PAC algorithm guarantees your epsilon optimal. So this is not necessarily fully optimal. In the third case, it is not guaranteed to have sublinear regret. So this is false. And the reason, again, for this is that if you just get an epsilon optimal policy, you might make epsilon mistakes for the rest of time. So epsilon times t, which would still give you a linear regret. The fourth is also false. So in general, you can think of just minimizing regret. Which is the difference between your policy and the optimal policy. Or maximizing expected cumulative rewards. And they're just the same thing. One, you just subtract from the other. Can either maximize cumulative regret or minimize-- maximize cumulative reward or minimize regret. And then in the fourth case, this will not necessarily be a PAC algorithm. So the only one in this case is only B is true. And the reason for this is a PAC algorithm has to make a finite number of mistakes. So it normally has to be polynomial. And so even this would say this algorithm would be consistently converging to the optimal policy. But you don't necessarily know how long it would take. So it could be very expensive. And then the final question has us think about which algorithms could generate the observed reward. Raise your hand if anybody wants me to go through that? I'm happy to, and step through it. Or otherwise, we'll move on to the next slide. Allow the people to just think about how different algorithms would run, and whether or not they could generate this observed data. Well, we'll release the solutions for the quiz over the next day. And we'll also release the grades. Yeah? So go back [INAUDIBLE] mean that we're only allowed to meet a finite number of mistakes? Or is it that your [INAUDIBLE] mistakes should be within some budget? Finite number of mistakes. So a great question. So in terms of PAC, what we normally require is that the number of mistakes made-- well, with high probability on all but a finite number, you will be epsilon optimal. And that finite number needs to be a polynomial function of your problem parameters, including like 1 over epsilon, the size of your state space, the size of your action space, et cetera. It doesn't always tell you when those mistakes will occur. They may be at the beginning, or they may be later. I was thinking if you could have some sort of [INAUDIBLE] mistakes that it would still be fine as long as there's some [INAUDIBLE] some form [INAUDIBLE] sequence [INAUDIBLE]. Yeah. I mean, you certainly could do that. It wouldn't be PAC, unless you could guarantee with high probability that total number of mistakes would be small. Yeah, it's a good question. One thing some of the work that we have done in the past is-- you may or may not know what epsilon you want to commit to in advance. And so we've also developed algorithms where you could think of this occurring for different epsilons. Maybe as you have different amounts of budget, you might want to be able to pick epsilon. If you get a lot of data, maybe you can get more optimal. Good question. Anybody else have questions about the quiz? All right, well, feel free to-- we have normal office hours this week, so feel free to come to our office hours. Next week, we will not have office hours anymore. But if you have any questions about the quiz or about your projects, feel free to come see us. So I think it's always exciting to go back to the beginning of the quarter and think back of all the things that we've covered, as well as looking forward in terms of the field. So when we very started the very first lecture-- the slide might look somewhat familiar-- we talked about how reinforcement learning is fundamentally the question of learning through experience to make good decisions in order to optimize our long-term reward. And so that's really the central question that it tries to start to answer. And we talked about there being a number of different learning objectives in the course. And so what I hope that people will walk away with in this class is to understand what are the key features of reinforcement learning. And how does that change compared to supervised learning, and AI planning, a lot of other areas, or unsupervised learning? To understand how if you're given an application problem, whether and how you should use RL for it, what algorithms might be appropriate. To be able to implement in code RL algorithms. And you have had lots of practice with that. And then to understand how we would compare and contrast about what it means to have a good RL algorithm. And what are the ways we should evaluate algorithms themselves as a way to help us understand if we're making progress. So thinking about things like regret, sample complexity, computational complexity, empirical performance. Does it converge to the optimal policy? Does it converge at all? And then also to understand the exploration-exploitation challenge in terms of data collection. And these sort of fundamental challenges we have between the data that we gather, allowing us to learn things about the environment and about different decision policies, versus using that information to actually obtain high rewards. And so throughout this, you've had a chance to think about this on the quiz, on the midterm, on the homeworks. And then now also in your final project. So what I'd like to do now is to, again, revisit the second question. Because I think, really, as you go forward, this will be-- when you use reinforcement learning, this is going to be one of the things that you constantly have to do. Which is to decide for any new problem you're looking at, is it appropriate to think about reinforcement learning as a tool to help you solve that problem? And so I think to do that, it's helpful to go back to the motivating domains from the first lecture. So three of the domains-- we talked about a number of different domains throughout the class. But here are three of the domains that we talked about on the first lecture. So the first one is AlphaTensor. This is AlphaTensor. And in AlphaTensor, the idea was to figure out a more effective algorithm for learning to multiply matrices. And the amazingly beautiful thing they did in that case is that they actually are doing reinforcement learning to learn algorithms, which I still think is really incredible. It's extremely creative. And so what they want to think about in this case is if you want to multiply two matrices-- this is just two by two, but they go beyond that-- what is the way we should operationalize that so that we can think about the particular products and sums that we're doing in order to reduce the amount of computational complexity we need to accomplish this in a correct way? And what the researchers at DeepMind were doing when they were thinking about this is they were thinking about a common task that comes up everywhere. We multiply matrices all the time for almost all of AI and machine learning. And so we're really relying underneath those constantly that cost that we're doing. And so they're thinking about, can we essentially invent better algorithms for doing some of these really basic substructures? So I think that was really exciting. And I think this is one of the domains that now you have some of the tools to be able to do the same types of algorithms is what they use to solve this. Bless you. So what I'm going to do right now is I'm going to revisit these three. And then I'm going to ask you to think about how given what you know now, how you would formulate them. And some of them, I've talked about a little bit more or a little bit less. But I'll first just give you a quick refresher so you can think about, given what you know, you might formulate this. So the second one was plasma control. And this is much more of a controls like, more like the MuJoCo type of task that we saw where they're trying to manipulate and control these different plasma. And they want to think about a control policy to allow you to achieve different types of configurations. And then the third one was thinking about, how do we figure out who to test given finite resources? So this is for COVID testing. So if you have a bunch of people coming off an airplane and you have a finite number of tests, who can you test to better understand who might be sick and restrict the flow-- restrict the spread of COVID. And this is a process that's happening every day as people are flying into Greece, into different airports. And then you would send off those samples to labs. And then a few days later, you would get the results. And those people that you asked to test could come out of quarantine. And so what I'd like you to do now, and I've posted a poll for this, is to think about the following. So I'll label this as well. This is the AlphaTensor. This is plasma, and this is COVID testing. If you can go on to the poll and say which domain are you choosing-- is it a bandit? Is it a multi-step RL problem? What type of problem is this? What setting are we in? Is the problem an offline setting or an online setting, or some combination? What do you think the state action rewards might be? And what algorithms do you think you would use to try to tackle this problem? And we'll take a few minutes to go through that. I will give a few more minutes, and then share some of our thoughts. And I think one good thing to think about in this too, is like, are there problems with distribution shift that might come up? Are there cases where we'd want to be conservative with respect to the results that are being generalized? Or do we not have to worry too much about distribution shift in these cases? Could there be unsafe states, or too risky, or things like that? So I think we actually have a nice breakdown. Raise your hand if you did plasma. Is there someone else here? So we do have another person doing plasma, but they are remote then. Raise your hand if you did COVID. OK, maybe you want to go near those guys so you guys can all compare your answers. And then did you guys both do AlphaTensor? OK, perfect. So why don't we take a minute and talk to your neighbor. And I'll also come around and see if you guys came up with the same formulation. [SIDE CONVERSATIONS] You can't be like, plasma, stop. Yeah, exactly. So this research or anything of that sort is ultimately-- [INTERPOSING VOICES] Yeah. So I think it would be a-- it's like you-- [INTERPOSING VOICES] I'm not sure how to-- like, what I put down it was like, total number of test cases of the country. But that doesn't represent the people being tested. It's like [INAUDIBLE]. But there has to be some way to include the effect. That would be the hope. Yeah, unless you really proxy it, and you could say like, it's the [INAUDIBLE] number of tests, a fixed number of tests, and then this number of [INAUDIBLE]. I would think it was closer to offline, because there's this batch setting with delay. You have to make decisions for [INAUDIBLE]. Well, then maybe like-- so what do you think in terms of algorithms? [INTERPOSING VOICES] That's right. Exactly. So you get Thompson sampling, these samples from the prior. So then-- yeah. This is a nice motivation for why [INAUDIBLE]. Yeah, because they really want-- I mean, you kind of get some exploration that they have features of people. So you get-- it's not like everyone would necessarily get the same. But in general, if there were two people that-- [INTERPOSING VOICES] I feel like Thompson sampling is kind of a cool way to do it without having to think about it. [INTERPOSING VOICES] Because if someone's going to go on a farm, maybe it doesn't matter as much. But if they're going to go to Seoul, then I don't think any of that [INAUDIBLE]. Let's come back together. I really like these domains. I think that they're really interesting domains to think about, what is the implications for how we model these. And the choices that we have to make. And what are the algorithms that would work? So let's go through some of them, because I know that a lot of people pick different ones. So AlphaTensor-- because I think nobody-- because I know there's some people who are watching online remotely, we have more answers as well. So I think it's really interesting to see the perspective. I'm not sure that-- there was very few people that mentioned Monte Carlo tree search. But actually for AlphaTensor, it is something-- I guess maybe the alpha should hint at that-- like AlphaZero, et cetera. It is something where they're using reinforcement learning and policy networks, et cetera. But they're also combining it with AlphaZero-like technology. So they use Monte Carlo tree search in this case. So they have Monte Carlo tree search. Carlo tree search. So this is a reinforcement learning problem. It's multi-step, because the idea is that you want to take a series of steps until you solve the multiplication problem. And what the steps are, in this case, is you could think of it as like algorithmic steps. So like, which parts of your-- if we go back to here. Let me just make this big for a second. If you think of this, when you do matrix multiplication, you have A1 times B1, and A2 plus A2 times B3, A1 times B2, A2 plus B4, et cetera. You can think of there's all these different products and sums. And you could do them in different orders. And you can kind of refactor that. So when you think about all the operations you could do, you're going to have to do a series of operations such that-- remember, what they're trying to learn here is not how to multiply two particular ones. But they're trying to learn the algorithm that will always correctly solve in the minimum number of steps. So here, their reward function is the number of steps, the number of computations that you have to do. And of course, it has to be correct. And to me, one of the brilliance of their ideas is, how do you make sure that you're only searching within the space of correct algorithms? And so there's some really nice properties for this particular problem that allowed them to do that. Some other people had also noticed this in the past. And then what they said is, oh, given that we have that-- given that we have a way to verify and only search in the algorithms that are correct, now what we can do is just optimize for length. And so the way that they do that, in this case, is they're going to-- very similar in certain ways to AlphaZero-- be able to search through using policy networks and value networks. So you can see here, they have a neural network with both a policy head and a value head. Similar to what we saw for AlphaZero. But they are going to do this forward search. Now, one of the interesting things about this is that compared to what we saw for AlphaGo, in AlphaGo-- because I saw some of-- we talked about this with some of you and saw that in your notes-- at runtime, they're not going to do search anymore. What they're going to do at this point is they're just trying to find the best possible algorithm. And then in the future, they're not going to do any additional Monte Carlo tree search, unlike what we do with plain Go. Because the assumption is, at that point, they have the algorithm. And they'll just apply it to multiplying. So they don't continue to do Monte Carlo tree search kind of at runtime. This is all something done just to find that best algorithm. So this is a case where we would have Monte Carlo tree search, and we would also have policy networks. Policy and value network. And where they're sharing-- again, this is a single neural network. So you can get shared representations here, very similar to AlphaZero. And then they can play in this case. Yeah? How do they overcome distribution shifts? How do they overcome distribution shifts? [INAUDIBLE] So they are trying to have-- so all of the algorithms they search through are correct. So there's no distribution shift in that. They will always be correct for a future problem. It's just that it may or may not be that they found the very most optimal one. So there's not the same problem that you might write into different states. So the nice thing here is it's just a series of operations. It may be that the search is stuff that they didn't find the optimal one. So there might be still better algorithms that are shorter. I don't think they prove this as a lower bound, to my knowledge. And so there's not-- it's a great question. There's not going to be a problem with like, when you deploy this on a new matrix multiplication that you might get something wrong. It's just that it may or may not be the most optimal way to multiply that particular two matrices. So I think there, that's the cleverness of having the policy network be-- or having the space they search over always has fine correctness. Yeah? Were they able to interpret the algorithm or learn something [INAUDIBLE]? Yeah, it's a great question. So was there some kind of high-level insight-- and in particular, high-level insight you could translate to other problems? Not that I remember. I think that they-- I don't remember there being any sort of particular like, aha moment, now, this means for all these other types of problems, we can do this. It'd be interesting to go back to the paper and see if there was anything that I missed in that case. So I think what they found in this case, to what I remember, is that they relearned a couple different well-known algorithms for trying to-- during the search process, they learned a couple algorithms that are known to be good and more effective. And then found some others that hadn't been discovered before. And so I think there's also an interesting question, because there may be other utility functions for downstream use of these algorithms. And so in that case, you might want these approaches to provide you a set of solutions, a set of algorithms. And then people could pick which ones they thought were best. All right, so this is a multi-step RL problem. And here, the state of the system would essentially be, what are the operations you have so far? So what are the operations that you've done on the input two matrices specified as tensors? And then how far do you need to go until you can get the complete solution? And the reward in this case, assuming that you've conditioned everything on being correct, this is just length. So next, let's go to Learning Plasma Control for Fusion Science. And I think this is a really interesting one. I appreciated that I saw for a lot of people saying like, we don't want to do epsilon greedy on real RL with plasma. That's probably a bad idea for all of our health. So this need to be like some form of offline phase. And that's exactly right. That's certainly what they did in this case. I think it's interesting to think about how it's represented and the different types of controls you'd be applying in this case, which generally will be real valued. So it's a very different problem than AlphaTensor. Let's look at what their architecture was. So in this case, one thing that they also really emphasize in this is that they had to spend quite a long time-- they had to think really carefully about what is the objective. So this is an interesting one. It's not just minimize the number of computations to solve two matrices. It's saying, we want to be able to manipulate plasma into particular configurations. And so you could imagine, in this case, you might have lots of different reward functions. And you want to be able to quickly learn policies for those. So what they do to ameliorate the offline safety issue is they build a simulator. And I was just coming to someone that-- on a recent panel I was on, I was talking to a mechanical engineer that said that's one of the reasons they were really interested in AI and machine learning, is they like to make simulators a really computationally expensive physical processes. And so they here have a simulator that is fairly high fidelity, but not perfect. And is high fidelity enough that they think it'll be useful, but low fidelity enough that you could do optimization over it. So what they're going to do in this case is they are solving the offline case by constructing here-- not necessarily from data or maybe from a physics model-- a simulator. So we're going to do model-based RL. Model-based in the sense that we have to have a model or a simulator. But then what they're going to do is an actor critic method. So they are going to do actor critic, in this case, where they have a control policy. And they also are going to learn-- so we have the actor here. And they're also going to be learning a critic. So I thought this was pretty interesting for why they took this particular architecture. I'm just going to read you a little bit about that part. Let me go down there. So a couple of things. So I thought one of the things is they use an actor critic method that is related to something else we saw, but not exactly the same. It's called MPO. And I'll write that out in a second. But one of the things that I thought was interesting is they said, in our simulating period, we can do sort of whatever we want, and we can have a really complicated critic when. We are deploying this, it has to be real-time. So some other people, I think it was-- brought this up when we were chatting about it. This is like self-driving cars. And you have to have really fast controllers. You can't do Monte Carlo tree search and wait for us to decide-- like, the plasma's going to do something. And so either you're controlling it, or it is doing something else if you're not making an active control. And so they needed an actor, a.k.a. a policy, that is really computationally fast. And so what they said is that, inside of their actor critic architecture, one of the reasons they wanted to do that during the training is they could require their actor to be pretty low-dimensional. And so have a pretty small network to specify the actor or the control policy, which is what they're going to eventually deploy. But they could have a really complicated critic. And so they can leverage the fact that in the offline setting, they can really, in a complicated way, many parameters, specify their value function. Because this is all offline. And so this is of a nice, interesting asymmetry between computational efficiency, and what are the affordances you have offline compared to online. So they have a very complicated critic. And they have a very simple actor. And so then they train the actor to try to find a good point in that policy space using their really complicated critic. And so they said, the representation of the control policy vector is restricted, as it must run on TBC with real-time guarantees. But the critic is unrestricted. So I thought that was pretty interesting that they had this. Now, another thing-- and this came up in some conversations-- is, as you might imagine, if we go from offline to online, there is always the problem that it might not translate. And again, we're dealing with plasma. So we want to have some sort of safety guarantees. So here, the ideas we've talked about before about having more trusted regions or having pessimism come up. And the way that they handle this is by putting it inside of the reward function. So they essentially define areas which they think could cause bad outcomes. And then they put that inside their reward function to lead to a policy that veers away from that area. And I think, again, that's a pretty common idea that if you have safety-- this comes up in robotics and other ones, too. Claire Tomlin up at Berkeley does this, too. A number of others. You put that inside of the reward function, so the resulting policy avoid those. And so here, they're doing that not necessarily because reaching that particular part would be bad, but because you're getting close to a part where it might be unsafe, or where you don't trust your simulator. So let's go back to here. So in this case, it's an actor critic. Actor critic. This is complicated. This is simple. It has to be simple for speed. And we all do this with a simulator. We put penalties and the reward to avoid inaccuracies in simulator, or unsafe outcomes. And so this is very similar to this pessimism over the places where we're uncertain, whether because of data sparsity, or because of known problems in our simulator. [INTERPOSING VOICES] Or how do you double check that? I assume they really don't want to be making [INAUDIBLE]. So a great question. So my guess, in this case, is that it ends up making you just pretty conservative. I think of just how far away-- no, I assume in this case, maybe because some of the physics simulators that they have access to, that they could play with some of saying like, if you-- how negative do you need to make some of these? Or how out of bounds? Or how hard of a constraint is that? So that you could be very confident that before you deploy this, you make sure that this doesn't reach there. At least in the simulator, you could see whether or not you're violating those constraints. Or if you have these penalties, if you're sufficient not to reach parts of the area that you think you might want to avoid. Whether that will translate to your real system is an important question. So, yeah, it's a great issue of how you-- no, I think it also introduces the really interesting question of whether you can verify. So there are other methods. This is not some-- most of what we've talked about is not those, but where you could verify that you're not going to reach unsafe regions. And this would certainly be an area you might want to do that. The third one was efficient and targeted COVID-19 border testing for-- I should have also mentioned-- so this is also a multi-step RL problem. So absolutely, the controls you're doing affect the next state. And that's the whole point. And then you want to manipulate the plasma into a particular occasion. So it's definitely a multi-step system. This one is thinking about how do you do efficient and targeted COVID-19 border testing? And even though it's via RL, it really is a bandit problem in this case. So it's a repeated bandit problem. It's a batch bandit with delayed outcomes. So let's make this a little bit bigger. So again, remember to think back what happens in this case. People come in. Greece has some information about those individuals before they show up. We have finite numbers of tests we can run and process. We have a policy for each individual coming off that plane, whether or not they're going to be given no test or they're tested. You get the results 24 hours later. And you use that to update your policy. So I think this is a really nice example of this batch bandit process. Who you test today does not affect who arrives tomorrow on a plane. So it's a bandit problem. But we have this delayed outcome problem that you don't observe the outcomes of who you just tested for a while. Which means that algorithms like Thompson sampling may be helpful. And then in addition, some of the other really big challenges in this case is that you have a lot of constraints. You have constraints for multiple reasons. So we have constraints over the number of tests we can run. You also can have different constraints depending on where you're arriving in Greece, and where you can send things. So there are different testing sites which might have different capacities. And in some cases, also, you might have-- I don't think they dealt with in this paper, but sometimes you might have fairness constraints, too. Like, maybe it's best to test all the women, but maybe that's considered unfair. And so you may have a number of different constraints that you can think of as restricting your policy class. So it's a pretty interesting interaction problem here. And also because of the fact that it's budgeted, it means that a lot of your outcomes are coupled in a way that they might not be. So, for example, if you give me a test-- if we only have one test that we can do in this room and you give me the test, then you can't give it to any of you. And so there's this interaction, too, in terms of the data that we get to observe for the right. So I think this is a really interesting case. And it is really interesting that it ended up having a significant benefit. One of the things, too, that's interesting about this is how we define the reward. One thing that we were talking about in our smaller groups is that, really, would like to understand how this is impacting downstream COVID outcomes. And you can measure those, but you can measure those really late. You can use those as a way to evaluate how effective the overall program was, but not necessarily a reward you can use to optimize. And that's often a really common challenge. The rewards you get immediately that you could use to change your policy may be different than the downstream outcome you care about. And on Friday, I was at an experimentation workshop at the business school here, and I was giving a talk. And I was really excited and interested to see how many other people were also thinking of this challenge of short-term outcomes versus long-term rewards that you really care about. And I think this comes up a lot in advertising, and other areas, too. Companies like Netflix, and Spotify, and others we're talking about this common challenge where you have to make policy decisions-- or update your policy way before you can maybe observe those outcomes. And so if you have to wait a really long time, it limits how quickly you can experiment. And so in this case, too, you might really care about these downstream ones. But one of the points of this paper was to argue, looking at that lagged information was allowing people to make not as good decisions. And so you need these sort of shorter term outcomes. So do we have any questions about this one? So I encourage you to-- if you haven't read any of those papers, they're really beautiful papers, if you want to read any of them or all. And then just finally, if you remember all the way back, we talked about ChatGPT at the very beginning of the class. And I think you should feel excited now that you really understand this whole pipeline of what's possible. The first is sort of training a supervised policy, which we could think of as behavior cloning. The second is doing direct preference elicitation. We did it with two pairs, and then doing PPO. And we also, of course, did DPO as well. So I think now, even though we didn't do with large language models, you really have a sense of the whole process you could use if you were to train large language models and do the fine tuning. So now we're just going to wrap up with some of the main ideas, and then looking forward. So if we think about the main characteristics of reinforcement learning, this idea of learning directly from data to make good decisions. We've been thinking a lot about optimization, delayed consequences, exploration and generalization. And I think a key thing just to remember, if you didn't remember anything else from this class, is that one of the big differences of reinforcement learning is that, in general, the actions impact the data distribution. Certainly, of the rewards we observe. But often, also of the states we get to reach. And that's just very, very different than supervised learning or unsupervised learning, where the data we get doesn't-- you always see the label, or you just have a static-generated distribution of data. So this is both a huge opportunity and a huge challenge because we have to think a lot more about distribution shift. So in terms of the standard settings we've seen, we've talked about bandits, where the next state is independent of the prior state and action. As well as general decision processes, where the next state might depend on all the previous actions and states. Or it might be Markov, and it only depends on the immediate state and the immediate previous action. We've also talked a lot about the online/offline settings, where either you have historical data only and you're trying to learn better policies from that. Or where you can actually actively gather your own data. And I will highlight there that I think many real-world settings are often between these two. Many. So in many cases, you might have a large pool of offline data, and then you might be able to get a small amount of new online data. This comes up in robotics. It comes up in some of our work. We often call this sort of experimental design. So that you might have offline data, and then you can design an experiment to gather a small amount of data to try to learn a good decision policy. So I think, in general, we can think of this as an entire spectrum between these two extremes. Now, what are some of the core ideas we've seen? Well, of course, we've seen a lot of different ideas. But I think it's nice to pop up a level and think about the common themes. And Chelsea Finn, who teaches Deep RL, also had a really nice slide on this. So I found that my thoughts were aligning with a number of hers as well. So one thing is just to be really familiar with the fact that when we have function approximation-- which we're almost always going to need because we want to handle hard problems-- hard, complex problems. And we want to do off-policy learning that, honestly, we often want to do, whether we're online or offline. And just remember, off-policy learning just means that we want to take some data that was generated from one decision policy and use it to think about how another one might work. Whether in terms of gradient steps, or in terms of fully offline learning. And this is generally just really hard. So you could argue that a huge number of papers in reinforcement learning just think about this problem. It's just incredibly hard. And the reason is that whenever we have a new policy, we're going to get a new distribution over state action rewards. And that means that it may not match our current data. We have a data distribution shift. And the reason we want to do this-- the reason we want to have use the offline data is because we want to be data efficient. And this is true even if you can be online. Because as we saw for things like PPO, if you follow the theory, or if you follow this, you often have to really be incredibly conservative, or just have bad performance for a very long time. But the problem is that when we combine these two, in general, we're going to be doing generalization or extrapolation. And whenever we do that, we need to be worried that, like the values that our predictions of how good a policy will be, will not match its actual performance. And so over and over and over again, we've seen, how do we try to mitigate this in different types of methods? So in PPO, the way we control this-- and this is an online method-- is we control it with clipping. We just can't take too big of a step inside of our gradients. And that allows us to make sure that we are limiting this extrapolation problem. In the DAGGER case, we mitigated this by getting more expert labels. We knew that there could be a data distribution shift when we started to follow our behavior clone policy. And so we just try to get more labels when we get into states where we make decisions different than the expert. So we can cover the distribution of states we reach under the learned policy. And things like pessimistic Q Learning, which came from my lab. CQL, which came from Berkeley. And MOPO, which came from other colleagues of mine here at Stanford, all introduce pessimism into offline RL. Again, exactly to limit this extrapolation problem where you're overly optimistic about what will happen. So I don't think you should think of these as being the only ways to solve this problem. I think what they should inspire you to is to think, wow, this is a problem that comes up really throughout all of reinforcement learning. And we have some methods for trying to handle this. But this is certainly not a solved problem. Some of the other core ideas that we saw a lot was this idea of there's different ways we could think about the main objects in reinforcement learning. So we had this sort of models, values, and policies. Sometimes people ask me like, do we really need all of these? Or are these all useful ideas? I think some of the application areas we were just going through illustrate why these might all be useful ideas. So models are often easier ways to represent uncertainty. So if we only have finite data and we're training something about a value, or a model, or a policy, often, it might be easiest for us to represent that uncertainty with a model. So we have an idea of why that might be. Why might it be easier to represent uncertainty for a model rather than a Q function or a policy? You could disagree with me, too. But I can give you why I think this might be the easiest. We're building just like a dynamics model or a reward model. Why wouldn't that be an easier place for us to represent our uncertainty about? How the world works compared to trying to represent our uncertainty over the Q function or the policy. Isn't it just because when you're making-- when you're just like [INAUDIBLE] uncertainty in your policy, there's uncertainty both [INAUDIBLE], but also uncertainty about given your assumption of the world of what you think the best action is. So you're just dealing with a joint uncertainty, whereas the model of the world is kind of like-- it's a more specified problem, like one source of uncertainty. Yeah, I think that's great. That's a great intuition. That's what I was going for here. So to repeat what I said here, when you think about policy uncertainty, there's-- that kind of combines and wraps up this idea of uncertainty over how the world works, and uncertainty over what you should do to make good decisions given that world. And same for the Q function. And there are ways to directly represent your uncertainty over the policies and the Q functions. But models, it's a prediction problem. And so we have lots of tools from supervised learning, and from statistics and data science to think about modeling our uncertainty when it's just a prediction problem. Like, what state will happen next? Or what reward will I get in this state? There's no planning or decision making yet. It's just prediction. And so it's sort of a nice place for us to reduce or leverage the beautiful history of work in all the other fields of how we can do this easily, instead of then having to propagate that through. So I think, often, this is an easier place to represent our uncertainty. Of course, there's no free lunch. If we have it there, we want to think about our uncertainty over policies and value functions, we still have to propagate it. But it may be easier for us to represent that and drive ourselves towards it. They're also really useful for things like Monte Carlo tree search. You can use models for simulators, or for plasma. You may be able to use these ones as a place to think about risky domains, or to be very data efficient. The Q function, in some ways, is kind of the central part of RL. In the sense that it just summarizes the performance of your policy. And you can use it often to directly act, because you just take an argmax with respect to the Q function. So it's a good way to summarize how good things are. And policies are just ultimately what we want to have. We want to have good decision making. We often want to know exactly how good that is. And that's maybe where the Q function is one particular nice thing. But ultimately, we want to try to make good decisions in the world. I think another thing that's come up repeatedly is this question of computation versus data efficiency. And I think one thing that it's really useful to remember is that in some cases, they are the same. So in this class, I've often talked as if they're sort of totally different. But in many situations, if you have a simulator, data is the same as computation. You're either using your computation to maybe do more planning, or try to get to a better policy before you simulate the next step. Or you're just simulating more steps. And so I think when you look at papers, if they have a simulated domain and they're trying to do something really fancy in the back, it's useful to remind yourself that if it was a real problem you want to solve, you could either take that same computation and just have maybe 10x more samples. Or you can do 10x more computation between each sample. Now, in some other cases, we really do have limited data. We just fortunately do not have 7 billion people with COVID. There's just a finite number of people. And there's a finite number of students. And so sometimes you really want to be data efficient. When you do that, it's often trading off for computational cost. So we're going to try to squeeze everything we can out of the data. And when we do that, we often are going to rely on methods that are much more computationally intensive. And also, as you've seen in some cases, you have real constraints on this. Like in plasma. Like in self-driving cars. Like in robotics. There are sometimes cases where you have to have fast computation, because otherwise, there is a default. There's kind of a hidden action, which is, you have to make a decision at every time point. If you're not doing something optimal, something else is happening. There's some default action that's always occurring. Now, what are some of the open challenges? I think there's a lot of open challenges. I think RL is a fascinating area. But RL has not yet had the applicational impact that we've seen in some other areas of AI and engineering. And I think this is for a number of reasons. But one of them is that you really want methods that are off-the-shelf and robust and reliable. And many RL algorithms have hyperparameters. You have to pick the learning rate. You have to pick-- some of these are the same as normal machine learning, and others of them are different. And one of the challenges here is if you're online, even though in our world, like when we're doing a homework, you might be able to try it with different hyperparameters. In a real-world setting, like for health care or for customers, you would just have that one trajectory. And so in that case, or one deployment, you can't optimize those parameters. And so I think that there's this real need for automatic hyperparameter tuning, model selection. By that I mean, how do you figure out what architecture you use. How do you even write down the problem? And generally robust methods, model selection terms like the size of your neural network, et cetera. And just general sort of robust guarantees that we're not going to suddenly have one run where your performance is really bad. The other is that we often need things that are going to be able to span this data versus computation efficiency. And we don't normally have very good ways to allow a practitioner to say like, OK, well, this is how much I care about this or that. It'd be really nice if we could have sort of like Pareto frontiers. And you could say, well, if this is computation and this is data, you might say, OK, I want to have things that are always somehow optimally trading off between those two. And depending on my application area, I can pick where I want to be on this curve. And I also think this hybrid offline-online case is a really important one. Where many organizations might be willing to do a little bit of additional data collection, but not fully online learning. I think there's also some just really big questions for reinforcement learning. We focused a lot on the Markov decision process formulation. That's where it comes out of the 1950s. And Bellman, that's how I learned about it, many people learned it. And it has some really nice intellectual properties. But it is not clear that this is the right way to solve data-driven decision making. This is one framework. So I had a professor when I was a grad student, who said that the whole world is a multi-agent partially observable Markov decision process where you're doing learning. But it doesn't mean you want to solve it like that. And so while, in many times, we might be able to model things in these kind of stochastic Markov decision process ways, that may or may not be the most efficient way to represent the problem. It's just like how you could always represent a bandit as a really complicated RL problem. But if your next states are independent of your previous one, why would you do that? So I think there's some real questions over like, are there better formulations? I think a second thing is that, historically in reinforcement learning-- and even throughout most of this class-- we focused on, I'm going to learn from this one task from scratch. But of course, that's not what humans do. We constantly are building on our prior experience. We are sort of imperfect agents for learning across many, many, many tasks. And what we've seen from generative AI-- sort of large language models, et cetera, is that doing many, many tasks might be really powerful. And that's been relatively understudied in the RL setting. And it might be much more effective. We've seen even in like AlphaZero, and AlphaTensor, and others, that these shared representations can have huge benefits. And so those might be really productive ways to think about accelerating the speed of decision making and learning good data driven policies. I think a third thing is thinking about alternative forms of feedback. Assuming you get single scalar rewards, it's pretty limiting, particularly now that we have large language models. You could imagine having really rich feedback or really sparse feedback. Like, thumbs up, thumbs down, or preference pairs. Or really detailed examples about how something is wrong or what your preferences are. And now that we can start to have language as rewards, I think that's a much richer opportunity. And people are starting to explore this already. Another sort of just what settings we're in. Most of this class, we thought about stochastic settings. Take an action from a state. You get to some next state generated sort of stochastically from some indifferent process. But that's not very common in real-world settings. In many real-world settings, there are other stakeholders or multi agents that might be adversarial, or might be cooperative. You might have a teacher that's helping the agent learn something. Or you might have an adversary that's competing with that agent. And so those settings are also really important to consider. And I think another question, too, is, throughout this class, we've been thinking about integrating, and doing learning, and planning, and decision making all at once, everything. And that's wonderful and elegant. But there are many approximations to this. So in some other fields, they often do system identification. Like you might learn how the Markov decision process works. You learn your dynamics model, you learn your word model. You stop, you plant. And so while this offers some flexibility, it also introduces a lot of complexity. And again, in some areas, there might be some really good alternatives to this. And finally, this is one that's perhaps closest to my heart, which is, I think that there's just an enormous room to do better data-driven decision making in domains that could benefit. So I think there are lots of application areas we've talked about in class. But there's so many areas where I think our society could benefit from better decision making. And so it'd be incredible to see more of that impact, whether it's from the frameworks we've covered in class or from others. And I think one of the wonderful things is that you guys are very well equipped now to go out and start answering these questions, or other ones that you think are important. All right, I'll just close with two more slides. One is that if you like reinforcement learning, there is a lot of people at Stanford who think about reinforcement learning. There are lots of classes. There's at least another five. So there's Deep RL with Chelsea. There's Decision Making under Uncertainty with Mykel. Mykel and I both offer advanced courses in needed decision making or RL. And Ben Ryan Roy often also offers an advanced RL, or bandit class. So there's lots of places to learn more. And finally, thanks for being part of the course. It's great to get to meet everyone. And we're really excited to see your posters on Wednesday. Thanks.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Policy_Search_3_I_2024_I_Lecture_7.txt

Hey, everybody. Welcome back. We're going to be talking more about policy gradient methods today, and then starting to talk about imitation learning. But we'll do a quick refresh your understanding to start. I think everyone agrees that it will not necessarily converge to a global optima. So that's great. There's some differing opinions about some of the other ones. So maybe turn to a neighbor and talk about whether a baseline term can help to reduce the variance and whether or not after one step of policy gradient, the resulting policy can get worse. [SIDE CONVERSATIONS] OK, great. All right, so I think a lot of people remembered from last time that, in general, a baseline term does help with the variance. And that's one of the reasons we are adding it. You can initialize it with-- you can't initialize it with a deterministic policy. Does somebody want to say why? What's the problem with deterministic policies? [INAUDIBLE] Yes. And remind me of your name. [MUTED] Potentially, there's an action that the policy will never take. So it's not able to reach a local minima or optima. Yeah, so if you're. It's like [MUTED] said. So if you are only taking actions deterministically, you're never going to know about what other actions are in that state. So if your current policy is suboptimal, you won't get to the optimal state. And then, the last one is something we're going to talk more about today. So in general, it can get worse. This is true. In general, we're not guaranteed to have monotonic improvement. We would like to have monotonic improvement. But in general, policy gradient doesn't guarantee that. But last term with PPO, we saw things that are trying to get more towards that kind of monotonic improvement. OK, great. So what we're going to be doing today is we're going to talk a little bit more about PPO and some of the theoretical underpinnings of it, as well as another feature about it that we didn't talk about last time. And then, we're going to start talking about imitation learning. So first, and all of you guys are going to be implementing PPO as part of your homework, as well as reinforce. So you'll get a chance to practice with this. We're first going to talk about Generalized Advantage Estimation. So first, let's just refresh our memory of some of the challenges with policy gradients that motivated PPO and a whole bunch of other research on better policy gradient methods beyond reinforce. So in general, remember, we're using theta to parameterize the policy space. And we're just going to do stochastic gradient descent to try to get to a good value, a policy with a good value. The challenge is that, in general, when we did reinforce, the sample efficiency was poor. We had to run, get data from one policy, take a single gradient step, and then get more data from the new policy. And as we were just discussing, I think [MUTED] was mentioning this or maybe [MUTED] that the distance in the parameter space is generally not equal to the distance in the action space, so sort of the policy space. So when you make a small change in the parameters, it might really change the type of actions you take. So in proximal policy optimization, we saw two different ways to try to make it so that we could essentially take bigger steps in between each run of when we execute a policy, but do so in a way that would try to encourage monotonic improvement. And so we saw this bound, and we're going to come back to that very shortly, which looked at how we could try to approximate the performance of a new policy under-- only using the data that we have right now. So that's an instance of off-policy estimation. And the bound showed that this relates to the KL divergence between the actual actions taken under the new policy versus the old policy. And in PPO, you could either do this adaptive KL penalty, which says don't go too far from your previous policy in terms of the actual actions it takes, or a clipped objective, which is going to do something similar. All right, so one thing that you probably noticed, and particular, if you started implementing this already, is that we talked last time a lot about using the advantage function. So we talked about how we are going to be doing and, in general, for sort of policy gradient, we're often going to want an advantage function. And you might wonder what we're going to plug in for that. There are a lot of different choices for what the advantage function could be. So what we're going to talk about today is a particular choice that was used in PPO and that can be pretty powerful. So let's go back to last lecture before we introduce PPO and talk about the N-step estimators. So in general, in class, since the first-- probably since the second lecture or so, we've been talking-- second or third, I guess. Probably third, lecture 3. We were talking about this trade-off between methods that bootstrap and use the Markov property, such as temporal difference learning, and methods that don't leverage anything about the Markov property, like Monte Carlo. So in particular, we've talked about cases where this is a temporal difference estimate, where you just have the immediate reward plus gamma times the-- you immediately bootstrap, you plug in your estimate of the value of the next state versus ones like A infinity here, where basically we have a Monte Carlo estimate. Obviously, you can't actually go out to infinity. You'd have to have episodic cases. But we, generally, have been focusing on episodic cases so far in the class. Minus the value. So the blue there is just us subtracting off our current value. So we have this advantage estimate. So we talked before about the trade-offs between these different estimates and how some of them would have higher bias and some of them would have higher variance. And what I'm going to talk about now is sort of a way to use these to try to get to a new form of advantage estimation. And it also involves a technique that comes up a lot in reinforcement learning. So it's a useful thing to be aware of. So what we're going to do is we're going to define something called delta Vt. Let me just highlight this. And what this is is it's just essentially here, the TD backup. So this is just what we've often seen where we have our immediate reward plus gamma times V of the next state. So notice here that we've got a time step t. So that's going to be important. And then, this is just the value, our current estimate of the value of the state. So this should look like normal. This is the same advantage as up here. But note, I could plug in different ts here. So we've defined this new thing called delta. So when we use this new delta, then we would say that our advantage with-- the advantage function we've seen before where we use a TD estimate is just exactly equal to delta Vt. Why is it V? Well, V is specifying here what value function we're going to plug in. And this is just exactly equal to this. So same as what we saw before. Now, the next thing that we can say is, well, actually, what is the advantage for if we use this two-step estimate. That is this. So we've got this expression. And that is exactly equal to delta Vt plus delta Vt plus 1. So I'm just going to write that out, so we can see it for one second for why that's true. So we had rt plus gamma V st plus 1 minus V of st. That's this term. Plus gamma r of t plus 1. Because notice here, this is t plus 1. Plus gamma V of st plus 2 minus V of st plus 1. So when we do in this way, what will end up canceling here is the V of st plus 1. Let me just-- so now, we have a gamma here in the front. So we have this term cancels with this term. Oops, sorry. Let me do that. This term cancels this term. Let me just be a little careful here. We're going to have this term cancels with this term. Good. We're going to get the two rewards, rt and rt plus 1. The second one, so that's here and here. And then, we have gamma squared times V of st plus 2. So I just wrote out exactly what the definition was of delta and delta t plus 1. And essentially, the important thing to see here is that one of the terms canceled. And so that's why we ended up getting exactly the same expression for a 2 as we had before. And you can repeat this. And what will happen is all of those intermediate terms, these things where you were bootstrapping will cancel along the way. So this is why it's called a telescoping sum. Because here, we're adding something that at the next round we're going to subtract from this-- the next one. And so those are going to cancel, and so you just end up getting all the discounted sum of rewards plus the last term. Who's seen telescoping sums before? OK, so maybe about half of people here. So for some of you, this is really familiar. For some of you, this might be new. It's a useful technique to know about, because it comes up in a lot of the reinforcement learning proofs. Yeah. Can I ask, just in comparing this to the equation for a hat sub t 2 above-- Yeah. --are we basically saying that gamma squared V st plus 2 is equal to gamma. V st plus 1? No, we're just saying-- we're just actually literally canceling it. Good question. So when we write out this expression, there's a gamma in front. And because there's a t plus 1 here, this will become s of t plus 2. This will become s of t plus 1. And there's a gamma. So this will be a gamma, gamma times a minus V of st plus 1. And that will cancel with the V of st plus 1 in the previous one. OK, got it. Yeah. Yeah. So they had 2t on top and on the bottom. Are they equivalent, though? They're identical. Identical? OK. Yeah. Yeah, that was a good question. So yeah, I've written in this notation now with the delta notation, but this is exactly equal to this, which is the same. So these are identical. Ooh! Yeah, so thanks. So there's a typo there. That might be the question, yeah. Let me just highlight that. So this should be t. Yeah, in general, we're always bootstrapping with the final time step. Thanks for catching that. Yeah, so these all end up being exactly equivalent. And we've just rewritten it in terms of this delta notation using a telescoping sum. So these are just different end step estimators. We haven't done anything new yet in terms-- I mean, we've rewritten things, but we haven't introduced any new type of estimator. These are just different advantage functions. And as you might imagine, the first one is going to be low variance, high bias. The last one is going to be low bias, high variance. So Generalized Advantage Estimation involves taking a weighted combination of k-step estimators. So we had here, this was just lots of different estimators. And you might say, well, how do I pick among them? I'm not sure how I'm going to pick among them. I'm just going to take a weighted combination of all of them. And in particular, you could just take an average weighted combination. So let's just step through this a little bit, just to see how we do this. So what this is saying here is I'm going to take the 1-step advantage estimator plus lambda-- I've introduced a new parameter here, lambda. Times the 2-step 1 times lambda squared plus the 3-step 1. I'm just saying like, OK, well, why don't you use all of my estimators? And I'm going to weigh my different estimators. So now, what I'm going to do is I'm going to-- I've next just written this in the delta notation that we saw on the previous slide. And now, what I'm going to see is that some of these terms appear a lot of times. So there's this. This term appears in all of the terms, in all of the advantages. The second one only appears in the second to the last one, et cetera. So I'm going to collect terms. So I'm going to write this as follows. And this was introduced in a previous paper to PPO, and then PPO builds on it. And just notice what I've done there. I've noticed that I had this term. So I'm just taking all of those terms, and I'm noticing how many lambdas I had in front of them. And then, I'm going to have delta t plus 1 V times lambda times 1 plus-- squared. Or I'll write it differently. So this term is going to start with lambda plus lambda squared, because it's in the second through all the rest of the terms. Plus-- OK. All right, so I'm just rearranging the sums. And now, when I look at this, I realize that I've got a geometric series. So this is just going to be equal to 1 minus lambda times lambda t V divided by 1 minus lambda plus-- let me just make sure I've got it. There was-- I'll put it on the next page just to make sure I made a-- there should have been a gamma here. Let me just put gamma. I'll write out cleanly in the next slide, so this will be clear. So we also had a gamma here from before. So gamma squared. And then what you do is you realize this is a geometric series that goes to 1 over 1 minus lambda. And then, this is gamma times lambda 1 over 1 minus lambda. And this is gamma squared times lambda squared. I'm just using the fact that this is a geometric series. It's fine if you haven't seen this. If you've done real analysis, you've seen this before. And that means that the term below just looks like the following. And this was introduced by a previous paper. And the idea there is to say, well, why don't we just take kind of a weighted average of all these different terms that have different biases and variances, and then we can re-express it compactly. We don't actually have to compute all of the advantages separately and track them. We just are going to keep track of these deltas. And these deltas are pretty easy to keep track of, because those are just like these one-step differences between-- so just remind ourselves what the deltas look like. The deltas are pretty easy to keep track of, because they're just the difference between your previous estimate and your new reward plus gamma V of st plus 1. So you can just keep track of those over time, and then you're just weighing them. And our derivation just followed it. And so then you just sum these up, and you essentially have different weights. All right, so let's think about what this means in terms of bias and variance, as we often like to in terms of the estimators we're using. So this is trying to be an estimate of the advantage. And we'll do a trickier understanding now about how different choices-- so this is a discount vector. So this is the discount factor, comma, your choice of lambda, which is how much you're weighting earlier ones versus later ones. So GAE is generally a function of these two hyperparameters. And let's think for a second about what this does for bias and variance and how it relates to t. Can you select multiple or no? You can? OK, good. I was going to say, otherwise, TA that's helping make these, I can just check with them. And feel free to go to the previous slide to look at the definitions. And the reason these all are really important is because if you get better estimates of the advantage, you're going to get better estimates of the gradient. If you get better estimates of the gradient, you can hopefully use less data to get to that really good policy. So that's why people spend quite a lot of effort thinking about-- with using deep neural networks, both either for the advantage or for the policy, how can we really quickly get good estimates? Is [INAUDIBLE] comma 0 even defined? [INAUDIBLE] 0 to 0. There's a 0-- is this defined? Yeah, because in the first term, there will be 0 to 0 for the x term. 0 to 0. There shouldn't be to the x-- oh, you mean here? In that case, you would just plug 0 in up here, and then that would disappear. And you would just get this number. All right, why don't you turn to your neighbor and see if you got the same answer? Because the definition of 0 to 0 is 1. Yeah, it's easier just to look at this one. [SIDE CONVERSATIONS] OK, so thanks for the good question. I'll make sure to clarify the notation. If lambda is equal to-- whoopsie. I'll make sure to clarify that-- let's just say if lambda is equal to 0, look at first line. OK. And I'll make sure to clarify that in next year's slides. So in that case, everything drops off. So if lambda is equal to 0, all the other estimators go away. Basically, you have no weight on all of the advantage estimators that are 2 or more. And so it just becomes the first term. And the first term is the TD estimator. Yes? When lambda is equal to 1, shouldn't the entire thing becomes 0? If what? If lambda is 1. If lambda is 1. Yeah, so if the lambda is 1, well, then you're also-- you're summing from an infinite number of terms, too. But yes-- well, so this is true. So b is true. And the second one, we'll see on the next slide. It's a little weird to write down in this fractional infinite horizon, because you can't ever do Monte Carlo returns with infinite horizon. But it's a good-- you guys have good questions. I'll make sure to clarify why what happens in the age equals infinity [INAUDIBLE] and lambda equals 0 cases. Just so that the infinities are clear. But this is certainly false, because this is not TD 0, because we'd have a whole bunch of terms here. And there'd be this weird waiting in that case. And then, this-- because then you have to weigh how much is this term versus the infinity of the other-- like, the 0 versus the infinity of the other term. So I'll make sure to clarify that. D is also true, because once this is a TD estimate, then we generally know TD estimates have higher bias and lower variance. So sort of true. Now, note in general, you would think, therefore, you want to put lambda somewhere in the middle. Because it will be balancing between bias and variance. But what they do in PPO is a little bit different, but it's related to this. So this is what the Generalized Advantage Estimation is. You do this like exponential weighting over lots of different advantage estimators, but without actually having to have separate copies and memory of all the advantage estimators. So that's why this is nice. So what we're going to do now is see what we actually-- what they actually did in PPO. Which is instead of doing all of these, we're just going to do a finite number. So what they're going to do is a truncated version, where they use this, but they only go up to a certain point. So they're not going to go out for to forever. There's multiple benefits to this, including the fact that they're going to be in episodic domains. And what this means is that, let's say, your horizon is very long, but not infinity. So your horizon might be something like 2,000 steps for your Mont Car or something like that. You might pick t equal to be 200. And what that would mean is-- so remember the benefit-- one of the benefits of temporal difference learning compared to Monte Carlo is that you can update your estimate after every step. The problem with the advantage estimator that is defined here is you still have to wait till the very end to update your estimator, because you need your advantage near infinity, and then you're going to weigh all of them. So you don't actually want to do that in practice. So one thing that PPO proposes is to say, well, why don't we just do a truncated version? And that means every T-steps, like big T-steps. So let's say t is 200. Every 200 time steps, you can compute this. You compute your new sort of weighted average-- advantage estimator, and then update. So you can think of the big T here is determining how long you have to go before you can make an update. So that's what they do in PPO. They use this truncated Generalized Advantage Estimation in order to get better estimators. OK. Anybody have any questions about that before we move on to-- going back to this question of monotonic improvement? OK, so now, let's go on to another important feature of PPO, which is it's really sort of in some ways going backwards. But I wanted to make sure to go through the algorithm for a last time, so that you guys could start working on implementation. But I think, and as in many papers, the theory is a little bit decoupled from what's actually done, but it sort of serves as motivation. So I think it's useful to go back to the bound that was proposed there that helped inspire their algorithm and think about what it actually implies about what happens when we do updates. So remember that-- and as you are proving right now for Homework 2, remember that what we do in-- what they were thinking of doing in PPO was to say we want to be able to use our old data from a policy pi to estimate the performance of policy pi prime. But the problem is that in general, that's going to induce a different state distribution. And so we played this approximation and said, let's just ignore the difference in the state distributions. And that's great, because now we can use our old data to estimate the value of our new policy. Only our old data. Because we always know what the actual policy parameter is, but we don't actually have to gather new data from it. And we called this sort of L pi pi prime, because pi prime is here, but everything else is being used by pi. And what was proven was that if your two policies have a close KL-divergence in terms of the actual actions they take, then you get this bound on performance. OK, so it said this approximation is not too bad. And in particular, we get this thing of this monotonic improvement theory, saying that the value here-- so I'll just write down that J of pi is equal to V of pi. Some people use J. Some people use V. We mostly use V in the class. OK, so the value of your new policy pi prime minus the value of the old policy is greater than or equal to this term that we had on the previous slide, so this L term, this whole thing. I'll just draw [INAUDIBLE]. Minus this sort of error that we get from the fact that we are approximating the state distribution by something that's not true. So we have this nice bound. And now, what we're going to go through now is to show why if we maximize with respect to the right hand side, that we are guaranteed to improve over pi. That shouldn't necessarily-- well, I'll ask first. So who has seen this sort of majorize, maximize algorithm before? I wouldn't expect you to. [INAUDIBLE] So this kind of goes back-- I think we've seen ideas related to this in policy improvement from the very beginning. But this is different, because we've got these bound sets. So what this is saying is this is a lower bound. This says that the difference between these two policies is at least as big as this term minus this term. But it shouldn't-- and what we're actually going to propose to do is to say, all right, well, if we try to pick a pi prime that maximizes this lower bound, does that actually mean that we're going to be guaranteed to improve over pi? And it shouldn't necessarily be immediately obvious that would be true. But it's going to turn out that that's the case. So let's just go through the proof for that, which is pretty cool. All right, so we're going to prove that if you do this, if what you try to do is pick a policy, pi k plus 1, which is the argmax of this lower bound, that you will, in fact, get a new policy that's either at a local optima or is actually better than your previous policy. So that's the idea of what we're trying to do. So note a few things. So pi-- so we're going to assume that we have some pi K. That's our previous policy. And that it was feasible. So it's a well-defined policy. Sums to 1. It satisfies all of those constraints. OK, so now, let's just write it in terms of-- OK, so now, recall that-- let's just do something a little silly, but it's going to be useful. OK, so we're going to look at what L pi of pi k of pi k is. That's this term. Let's just see what that is if we just plug in, if we try to evaluate what that term-- what that sort of expression is when we plug in the same policy as what we actually used to gather our data. All right, so remember that would just be equal to 1 over 1 minus gamma, expected value over s according to d pi k. Just writing down the definition of what L is. And this is going to be pi k of a given s divided by pi k of a given s times A of pi k. All right, well, this is just one. So this cancels. But the important thing to remember here is that the advantage function of a policy with respect to itself is 0. So if I take actions according to the current policy and compare what the value is to taking actions according to that current policy, and then acting according to the current policy, minus f first taking actions according to the current policy, that difference is 0. So that's the-- I can just write that out, too, in case. So just remember, what we have here is we're going to have Q pi k of s, a minus V pi k of s. But notice what we have here is that what are we taking-- what's the distribution? We're taking these actions. It's exactly pi k. So Q pi k, if you first follow-- like, I can just write that out just in case it's helpful. So this is like sum over A pi k of a given s, Q pi k s, a, which is just equal to V pi k. It's like if you start taking this action and you follow the policy, and then you follow the policy from all future time steps versus if you just follow the policy from now till forever, that's exactly the same. So that means that this is 0. And that's good. And that means that because-- if we think back to what this looks like, that says that the difference between the value of the policy and the policy itself is 0. So this bound is tight if you are evaluating it with respect to itself. There is no difference between the value of the policy and the policy itself, because the-- oh, I'll say the next thing. So then, because also D KL of pi k over pi k is equal to 0. There is no KL. The KL-divergence between a distribution and itself is 0. All right, so now, let me just label these two. So let's call this term 1 and this term 2. So what we have here is we have that term 1 is 0 for pi k and term 2 is 0 for pi k. All right, so that means 1 minus 2 has to be at least as great as 0. Does somebody want to say why that is? I made it-- that's not immediately obvious from these steps yet. You have to make one more step. And it has to do with the argmax. Anybody see why that is and want to share? Why is 1 minus 2 always have to be greater than or equal to 0? Given the argmax. [INAUDIBLE] because we know of the policy that's going to be 0. Achieve 0 for [INAUDIBLE] Exactly. Exactly what [MUTED] said. Yeah. So pi k is an existence proof, that there exists at least one policy for which the right hand side is 0. We're taking an argmax over the whole policy space. That means the argmax has to have value at least 0, hopefully better than. And so that is exactly why. So because argmax is at least as good as pi k because we're trying to maximize that. OK, so what that means then is that-- so remember, all of this term here on the right hand side was what we had here. So this whole-- so we had J pi k plus 1 minus J of pi k is greater than or equal to term 1 minus term 2, which we just showed is greater than or equal to 0. So what we just proved is that by maximizing with respect to our lower bound, we got a new policy that was at least as good as the old policy. Which is really cool. So that means that using a lower bound on the gap between the performance of the policies is sufficient to allow us to make monotonic improvement. So that's super elegant. So now we could have something if we actually did this-- most policies do not do this, and we'll talk about that in a second. But if you actually did this, you would get monotonic improvement. And there's certainly a number of domains where it'd be really cool to get monotonic improvement. So I think I've mentioned education before, but you could imagine health care as well, like there are a lot of cases if you're doing stuff in the intensive care unit, et cetera, you people might be kind of worried about doing random exploration or epsilon-greedy. But if you could, say, we're, only going to improve when we know that the new policy is at least as good as the old policy, that's likely to be a scenario that's much more palatable. All right, so I wrote this out a little bit more here. And one of the elegant things about this is that we can restrict ourselves to parameterize policies. This doesn't mean we have to have completely-- we can think about any sort of policy class. And as long as we initialize-- so our initial policy is in that class. It could be a Gaussian. It could be a deep neural network. Then, you will-- and then, you keep doing argmax over your policy class. You'll get this monotonic improvement. So it's really nice. It's really elegant that you could do it in this case. But unfortunately, like many beautiful theory things, it has some limitations. So if you look at the actual-- so C is a constant. And we haven't went through what the constant is in class, but you're welcome to look it up in the paper. When gamma is near 1, and what gamma near 1 means is that we care almost as much about long horizon rewards as we do about immediate rewards. When it is close to 1, gamma is pretty large. And so what that means is that, in general, that second term can make you be very conservative. So why is that? Well, that means you've got-- if C is really large, that means that if your new policy takes actions that are quite different than your old policy, you're going to have a really big penalty. So what that basically does is it shrinks your step size. It says this is going to be a term that is weighed a lot. And unless you only make very small changes, you could get a big penalty. Essentially, because you're saying, I'm really not sure. It might be that when I change my policy, I end up with very different state distributions. And I don't know what the rewards would be there. So what that means is that in practice, if you actually try to use this equation directly, just straight from the theory, the step sizes are too small. Now, when people say they're too small, that doesn't mean that there's anything wrong with them. It just means it's going to take way too long. It just means that people are impatient, but they're impatient or were being very sample inefficient. So it means that this is reasonable. It will hold. You will get monotonic improvement. It's just going to take a really, really long time. And it's not going to be feasible for a lot of things, or it's not going to be practical. And so that is what sort of helped motivate why you might want to tune the KL penalty, which we saw last time, where you sort increase or decrease how much you care about this penalty, or use a trust region, or use the clipping. And so that's why we see a difference between what's formally guaranteed by if you were to just directly use this lower bound versus what's actually done in practice. But I think in terms of the take-homes from this part on policy gradient and PPO is that it's really useful to know that you don't just have to take one gradient step. You can be much more data efficient. You can play this trick of pretending there's no change in the state distribution in order to take several gradient steps and that you can do that while still trying to maybe approximately get monotonic improvement-- PPO does not guarantee monotonic improvement, but it can be pretty close-- by thinking explicitly about these lower bounds, and how much your performance might change, and how much essentially your state distribution might change, so that when you're not confident in these approximations. It also uses generalized advantage estimation, which can be helpful. And as I mentioned before, it's extremely popular. You can use it in many, many places, in part, because also you don't need your reward function to be differentiable. So people have used it in lots of domains. And the other thing that I think is just useful to remember when we think about policy gradients is that you can also use them with actor-critic methods. So you can have deep neural networks to approximate your value function, and then use that for your advantage estimation and combine them. And so that's what most people do, is that they have some sort of critic, a.k.a., your value function estimate, and a policy. And these are only to-- reinforce and PPO are, of course, not the only policy gradient algorithms, but they are the backbone to-- well, they're still used empirically a lot. And then also they're the backbone to many of the other ones. So if you read other papers, they'll be really useful baselines that you often see or that people are building on. All right, we're now going to go into imitation learning. But does anybody have any questions before we start there? Yeah. On slide 22, just a general question-- or, I guess, sorry, the one before this. Yeah, so this-- does that mean when the policy is more myopic and gamma near 0, then your step size will be like-- you'll be able to improve more to a greater extent? Yeah, that's a great question. So like, is the converse good? So if gamma is near 0, is this practical? I don't actually know off the top of my head what C looks like for gamma equals 0 or not 0, but near 0. So I don't think anybody uses this in practice. I think they always use the clipping or the KL trust region. So my guess is that it's still not practical. Oftentimes, the C constants will often be a function of V max, like your maximum value, often scaled by 1 over 1 minus gamma. So it can really be quite enormous in many cases. So it might be that here it was particularly-- they might be interested in cases where your horizon is pretty large or where you-- I think one thing here, too, is that if we're in the episodic case, there's not really a good reason to think that the discount factor shouldn't be near 1, because you probably actually do just care about all the rewards. So they're probably mostly interested in domains, where they didn't think it was reasonable. But yeah, that's a good question. All right, let's talk about imitation learning. So as we've said before, in general, in computer science, we like to try to reduce things if we can. We like to reduce them to other problems that we know how to solve. And so imitation learning is going to be our attempt to try to do that, at least in certain ways, for all of reinforcement learning. And some of these slides come from some of my colleagues at Berkeley and at CMU. So in general, we're going to now be thinking about the case where we're not going to be gathering data online. So we saw in PPO that we tried to reuse our data little bit more to take bigger steps. But one thing you might wonder is, well, why do I need any more data at all? Couldn't I just gather some data, and then just use that? And maybe I don't need to gather any new online data. And we'll see more ideas about that shortly. But one case where you might think that would be reasonable is, what if you have great demonstrations? So you have instances of doctors making really good decisions in the intensive care unit, or you have people flying planes, or you have people driving cars. Why couldn't we just use those examples to directly learn decision policies? And so the hope would be there is if we just have those recordings, any time someone's driving like a Tesla or someone's driving an airplane, could we just get those sort of state action pairs and tuples and use that information to try to learn a policy directly? Now, one thing you could do instead is to say like, well, you'd have a human in the loop, but that's going to be pretty expensive. And so the hope would be that instead we could just use the demonstrations people are already doing and that might be much more reasonable, too, in terms of people's time. So one thing in this case would be, all right, now, maybe we're going to try to just look directly at demonstrations, and that means we're not going to need to have anybody to label things. This is an example from trying to understand what the reward function might be for driving. So I guess, I should say, in addition to the fact that we often have data about people doing these sorts of complex tasks that we'd like to imitate, it also might be in those tasks that it's really hard for someone to write down a reward function. Like, maybe in this sorts of setting, you want to avoid the water unless it's really, really steep or really gravelly, in which case maybe your truck or a train can go into the water. Or maybe like in general, you want to avoid trees. But again, if it's really slippy and muddy, it's actually better. And so it might just be that it's really hard for people to write down a reward function in this case. But they could drive it and indicate that implicit reward function. And so again, that might be easier to gather. This comes up in a lot of different cases. And people have thought about it certainly a lot for manipulating heavy machinery or manipulating cars or things like that. But for things like driving and parking and stuff, those are a lot of cases where people provide those sorts of demonstrations where it might be hard to specify that reward function. So the idea from learning from demonstrations is that you're going to get a number of expert demonstrations. So experts will demonstrate things, whether they're flying a helicopter or manipulating something with a robotic arm or stuff like through teleoperation. Dorsa Sadigh's group does a lot of this. And it will give you a sequence of states and actions-- not rewards. So you just are going to have trajectories of state action s-prime. So we're not going to have any rewards anymore. Everything's just going to be implicit in this case. And now, we're going to assume that it's easier for people to do this. So they're just going to, hopefully, be able to provide these demonstrations or they maybe already have. So what's the setup for the rest of today? The setup is that we still have a state space and an action space. We're going to assume there's some transition model. And there's a reward function, but we don't know it. So there might be a reward function, but we don't know. So there's nothing explicit here. There's no explicit rewards. And that we have these set of demonstrations. In behavior cloning, what we're going to do is just reduce this to supervised learning and try to learn a mapping from states to actions. I'm just going to try to clone the behavior. And then, we're going to also see some about, can we actually recover the reward function that people might be using to generate their behavior? And then if we have that, can we actually try to get a new good decision policy? But the first one is just to try to directly learn a policy. So this is called behavior cloning. And essentially, once you decide to do this, this is just off-the-shelf supervised learning. So now, you treat it as you have a sequence of states and actions from your expert trajectories. And you can use whatever tools and supervised learning you want. So just anything can be done there. It's just to reduce. This is strictly now made into a supervised learning problem. And there were some really early successes. So like ALVINN from a very long time ago, and then in 1993, learning to fly in a flight simulator, really early on in the history of reinforcement learning or the modern history of reinforcement learning, people thought like, could we just reduce this problem? And we'll see in a second what's one of the challenges that comes up when we do this. But it certainly can be really helpful. So it's kind of fun to look at ALVINN. This was-- yeah, late '80s. But I think this must have been kind of amazing. They were already thinking about cars then. They're already thinking about not-so-deep neural networks, but they were thinking about neural networks. I think this came out of CMU, if I remember right. And they had this tiny 30-by-32 video input. And they used this rangefinder. And so they were trying to use not-so-deep neural networks to do behavior cloning for driving in the late '80s, which is pretty awesome. So it can be done pretty well. In reality, this is something that still people try a lot. It's a really good baseline to try if you have good data. And I'll talk about some of the challenges with doing behavior cloning. But I think one thing now is like if you have a lot of data, like a lot, a lot, a lot of demonstrations-- like imagine you have all the data from all the pilots, like you have their actual what they're doing, all of the different sort of input actions they're doing, and you have that for, I don't know, all of United or something like that. So if you have an enormous amount of data and you have a pretty sophisticated supervised learning technique, it can work really well, particularly if you use behavioral cloning with an RNN or something that takes a track of the history. So while what I wrote here involved just states, and then like the last state, like a Markov assumption, like the state and the action, you don't have to do that. You could also say I could have my state, action, state, and then go from there to a1, or state, action, state, action, state. So you, in general, could use something that is like a recurrent neural network or anything that keeps track of long term histories. It does not have to be a Markov representation. And that often can work very well. Again, it depends a lot on your domain. I think that there's a nice paper a few years ago in CORL, which is one of the robotics learning conferences, where they looked at what were some of the important factors when you're doing offline learning and for robot applications. So it doesn't always work well, but it can work really well, particularly if you use the history. What domains might have a history [INAUDIBLE]? Because imagine if you're flying or driving, really what matters is just the current moment. So when is actually helpful? I actually would debate that. So I think even-- it's a great question. I think maybe it partly depends on how you're thinking of the state space. But I think if your state, say, for-- let's say I'm driving. If my state is just my immediate position, that's probably not enough. I probably need at least my last few to get to velocity and acceleration. So you might already be thinking, oh, in my state, they already have those. If that's the case, if your state already incorporates like something about the first or second order derivatives, that's probably OK in some cases. But in other cases, if it's just your immediate sensors, then you want the longer history to capture that. And same for planes and stuff. Yeah, it's a good question. So this is always just a really good thing to try. It's a really natural baseline. It's generally really easy to do. People often report it in offline [INAUDIBLE]. It's extensively used. It does not always work. Let's see why it might not work. And I think one of the themes that you're seeing now with the policy gradient work right now is this challenge of what are the states you reach and how, when you use different policies, you're going to end up at different states. In general, that's the definition. If your policies don't ever reach any different states and they never take different actions, they're the same policy. They generate the same trajectories. So DAGGER was a paper in 2008. I'm trying to remember. It came out in the first decade of the 2000s to try to address some of the challenges with behavioral cloning. And I think what they were noticing is this challenge of if you do behavior cloning, sometimes things go badly. And essentially, that's because the decisions that you make over time can have cascading effects. So let's see what that might look like. So if you do something like supervised learning, you-- and this is what we do when we would be reducing our problem to imitation. In behavior cloning, we just reduce it to supervised learning. In general, we assume that our pairs, our data points, our x, y pairs in supervised learning are IID. So they're are Independent and Identically Distributed. And they're ignoring temporal structure, because they just assume they're totally independent. But in our case, they're absolutely related. In fact, if you assume a Markov structure, then what happens is you have s0, a0, s1. And so whatever you did here exactly helps determine what is the next state you do. So they're not-- the states are definitely not independent at all of your different time points. So one of the challenges with that is that if you have independent and time errors-- and generally, that's not too bad. And that's what most of our supervised learning guarantees are for, is that you assume your data is all IID, and then you can think about how much sort of error in your estimates, what sort of error you get. So in general, if you have an error at time t with probability less than or equal to epsilon and you have T decisions-- so let's assume we have T decisions, then your expected number of total errors, if all your decisions are independent, is just epsilon times T because they're all IID. And that's not too terrible. But that's not what we normally have in-- oh, I see what happened there. OK. OK, let's think about something else. I'll add a different picture later. Let's think of a racetrack. All right, so in this case, you have a racetrack. And your car is driving, except for your supervised learning thing isn't perfect and so it makes a small error. So what you actually do-- you should maybe-- maybe you actually should have went this way, but you went to the black part. And now, you, again, make a little bit of an error. And now, you're off the track. And now, this is really tricky, because you may have almost no data in the part of-- because your humans never decided to drive off the track. And so now, you're in part of the region where you have very little data and very little coverage, and you're even more likely to make mistakes. And so what you can see in this case is that if you make small mistakes early on, those can compound and get you into parts of the state space where you have even less coverage and you generally have even less accuracy. And so in general, you can actually do much worse. And this is because you have a data distribution mismatch. If the policy that you compute gives you a different distribution between train and test, then you don't necessarily have the same guarantees. And we're going to get a different distribution here, because the policy we were using to gather the data is not exactly the same as the policy we get now. And we saw that in PPO, too-- that when the policy changed, we're going to get to different states and actions. What's causing our policy to change here is the fact that we can't perfectly imitate the expert. OK, so let's just see what that looks like. So in this case, we had-- in our training set, we had pi star, which we assume to be our expert. And we were generating states from pi star. In our test set, we have learned a policy by trying to match the state action pairs we saw in our training set and we're getting a different distribution of states. In general, this is going to be different. And we're going to get worse errors in this case. Sorry about the-- I see what happened in this case. So I'll just draw it. So what can happen in this case is let's say this is the error you make now, and then you can make another error. And it keeps compounding. So if you make an error at time step t with probability E, essentially what can happen there is that you may then make errors on the remaining time steps. So could cause you to get into parts of the state action space for which you make lots of errors, and then you incur lots of regret or costs through the end. So in general, and I'm not going to step through all of the proof today, the error can actually compound instead of linearly with the number of time steps. It can compound quadratically, which means that essentially your performance is much worse than supervised learning would predict. Supervised learning said, oh, I've got an epsilon optimal or epsilon accurate policy. Great. And what this says is because all of those decisions are being made across an entire trajectory, you can actually end up with epsilon times T squared errors instead of epsilon times T. So this is what motivated DAGGER. DAGGER said, OK, what's the problem? The problem that's happening here that we'd like to address is whenever we make mistakes, we go into a different part of the state space. Once we're there, we maybe have very little guarantees that we're going to do anything reasonable. So essentially, what we want to try to do is figure out how we might correct or adjust to those states that we reach that weren't in our original training set. So the idea in this case is that your-- this is going to be an iterative approach. So you get a data set where you take a current policy and you execute it in the environment. So it's like you drive your race car around a track. And hopefully, it's similar to what the expert would have done, but probably not perfect. And then what you do is you go to your expert, and you say, OK, this is what I did when I went around that track. What should I have done there? They're like a coach. And so then what the coach does or the expert does, they say, ah, in each of those states, this is what you should have done. So it would say, hey, if you went like this, and then you did this and did all these other crazy things after that, it would have said, OK, no, first of all, here, you should have gone here. And then once you reached here, you should have went down to try to get back onto the road. So essentially, what you're having a human do is you're having them label at every time point, at every state in that trajectory what they would have done. And when you do that, that gives you a new set of data to learn from. So it's like your expert pilot gives you feedback on every place you made a mistake when you just did your last flight run, and then you integrate that. You're like, oh, OK, when I'm feeling this form of lift, next time, I got to do this. So it gives you a whole bunch more data, and then we aggregate that data, that's why it's called DAGGER. So we're aggregating the data sets of the old data we had and the new data that we just got from our expert. We then do behavior cloning again on our new data set, which now includes more of the states in the environment. And then we repeat. And I think part of the motivation for this. And this is why I said behavior cloning can work really well when you have enough data, is that the problem that's happening here is that we're assuming we don't kind of have full coverage over the whole domain of what the expert would do at any place inside of the, say, race car track. And what this is allowing us to do is to better figure out over the whole space what the expert would do, and make better decisions, and correct, in case, we end up in those. So in DAGGER, we do this over and over and over again. And there's some nice theoretical guarantees of what you'll converge to when you do this. And what they did is to show this for things like driving, driving in a simulated domain, like a Mario Kart or a video game, and show that they could learn quickly how to get a very good policy that didn't suffer from these kind of compounding errors. Can anybody think of what a limitation might be of doing this over behavior cloning? Yeah. [INAUDIBLE] Exactly. It's super expensive. Yeah. So you have to-- basically, it's like you have to have that coach, or your teacher, or your expert with you the whole learning. So the nice thing about behavior cloning is you get data once, the data might already be available, and then you can just learn from it. Here you have to have constant supervision. Now, in some cases, that might be reasonable. But in most settings, that's going to be really useful. So this is very human in the loop. Human has to supervise. And so I think for those reasons, that's one of the reasons that in robotics and some other areas, people certainly have built a lot on DAGGER. But I don't think it's as popular as behavior cloning, because it really does require a lot more work of the human. All right, so a second thing you might want to do is learn a reward. So you might say, all right, there's-- I'd to actually figure out what the reward is. You might want this for several reasons. You might want to learn the reward, because you want to understand something about human decision-making. Like, you might say, all right, I want to understand how surgeons are making trade-offs when they're dealing with really complicated situations of like, how do I trade off time or risk or things like that. And maybe it's really hard, or just their time is really valuable to ask them lots of questions. But you really would like to understand that preference structure. So that's one goal. And another is that you might want to use that then to learn a policy. You might say like, if I can extract that from the data, then I can learn a policy from that. And you'll see that in Homework 3, because we're going to be doing RLHF as part of that. We're going to try to learn from preferences. So there's lots of reasons you might want to be able to learn a reward function. So in this case, we're going to be in a similar setting. We're going to still have a state space, and action space, and a transition model, but still no reward function. It's still going to have some expert demonstrations. And what we want to do is infer the reward function the expert was using implicitly to make their decisions. And what we're going to assume for now is that the teacher's policy is optimal. So you can call it expert-- the teacher, the expert's policy is optimal. So let's think about what we can infer from that. So if you see someone's demonstrations, and you know that their optimal-- so teacher, I'll use teacher, equals to expert for this thing. If you see these, and you know it's optimal, is there a single unique R that makes teacher's policy optimal? Are there many? Does it depend on the Markov decision process? Are you not sure? Now, remember, we know that the actual policy is optimal. If you think there are many, I'd like you to give me an example. A simple one, which would make things optimal. I mean, not in the thing, but I'll ask in a second. All right, why don't we just do a quick check. And talk to a neighbor, and see what you find. [SIDE CONVERSATIONS] OK, good. So almost everybody said the answer is B, which is true. There is many. Does anybody want to tell me kind of a silly one that any policy is optimal under? Yeah. To scale by a constant factor. Yeah, that's great. And I was hearing that, too, over there. So if you scale-- if you take a reward function and you multiply it by a positive constant, then that can't change the policy. 0 works, too. So you can just use 0, and any policy is optimal if you never get reward. So I bring this up not to trivialize it, but just to highlight that this is a huge identifiability problem. There is not a single R. Even if you know that the demonstrations are expert, there's not a single reward function that's compatible with them. So that's a problem. And that's something to keep in mind when we start getting into RLHF and DPO shortly, that this is either you need to be making other sorts of assumptions to constrain your reward function or, in general, we're going to have to make additional choices or constraints, because otherwise this is not identifiable problem. OK, great. So one thing some people do to try to think about how we might do this is to think about-- [INAUDIBLE] What happened? I was editing two sets of slides. And I think the other one is now well updated. But this one was not, unfortunately. In any case, we talked briefly about value function approximation through deep Q-learning. Deep Q-learning naturally implies that we would use a deep neural network, but you could use a linear value function just like a very shallow network. The idea here-- and this is all-- this work predated deep Q-learning-- is to think about, generally, where your reward is linear over the features. So your reward of s. So here, we're just doing reward respect to states is w. W is just going to be a feature vector. W is just going to be a vector. X of s. And x of s here is just a feature representation. So this is just features are xs. So that, for example, could be like if I'm a robot, if this is my current location, what's the distance to that wall? What's the distance to that wall, that wall, and this wall? That would be a set of features. And then, I could have a weighted combination of those to give me the reward of me standing here. And the goal is to identify the weight vector w given a set of demonstrations. So in that case, you can also express the resulting value function for a policy as a combination of these weighted features. And let me just write it out. So let me just write it out, particularly, because we didn't do it in class very much. I'm going to write out what that looks like. So it's the states we reach under this policy. This time step t equals 0 to infinity gamma T of our weight vector, it's unknown, times our feature representation for that time step given we start at s0. But note here, that w is always the same, so we can just take this out. So we have wT, the expected value of s pi. And this should start to look somewhat familiar, because it's going to look like these weird discounted features that we've seen sort of before. So we can also call this wT mu of pi where this is the state distribution under discounted state distribution under pi. OK, we've seen this before where we go sort of back and forth between thinking of there being time steps and thinking of us as sort of saying, well, over all time, how much time do we spend in each of the different states? So in particular here, I've defined mu to just be the discounted weighted frequency of state features starting in a particular state. So why have I done this? Well, I've done this to say we can relate what the value is to just a linear combination under this linear reward function, a linear combination of my weight feature, which I don't know, times my feature distribution. And that's good, because I have access to features. I have access to trajectories that were demonstrated by my experts. And I can use that to extract the features of those states and compute something like mu. But we don't know what w is yet, so let's think of what we could do. So the goal here is that we want to identify the weight vector w given a set of demonstrations. We've just seen that we can rewrite the value of a policy pi if these rewards are linear as wT mu of pi, where it's the discounted state frequency. All right, so what we know is that V star for the optimal policy is greater than or equal to V pi for any other policy. And that means that w pi of mu pi star has to be greater than or equal to w pi of mu pi for all pi where this is observed. So [INAUDIBLE] experts. So what it means is that if I pick any other policy and I generate what are the state features you'd get under running that policy in the world, that distribution of features has to have lower reward than the features I've actually observed in my data, because I've assumed my expert is optimal. So my experts demonstrated things. It's optimal. And when they demonstrated things, like let's say they're controlling a robot and the robot spends all this time over in this part of the room. And if they spend time over this part of the room, then all my features are going to come from over here. And that means that any other policy that I use, its features have to have a lower weight if they don't match what the features are of the expert. Regardless of what w is, right? Because this has to hold. So this-- for the w that we pick, this has to be true. So you can rewrite that as saying that the value of V star has to be greater than or equal to V, which means that the value under-- yeah, we can just write it down in terms of this and the resulting frequency. So therefore, the expert's demonstrations are from the optimal policy to identify sufficient to find a w star such that this holds. So we know this has to be true under the true expert that-- under the true w, it has to be that features we get under the expert policy have to have a higher reward than the features we get under any other policy. So this gives us a constraint. It says when we are searching for what w is, because remember, w determines our reward function, this has to hold. [INAUDIBLE] And then what we can do is-- so it's sufficient to say, well, what would be one thing we could do to be optimal if we wanted to get to a policy? Well, we just need to match the features of the expert. We need a policy that induces a same distribution of states as the expert. So in general, if you have a policy such that the features you generate under that policy are really close to the features you get under pi star, then for all w with w infinity less equal to 1-- this is using Holder's inequality. You're guaranteed that the reward of this policy is very close to the reward of the optimal policy or your expert policy. And all of this is just to say you can reduce the problem of reward learning and policy learning, in this case, to feature matching. That's kind of the high level idea, is to say, in the case where you don't observe the reward directly, but you have access to optimal demonstrations by an expert, all you need to do is to find a policy and a reward function that allows you to match those features. Because those are the features that we know have high reward. Now, as we've already talked about, there is still an infinite number of reward functions with the same optimal policy. So even when we think about this mapping to features, it doesn't solve the issue we just identified. And there are many stochastic policies that can match the feature counts. So I haven't told you anything yet to solve that big problem. I've just told you sort of another way to think about it. And so there's this question of how do we pick among all these different options. So there's a number of different ways to do this. Some of the largest and most influential ideas are these two. Maximum entropy inverse reinforcement learning and GAIL. And what we'll do next time is to talk about maximum entropy inverse reinforcement learning, which has been very, very influential. So this is in 2008. So we'll pick up on that on Thursday-- on Wednesday. Thanks.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Q_learning_and_Function_Approximation_I_2024_I_Lecture_4.txt

All right. Come back. We're going to start lecture 4 in reinforcement learning. So we're going to be covering today Q-learning, and we're going to cover Deep Q-learning. This result came out in roughly 2014. And I remember it being a really big deal, because one of the big conferences, Neural Information Processing Systems, DeepMind came and had this amazing demonstration that they were able to now have an agent that could learn to play video games really well. And an important thing to note here is like they are doing video games from pixel input. So like they're just getting the same input as what we do. And what the agent was learning to do is to control the game through this and through reinforcement learning. And so we'll talk today about the algorithm that they did to do that, and we'll build up to that point. And this is a short video they show to just illustrate how the agent is learning through direct experience to try to optimize the score. And so what it learns in this case is it starts to learn particular strategies that allow it to do really well, which may or may not be the same ones as what humans would use. And so it was pretty incredible. This was one of the sort of most impressive successes of reinforcement learning at this point, particularly at trying to do tasks that humans can do as well and from pixel inputs. So we're going to see today how that algorithm works. All right. But before we do that, let's start with a quick, check your understanding. These are posted inside of Ed. And this asks you to think about the policy improvement stage. So we're going to be talking today a lot about learning through direct experience and scaling up towards function approximation with doing that. But first, let's think about when we're doing this, what sort of form the policy has? And then as we do this evaluation-- we do this sort of repeated evaluation and policy improvement, what happens in this case? These are the first two questions on the polls. Sorry, I just joined the class. The poll? It's on Ed. Yeah. What's your name? Thanks, yeah. So if anybody is not-- is new to class, you can go to Ed. You should be able to get to that through Canvas. All right. We have good agreement on the first one. This policy is stochastic under the assumption that for each state, there's a unique max. And it means that the new policy will be deterministic. So almost I think, everybody said that correctly, which is great. So now-- so this is the-- it's the answer for this. But there's some disagreement about the second one. So why don't you turn to a neighbor and compare what you got for whether you can compute Q pi i plus 1 by using this to generate new trajectories? And remember, what I mean by this is I want to know whether or not you can get the state action value for every state and action pair under this new policy. So I want to know if you can compute Q of s, a under this new policy. So I'll give you a hint. If a policy is deterministic, how many actions does it take in the same state? What? Right. So are you going to get any data about any other actions in that state? So can we compute the Q value of all actions in that state? No. That's right. Yeah. So this is false. We can't compute it, because if we have a deterministic policy, then we only ever take pi of s. So we would only take pi of i plus 1 of s. That would be the only action we'd ever take in that state. Because the policy is deterministic, it only takes that one-- that one action. And so that means you're just not going to get any data about what it would be like to take other actions in that state. And so it's useful to know, because it means that if we had models of the dynamics or if we had models of the reward and we could do some other things, then we might be able to compute these Q values. But here, if we're going to start thinking about just learning this from data and from direct experience, that if we have a deterministic policy, it's not going to give us any data about trying different actions in the same state. And so that's going to introduce some important challenges that we have to tackle when we're trying to get data about the world in order to learn an optimal Q-function. Great. So what we're going to be doing today then is try to think about building on what we learned last time about policy evaluation, where we're trying to learn directly from experience, to be able to evaluate how good a particular decision policy is. How do we leverage that information to then actually learn an optimal policy, to actually learn a good decision, you know, a good policy without having to model of how the world works? So we don't have access to an explicit parametric representation of the dynamics model or the reward model. And then we're also going to talk about value function approximation. And in particular, we're going to talk about Q-learning with deep neural networks, a.k.a. DQN, which led to this really seminal result in having machines that can just play directly from vision to learn how to play games like Atari. But I'll just pause here in case anybody has any questions or logistic questions before we dive into this. All right. And we're going to cover a lot today, because next week, we're going to start policy gradient methods. And we're doing that, because we think that that's a really important thing to focus on. So-- but there will be quite a lot today. And you're welcome to reach out. I put a bunch of work examples at the end, in case people want to step through some of those with Mars Rover and others. All right. So these are we're going to assess a bunch of things. And we're going to start by thinking about staying in the tabular land, so staying where we can write down the value function as a vector, and then trying to learn how to make optimal decisions in that case. So let's first just talk about the idea of generalized policy improvement. So we've seen before this idea of alternating between policy evaluation and policy improvement. And now we're going to think about that for slightly more general cases of policies. So what we just said here is that if the policy is deterministic, we can't compute the state action value for any action that's not the policy. And so what we'd like to be able to do now is to have kind of more coverage. And to do that, we're going to have stochastic policies. Because if the policy is stochastic, then we'll try multiple actions in the same state. And we can use that data to estimate the Q-function. So we're staying in what we're calling model-free policy iteration, meaning we're not trying to explicitly build a dynamics or reward model. We're just trying to directly estimate a Q-function. And once we have a Q-function, then we can extract from it an argmax policy or something else. Yeah. And we're now going to be using an estimated Q, because we will be estimating Q from data directly from experience. All right. So this is going to introduce this general challenge of exploration, which is we can only learn about the things we try in the world. So this is just like the-- can't know how much better or worse your life would be right now if you're drinking coffee at Coupa. Same thing. Like we can only learn about the actions that we take. And so we need to learn about actions by trying them. So we need to explore. But the downside in general is if we try new actions, we are spending less time using our knowledge to make good decisions. So you might imagine that you can act randomly always. And that would work for like learning a lot about the world and learning a lot about Q-functions. But you wouldn't be finding-- you wouldn't be acting using that knowledge to try to gain high reward. So this is known as the general challenge between exploration and exploitation. How much time do we spend exploring and getting new data about things that might be good? Versus how much of the time do we exploit our knowledge of how the world works, according to the data we have so far, to try to make good decisions? And this will come up a lot. There's really deep questions around here about thinking of, how do we quantify our uncertainty in our knowledge? And then how do we propagate that uncertainty into the value of that uncertainty for downstream decision making? So we'll see a lot more about that later in the course. And this continues to be a really active area of research. This is not at all solved. But here, we're just going to start to see some simple methods to try to tackle this challenge of balancing between these two things. So one of the simplest things you could imagine doing is what's called epsilon greedy. And the idea with epsilon greedy is you're going to just spend some of the time doing things randomly and some of the times, doing things the best way you know how, because you're kind of exploiting that knowledge. So if we just have a finite number of actions, because right now, we're still in the tabular case. So we just have a finite number of states and a finite number of actions. Then epsilon greedy policy says, with high probability-- so we have some epsilon here. Epsilon is going to be less than 1. It could be like probability-- it could be 0.1, for example. So with high probability, you're going to take whatever action maximizes your Q value in your current state. So you're going to kind of exploit your knowledge for whatever your state action value says. And you're going to do that with probability 1 minus epsilon. And then otherwise, you're going to take an action at random. And so when you pick an action uniformly at random, it might be one of-- the same one as the argmax, or it might be a different one. But either way, the main idea is that essentially, you spend 1 minus epsilon percentage of the time being greedy with respect to your knowledge and epsilon percent spend time acting randomly. So it's like maybe you say, OK, I'm committed to trying out new things at my restaurant. So once a week, I will try a random dish. And the other six days, I'll pick whatever I like-- like whatever I've liked in the past, and it's always been good. So this is a pretty simple strategy. This is not trying to have a deep notion of uncertainty or trying to quantify that. But nevertheless, this can be pretty effective. So in particular, we can prove things about policy improvement with epsilon greedy policies. So what we proved in the past is that if you do policy iteration, when you know the dynamics and reward models, you are guaranteed to monotonically improve. So each round of policy iteration, either you would stay the same, in which case, you'd found the optimal policy. Or you wouldn't change it, and in that case-- or you would improve. But when we did that proof, we assumed policy improvement using a deterministic policy. And it turns out the same property holds with epsilon greedy policies. So if your policy is always like an epsilon greedy policy, you can also get this kind of monotonic improvement. So in particular, and I'm not going to do the full proof today, but I'll leave it in just for time. But what this shows here is imagine that you have a Q-function, you have some policy pi i. And you have a Q-function, which tells you the state action value for that policy pi i. And pi is e-greedy, which means some of the time, it acts greedily with respect to that Q-function. And some of the time, it selects an action at random. So that's what it means to be an e-greedy policy with respect to that Q. It's making those decisions when it's being greedy with respect to that Q-function. So what this says is that pi i plus 1 is a monotonic improvement, so that V pi i plus 1 is greater than V pi i. And we can prove this here. So essentially, we're trying to prove in this case that the new policy that you extract through doing policy improvement, which is still an e-greedy policy, is going to be better than your old e-greedy policy. And the main idea is just to say you can kind of also do policy improvement when you don't have deterministic policies, but you have these kind of e-greedy policies. And you could still get monotonic improvement. And I'll leave that-- I'll put that at the end for later post proof. So this is just to highlight like, here's one thing we could do. And we're going to see that this is actually going to be a pretty helpful thing to do. This is one thing we could do to try to get data about other actions. So we're not just taking a single action in a single state, but we actually have some probability of drawing out multiple actions. And just to make that concrete, if you think back to our Mars Rover example, there are only seven states. So if you act in it for a long time, you'd repeatedly reach the same states. What this e-greedy policy is doing is saying like, even when you get to the same state, you might take different actions. So over time, you're going to get data that allows you to estimate the Q value of that whole policy. So now we're going to see is how we can use these ideas of e-greedy policies to actually do control. So what I mean by that is that we're going to try to learn optimal ways of acting in the environment. And we're going to start-- we're going to have the same scenario as last time. So we're going to either have Monte Carlo approaches, where we simulate in the world, and then we use that to try to improve, or temporal difference approaches, which more directly try to use the Bellman and Markov structure. OK, so let's start with Monte Carlo. So remember, what we had before. We used to have this Monte Carlo policy evaluation algorithm, where we would repeatedly loop, we would sample the kth episode. So we'd just like sample a series of states and actions under a particular policy. OK. And then you could compute the return from each step till the end of the episode. And then what you would do is you would-- bless you-- you would update-- for the first time you visited a particular state action tuple, you would update the Q value by a weighted average between your old estimate and then your new target, which was just the sum of rewards you got starting in that state and action till the end of the episode. OK. So this is where we often call it like our target. And we were using that, because we knew from Monte Carlo that what we want to do is really estimate the value of starting in this state, taking this action, and following this policy to the end-- to the end of the episode, that we can get a sample of that by doing this. And this sample is an unbiased approximation to the true expected sum of rewards you would get starting in this state and action and going till the end of the episode. Yeah? We can apply epsilon-greedy [INAUDIBLE]? We're going to see that. Yes, exactly. Yeah. So when we thought about this before, we thought the policy was like a deterministic policy, or that was the easiest way to think about it. But now the policy could be stochastic, and so it could be e-greedy. Yeah, great question. OK, so now, this policy-- good. Well, here, we'll go on to the next one. OK, so this was Monte Carlo policy evaluation. Now, what we could try to do is Monte Carlo online control. So what I'm going to do here is I'm going to introduce a different-- an additional line here at the bottom, which says after I do an episode, I'm going to potentially change my policy. So you can think of this as like my policy evaluation part. And this is my policy improvement. And again, I'll just write out what this means. So what this means is that for each state, for each s, the policy for s is going to be equal to argmax Q of s, a with probability 1 minus epsilon, else random. So that's what I mean by say, we're doing the policy improvement step is we take our Q-function. We say either you would take the argmax action, or you would act randomly. Sorry. What are we looping over in the outermost loop? Is it a k or? Yeah, this would be-- yes, this would be k. Yeah. So this is just-- you can think of the loop here. And I'll write that down. Loop over the episodes. So it's like I play one game of Atari. And then I update my policy evaluation. And maybe I change my policy. And then I do another round of Atari. So I like play Breakout a million times, sometimes more than that in some of these cases. Yeah. Ana, right? Is it-- Yeah. Yeah. I'm getting confused by this last line that you have out there. So, I mean, isn't it implicit that you are using the new-- you're using a new Q in the-- let's say, you're done with iteration number k. You moved on to iteration number k plus 1. When you're sampling the next episode, you're using the updated Q, right? Great question. OK. So maybe I should-- so what this says here is that initially, you construct-- so your Q initially is 0 everywhere. You could initialize in some ways, but your queue is 0 everywhere. And you're going to select something that's e-greedy with respect to that. Now, if your Q value is 0 everywhere, it means that all of your actions are tied. You have no information. You basically are just acting randomly. What this says is that the way we act is always with respect to our current policy. So the first time-- or you can think of as like motor babbling. Like your agent will just randomly press buttons. It'll move over the screen. It'll do that till it wins or loses the game. And then it will update its Q value. And what this is saying is that the next time, you're going to change what that policy is that you're using to act. So hopefully, it won't babble quite as much. It's just like, oh, well, sometimes I hit something and then I got an increase in the points. So maybe I'll try to do that action again. Is the policy guarantee to one of the new [INAUDIBLE]? Great question. OK. So I've not said anything about that yet. I haven't said anything about what the properties are of this. Guy in the back. I can't remember your name. Is it required to do this on-policy? Or could you do this off-policy that collects a number of demonstrations in an update later? Great question. Yes, you can definitely do off-policy. And we'll see that in a couple of slides. Yeah. OK. Any questions? These are all great. OK, so you should be skeptical that this is necessarily going to do anything reasonable. But it's certainly something you could run, like something that you could write down in a computer. So this is a process. So then, a question would be-- and I put some-- this is an optional worked example. You can go through it just to think about how it would actually update these. So some important properties are, how expensive is this? Does it converge to the optimal Q star as well as what is its empirical performance? Let's think first whether or not we think this is a good idea and whether or not we think that this procedure here is guaranteed to become a good estimate of the optimal Q star. So this is another check your understanding. It's on Ed. But what I would like you to think about here is that given the process I've just shown you here, do you think that the Q value we're computing is an estimate of the current policy? And do you think it will ultimately become Q star? And if you think it might or might not under some conditions, that's fine too. You can put that in there. Yeah. k-- like k changes with these two points? That's right. I was a bit confused about [? innovation. ?] What is Q pi k again? Q pi k is the true state action value function for the pi k policy. So what is the expected discounted sum of rewards? If you start in state s, take action a, and then follow pi k. Yeah, thanks for the clarification. Why do you vary the-- I don't remember your name. Why do you vary the epsilon? I have not said anything about whether I am varying the-- [INAUDIBLE] You're setting a different-- Oh, yeah, yeah. Here I am. Yes. I know what you're talking about. Why are we doing that? We'll talk about that. I forgot that I had already put that in there. Yeah, we can talk about it. OK. And one thing just to note here-- and I think this is a question 2. So just to be clear here, as you're thinking about this, so this is like an approximation of policy iteration. So we're kind of doing policy evaluation and then policy improvement. But it's helpful to think about kind of how much time we're spending doing policy improvement versus policy evaluation. So what this is saying here is that, you're going to sample one episode. And then you're going to do policy evaluation. OK. This is all just one episode. So it's like I'm going to play one-- I'm going to play until I win or lose at Breakout once under a particular policy. And then I'm going to change my policy. And then I'm going to play with my new policy once. And then I'm going to change my policy. And some of those games might be really long, or some of them might be really short. Yeah. So this is-- just to clarify, this is like representing what you do after playing one game in a breakout? So then I think I might just be confused, like, [INAUDIBLE]. When they're just like-- if I'm playing a game, the episodes just follow one after the other. So isn't there just-- there's just one kth episode? There's one kth episode. Yeah so like k is like I play-- so if k is 1, I'm going to play my first game. And I'm going to play it until I win or lose or until the game ends. So maybe Breakout finishes. Or maybe I'm playing Tetris, and like I fail, and I die. And that is one episode. And I'm going to use that to then update my Q-function. Then I'm going to change it and say, OK, well, my next round, I'm going to play differently. And then I play Tetris again until I fail. And then I see what the total points are. I update my Q-function, and I repeat. And some of those episodes might be really short. So maybe the first time-- particularly for these agents, the first time they play Tetris, maybe they lose in like 10 steps, might be a really short 10 step. Later, maybe they play for a long time. But in general, I've not told you anything about how long these episodes are. They might be really short. Or they might be really long. OK. And I think one useful way I find to think about this is that think about if they're really short, like really, really short. Like I take two steps, and I just fail. I did something really dumb. So in that case, think about whether your Q would be a good estimate of Q pi k. Like, would it be good if you've only seen two states? Or would it be pretty bad? So why don't you turn to someone near you? I think most people have voted if you have written something you have. But why don't you check and see what you think. [INTERPOSING VOICES] OK. Awesome. I'm hearing a lot of really good discussion. But I'm going to interrupt you, because I want to make sure we get to DQN. So this is where-- so one of the reasons that I bring up this particular example is that here, it's tabular. I mean, these are a little bit smaller, so it's a bit easier to see. But essentially, what I kind of want you guys to get out of today is that it should be sort of shocking that reinforcement learning works. [CHUCKLES] And we're not going to have time to go through all the deep mathematical reasons for why it does work sometimes in this class. But I'm happy to give people pointers. But so there's several things that are really kind of odd, if you start to think about this, when you go through this. So first of all, Q is not an estimate of Q pi k. It is not, because it is averaging over policies that are changing every episode or potentially changing every episode, right? Because in fact, in general, it will be, right, because we're decaying epsilon. So we're changing epsilon each round, which means we're making things more and more deterministic. But in addition to that, our Q might be changing. So essentially, I'm just trying a policy one round. And then I update my Q, and then I try something again. And sort of extreme example of this would be like flipping a coin once and deciding whether-- what its bias is or something like that. That's just not very much data to do this evaluation. And also, you're averaging this over many, many different policies. So Q is not an estimate of Q pi k. It's this weird weighted average of all the previous data and all the policies you've done before. Q should not be an estimate of like the [? big ?] changes in the latest k, k minus 1? Well, but not really. Because I mean you've averaged in that part. That part's from pi k plus 1, but this old thing was over all of your-- is like this weird weighted average of all the other policies you've tried. So yes, it is that, but also like that plus all the other policies. So it's this weird thing, right? The second thing is that we're only doing-- and I was talking to some people about this. We're only doing one rollout to try to evaluate a policy. And you might imagine there's a lot of stochasticity like even in something like some games, there's like random rolls of the dice and stuff like that, which means even with the same strategy, you might get different outcomes each time. So it'd be like if you drove to SF, and you did it once. And there was no traffic. And so you're like, I can always get to SF in like, I don't know, 20 minutes on the highway. But for those of you that drive to SF, you would know that often, there's lots of traffic. And so you would need to average over many rounds of doing this to see how good a particular route is. So the weird thing here is that we're just doing kind of one rollout. We're averaging into this weird Q thing, which is now going to be this weighted average of all the policies we've done. And we have this weird epsilon thing. And it should not be clear yet that we will necessarily converge to Q star. Like we are getting more and more deterministic over time, because we're reducing epsilon. So we're reducing epsilon here towards zero. Eventually, we're going to converge towards something deterministic. But you may or may not be convinced yet that the thing we're going to converge to is actually Q star. So fortunately, there are some sufficient conditions under which we can guarantee that this sort of thing will converge to Q star. And really, it's quite beautiful that this works. So one is what's called greedy in the limit of infinite exploration or GLIE. So the idea in this case is that if you can ensure that all state action pairs are visited an infinite number of times, meaning the number of counts that you have for a particular state and action pair goes to infinity for all states and actions-- this is for all. And the behavior policy. And what I mean by the behavior policy is, this is the policy you're actually using to make decisions in the world. And it will be important. There'll be distinctions between this and other policies soon, which is why we call this behavior policy. If the behavior-- so if you sample state action pairs an infinite number of times and your behavior policy converges to the greedy policy, which means that asymptotically, the action you select in a state is exactly equal to the argmax of your Q-function with probability 1. So you're just getting more and more deterministic. So then you were being greedy in the limit of infinite exploration that says that you're exploring everything an infinite number of times. You're always continuing to try all actions in all states. But you're getting more and more deterministic. So this is what it means to be GLIE. If you have a GLIE algorithm-- and I'll just note here, like a simple way to do this is to do e-greedy, where epsilon is reduced to 0 at the following rate. Yeah, so we'd have this-- so that's a simple one. And visit all states. And that should hold-- as long as you have an e-greedy strategy, then you will be able to visit all states and actions. So you're going to be visiting all states and actions under this GLIE strategy. Then under that, the Monte Carlo algorithm I just showed you for tabular representations will converge to Q star, which means as long as you decay epsilon at this rate, you are actually converging to Q star. You're getting more and more deterministic. You're still visiting all states and actions an infinite number of times. And this procedure is guaranteed to asymptotically get you to the optimal Q-function, which is pretty cool. And it should be somewhat surprising. All right. So that is GLIE. And that is of the reasons why we like to think about e-greedy algorithms, because they have this nice property that we can prove that we are going to get an optimal policy, even though all we're doing is we're acting in the world. And we're getting this data. Now, what you should be thinking about at this point is that, all right. Here's the Monte Carlo approach to doing this. There's probably going to be a temporal difference approach to doing this. And that's what we're going to see now. So now we're going to look into temporal difference methods for control. OK. So one of the interesting things is that there's going to be two different types of algorithms that we're going to focus on for temporal difference for control. And the idea in these settings is that we're going to alternate between two steps, again, this policy evaluation versus policy improvement. And one of the key things to think about in this case is how much time are you spending doing evaluation versus improvement. And what are we trying to evaluate, and what are we improving with respect to? So the idea now is that we're going to compute Q pi using temporal difference updating with an e-greedy policy. And then we're going to do policy improvement in the same way that we saw before for Monte Carlo methods. So we can do this e-greedy thing, where we are greedy with respect to our current Q value. And the first algorithm we're going to see is called SARSA. And the reason it is called SARSA is it is state action reward next state next action. It is short for that. S-A-R-S-A, SARSA. That's an easy way to think-- to remember why this method would be called SARSA, because those are the tuples we need in order to do updates. We need s, a, r, s prime, a prime to do an update. And this is going to be an on-policy algorithm. And this is related to what was suggested in the back. Remind me your name. Yeah, exactly what [INAUDIBLE] said. So can we also use off-policy data? And we'll see that really shortly. But SARSA is going to be on-policy. And what we mean by that is that it's going to be computing an estimate of the Q value of the policy we're using to act or what policy we're using to make decisions in the world. So let's see how it works. So in general, the form of SARSA is the following. We are going to iterate, our loop is going to be such that we start off. So this is the-- we start in some state. This is the s. We take an action a. We observe reward in the next state, and then we loop. And we take the next action still according to the same policy. And then what we're going to do is we're going to update our Q-function, given this tuple of SARSA, essentially. And what we're going to do in this case is going to look similar to what we saw before. So we're going to have our updated one is our old value plus alpha. So this is like our learning rate, our target. s t plus 1, a t plus 1, minus Q of s t, a t. So this is the target. And it's going to look similar to what we saw for TD 0, where we plug-in our immediate reward, plus our estimate of the expected discounted sum of rewards starting in that next state. And one of the important things to notice in this case is we are plugging in the actual action we took in the next state. So we're saying what is the expected discounted sum of rewards starting in this state and taking this action? Well, when-- estimate of it is the immediate reward I got plus gamma, times the Q value for the state I reached, plus the action I would take under this policy next. And that's one of the reasons why it's called on policy, because it's been specific to the action you would actually take under this policy. All right. And then after-- then the next thing we do is we do policy improvement. And what we would do in this case is, again, similar to what we saw, [? new one. ?] So for all s, this just means for all, for anybody who hasn't seen this notation. Pi of s is equal to argmax over a, Q of s, a with probability epsilon. And then what we do is we update our timestep. We update our epsilon. And then what we're going to do is just repeat. So then we're going to go-- then we're going to go to the next-- we're going to take our next state, take an action, and repeat this updating. So this is called-- yeah. Quick question. Like do we-- I have a bit confused about setting pi of s. Do we say pi is a deterministic policy that is one of this with this probability and the other one with the other probability? Or are we saying it's a stochastic policy that can-- It's a stochastic policy. Yeah, so it's a stochastic policy. At the very beginning, it's totally random. You just take any action in any state. Later, you're defining it with respect to your current Q value. And you're either being greedy with respect to that Q value or selecting action at random. Yeah. So one concern that I had was that what if we reach a terminal state and then just end? Good question. And this actually came up in another conversation earlier this morning. Yes, so if you reach a terminal state, then you just reset. So if s t plus 2 is terminal, reset the episode and sample s. So if you ever reach a state where it's terminal, what would happen next is then your whole episode just resets. You sample this initial state from the world, and then you repeat. So just like if I like finished my game, I failed at Tesla, it reinitializes the world. So these are still sort of assumed to be continuing processes. Yeah? I'm wondering-- what will [INAUDIBLE] to athletes? Great question. So what best and what we're going to see in just like a slide or two-- and you guys are-- probably half of you at least have probably seen this before. We're going to see Q-learning, and that's where it's going to be off-policy. OK. Really quick question. Like when it says t plus 2, do we do the step seven? Or do we skip it for one step and do it in the next one, because-- If it's terminal or in general? Like if it's terminal. If it's terminal, you would halt here. And then you would reset the whole thing. Then you would need to take an action, observe, or next state, and then jump into five. So you have to reset to two. Yeah, great question. All right. So let's see. Well, first, let's talk about whether this is guaranteed to do anything reasonable, and then we'll get going. So I've written this up neatly here. And then there's a worked example for the Mars Rover at the end of the slides. OK. So one thing to note here, too, is that now we've defined a general learning rate, so that we have a general learning rate here. OK. And we also have-- let me make sure I keep this in here. We're going to keep updating our epsilon. OK, so is this a good approach? So we can think of a couple of different things here. We can think of the computational complexity. So here after each tuple, we're doing an update. And in fact, we know that that's in general only going to change the Q value for the states and the actions that we're updating. So we just are doing that small update each time. We don't have to sum over all the states. So there's nothing that depends on the state space size per update. But of course, we're doing this many, many, many times. Does this converge to the optimal Q-function? So what we have here in this case is we have this weighted combination between our last Q-function and this new target. And again, Q is an estimate of the performance of a policy that might be changing at each time point. So it's similar to Monte Carlo. Like we're just like we're constantly changing the policy in this case. And so that should feel a little bit concerning. And empirically, it often does quite well. But Q-learning is more popular. OK, so what are the convergence properties? So it turns out that in terms of some of the mathematical formulations, this relates really strongly to stochastic approximation. And this is a deep literature with lots of really amazing results. In the finite state and finite action case, it's going to converge to the optimal Q star for SARSA, if your policy sequence satisfies the condition of GLIE. So we're going to visit all states and actions an infinite number of times. And we're getting greedy and greedier over time. And we have to put in a condition about the learning rates, the step sizes. So in particular, they have to satisfy the Robbins-Munro sequence. So they have to satisfy these two things, which is their sum goes to infinity, and their square is less than infinity. And we've seen this before. And an example of this would be a t equals 1 over t satisfies these conditions. So these results really sort of rely on these really nice results from stochastic approximation, because it should be a little bit surprising. You can think of this as kind of there's these different-- there's these mixing processes that are going on, because our policy is changing, our estimates are changing. How can we be sure that it's essentially going to be stable enough that over time, we're actually going to converge to something that's both fixed-- like we're not just going to oscillate forever-- and that it is optimal? So it should not be at all clear why this would necessarily work. And this is where we rely on those results from stochastic approximation, that also had to be extended to think about these particular cases during a number of really beautiful papers from the 1990s. So those are the 1992 and 1994 papers that show this. OK, so there's some really cool results that illustrate why this is possible. OK, SARSA for tabular settings under some mild conditions is guaranteed to converge to Q star. So now let's see if we can do off-policy learning. So off-policy learning is the idea that now we're going to be trying to estimate and evaluate a policy using experience gathered from following a different policy. So, so far, we've been thinking about Monte Carlo methods and SARSA, where we're at least sort of trying to always approximate the value of the most recent policy or averaged over all those policies. But now we're going to explicitly be trying to estimate Q star at all time points. OK. So in Q-learning, we are going to try to directly estimate the value of pi star-- which remember, we don't know, because if we knew what pi star was, then we wouldn't have to do any of this learning-- with another behavior policy. Pi b. So we're going to be acting in one way. And we're going to be trying to use that data to estimate the value of an alternative policy. And that's what Q-learning does. So in Q-learning, the key difference is that instead of trying to think about what is the action we actually took on the next time step, we're just going to figure out what is the best action I could have taken, because we know for the Q star value, it is the estimate of the optimal expected reward you could get if you take the current action and then act optimally from now on. So really, you would normally like to have something like this. So sum over s prime probability of s prime given s, a, times V star of s prime. So you would have in the Bellman equation. And what we're going to do here, what Q-learning does is it approximates that by this max. And that is different than what SARSA does, because SARSA used the actual action. And Q-learning says, I don't really care what actual action you took. I care about what is the best thing you could have done there, because that's giving me a better estimate of the maximum expected discounted sum of rewards I'd get from that state till the end of time. So that is what Q-learning is doing. So it looks really similar to the SARSA update. But our target is going to be the reward I got, plus the best reward that I think I could have achieved from that next state. All right. So then we get an algorithm that looks extremely similar to what we saw before. But we have this max over the next action. And then I'll just make sure-- I think I forgot to write down here. So whether we're doing Monte Carlo or SARSA or Q-learning, in all of these cases, we're interleaving gathering some data under our current epsilon greedy policy, and then using it to update a Q value. And because we don't know what the actual Q-function is, we're sort of doing this weighted approximation between our current estimate of the Q-function and the target that we just put in. And we do this over and over and over again. So similar to SARSA, the conditions to make sure that Q-learning in the tabular case-- so things get a lot more complicated once we go into the function approximation case. But in order for tabular Q-learning with e-greedy exploration to converge the optimal Q star, you again need to visit everything infinitely often. Your step sizes has to satisfy the Robbins-Munro sequence. And one important thing to notice here is that you can estimate Q star without being GLIE, which is different than SARSA, because you're always doing this max. So even if you act completely randomly, so just like infinite exploration, not being greedy, you could learn Q star, because in your Q star estimate here, you're always doing this max a. So that's an important difference compared to SARSA. But if you actually want to use that information to make good decisions in the world, you need to become greedy over time and be using that information to actually select the best action according to your Q-function. And for e-greedy algorithms with Q-learning, you normally do k over epsilon over time. So you're getting more and more deterministic. And you're taking your estimate of what Q star is and using it to make decisions. Yeah, we're now going to go into function approximation. I'm just going to pause there in case people had any questions. Yeah. So for either using SARSA or Q-learning, will it converge-- does it converge to a stochastic policy or a deterministic policy? Great, great question. So if there are no ties in your Q-function, as in like for any action there or any state, there is a uniquely best action, it'll converge to a deterministic policy. If there are ties, it'll generally pick between those arbitrarily. There'll be an infinite number of optimal policies if there are ties in your Q-function. Great question. All right. So now what we're going to do is we're going to layer on function approximation on top. So this was all assuming that we just had this table where you could write down the value for every state and action separately. And now we want to use function approximation. So we can start to do problems like Atari. So the motivation for doing this-- and I know for those of you who've taken machine learning, this is probably clear, but it's nice to think about what this means in the context of reinforcement learning. So what are the things that we might be storing or trying to manipulate? That might be the dynamics or reward model, the value function, the state action value function, or the policy. And if you were thinking about pixel space, you do not want to write that down as like one different value for every-- bless you-- for every different possible image in the world. So we're going to want compact representations like what we can do with neural networks, so that we reduce the memory we need to write down those dynamics models, the value function, or Q or the policy. We reduce the computation. And ideally, we might even be able to reduce the experience. And I think this last point maybe is a particularly interesting one to think about. So you can imagine, if your agent is learning to play an Atari game or play Breakout, it might want to know that, oh, well, if these pixels are slightly different here, most of the time, you might still take the same decision. And so then instead of having to learn from scratch what to do in each state, you can get this sort of generalization. And that could be really important in terms of reducing the amount of data we need to learn to make good decisions. All right. So how do we do this? What we're going to try to do is we're going to essentially do the same thing as what we did before. But we're also going to have to incorporate a function approximation step. So let's just think about how we would do this if we had an Oracle. So what I mean by this is we're not thinking yet right now about all the learning and like gathering data. We're just assuming how do we fit a function to represent our Q-function. So let's imagine that you had an Oracle that for any state and action, it would give you the true value for a particular policy and that state and action. So it would tell you like that's three or that's seven. So then you could say, OK, now I've just got a supervised learning problem, I've got input tuples of states and actions. And I have output values of my Q-function. And what I want to do now is just learn a function to-- a regression function to say, given the state and action, what is the output? So imagine that you're in a case where we have a continuous set of states, and we only have one action. Then you might just have all these different points. And maybe you just want to learn a function that predicts the Q for every single state. And you just learn like a parametric function. Or it could be a deep neural network. And in general, just like in supervised learning, the objective is going to be to find the best approximate representation of Q, given some weights or given some neural network architecture. So we've got some neural net. And we're just going to fit this to try to-- if we had these points. But of course, we don't have these points. And we're going to see how we're going to handle it. We don't have these points. But this is the intuition is that if you had these, then you could do the function approximation step by saying, OK, well, how do I-- I'm going to handle generalization by using a linear function or a deep neural network to say, for each of these states and actions, what is the output? So just to highlight here in this class, generally, we will be focusing on methods that use stochastic gradient descent to try to fit these functions. And again, I expect most of this is familiar for you guys if you've done machine learning. If you haven't, you can come talk to me or any of the TAs. Generally, we're going to just use mean squared error. And we're going to try to fit a function that minimizes the mean squared error. We're going to do gradient descent to find a local minimum. And we're going to do stochastic gradient descent just to compute an approximate gradient. Right. So in this case-- and I have here is just to write that out really quickly. You would have something like this. You'd have w, J, and-- it's just the derivative of this. I'm sorry. Like that. So I'm going to take this equation star. And I'm just going to take the derivative of it, which is going to be two. So we're just going to take the derivative of this. And essentially, that just means we're going to have to take the derivative through our Q-function representation, like using autodiff for deep neural networks. And then we can use this to update our weights. All right, so we'll do stochastic gradient descent to do this. And the main thing is that that's what we're going to be doing to plug-in in order to do policy evaluation or to do control. So of course, in general, we don't have those. We don't have for each state and action, what the Q value was. If it was, we wouldn't need to do any learning. We need to learn that from data. And so the idea is that we're going to do model-free state action, value function approximation. So just like what we've been seeing before, we're doing model-free state action value function. Now we're going to actually do that, but just do an approximation, where instead of writing it down as a table, we're going to write it down with these parameters function-- parametrized functions. OK, so the idea now is like similarly, we just saw before all these methods, either where we use Monte Carlo methods or temporal difference methods to try to do these approximations. Now what we're going to do is that when we do the estimate update step, we're also going to fit the function approximator. So just like in the algorithms we saw before where we do like policy evaluation and policy improvement, now when we do the policy evaluation, we're also going to just refit like our whole Q-function, for example. OK, so let's see how that could work. So for Monte Carlo value function approximation, we're going to remember that our return G is an unbiased but noisy sample of the expected return. So we can think of us having this state, action return, state, action, return, et cetera. And so you can substitute in those Gs for the true Q pi when you're doing your fitting. So let's see what that would look like. So in this case, remember what you would like here when we're doing our function approximation is that this is the real Q of the policy, but we don't know what the real policy, real Q value is. So we're going to plug-in r, observed return. So, we want-- would like Q of s, a. But we don't have that. So we're going to plug-in the return that we just observed. And then we'll just do the derivative-- we'll be plugging that in for our derivative. And then update our weights using that derivative with respect to minimizing the mean squared error. OK, so this would just be for policy evaluation. If you have a fixed policy, you would just do this at each time point. So after you see-- after you get a return, then you would update your Q-function. And you would do this many, many times. And for some of you, this might start to look redundant. But I think it's just useful to see that essentially the structure of all of these algorithms, whether it is policy evaluation or tabular or function approximation, is extremely similar. We are just either sampling an episode or sampling a tuple. We are going to do one step, which is like policy evaluation, where we update our estimate of the Q-function, maybe optionally do function approximation fitting. And then we're going to use that to figure out how to act next if we are doing control. We'll see an example of that shortly. OK, so that is Monte Carlo. OK. Oops. OK. All right. For temporal difference learning, it's very similar. But now we are going to have this weighted sum where we plug-in-- we bootstrap. So we plug-in our current estimate of the value of s prime. So this is the same update we saw before. This was for tabular cases and now we're going to do it for function approximation. OK. So let's first just see how we do it for function approximation. It's just useful, I think, when we look at this, to think about all the different ways we're doing approximations. We are sampling to approximate the expected value over the next state. We are bootstrapping to plug-in what the value of those states are. And now we're also going to do function approximation, because we're going to represent the value of a function with some weights. OK. So we're going to have these weights. All right. And again, we can just do stochastic gradient descent to fit our weight function to represent that value function. OK. So you'll get something like this, where as long as if you're in a terminal state, you'll restart the episode. Otherwise, you'll just be doing these computing the gradient with respect to your minimizing your mean squared error and updating your weights. And then we'll see how we do this for control. So for control, it's going to be very similar. Now, what we'll make sure to do is we're always going to be using the Q-function instead of the value function. And we're now often going to be doing off-policy learning again, like Q-learning. So we'll again do stochastic gradient descent. With respect to our Q-function, we're going to sample the gradient. And we'll have very-- an algorithm that's very similar to the one we've seen before. So we can either use SARSA where we have our Q-function, where we always plug-in what is the actual action we took next. Or we can have Q-learning where we plug-in a max over the next Q-function. And raise your hand if you've implemented deep Q-learning before. OK. So one person, but most people not. Yeah. OK. So you can imagine in general this is any form of function approximator. But often this is going to be like a deep neural network. OK. Now, one thing I just want to highlight here is that, again, just being in terms of being concerned whether all of this is going to work, there's a lot of approximations that are happening here, so particularly for Q-learning. And it's led to what Sutton and Barto, the authors of the book that is the optional textbook for the class called The Deadly Triad. And what they say is that if you are doing bootstrapping, meaning that you're plugging in an estimate of what is the value of the next state and you're doing function approximation, like you're using a deep neural network or a linear function, and you're doing off-policy learning where you are acting in a different way than the data you're getting, under those cases, you may not converge at all. Like just your Q-function may oscillate. You may not converge to anything. And you are certainly not guaranteed to converge to Q star. So it's just good to keep in mind that that could occur. I think for some intuition for why this can occur-- the Bellman operator, if you think back a couple of lectures ago, we proved as a contraction, meaning that as we apply it repeatedly, we went to this fixed point in the tabular setting. But the problem is that when you do a Bellman backup, that operator is a contraction, meaning that if you apply the Bellman operator to different things, their distance gets smaller afterwards. Value function approximation fitting can be an expansion, which means if you take two things and then you try to do value function approximation, like you align to this one and align to this one, the distance between two points afterwards can actually get bigger than before you did the value function approximation. So there's a really beautiful example of this in a paper by Jeff Gordon from 1995. I will just-- Jeff Gordon 1995 has a really nice example of this where you just can kind of visually see, when you have these two functions and these points, that after you do this value function approximation, you've actually made the distance between them bigger. And so that means that you have this thing, where you're kind of alternating between something which you know is a contraction and driving you towards a fixed point, and something which might actually amplify differences. And so because of that, it's not always the case that you're guaranteed to converge to a fixed point. So this is something important to know. However, I think it's also kind of a-- it's an important part of the history of the field in that, in the 1990s, there was a bunch of work showing that this could occur that even with some really simple settings like linear value function approximators, we just approximate things with a line, and that sometimes you could get these kind of oscillations or lack of convergence. And so people were really concerned about using function approximators with reinforcement learning. But then what happened is that DeepMind showed well, actually there are some ways to tackle this. And we can do really amazing things with it. And so I think it's a useful-- like a useful lesson from history over the difference between, well, what can occur and maybe some sort of not ideal cases versus what actually occurs in practice. And so we shouldn't let some of the negative examples limit us from considering what might work in some other scenarios. So let's see that now. Let's see DQN. OK. So the idea with DQN is we're going to use these ideas to actually play Atari. So we're going to take in images of the game. We're going to use convolutional neural networks. And we're going to have a really big, deep neural network to represent the Q-function and do Q-learning with it. So the idea was, well, we knew that sometimes, like Q-learning with value function approximation can diverge. And there's a number of different issues, but one of them is kind of this stability thing. So we know that there's correlations between samples. Your data is not IID, which is what you would normally want for when you're doing function approximation. And the other is that you have this kind of nonstationary target thing, which is like when you plug-in, say with learning, you're plugging in gamma plus-- sorry, r plus gamma times the value of your next state. And that value of the next state is constantly changing as you get more data. So what DQN did is they said, well, look, what we're going to do is we're going to use experience replay. In particular, we're going to reuse tuples over time. And we're also going to get fixed Q-targets. And both of those things ended up making a really big difference, particularly one of them. We'll see in a second. So the idea of experience replay is to say in general, if I think about states that are nearby, their Q-function might be pretty similar. And if I'm doing lots of updates, that's breaking my IID stuff that I want for my function approximation. So another thing you could do is just have a replay buffer of lots of all the different tuples you've seen in the past. And you could just sample from one of those and then compute a target value and then do stochastic gradient descent. And this might be really helpful anyway just in terms of data efficiency, because it means that instead of taking your data and using it once and then throwing it away, you keep it and then you can replay it. Just like how we talked about batch learning last time. So an experience replay can be useful, because we're both replaying our data. So we can squeeze more information out of it. And also we can select from very different parts of the past history, which makes those updates more independent. OK. So this is-- and in general, we're not going to keep the buffer for all time. We might keep the last million episodes or things like that. OK, So that's one thing we could do. Now, the other thing is that if we think about what's happening in this case, the way we change the weights is going to be-- in general, the weights appear here and here and here. So this target value is a function of the weights itself, because you're using the value function approximation to represent the value of your next state. And so the problem is that in general, this is going to change on your next update, because you've just changed your weights. And this can also lead to instabilities, because if you think of supervised learning, your x, y pairs, your y is changing even for the same x over time, because you're changing your Q-function. OK, because this is a function of the weights. And so as the weights change, this target value is going to change even for the same input. So the second idea is to have fixed Q-updates. And what the idea here is-- and so remember, this is like when we say the target weights-- this is going to be what we're using for the target weights-- is that the weight-- the weights or the parameters we're using to estimate the value of the next state we reach, we are going to not update those as much. So we're going to have our target network using a different set of weights and the weights that are being updated. So you can see here that we have a w minus, meaning that we're trying to make this more like supervised learning where we have a fixed output y that is not changing, while we're trying to update our w. And so if you think about the example, we want to just draw it like this. Here's our states, here's our Q-function. Right now, we'd like to make sure that these points, when we're trying to fit a line that those y's are not changing a lot where we're trying to fit the line. And in general, because they're a function of the weights themselves, they might be moving and perturbing. And so what we're saying is, no, we're going to fix these. So you can think of this as just being a fixed number for a while and then do multiple updates on this w to try to fit that function. And so what that means is we just have to we keep around these target weights, and we keep around the other weights. And this allows us to do-- this is what is called the fixed Q-updating. So if you think about what this pseudocode would look like in this case-- let's see-- is it's going to look pretty similar to the things we've seen. You're going to sample an action. You're going to observe a word in the next state. You're going to store the transition in a replay buffer. So you're going to keep track of it. Then you're going to sample a random mini batch of tuples from the past. You're going to do something, keep track of the episodes terminated. Otherwise, you're going to say my target y that I'm going to try to fit in my function approximator is my immediate reward, plus the maximum of my actions of my Q-function with my target weights. So I use my deep neural network to predict the value of that state action pair. And then I'm going to do gradient descent on the difference between these predicted y's and my current estimate with my current weights. So this is just the function fitting part. And then you repeat this. And then periodically, you update your target weights. OK? And I just want to highlight here, there's a bunch of different choices to be made. You have to decide, what function approximator are you using? Are you using a deep neural network? What's your learning rate? How often to update the target weight? How big should your replay buffer be? There's a lot of different choices that you have to make. All right. Let's just take a quick second here to see if this part's-- like a chance to think about this part. You may have just seen the answer, but that's OK. Which is-- OK, in DQN, we're going to compute the target value for the sampled state action reward next states using a separate set of target weights. So does that change the computation time? Does it change the memory requirements? Are you not sure? Put that in here. We're now going to maintain two different sets of weights to do our function approximation. All right. Yep. I see almost everyone converged to the right answer very quickly. It is doubling the memory requirements. So you have to keep track of a second set of parameters. It does not change the computation time. It just changes the memory requirements. So we just keep around two copies of your deep neural network, one with the old weights, one with the new ones. And then the Q-updating with respect to that is the same. All right, let's see what that actually does. So the kind of the key innovations for DQN where we are going to use deep neural networks that had been done before, but not with-- I think this is the first really big example with convolutional neural networks. It's going to maintain these really large episodic replays. And then it is also going to have these fixed targets. All right. So what they have here is they're going to do these series of convolutions, output. And they're going to output a Q value for each action. And they're going to use that to make decisions. And I think one of the things-- well, there's multiple really remarkable things about this paper. One is that they got extremely good performance across a really wide set of games. So instead of only having a few benchmark tasks, they looked at the whole suite of performance. They are learning a different policy per video game, but it is the same neural network architecture, and I believe, all the same hyperparameters too. So the idea with that is to say like, could we actually have the same type of architecture in the same way that we don't swap brains when we do different tasks, but have the same learning algorithm, learn to be able to do many different types of tasks? And so I think that was pretty impressive that they showed that that was possible. So you have the same algorithm, same hyperparameters, but it could learn to do well in many different tasks. I think one of the interesting things about the paper is to consider what were the aspects that were important for success. So here's just a subset of algorithms-- or sorry, a subset of the domains. This is a few of the games. And they also compared to using a much more simple function approximator. And what you can see here is that the deep neural network is not actually better, right? Like the deep neural network does not look better than the linear case. So it's not clear that just using a more function approximate-- like it wasn't just that they used a much more careful function approximator. And the second thing was whether they use this fixed Q, and that helped. So you can see now that they are exceeding the performance of using a more simple function approximator. So this idea of keeping things stable is helpful in terms of oscillations. But using the replay was incredibly helpful. So they went from 3 or 10 up to 241, or in some-- something from either roughly three times as good and sometimes even more like a couple orders of magnitude. So it was incredibly helpful to use an experience replay buffer. And maybe this isn't so surprising, because it means that they are just reusing their data lot. But it was really, incredibly important. And I think that's really helpful to motivate why thinking about sample efficiency and reusing your data is helpful. And then combining these ideas led to even bigger benefits. So it was helpful to have both the fixed targets and the replay buffer. But if you could only pick one, the replay buffer was just enormously helpful. All right. So as you guys know, there's been an enormous amount of interest in reinforcement learning and deep reinforcement learning since. There was some immediate improvements kind of within the next year or two. One is called Double DQN. And that also is a very simple change. It's maybe one or two lines. And it does increase some of the requirements. But for memory, but it is a really helpful approach. So it tries to deal with the fact that you can get some interesting maximization bias issues. And happy to talk about that offline. So there is a few different immediate next algorithms. But then there's been an enormous amount of work since. And I think it really led to a huge excitement in how we could couple these with really impressive function approximators. So just to summarize, the things that you should understand is to be able to implement TD 0 of Monte Carlo on-policy evaluation, so things like we talked about last time. You should be able to implement Q-learning, SARSA, and MC control algorithms, again in tabular settings. You should understand, what are the issues that can cause instability-- so things like function approximation, bootstrapping and off-policy learning, and have an intuitive sense for why that might be concerning. And then also you should know some of the key features in DQN that were critical. And then next week we're going to start to talk about a very different way to do things, which is just policy gradient methods. It is similar again to this. You can see how important policy iteration is. It's going to be similar to policy iteration and kind of similar to policy iteration of Monte Carlo in certain ways and directly trying to work with the policy. I'll see you then.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Offline_RL_3_I_2024_I_Lecture_10.txt

--up here asking you about DPO and RLHF. OK, great. Why don't you turn to somebody and compare your answers? [SIDE CONVERSATION] OK, so there's still pretty good-- there's a lot of disagreement on one of these. For the first one, it's false. Does somebody want to tell me which one does not learn an explicit representation of the reward function? So they do not both learn one. One of them does, and one of them doesn't. Which one does? Yeah. RLHF learns, and then DPO doesn't learn. That's right. Yeah, that's exactly right. So this one does learn. OK, so this is false. Now, it's true that DPO assumes a particular parametric representation for the award model. Both of them do. But DPO then inverts that. So you can directly do policy learning. It never has to explicitly learn a reward function in the same way that RLHF does. What about the second one? What do you think? Is it constrained to be as good as the best examples in the pairwise preference data? So I think this is false. Does somebody who also said false want to say, why is this false? Yeah. Maybe because we're using policy approximate, using a function to approximate it. So it could become-- Yeah, it could take a step, which is more positive than. Yeah, exactly [INAUDIBLE] I said. So you're going to-- at least if we think about the RLHF case-- we are using this information to learn a reward model. If that reward model is good even and can extrapolate beyond and generalize beyond the samples that we have, when you do PPO using that reward model, you can learn a policy that's better than your demonstrations. So this can, in fact, go beyond the best sort of performance that's inside your data. Or if you think of it in terms of the reward, maybe some of the examples you're showing aren't that great, but then you can use that to actually get a better policy. And in fact, you might think that's probably exactly what's happening with ChatGPT, because for ChatGPT, they initially got the fine-tuned model from supervised learning, and then they showed those examples to people. And people would pick between them. And then it learned a reward model, and then they got a policy that was better at generating those sort of responses. So you could argue that ChatGPT is an example that suggests, yes, this often can be true. We can learn a good enough reward model such that if we do PPO, at least a little bit of it, we can actually outperform the training examples. PPO and DPO does use a reference policy. Both of them do. And this idea will come up. We've seen it a few times already, and it'll continue to come up today. This idea of thinking essentially of how far can we extrapolate, or how far can we interpolate from our data? And when do we need to constrain ourselves to be fairly close either in the policy space or something else so that we don't generalize to parts of the domain where we might have really bad performance? We saw that in imitation learning. We saw that in DPO. We've seen that in PPO. In all of these cases where we're thinking, given the data that we have, how can we generalize as much as possible, but not further? All right. OK, so we're getting into a part of the class, which is probably my favorite part of the class, though I like-- of course, I'm biased. I like all of it. But we've been talking about learning from past human preferences. We first saw that sort of learning from past human demonstrations. Then we saw learning from past human preferences. And today, we're going to think just generally about learning from past data. So that could be generated by humans, or it could be generated by your robot or something else. And then next time, we're going to start talking about fast or data-efficient learning, and that's going to be useful for doing homework 3 as well, because the theory question for homework 3 is focused on data-efficient learning. All right, so we'll focus on that now. So in particular, for today, we're going to discuss, like we often do, thinking of separating things into a policy evaluation question and then a policy learning question, because we've seen repeatedly that if we think about can we evaluate how good a particular policy is, that we can often combine that as a way to bootstrap improving our policy optimization. All right. But I want to start with just a question, which is, can we do better than imitation learning? And of course, this relates to the question I just asked you in the refresh your understanding. So I'm just going to give up sort of an example. In my lab, we often think about education data or health care data or other cases where decisions are being generated by humans or automated systems where you might have, say, a series of patients. You could think of this as medical record data. And each of those people are getting a series of interventions. Maybe it's some medication. Maybe it's a medical checkup. Maybe it's a vaccine. And then we observe some sort of outcome. And in imitation learning, we saw the idea of saying, well, could we try to mimic the best human, or could we try to mimic expert data? And so an important question is whether or not we can go beyond that. And we just thought about one example where we might be able to go beyond that. But I think that there's a huge number of places we'd love to be able to go beyond the limits of at least the average human performance. Health care is certainly one of them. In America, we pay a lot for our health care, and we don't have particularly good outcomes compared to how much we are paying. So you would hope that maybe we could learn through reinforcement learning or others. Are there better sequences of decisions we could make in order to better assist, say, a new patient? So I'll just give a little bit of backstory of why I started thinking about this question. So about a decade ago, I was collaborating with Zoran Popovic and his lab and my grad students. He's at University of Washington, and he had this game called Refraction. And Refraction helps teach kids about fractions. It's one of the concepts kids typically find really challenging when they start to learn math. And so in it, you have a spaceship, and you're trying to fuel a spaceship by splitting laser beams in certain ways. So that you create fractions or subparts of laser beams to fuel spaceships and to save the-- save the agents. And in this case, so I think roughly around maybe 500,000 kids have played this game. And what we were thinking about is, how could we customize it to make it more personalized and adaptive to students? So in particular, there are all these different sort of game activities and game levels. And we wanted to understand, how could we use information about how the student was working in one of the activities to adaptively select which next activity to do? So this is a decision policy and you can imagine conditioning on all sorts of state features. So state features could be like how long they took. But it also could be things like where did they put down laser beams, or what series of mistakes did they make, you imagine? It could be generally a really, really rich context or state space. And then there were lots of different next levels we could do. So that was the question we were interested in. And in particular, in this case, we had access to about 11,000 learners who had been giving activities in a random order. Now, that was because there was a human designer who had designed a specific sequence through the game, but we weren't sure if that was actually optimal or close to optimal. And what we wanted to do is to see whether or not we could find using reinforcement learning an adaptive policy to help students persist at the game for longer. So this game was offered on something called BrainPOP, which some of you guys might have seen before. It offers lots of educational games for kids. And a lot of kids use it for a little while, and then they stop. So it's an optional game. And we had some evidence that suggested that if kids played the game, they were likely to learn things. But if they don't play the game, they are not. So we wanted to think about increasing student persistence in terms of the number of levels. And so we really wanted to go beyond expert performance in this case, like beyond what the experts had done. And so what we did is we used reinforcement learning, and we wanted to see if we could outperform essentially behavior cloning. And to give a spoiler of the types of ideas we're going to see today, in this case, we found we could learn a policy that increased persistence by about 30%. And so that suggests that in some of these domains, there may be essentially be enough data and evidence to find new decision policies that are substantially better than what is being currently done. And so that's what inspires me in my lab a lot, is to think about where can we use natural variation in the decisions that are being made or past experiments that were run in order to find substantially better decision policies than are currently being used. Yeah, not a super relevant question to the subject matter, but just out of curiosity, was the 30% distributed uniformly? Or was it just like the people who already played played longer, or the ones that stopped early would actually continue? This is a great question. So this is a really big challenge often is whether or not who are you moving inside of this distribution? So this is just an expectation, like most of what we've been doing for Q-learning and others. We did not analyze that too much in this case. In another much more recent paper we have, which I think came out in January or something, we did exactly that analysis to try to see who was actually impacted. And there, we were really excited that it was the lowest performers that were most impacted. And that was exciting because one of the big concerns is that a lot of these systems are just increasing the inequity gap. And this is particularly a problem in these optional ones because it's normally the kids that are furthest ahead that have the highest usage. So great question. But it also raises in terms of on the technical side, these questions around understanding-- sort of predicting estimates for subparts of the population and doing heterogeneous treatment effect analysis to figure out which groupings of contexts have different forms of Q values. Yeah. In terms of the policy, what was sort of being changed, like the simplest level, is it like the difficulty of the fractions? Is it how hard it goes as they're going up? Yeah, it's a great question. So in this case, I can't remember what the final exact policy was using, but the type of things that we're varying in this case is things around the fraction-- so like changing the numbers-- as well as different things of how tricky it is graphically to do that. So there was a couple of different things that we could manipulate as well as-- you can see, just like visually here, these look quite different. So one thing that we found in some other work in a game called Battleship Numberline, which I was excited because recently, my son uses BrainPOP. And it just popped up. And I was like, we worked on that. So that was exciting. In Battleship Numberline, which is another thing to do with fractions, we found there that variability was incredibly important for persistence. And so just changing how things look in that case, how big the battleships were, also makes a very big difference to persistence and engagement. I think that's actually an interesting question too, in terms of including the history and the state features to try to capture stuff like people caring about variability. So great questions. We'll talk a little bit more about this example later to talk about some things that we tried or that didn't work in this domain. But this is just to highlight that, I guess, we shouldn't set our expectations too low. So I think that imitation learning is amazing. And of course, if you're trying to imitate the best surgeons in the world, that's incredible. But there are many cases where we think we can go beyond human performance, particularly in cases where our high-level principles don't inform what we should do at a more micro level. So, for example, here we might have general principles of learning science. But it doesn't say which activity to exactly do when, and that's where it being data driven can be really helpful. OK, let me give you another example. Another place thing that we think about a lot is health care. We've collaborated a lot with Finale Doshi-Velez at Harvard and her lab. This is an example, thinking about hypertension and trying to optimize different policies for that. There is a really amazing data set called Mimic that comes out of, I think, MIT and MGH, Mass General Hospital, which has lots and lots of electronic medical record systems. And so what these guys did in this particular paper is to look at behavior policy-- so that's this flat line-- and to see if they could learn policies using a method called popcorn that they thought would be much better. And again, here the results depend on the method and some of the hyperparameters they're looking at. But the important thing just to notice here is that a number of these policies are substantially better than baseline, suggesting again, that there may be domains where we can leverage the intrinsic variability in the data and identify things that are working much more successfully in this systematic way. So when we think about doing this, generally, I would call this sort of offline or batch or counterfactual. And it's counterfactual because what we're trying to do is to estimate or learn policies that don't exist in the actual data collection strategy. So we have this setting now where we'll assume, like in imitation learning, that we have a data set of n trajectories. So we're going to assume now we're going back to the standard MDP setting. And it's not pairwise preferences. We're just back to having sequences of states and actions and rewards. So all right, so in particular, we may have things like this where we have data from one policy and data from another policy, and we want to think about how we can learn from that, thinking about the state distribution of what's actually best. Now, I'll just highlight here two reasons why this is hard. So we're always trying to estimate a counterfactual here over what might have happened that wasn't tried. So in this case, we don't know for this patient group what would have happened if we gave them that treatment or vice versa. So just a reminder, this is the fundamental problem of causal inference. And this is going to be a big challenge for us here, particularly when we try to go beyond the performance of the policy we saw in the past. So data is censored. And of course, in general, again, we're going to need generalization because we don't want to have to enumerate all the possible policies. And I do just want to highlight here that in addition to education and health care, you want to think about climate change or many other areas. There's just a huge number of scenarios, including robotics, because it's often really expensive to do robotics experiments where these types of ideas are helpful. Now, one thing you might be wondering about is, when I'm talking about this and I'm talking about trying to understand the performance of a new decision policy that was not used to gather the data, you might start to think back to Q-learning. There's a lot of work on off-policy reinforcement learning from really the very beginning of reinforcement learning. So you might say, why don't we already have the tools that we need to try to tackle this problem of learning better policies? And in fact, as we saw, ChatGPT, if we learn a reward function and we do PPO, that is doing off-policy learning. So that's one example. So why can't we do this? Why can we just do Q-learning or some of the other methods we've seen? One thing to remember is a little while ago, I said, sometimes we have this deadly triad of bootstrapping function, approximation and off-policy learning. That sometimes when we combine all three of these things. Things can fail. And that was part of the motivation for PPO, is that we don't want to go too far from the distribution. Let me just talk a little bit about what can happen here in the context of Q-learning sort of model-free learning. So this is the BCQ, Behavior Constrained Q-learning, from Scott Fujimoto. And what this shows here is that-- so these are a bunch of different methods. This is DQN, Deep Q-learning. This is behavior cloning. This is the behavioral policy. So what they did is they gathered some data, and then they tried to use different methods to learn a policy from it. And this is DPG. And what they found with this is that some of the methods did really bad, even given the behavior data. DQN does about the same as the behavior data. But what they found here is that by being a bit more careful and using methods that were explicitly designed to handle this offline data-- in this case, BCQ behavior, constrained Q-learning-- they could do substantially better. And so that suggests that we don't probably just want to use-- if we know our data is fixed, and we know we're not going to get additional data, that it may be worth it for us to use different types of algorithms in order to handle the fact that our data is constrained and we're not going to be continuing to get fresh data. So that motivates why we're going to need new methods. All right. So now what we're going to do is dive into policy evaluation, and then we'll talk about policy optimization afterwards. So in batch policy evaluation, what we're going to be thinking about is we have a particular policy of interest. And we have a data set, and we'd like to be able to use that data set to estimate how good that policy is for one state or on average over a set of starting states-- so similar to what we've seen for policy evaluation before. One thing I want to highlight-- this is by Phil Thomas, who I had the privilege of having as my postdoc a few years ago. He is a professor at UMass Amherst. We generally want to think about sample efficient methods for doing this. So in this case, he was working with Adobe, and they have 10 million to 20 million trajectories. It doesn't matter too much what these lines are. The key thing here is that this is the behavior policy, and you want to be learning policies which you're confident are better than your behavior policy. And this is just to highlight that depending on the methods you use, you may be confident at very different points, so just meaning that data efficiency and having good algorithms is going to matter a lot. Yeah. By behavior, you mean the policy that was observed in the current set? Exactly-- behavior policy. Great, great clarification question. When I say behavior policy today, what I mean is the policy that was used to gather the data set that you have. So I'll just write that out on here. So we're going to assume behavior policy is the one that was used to gather your data. All right. Let's first think about using models. So this is actually the first thing we tried to do with Refraction. We thought, OK, great. Travis Mendel, who is the grad student leading the project, we have all this historical data. Let's just try to learn models from it. So we're going to look at, and we're going to represent the state space in some way. And then under different actions-- here in this case, that's just different levels. And there's only a finite number of levels and activities. Let's learn a dynamics model. So in this case, the idea is that we have that existing data set, and we're going to learn an explicit dynamics model. And we can learn an explicit reward model. Now, in our case, our reward model was known because it's persistence. So we could essentially get a reward every time the student didn't quit the game, but we didn't know the dynamics model. And so that's what we're using the data to learn. Now, as you might imagine, we had to make a lot of choices here about what state representations you would use. And so we thought about lots and lots and lots of different state representations. But once you have that, then you can treat this as a simulator. So now you have your simulator of the world because you have a dynamics model or a reward model. Either you can do this analytically, like in some of the methods we saw in some of the first few classes, or you can use dynamic programming or Q-learning, Q-evaluation with those to explicitly learn what the value is. But really, can use anything. You can even use like Monte Carlo methods because you can try to learn from this simulator an optimal policy. So you can either try to learn an optimal policy or you can evaluate a specific one. You can do either of those. So I'll just write that here. So you can either evaluation or learn a new policy with any other oral method because now you have a simulator. OK, let me show you what happens. All right. So the first thing I'm going to show you is the following. What we have on the x-axis here is different state representations of this environment. So these are obviously really small state spaces. Like we don't actually think that human learning is encapsulated in terms of five states or 10 states, but you can just imagine sweeping this. So these are some of the state spaces we consider where we use really, really condensed state spaces or much more complicated ones. What this is showing here is normalized score, and this is log likelihood. And this is held out. So what this is saying is as you might expect, as you increase your state space complexity, you get a better fit on the data. You can better predict the next state of the student if you use a more complex state space. And that's not totally surprising, right. Because we think that human learning is complicated, and so we really think we are getting a better dynamics model. And again, just to emphasize here, this is cross validation. So this is on a held-out set. It's a not training error. So we're doing better in terms of this. Now, what are we doing with these once we have these? Yeah, go ahead. [INAUDIBLE] Yes. Yeah. So the data set size is fixed here. What we're trying to do is given the data that you've seen before, there are all different ways. We just have clickstream data. There's tons of ways to model that as state space. And so we're just doing model selection. Now what we were doing here then is once we had that simulator, we were trying to learn a good policy, and then we were evaluating the performance of that actual policy. Now I'll tell you how we actually evaluated that policy shortly, but this is the important thing. So this is saying that the models that we're getting are actually better. But here's the problem. If I take this policy, which really if I take this model, which really is a better model, it really does fit the data better. And then I do say dynamic programming with it, and I extract an optimal pi star. So that's the procedure. I take my model. I learn an optimal policy, and now I want to know how good that actually is in the real world. If I evaluate that in the real world, even though the model itself was actually better, what you can see is the actual value of that policy is getting worse. OK, so I've got a better simulator, but the policy I get by optimizing for that better simulator is worse. OK, so this is the actual unbiased reward estimator. And I'll tell you shortly how we do that because of course, under the model's opinion, the model thinks it's-- the policy it's helping produce is great. Let me just make sure that the pipeline of what we're doing there is clear. So what we do as we are getting-- we're going from data to a model of the dynamics model. And then we add in a reward function. And we extract a pi star for that estimated dynamics model. But that's just under the simulator. And then what I want to know actually is what the true value is of that policy I've computed. And what this graph is showing is that even though my model is getting better, the actual performance of the value I'm getting out is getting worse. Now, when we first saw this, we were kind of confused. We weren't quite sure why this was happening. And in fact, there had been some work a few years prior to this in the educational data mining community that suggested doing exactly what we were doing here, which was build a model, then use it to simulate and learn a good policy, and then deploy the policy that looked best. But what our work here suggested is that it was not a good idea. Now, the reason for that is because the model is misspecified. Now that means that under this model misspecification, the value it's getting when it computes the optimal policy. So you can think of there being two things here. There is one thing which is v hat of pi hat star, which is its own estimate of how good its value is. And then there is the true value of it. And these in general are going to be different, and these in particular are going to be different if your estimated model is bad. So it's going to think I'm doing great. This is going to help students persist till the end of time. But if the model is misspecified, meaning that even with infinite data, it will not converge to the true model of student learning, then that estimate will be wrong. And as you might imagine here, 20 state model of human learning is not that great. Yeah. [INAUDIBLE] 10 state. Yeah. Yeah. So it's not saying that-- it's not saying that some of these policies might not be good policies. What this was arguing to us-- so it's a great question. It's not that inside of these there might not be pretty decent policy classes. You could argue that education works because there's decentish policies. I mean, I don't have perfect models of all of you guys' learning, but it's still sufficient for us to be able to learn and communicate. What I'm arguing here is that we should not just use the accuracy of the dynamics model as like a proxy for which of the values or which of the policies to pick. This is arguing that we need separate independent estimates of really-- we want to basically in some ways kind of like what we saw with PPO and policy learning. We would like to directly evaluate the performance of a policy instead of using as a proxy, how much our Q-function is changing or how accurate we think our dynamics model is. Yes. So when we evaluate the policy, we execute it under the real environment or estimate the policy performance using our estimate. So there's two things. We can do it under our simulated model, or we can do it under our real model. We don't want to have to do it under our real model because we want to know which policy to deploy before we actually deploy it. Otherwise, we could kind of be doing online. So what I'll shortly be giving you is a way to get an accurate estimate of how good the policy is before we deploy it. I haven't said how to do that yet. I've just argued that using models alone might not be good. Do you have a question? OK, yeah. So about model misspecification is one way to think about this just like you're kind of overfitting your dynamics model by increasing the number of states that you represented? It's a question. We're not overfitting here because it really is a better fit. It's still just not a perfect fit. In other ways, you might say this is it's not realizable. This is not the real model of student learning. And under this, that means that there's still essentially significant bias when we do this learning. Now, one thing I just want to note is model-based learning can still be helpful. One thing that we may want to do in this case is explicitly build different models when we know we want to evaluate different policies. So normally, when we fit a model, we try to minimize the loss under the data distribution of the behavior policy. So if you have a bunch of data and you fit your dynamics model, you're essentially trying to optimize for the accuracy over your behavior policy. But if you know that the policy you want to evaluate is different, you can actually change. So you can weigh your errors separately. So this is a paper that we did a few years ago with [INAUDIBLE] and Omer Gottesman and others, which just highlighted this, that you could change your loss function and essentially up weigh your accuracy over the state and action pairs that you think you will encounter under a different policy. And that can help a lot. So you can see here this was for a medical domain. And what you can see is that this green here is ground truth. And what we found in this case-- so ours was our model here. And this is just if you fit for the behavior policy. And what you can see is by essentially reweighting your data, you can fit dynamics models that are much better fit the type of dynamics you'd see in the future. But now I'm going to introduce sort of model-free methods, and then we're going to get into importance sampling as other ways to try to do this policy evaluation that hopefully have different limitations or less limitations compared to the model-based method. So one of the first methods that I'll talk about here is fitted Q evaluation. So fitted Q evaluation is going to look pretty similar to deep Q-learning, but there's just going to be a couple important differences. So our data set here is a bunch of just different tuples of state action reward next state. Recall that our Q-function Q pi is just going to be the immediate reward we got from being in that si, a tuple-- so whatever we saw in our data set-- and then we'll put in plus gamma times V pi of S I plus 1. And then what we do is we try to minimize the difference between this under a parameterized function, just like what we saw with deep Q-learning versus the observed data tuples. So you can think of this as our target. And this is called fitted Q evaluation. It's closely related to something called FQI, which is fitted Q iteration, which I think was around 2015, 2005ish. And so this is very similar to what we've seen with deep Q-learning before. We just fit this function. The key thing here is that we want it to be for just a single policy pi. So we're not doing an argmax. So this is how the algorithm works for fitted Q evaluation. We sort of initialize our Q function randomly. It could be a deep Q Network. It could be something else. We compute the targets where when we put in the next Q, we have to use the policy we're interested in evaluating. So we're only doing this for the actions we would take under the policy we care about. We build our training set of Xs, actions, and our output Q target, and then we fit our Q function. And so, again, the key difference here compared to DQN is there's no max. We are fixing this part only for a fixed pi. But aside from that, it should look really similar. And so one of the-- so this was something that was very closely related to a common algorithm for doing off-policy learning, which is fitted Q iteration, excuse me, which is very related to deep Q learning. And one of the things people wanted to understand is whether this thing that was working in practice actually had some theoretical grounding behind it. Like, could we say anything formal about how good this approach was. So just to give you an illustration of the types of guarantees that we can get in this case, what we want to look at in this situation is to think about what is the generalization error. OK, let me put this in here. So I won't go through the whole paper. I just want to give you an illustration of the types of guarantees that you might get in this setting. What they would like to know in this case is to compare the difference between the value that you will compute under this procedure versus the true value of the policy. This is your normal discount factor. And then there's a whole bunch of things that are additional. Let me highlight some important things here. And here is the number of samples you need. So n tells you about how much data you're going to need in order to do this. So this is how much data-- much data. OK, epsilon is sort of your desired target accuracy. This is one of the really important things. So we're going to have something called a concentratability coefficient. Concentratability coefficient is going to be the difference essentially between the distribution of state action pairs that you have in your data set and the distribution of state action pairs you would get under your desired policy. So we saw this before with PPO of thinking about these divergence in the state action distributions-- state action distributions. And it's also related to what we'll call overlap later. So I won't go through all the details in this case, but I want to just give an illustration that people often think about trying to understand if you have a data set of some behavior data, how accurate you can hope to be of evaluating the performance of the policy depends on your discount factor, because that says how accurate you want to be and how much you care about long-term rewards, how much data you have in terms of your target error, and how closely related your state action distributions are from your training set to your test set or your desired policy. Now one of the challenges about this approach is that it generally still relies on the Markov assumption. So we're still assuming our data is all Markov, and it relies on our models. In these case, the sort of Q functions being well specified. So what do I mean by that? It means that we really can fit the Q function. There's some existing Q function in the world for our policy, and we can really fit it. And if you say, for example-- let's say that this is your state space. It's just one-dimensional. And this is what your true function looks like. You can imagine that it looks something like this. And let's say that you are restricting yourself to fit a line like that with just two parameters. So in that case, even if you had infinite amounts of data, you're still going to have a lot of error. You're not going to be able to fit the Q function. So these methods assume typically realizability, meaning that if you had infinite data, you could fit the function. The problem is that you don't have infinite data. OK, all right. So now we're going to see a really beautiful method called importance sampling, which allows us to deal with this. We've seen sort of brief ideas about this before, but I'm curious if anybody who's seen this in other classes. Who's seen importance sampling before? So just a couple of people. This is one of the favorite ideas in CS234 according to some past people. All right. So the idea what is the motivation? So importance sampling is an idea from statistics that we have imported over into reinforcement learning. Why would we like to do this? Well, we want a method that doesn't rely on the models being correct, meaning that we can actually fit things with a two-layer deep neural network or stuff, and that we don't have to rely on the Markov assumption in the state space we're using. We saw before that we could use Monte Carlo methods to accomplish this for online policy evaluation. And now we want to do this for offline data, meaning that we have data from a different distribution from the policy we want to evaluate. And the key challenge, as has often been, is data distribution mismatch. OK, so here's how importance sampling works. Let me just specify what this means. Let's say we want to try to understand the expected reward over a distribution of states. So for this part, you can just think of x is equal to states. And R of x is equal to the reward of a state. This works for very, very general distributions. But you can think of that here as just being a [INAUDIBLE]. All right. What we're going to do is the following. This is what we would like to evaluate. So you could think of this here as maybe being p of x could be equal to the probability of reaching x under policy. So you might really want this. You might want to know what is the expected reward I'm going to get under this policy where I know what my reward is for each state or I have samples of it, and then I have this probability distribution. The problem is that you don't have data from that. So we put no data from p of x. So that's the general challenge we're in. We want to see how well our alternative policy would work for helping students persist, but we have no data from that. So here's the trick. Let's multiply and divide by the same thing. I'm going to introduce a new policy and its distribution Q OK. So q of x is a different policy. This is a different policy. Maybe it's going to end up in different states with different probabilities. OK, so let's rewrite this. This is going to be equal to q of x times p of x over r over q of x r of x. OK, I haven't changed anything yet. This is exactly equal. But if I have data from q of x, I can approximate this excitation with samples. So this is approximately equal to 1 over n sum over I equals 1 to n of x sampled according to q of x, p of xi, q of xi, r of xi. This is super beautiful. What we've said here is that I really want to estimate the expectation of something over, say, policy, this policy p. I don't have any samples from p. What I can do is I can just take samples from my policy q, and I can reweigh them. So it says, if I was really likely to take-- to reach a particular x under a policy q but less likely under this one, I'll weigh that data less. If I'm much more likely to get to a state xi than I was under here, I'm going to upweight those samples. So this is beautiful, and it's unbiased. So this is an unbiased estimate. We'll extend it in a second to think about multi time steps, but just for single time step right now, this is how we can do this. It gives us an unbiased estimate. And as we'll see shortly, we can extend this to multi time steps. And we don't have to make a Markov assumption. OK, so this is a really lovely idea. So we can compute this expected value under an alternative distribution. And it is generally an unbiased estimator under a couple of assumptions. The first is that the sampling distribution Q-- so our alternative policy-- has to be greater than or equal to 0 for all x such that P of x would be greater than zero. What does that mean in practice? That means that if you could reach a state under your policy you care about with a non-zero probability. So let's say, I don't know. Your student could get to this particular level with non-zero probability under your target policy, then there has to be some probability you'd also get there under your training data set. This is sort of reasonable, right. So this says that if I want to think about-- I don't know-- a policy that like recommends restaurants versus coffee shops, I can't use that data to then estimate how good it would be to go to the movies, because I've never done that. For anything that we're trying to estimate here, we have to have non-zero probability for that x. The second thing is a little bit more subtle, but it comes up a lot in real empirical data, which is called no hidden confounding. And that means that essentially you have to know all of the features that were used to define this distribution. So this doesn't-- may not seem as clear in this part, but I think once we start getting into multi time steps and the sequences, it becomes really relevant. So let me give an example. OK, so imagine like a health care setting. So if we go back to that electronic medical record setting, we often are interested in what would have happened to a patient if we did a different action. So we want to know what that counterfactual is. One of the challenges there is that we will have certain features that are in our electronic medical record system. We will see an action, like someone was taken to surgery or some drug was administered, and then we see the outcome. In order for importance sampling to work, all of the features that were used to make that decision or pick that action have to be known. And that's called no hidden confounding. Now, why is that? Well, it might be, for example, you might see that there are certain patients that are sick, and then a particular action is taken. And maybe they die. And you might see other patients that look like they have the same features, and a different action is taken. And they live. And in that case, you might think, oh, maybe the decision was just bad. That's possible. But it's also possible that there are just hidden additional features that you don't have in your data. And that meant that the first person was much more sick, and that's why they got that particular treatment versus the other person. So it might be that there's important reasons that are not part of x that are being used and used to define what the action is in the data set. Excuse me. And in those sorts of confounding scenarios, and if you try to use importance sampling, you will not get an unbiased estimator. This is really important and really hard in practice. It comes up all the time. And in fact, one of the things we were just doing on a paper we just put online, we were trying to think really, really carefully about whether or not there would be additional confounding beyond the features that we had in our data set. So in that case, we had done an experiment to see whether or not offering students access to GPT-4 would increase or decrease participation in the class and exam scores. And only some people used GPT-4. A lot of people that were given access to it did not use it. And so an important question there then is, well, is there something intrinsically different about those students who were using it that also would confound their test scores? And so this issue of hidden confounding comes up a lot, particularly when actions are optional or being made by humans. Now, if you're in MuJoCo or something, this is easy because if you have control over the simulator, you don't have to worry about it. But it's important to know in practice. All right. Let's take a second and check your understanding. So we haven't really talked about bandits yet. Don't worry about exactly this. We're going to be doing policy evaluation. So let's say we have a data set for-- we'll just say samples-- for samples from three actions. OK, action one is a Bernoulli variable with probability 0.02. You get a really high reward of 0. The second one, you get probability with probability 0.55. You get reward of 2 else 0, and the third one with probability 0.5 get a reward of 1 else 0. Your data is going to be sampled from a particular behavior policy. So this is what we've been calling a behavior policy where with probability 0.8, it pulls this action. Else it pulls action two. The policy we want to evaluate pi 2 pulls action two. Excuse me, it pulls action one. This question asks you to think about what are true about the performance of those policies, whether or not we could use the data from pi 1 to get an unbiased estimator of pi 2 and whether or not the rewards being positive or negative might impact that. The third one is kind of hard and might require looking back at the equations on the previous slides. Water for a second and then [INAUDIBLE]. All right. Why don't you turn to a neighbor and see what you got? [SIDE CONVERSATONS] All right, let's go through this. So the first one requires a couple of nested expectations. So let's go through those and make sure I get my math right. So for the first one, pi 1, so there's two levels of stochasticity here. We have a stochastic policy, and we have-- we have rewards with stochastic actions-- or stochastic rewards. So let's first just figure out what the expected reward is for action one. This is equal to 0.02 with reward 100. So that is 2 plus else you get a reward of 0. So the expected reward for action a1 is two. I'll just write that as here. Expected reward for action one. We can do the same calculation here. So the expected reward for A2 is just going to be equal to 0.55 times 2, which is just equal to 1.1. And the expected reward for A3 is just equal to 0.5. So generally, policies that put more weight on action one are going to be better. Now let's see. Look at what the expected value is of pi 1-- so what pi 1 is. It's going to say with probability 0.8, we're going to get the reward of a3 plus 0.2. We get the reward of a2. The reward of a3 is 0.5. So it's 0.8 times 0.5 plus 0.2 times 1.1. So that's about how much that one is. So this is like approximately-- yeah, OK. So this one is like approximately like 0.42ish. I'll double-check the exact math. It's roughly like that. Now let's do it for-- so this is the reward for pi 1. We'll do the same thing for pi 2. This says with 0.5, it gets the reward, the expected reward for r of a2 plus 0.5. It gets the reward of a1. So that's equal to 0.5 times 1.1 plus 0.5 times 2. So it's approximately equal to 1.5ish. I think I was off by 2 when I was chatting to some people before. Yeah. I got 162 for first one, 0.65 for the second one. So I do my math wrong. I think this is-- I think it's going to be more than that because they expect your word for this one is 2. And so this one has to be-- Oh, 2-- OK. I thought you said it's 0.2. Yeah. Yeah. So I think this ends up being roughly 1.5. I can double-check my math, but I think that's right. So pi 2 does have true higher reward. So this is true. The second is we can't use pi 1 to get an unbiased estimate of pi 2. Why is that? So this is true also. Why can't we use pi one, data from pi 1? Because it never pulls. It never does action one. That's right. So it never does action one. So it's like saying you have data about all these restaurants. And then you ask it, OK, I also have a policy that's now going to go to this new restaurant, and you have no data from that. So we can't get an unbiased estimate of the average award. This one's hard. This is false. It turns out that you can still get an unbiased-- you can still get a lower bound on the performance of a policy using another policy which doesn't have complete overlap if the rewards are strictly positive. So if the rewards are always greater than or equal to 0, you can do this. Why is this? So we have a paper on this from a few years ago now, just for why this happens. Essentially, you can think of it as if your behavior policy doesn't include some of the actions that you want to evaluate, it's like putting 0 mass on those. OK, because if you think back to what is happening here, it's like you never sample them. So you have zero probability mass on some things that you want to evaluate. You want to include a policy that sometimes recommends movies, and you never do. So it's like putting 0 mass on that. If all your reward is positive, that's essentially just lowering your estimated value. So it turns out that if all your rewards are positive, you can use a behavior policy that doesn't have complete coverage with your target policy, but it will be a lower bound. The reason why that might be useful is because if it's still the case that your target new evaluation policy is better than your behavior policy, even though it might not have full coverage, you may still want to use it. So you're like, oh, it doesn't matter whether those recommendations it makes for those new movies is good or not. It's already a better policy. So we can do that. OK, great. All right. So it turns out that we can also do this for RL policy evaluation. So I just showed you a much more simple setting of this. And I'll highlight too here that importance, sampling like many things in stats and math, et cetera, goes by many different names. You'll often see things like inverse propensity weighting. So if you take econ classes, people often refer to these things more as like IPR or inverse propensity weighting. What I learned about them, I learned about them as importance sampling often also depends whether you're using these to design ways to gather data or whether you have historical data. OK, let's see how we can do this for reinforcement learning. So in reinforcement learning, we can do exactly the same thing. So I have what I want to have. These are now my trajectories. And as we've seen before, we can think of the value of a policy as just being an expectation over all the trajectories that could be generated by that policy from initial start state times the reward of those trajectories. So this is the reward of a trajectory tau. This is the probability of a trajectory under the desired policy. So what we can do in this case is the following. We can just multiply and divide by the same thing, like what we saw before. So we're going to imagine that we have data from a different policy. So I'm going to call this pi B. So I've now introduced my behavior policy. OK, so I'm just going to rewrite that so I can just have this weight. This is just reweighing what's the probability of me getting a particular trajectory under my behavior policy versus my target policy. OK, so we have that here. Write it out here first. So now we know. Let me put this from here. We know from before that if we have samples from our behavior policy, we can approximate this expectation by a sampled expectation. And we reweight these. The next thing is to make sure that we can compute what's the probability of a trajectory under our target policy versus our evaluation policy. And we've seen things like this before. So just remember, what we can do in this case is that the probability of a trajectory, given a policy in action, is equal to the product over equals 1 to the length of the trajectory, the probability, the transition probability. I'm just going to write it as a deterministic policy for simplicity-- deterministic for simplicity. But you can extend all of this times the probability that you would take that action s-- oops. Actually, yeah, I'll rewrite that. I don't want it to be deterministic. That will be misleading. I put it here. OK all right. So this is just the probability of us taking the action given the state under our policy and the transition probability for every single time step. The nice thing that we can see in this case, we can write that out for both the behavior policy and the target policy. And as we've seen in some other cases, this will cancel. So you don't need to know the dynamics model. So this is beautiful and incredibly helpful under similar conditions to what we just saw as long as you have coverage, which means that you will visit the same sort of trajectories, maybe with differing probabilities. All you have to do is reweight them so they look more like the policy that you want to evaluate. And we assume that this because this is just your policy probability. It just says, what action would you take in this state. And so this is known if you're doing policy evaluation. So this first introduced for RL, to my knowledge by Doina Precup, Richard Sutton, and Satinder Singh in 2000. Then there's been a lot of follow-up work and leveraging of this. It's super helpful. We don't need the Markov assumption or anything. OK, requires fundamental assumptions. It's unbiased, and it corrects for distribution mismatch-- so extremely helpful. I won't do this now, but you might want to look through this later just to think about, given everything you know about Monte Carlo methods, et cetera, like what might be some of the limitations of doing this? I'll just briefly say there's been a whole bunch of extensions. One thing is called per decision importance sampling. Similar to a policy gradient, we think about the fact that in terms of the reward and the decisions, later decisions that are made can't affect earlier rewards. So you can reduce the variance by being a little bit more strategic in where you put your weights. And we saw similar ideas to this in policy gradient. So this is called the per decision importance sampling, and it helps to have better properties in terms of-- particularly for long sequences. In general, the variance is pretty high, like for most Monte Carlo methods. One thing to know is there's concentration inequalities, like the Hoeffding inequality, that you can use that will generally scale with the largest range of the variable if you want to start to get confidence intervals over these values. And this can start to be pretty terrible for long horizons for importance sampling. So I'll post afterwards what the solutions are for both of these check your understandings, but it's pretty informative to think about exactly how bad this can become. OK to deal with this, there's a lot of different extensions. One thing is that you can if you do have Markov structure, you can think about state distributions instead of trajectories, and that can be very helpful. There's been a bunch of work in that direction. One work that we've done in others is called-- is taking ideas from statistics on doubly robust estimation and using these to reduce the variance in these methods, as well as trying to blend between methods that make a Markov assumption and methods that don't. All right. I want to finish now by talking a bit about how we can use these ideas and others to think about offline policy learning. So I think there's a couple important ideas we went through so far today. One is that you can just build a simulator from historical data, and you can use that to learn. But it may be biased, and that bias may be substantial when you're trying to use it to pick policies. We can do model-free methods, but we're going to want to be careful about those. And we're going to see more of that later. And you can use importance sampling to get an unbiased estimate, but it might be high variance. We're now going to think about those sorts of ideas in the context of when we're actually trying to pick a policy and do optimization. So I'm going to go back to this issue of coverage because I think it's important to emphasize. So let's imagine that you have antibiotics, mechanical ventilator, and a vasopressor, all things that are often used in an intensive care unit. And you might have different probabilities of these interventions. And let's say you want to evaluate a policy that frequently does mechanical ventilation. As we've been talking about, your data has to support the policy you want to evaluate. So if this is your behavior policy, that works because every single action you want to try, you have a non-zero probability of drawing that in the data. If you have this policy that doesn't work, that's the same as the example that we saw before. So if you never use a vasopressor in your behavior data, you cannot evaluate how good that would be in the future. Now, when I draw like this, or in the example that I gave in the check your understanding, it's pretty obvious because there's a finite number of actions, and it's pretty clear, if we didn't take action, the vasopressor action, that we can't evaluate it. But in real data sets, it often gets really hard to understand what does it mean to have sufficient coverage? So in general, this is going to be hard because we're going to want to say, well, is it OK? If it's 0, I definitely can't do it. But if it was like right here, is that sufficient? If like one in a million times I use a vasopressor, is that going to be OK? Does it have to be in my actual data set, or does it just have to be there was a chance of me doing this? So all these issues are kind of exactly how much data support you need come up. So up to around 2020, most of the methods for doing off-policy evaluation, kind of model based or model free, assumed overlap. So if you're doing off-policy estimation, it means for your policy of interest. But for off-policy optimization, it often assumed all policies. So every single policy you can imagine in your domain had to have coverage with your behavior policy. Now, if your behavior policy is random, that's fine. But if your behavior policy is, say, how physicians operate or how teachers operate or some sort of policy that's not completely random, that wouldn't always be satisfied. And in general, many, many real data sets don't involve complete random exploration. And this means if you assume this and use these methods and it's not true, then you might end up sort of going into parts of the domain or ending up taking policies that go into parts of the domain where you have very little coverage. So I'm going to introduce an idea. And it turns out this idea, there was a number of groups that all started thinking about this at the same time, and I'll cite a few others of them in a minute. We call this doing the best with what you've got. So the idea was, how can we leverage data sets where we only have partial coverage? Like we still want to do as well as we can, but within the support of the data. And this is similar to the KL constraint or PPO clipping that we've seen before, but this is all going to be entirely in the offline case where we don't manage to get any additional data. And the key idea that we're going to think about here is just being pessimistic so that when we don't think we have sufficient coverage, or we have high uncertainty over what the reward might be in a particular state or action, we want to be pessimistic with respect to that uncertainty. I want to highlight that just even when our paper came out here, there was sort of increasing interest in offline RL. But what we noted is that there was still quite a few challenges, and I just want to illustrate that with a really simple example. So this is known as the chain MDP, and we might talk about it more when we talk about data efficient exploration. This is not exactly the same as all the chain MDPs, but there's a number of them. And they're used to illustrate the hardness of learning good policies. So the idea in this setting is you have an initial start state S1 and then-- or sorry, S0. And then under one policy, mu, you have a probability of going to S1, S2, et cetera. And also with another probability, you have a probability of transitioning to S10. It's a really, really small MDP, just a very small number of states. The important thing to note here is that all of these states have deterministic reward except for this one. So in reality, this has an expected reward of 0.8. You always get 0.8 when you get to that state. And this one has an expected reward of 0.5. So it's a worse state. But if you go there because of stochasticity, some of the time, you'll get a 1 there, which means when you have finite data, you might think that state s9 is better than S10. That will just happen with your data. So what we could show in this case is that a bunch of the other algorithms for doing conservative batch RL-- and I won't go through all of them, but happy to talk to them offline-- had this weird behavior where as you this is your behavior data set. So as you increase the amount of your behavior data, you would hope in general that you get a better, better estimate of a new policy, and you get actually better and better performance. This is the success rate. And we found we had this weird behavior where for a lot of the other algorithms, they would start off, and they would learn the optimal policy for this domain, which is to go to S10. But then as you got more data from your behavior data set, they would get misled because sometime you would have seen S9, and it would have given you a one. And if it saw that, it would say no, I don't want to go to S10. I want to go to S9 instead. So with intermediate amounts of data, these other methods would get confused and learned a bad policy. And it was only as you started to get a lot more data that they would end up getting back and realizing what the best policy was. And so that was somewhat concerning, because you would generally hope that you get some sort of monotonic improvement as you get more and more data from your behavior data set. But here we were seeing that some of the previous methods had this sort of unfortunate behavior. And it turned out it didn't just happen in for these particular examples, but we could show some other types of examples where we got very similar types of performance challenges for other methods. So the key idea is pretty simple, which is just be pessimistic if you haven't seen a state action very much. So we defined a filtration function, which is just a simple threshold that says, let me check for this state and action pair. It's kind of what my density is, how much I've seen it. If it's greater than a threshold, this is going to be 1. OK, so what this threshold is doing is trying to account for your statistical uncertainty you have if you have finite amounts of data. So if you haven't seen things very much, this is going to become 0. If you've seen things a lot, it's going to be one. That's all we're doing. And then we can just combine this with Bellman backups. So just like for DQN or Belmont operator, we can just apply it. So that when we are looking at your reward plus gamma times your expected discounted sum of rewards, we look at the states you might get into. And if those states we don't have very much data for, then this whole thing becomes 0. And so it's like saying if I transition to a next state for which I don't have much data, I just pretend its reward is 0. And then I back up from there, which essentially means I don't want to take actions that transition to states for which I don't have enough data-- just pessimistic. And it's going to be a lower bound. If your rewards are all bounded by 0, it's just going to be a lower bound on your potential reward. So since we assume that our rewards are all positive and you can always just shift them, this is going to become a pessimistic estimate for all of those tuples. And you can do this for either policy evaluation or for-- so you can use this in like policy gradient type approaches or for Q-learning type methods. And it turns out this helps a lot. So let me just-- we call this marginalized behavior supported policy optimization. I'll just highlight what was-- because one of the key things of this paper was the theory that we showed with it. As I said, a lot of the previous methods had to make assumptions over coverage that like your data, covered any possible policy that you might want to evaluate. And under that, you could ensure that the policy that you learn is close to optimal. Ours does not make that guarantee. It only says, let's think about all policies that we could reasonably evaluate that have sort of enough coverage. We are guaranteed to find the best policy within that class. I'll skip through this now due to time. But under some assumptions that we can also give these kind of finite sample guarantees similar to what we saw for the fitted Q evaluation. All right. And I'll just highlight that those do include the function approximation. So these aren't for tabular. So this is what's pretty cool to see. So in this case, this is hopper. This is the behavior policy. So this is the behavior policy used to gather the data. What you can see here is if you use DDPG, that actually does worse than the behavior policy. If you use behavior cloning, it was a little bit better, about the same. It used a particular Vi. We then compare it to BCQ, which I mentioned briefly before, Scott Fujimoto's work, and you can see that and our approach in green both do substantially better. Again, highly in some of these cases, the data does support you learning a much better policy, and you should do-- you should try to uncover that by using these methods that explicitly think about your uncertainty. Now, I'll skip this just due to time. There's some interesting theoretical reasons why model based might be even better. At the same time, there were three papers that all came out at NeurIPS-- ours was one of them-- the same year with all basically very related ideas. Ours was a model-free based approach and work by some of my colleagues, Chelsea Finn, and [INAUDIBLE] and others, learned a model-based approach where they penalize model uncertainty during planning. And they had some very nice results in D4RL cases. Ours is a bit more theoretical and model free. Theirs was a little more algorithmic and empirical and also had some really nice-- and was focused on model-based approaches. I'll just highlight that another method that came out similarly around the same time was conservative Q-learning, and that has also continued to be very popular since. So that's another way to think about being conservative. We're almost out of time, so I just want to do share how do these different approaches compare. Pessimistic approaches in general do better than alternatives. All of these have some form of pessimism. These are model based. This is the sort of behavior constrained Q-learning, some nice work there from Sergey Levine's group from Berkeley and CQL, which is also from Berkeley. The different methods tend to do better or worse in different settings. I think that in general, the key thing to understand from this part is that it really can be beneficial to think explicitly about uncertainty and use that to penalize and constrain your function to be in the parts of the domain where you have support. And again, this is pretty similar. It should definitely make you think back to PPO instead of having constrained updates. So many of these different settings-- we're really trying to think explicitly about coverage and how far we can use the existing data we have. But particularly, here where we assume you don't get any additional data, you're just going to deploy a policy at the end, we want to think about exactly how much support we have. OK all right. I will skip the last part because we're going to be out of time. If you're interested, I just want to highlight that you can extend these ideas to think about there being constraints. So we had a science paper a few years ago thinking about what if you want to make sure your performance is improving compared to baselines. And in particular, we used like a diabetes insulin measurement simulator. It's a really cool simulator. It was approved by the FDA to replace early-stage animal trials. And you can learn new ways to do insulin delivery. And what we wanted to illustrate in this case is that by thinking explicitly about your uncertainty over the performance of new decision policies, you could quickly learn a policy that you were confident would be better than the existing policy. So I just highlight that to say that there are lots of cases where you'd like to do this offline policy learning, but do so in a way where you have safety constraints or constraints over the performance. All right. Let me just summarize this part. So in terms of things that you should know or be able to do-- excuse me. You should be able to define and apply importance sampling for off policy, policy evaluation and understand some of the limitations of these prior works. You should understand why offline RL might be able to outperform imitation learning. You should know this idea of pessimism under uncertainty and be able to have some application areas where you might want to be doing offline RL or offline policy evaluation. So particularly in high-risk settings, that can be important. What we'll be doing next is going to start to talk about how if we can gather our data, how we should gather our data in order to really efficiently learn policies? I'll see you on Wednesday.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Exploration_1_I_2024_I_Lecture_11.txt

Hey, everybody. Welcome back. We're going to start to talk about fast or data-efficient reinforcement learning. Before we do that, we're going to start with a refresher knowledge. One of the things that's fairly good evidence about in terms of learning is that spaced repetition is helpful, so I'll try to periodically bring up ideas that came up earlier in the quarter when we do the refresher understandings. All right. Why don't you turn to a neighbor and see if you got the same answer. All right. So we'll go ahead and get started. For those of you that just came in, feel free to vote. The first one is that importance sampling does not leverage the Markov assumption. Do not need this. So this is false. Important sampling can work with non-Markov systems, which is one of the benefits of it. It makes very little assumptions over the data distribution generating process. So it can be used in just similar to how we could use Monte Carlo methods to estimate the value of a policy through rollouts. This also is very general. Let's go through the next one as well. So let's now use the Markov assumption. For this one, the first one is true. So we can think of using the advantage function of one policy and samples from the other. The second is that we can importance weight between the two policies and get the samples from policy one. So it's not really an exact bound, but it turns out we can bound how off that is. And the reason it's not exact is because we're using samples of states from one policy, whereas in reality, the other policy might visit different types of states. And PPO uses these types of ideas. And the approximation error is bounded by the average over the states visited by one policy between the two policies. So this is trying to say, how bad is this approximation when we use just samples from one policy, OK? That was one of the really nice insights of that prior work, is to show you actually could bound what is the error in the app-- the approximation error that we induce by pretending that we'd get to the same states under policy two compared to policy one. Awesome. So last time we talked a bit about learning from prior data. And really, the last few lectures, we've been talking about how to learn from human feedback or from past demonstrations of people or historical data we have. And now we're going to switch, and we're going to think more about, well, what if we can actually gather that data? And of course, that's where we started at the beginning. We thought about-- if we are quite early on, we thought about how to evaluate policies, if we could gather data. But we didn't think a lot about how that data was gathered. We talked about epsilon-greedy, and we'll talk more about epsilon-greedy today, but we didn't think super strategically over the influence of the way we were gathering data. And so for the next few lectures, we're going to talk about that a lot. And that's really a critical part, particularly for online reinforcement learning. It's like, how do we actually gather the data we need in order to learn to make good decisions? And can we do this-- are there better or worse ways to do this? So one of the things I want to emphasize when we start thinking about this part of the course is, a lot of reinforcement learning, particularly if you have simulated environments, focuses on computational efficiency. So we think about things like any place where you have a simulator. So if we want to do Atari or if we want to do MuJoCo, in these cases, computational time is essentially the same as data, because you can either be using that additional computational time to sample from your simulator or to actually spend more time computing a Q function or policy. And so to some extent, simulators blend the difference between computational efficiency and data efficiency because it's all just computation. Like, you have a simulator and you can either give you data or you can use it to do Bellman backups or whatever else you want, but you could just count how much total resources you're using essentially in terms of computation. There are a lot of other domains where computation is really separate from samples, like, from actual data. So this is data. And these are a lot of the application areas that I tend to think about and a number of other people think about as well. So if you think about something like using mobile phones for health interventions, or if you think about consumer marketing, like, which ad to show to people, or you think about educational technology, or you think about climate. I'll do environmental policies. In all of these cases, there's a real world that's happening out there. There's a real students or there's real patients or there's-- where you're trying to decide, say, policies to encourage wildlife conservation or others. And so you have computers you can use to compute that policy, and then you have real world data. And the real world data I'll call samples or sample efficiency here, and you care often a lot about that real world data and ho-- squeezing the most you can out of it. So in particular, you might imagine that if you have, say, data from 500,000 patients or something like that, that's quite a lot, bless you, but it's not nearly as large as what you would normally have in the case of, say, Atari, where you can just run the simulator forever. Or in the case of things like AlphaGo, where again, you could just play the board game, go against each other-- against simulated agents forever. So a lot of the things we're going to be talking about over the next few lectures are just going to assume that we care about this because we can't get infinite data. So we have to be-- we're thinking about cases where like, these are coming from patients or they're coming from students. And so we want to be much more careful with the data we're gathering. And think about how we could maximize the information we get out of those to try to make good decisions. So when we start to do that, there's a number of different things we want to consider in terms of how good are the different algorithms we're going to consider? So one thing might be that if it converges at all-- and we've seen before that from the deadly triad, we're not always guaranteed to converge. So we've seen that for some settings where this is not guaranteed or it hasn't been proven yet. So you're not even necessarily guaranteed to converge to anything. It might chatter. It might oscillate. It might not go to anything stable. Second question you might ask is if you're going to be guaranteed to converge to the optimal policy. And then a third thing that might be really important is, how quickly? So in this case, it's going to be, how much data? And we're going to see different types of measures to evaluate different reinforcement learning algorithms. So let me just give you an illustration too of why these things might look quite different. So imagine that you have something like this where this is time, where this is a reward, OK? So you have really different algorithms. You could have algorithms that look like this, that might be one algorithm, or you could have an algorithm that looks like this, really smooth, and you could have algorithms that in general, maybe most of the time do great, but periodically make terrible mistakes. Versus could have another algorithm which never does awesome, but is always pretty good. And those are really different types of behavior. So if you think about that in terms of, say, an AI clinician, you could have an AI clinician that on average helps, like, let's say, 80% of your desired outcomes. Like, it helps you manage your blood pressure with 80% accurate fidelity. Or it could be that for 80% of the population, it helps them completely manage their blood pressure. And for 20% of them, it fails. So those are really different types of performance guarantees, and we'll think about whether trading off between those and what sort of algorithms guarantee us to have different sorts of performance. So we'll start to introduce different types of settings and ways to evaluate the quality of algorithms, and we're going to start with bandits. And we've talked very briefly about bandits in the context of ChatGPT and preference learning. We'll talk a lot more about them now, and then we'll move back into the Markov decision process case. A lot of the ideas from bandits will turn out to exactly or quite easily translate over to the RL setting. OK. All right. So let's dive in. So what is a bandit? So a bandit is a really, really simple RL problem. They've been studied since, I think at least, like, around the 1920s. There's a very long history of research on multi-armed bandits. It's been used for all sorts of application areas. So let's describe what it is. So the idea in this case is that there's no states. There's just a finite set of arms. And arms are the same as what we've been calling actions before. So as a concrete example, you might think of there being like 20 different ads you could show customers. And we're going to assume that there's a probability distribution over rewards for each arm. So maybe on average, this gives you 90% click through rate for this particular ad, and this other ad gives you 20% click through rate. But that's not known. That's not observed. And what will happen is that each time step you get to select one of the actions, and then the environment will sample a reward from that stochastic variable. So if the click through rate is 90% for that particular arm, most of the time, you'll get a 1, and sometimes people won't click on it. And the goal is to maximize your cumulative reward. So overall time steps that you get by most amount of, say, clicks. And this is a very simple setting, but it's been used extensively in a lot of areas. You could think about this for something like, how could I-- if I was doing something like a clinical trial, how might I randomize the next person over what treatment to get, a treatment or a control, for example, for ads, for many, many different types of application areas? So I'm going to go through-- I'm going to have some running examples for this part of the course, and we're going to have a sort of a silly one that's going to be illustrative. So let's imagine that we're trying to treat patients with broken toes. This is nothing to do with medical stuff, so this is not medical advice. Imagine you have three different options-- you could do surgery. You could buddy tape the broken toe to another toe, or you could do nothing. And your outcome measure is a binary variable about whether or not that toe is healed or not healed after six weeks. So that's our setting. We've got broken toes. We want to figure out, what's the best strategy for healing them? And we're not going to do a clinical trial. Instead, we're just going to say, well, sometimes people come in and they've broken toes, and I'm going to try to figure out over time, which thing is best. All right. So in this case, we're going to model it as a multi-armed bandit with three arms. The arms are the-- and we're going to model each arm as a Bernoulli variable with an unknown parameter theta I. So let's just do a quick check. Your understanding about the framework of bandits. OK. Great. I think most people are converging on these already. Yes. Pulling an arm or taking an action is just the action we're actually doing. The second one, this is a better fit to the problem than an MDP because we're only going to make one decision per patient. And we're also going to assume that whatever decision-- whether [INAUDIBLE] toe heals after we do this is independent of whether or not when Sophie shows up, what we do. So these are totally independent processes. The next person to show up. So we don't have any sequential dependence, even though we're making a sequence of decisions. It's like, each time point, there's a new person. We're just going to decide what to do for them. And yes, this is right. So if your theta I is between 0 and 1, meaning your outcomes are not deterministic, sometimes you'll heal, sometimes you won't. OK. So one thing that we could do to solve this would be to use-- yeah. So to confirm there is no time point dependence you have to probability distribution. It has to be the same in every single [INAUDIBLE]. Great question. We're going to assume for now that everything's stationary, meaning that reward probability distribution is the same at every time step. So there's lots of really interesting questions around nonstationarity. Our labs don't work on that. There's lots of other really interesting work on this, like with time point detection and change points. For now, we're going to see about stationary. And that would include the fact that we don't suddenly get a new distribution of people for whom different things work. Good question. All right. So one thing you could imagine doing is just to be greedy. So what we're going to do in this case, we're going to use Q today not to denote a state action or discounted sum of future rewards, or you can think of it like that, except for there's no state, there's a single state. And it's only over actions and it's only the immediate reward. So what Q here would denote is what is just the expected reward of RMA? And we can just estimate that by counting. We can just look up every other time we did surgery, what were the outcomes for that individual? And we can average. And what the greedy algorithm does is they just selects the action with the highest value and takes that action, observes the outcome, and repeats. So let's think about what happens when we do that. So if you have this setting-- imagine that this really is the true set of parameters. So surgery, in this case, in our fake example, is actually the most effective buddy taping the second, and doing nothing is not very effective. So imagine this. So you start off-- and this is pretty common with a lot of bandit algorithms. If you have a small, finite set of actions, often, you'll just start off and you'll sample everything once. Now, when you start to get into really large action spaces, like, all of the ads you could recommend to people, we'll have to do something smarter. But in this case, you can just sample all the actions once, and let's see what you would observe. So in this case, imagine that you get the first observation here is 0 for arm one, it's 1 for arm 2, and 0 for arm 3. So which arm-- this is not meant to be tricky. Which arm would you select next under the greedy algorithm? And which of them has the highest? [INAUDIBLE] Great. Exactly. So you would just-- there would be-- deterministically, the probability of picking a 2 would be equal to 1. You're just determined to take whichever one looks best. So would that be good or bad? Bad. And in particular, would you ever select the optimal action? No. So you actually couldn't-- so it will never find it, because you have a really low estimate of the true value of these two. Your average for a2 can never drop down to 0, because you've got at least 1, 1. And so even if you get 0s forever, which you're unlikely to get two for a2, you're never going to sample a 1 again. So what we would say in this case is that this means that you will not converge to the optimal action, and this algorithm is not very good. And we'll formalize what we mean by not very good in a second. So this just is to illustrate why you should not just be greedy, that you can lock on to the suboptimal action forever. This highlights why you need to do some form of exploration because you can, in fact, make an infinite number of bad decisions. So how do we quantify what it means to make an infinite number of good or bad decisions? We're going to use the word regret. And we're going to mean regret in the case of sequential decision making, OK? So the idea in this case is that we're going to think formally about, what is the difference between the decisions that our algorithm makes and the optimal decisions? And then we're going to score the algorithm based on what the gap is. So in particular, the optimal value, just like what we've seen in the past, is the maximum overall, the Q value, so it's whichever arm has the best, highest expected reward, and the regret is the opportunity loss. You could also think of this as the difference-- is the advantage. The advantage of the optimal action compared to the action that's taken. And so your regret, just like we often use it colloquially, is the gap between what the agent could have achieved and what it actually got. We're going to focus here of looking at these in expectation. Of course, due to stochasticity, there could be times where the particular reward you get for a suboptimal action might be higher than the action-- the reward you'd get for the optimal action because of stochasticity. But we're just going to focus here on expectations. So we're always comparing the expected reward of the optimal arm to the expected reward of the suboptimal arm. So that's regret. So how do we compute it? We're going to think about comparing this over all time steps, and we're going to maximize cumulative reward, which is equivalent to minimizing total regret. Because remember, this is unknown, but it's fixed. So we really want to maximize our total reward, and we can either think of that as you're maximizing the Q you got over all time steps or we're minimizing the total regret. And normally in bandits, we talk about minimizing total regret instead of maximizing total reward. All right. Let's see how we can think about how big the regret will be. So let's let Nt(a) be the number of times action a has been selected at time step t. So that means that if your agent has made t decisions, you count up and see, how many times did I take action a1? How many times did I take action a2? How many times did I take action a3? The gap for a particular arm is essentially its advantage of a star over a. So it's just the difference between what is the expected reward the optimal action would have gotten minus the expected reward you get under this alternative action? And we often call this the gap. I think the literature developed somewhat independently, and so I think that's why people don't commonly call it the advantage. In the case of bandits, they typically call it the gap. And the gap will turn out to be pretty important, because as you might start to think about intuitively, depending on the size of the gap, it's going to be easier or harder to learn which of two actions is better. So if the gaps are really large between action 1 and action 2, which means they have really different expected rewards, you're going to need less samples to figure that out. If the gaps are really, really small, generally, you need a lot more data, OK? So again, it's going to just be a function of the gaps and the accounts. So we can just think of the number of times that you took each action and the difference between-- and this gap. The difference between the optimal action you should have taken and the reward you actually got. And so our expected regret here is just going to be the sum of times you take each action times the gap. And so what that means intuitively is that we do not want to take actions which have large gaps very much, and it's more OK if we take more of actions that are close to the optimal action. And a lot of algorithms-- for a lot of algorithms, what we try to do is we try to bound this quantity. So we try to say, in advance-- in general, this is something that we can't know because this requires access to what is ever is the optimal action and its value. And we don't know either of those things. But what we can do is we can have algorithms where we can prove something about how the regret grows. OK. All right. Let's just see what I mean just to instantiate that. OK. So again, we can't do this in the real world, but we can do this for a toy example. Let's just think about what the regret would look like in this case. So this would be a series-- if you were running your greedy algorithm, so this is the actions. This is time. This is 1, 2, 3, 4, 5. So we first take all of our actions. In each of those cases, the true optimal action was a 1, and our observed reward was 1 and our regret was as follows. So a1 really is the optimal action, so we have 0 regret there. The second one, our regret was this. And for the third one, our regret was this. So this just shows you what the size would be. And so this here is actually the gap. It's the gap between the optimal arm and the arm that you're taking. So this just shows you how regret can grow. And as you might expect, if you make bad decisions forever, you're going to get linear regret. So for example here in the greedy case, if we now take a3 forever, our regret is going to be the total number of time steps t times 0.85, because that's how much we're losing for every single decision, and then we sum them all up, OK? All right. So in general, it can linear in the number of decisions. And so part of-- the main thing we're going to be trying to do is ideally, you would have constant regret or zero regret. What I mean by constant regret would mean that you make a finite number of bad decisions. So if you can figure out what the optimal arm is and then take that forever, then you'll have constant regret, because it just is going to be, say, I make 10 decisions, then I learn the optimal arm, and then I make the optimal thing forever. That's generally pretty hard to do. In the worst case, you'll be linear. You'll make a mistake on every single arm decision forever. And typically, what we're hoping to find is we're hoping to have sublinear regret. So it still might grow with the number of time steps, the number of decisions you're making, but it's not going to be linear. OK. And we'll see a lot more about that. OK. All right. So what we're going to think of next is-- we've seen these before, the epsilon greedy algorithms. So let's think about what sort of regret epsilon greedy will have. We've seen that greedy can be linear. Now, let's see if there's some better things we can do. OK? So in this case, we're going to do just to refresh our memories. With probability 1 minus epsilon, we're going to select-- we're going to be greedy. We're going to select whichever action is the arg max. And otherwise, we're going to select a random action. And that means that epsilon amount of the time, we're going to be making some suboptimal decision. Because unless all of your arms are optimal and your gaps are 0, in which case it doesn't matter what arm you're picking, if you select things at random, you're always going to be making some bad decision at each time point, OK? So what does this look like? So what this would look like in this case is, imagine, again, we sample all three arms to start. This is our epsilon. I'm just going to work out what it will look like. So with this case, we're going to-- 90% probability, we're going to be greedy. And in that case, we will take action a1 and a2, each with probability-- assume that you split ties, 45% probability. And then with 10% probability, we will take all of the other actions. So we'll have 3.3% a1, a2, and a3. So that's just to be concrete about what that would look like in this case. I'll skip through this. So the question here is, what will this regret look like? So now we want to try to compute this for epsilon greedy to think about whether we'll have sublinear regret for epsilon greedy. OK? All right. So let's assume that we're in a setting where there always exists at least one action such that the gap is nonzero. That means that not all arms are tied. If all arms are tied, again, doesn't really matter what you do because everything is the same. And so it doesn't matter what action you take. So this makes it a nontrivial decision making problem. So let's think about it in terms of our thing, whether or not epsilon equals 0.1 can have linear regret and whether epsilon equals 0 can have linear regret. As in, this is generally trying to think about, are there settings of epsilon for which you could get linear regret and maybe settings of epsilon where you couldn't? I don't know if this is actually on the-- there are [INAUDIBLE]. You don't know what? I don't know if this one is actually on the post. We've answered all three different. I wonder if that was missed. All right. Well, if it's not on Ed, fell free just to-- I have it. Oh, OK. I can check and see-- I wonder if something's got missed in the last one. Ooh. OK. Hold on. I'll post those in. But feel free just to think for a second, and then I'll ask you to talk to your neighbor. Let me see. I think something's got mangled. Which ones are mangled? This one. OK. I think this should be there now. You can check. I just updated Ed Don't tell me that didn't work. Does that work? Looks like? Great. I think most people agree on this, but maybe we'll just do one minute and check with the neighbor and just check you got the same thing. All right. I'm going to interrupt you for a second. So I think one way that's useful to think about this is when we think about how many times we sample things, all of the arms are going to have a lower bound on the number of times we sample them, which is at least epsilon divided by the number of actions times t, where t is the total number of decisions we make. And so I think that can be a helpful way to think about this, that you see there's a T here times some constant. Because there's a big T here times a constant, that means you're going to have at least linear regret. So if epsilon is greater than 0, you will have linear regret. And if epsilon is equal to 0, you're greedy. And we just saw that that can have linear regret. So in either of these two cases, unfortunately, both of these-- both are true. Somebody have any questions about that? Now, it turns out there are certainly better and worse ways of setting epsilon. But if you just set epsilon in a static way, it can be pretty bad. And as you might remember from a long time ago, sometimes we talked about decaying epsilon over time. And so that can matter a lot, too. But static epsilon is not great. All right. So let's look at what this can look like. If you think about how regret is growing over time steps, these are very common plots. When you look at bandits or some other approaches, we'll see if we consider what total regret is, you'd like a regret to be 0. If you make it greedy, it can be linear. If you make it epsilon greedy, it's normally a little bit better, but it's still linear. If you decay it, it can get a lot closer. And it is going to be possible to be sublinear for good choices of algorithms. One of the challenges for this is that it can turn out that there can be some pretty good choices of epsilon, but it often depends on dependent properties that we don't in advance. So we need to have an algorithm, which before knowing how anything about the problem in terms of the gaps or anything like that, can be guaranteed to have sublinear regret. So first of all, let's think about what type of regret bounds we might get and is there reasons for hope? So a problem-independent bound talks about, how does the regret grow as a function of t for any possible problem that you might be given? So what this might say is, I might give you an algorithm which is guaranteed to be sublinear in t no matter what bandit problem you put me in. So that's just an algorithm that will work well for any potential domain you put in, and it'll make a bound on its performance. Instance-dependent or problem-dependent bounds bound things as a function of the gap. And one of the really elegant things of problem-dependent bounds is that it doesn't mean the algorithm has to know the gaps. It just means that if it turns out the problem is easy, like, there are really large gaps, you will have a much better regret. And so some of my labs work and a number of other people too, were often very interested in this. And I think at a high level, what this means is you have an algorithm that's adaptive to the problem. So it means that your algorithm will be guaranteed to do really well on the problem if the problem is easier to learn in. And if it's harder, well, then you can't do well anyway. It'll do as well as it can. So we'll talk about bounds that are both types of these. In general, is the gap usually less than the value if we're considering only the words? And we usually consider only rewards between 0 and 1? Great question. Totally depends on the domain. So if you're looking at Bernoulli's, then it's naturally between 0 and 1. Other domains might be very different. You can always normalize it. I think whether the domain has really big gaps really depends. So if you think about something click through rates for ads, click through rates are really, really hard to optimize. It's often like 0.01 versus 0.011. Nobody likes ads. So in those cases, the differences, the gaps we're looking at, could often be really tiny. And so you'll generally need a lot of data, and having smart data efficient algorithms will matter a lot. There might be other cases where there's really big gaps. If the problem has really big gaps, it's really easy. And so it tends to not matter too much what you do there, because you can quickly estimate them. Great question. OK. All right. So here's a reason for hope. So there's a nice lower bound by Lai and Robins-- I think this was around 1950s, it's been a long time, which tries to think about, what's the minimum regret you're going to get as a function of the problem? And so this means that any algorithm is going to suffer at least this much in terms of regret. So it says, you're going to at least be log t, like, the number of time steps, number of decisions you've made. And for any arm for which it is suboptimal, you're going to suffer this in terms of a KL difference between your distribution of rewards you get on your arm on that arm versus the optimal arm and with the gap on the numerator. But this should be promising because it's sublinear. It's log. It's not linear. Which means that the lower bound says, according to this, it is not yet impossible to try to have sublinear regret, OK? And this would be considered a problem-dependent or incident-dependent bound because this holds based on the unknown gaps. OK. So now we're going to see one of my favorite ideas in the course, which is optimism under uncertainty, which gives us-- I think it's a lovely principle because it shows why it's provably optimal to be optimistic about things, which is beautiful. And it's going to be one of the first things we're going to see that's going to allow us to have sublinear regret. OK. So why is optimism good and what do we mean by optimism in this case? What we mean is, we're going to choose actions or arms, some typo there, that might have a high value. Well, what happens when we choose things that are good? So one thing that can happen is we actually get high reward, OK? So that's good, because that's our goal, because we want to get high reward. We want to maximize reward/minimize cost. What's the other thing that can happen if we pick something that might be good? Might have high reward. Low reward. Low reward. Exactly. OK. Yeah, those are the only two things, you can either get higher or you can lower award. What happens if there's low reward? I mean, of course, there's that. But aside from that, what happens, do you think, probably to our estimates, those Q estimates if we get low reward? Yeah. [INAUDIBLE] Exactly. Yeah, exactly. Remind me your name. Yeah. So what you said is exactly right. So basically, either you get high reward or you learned something new, OK? So the other alternative is you get low reward and you learn something, and you're going to improve your estimates. And from the point of view of a reinforcement learning algorithm or abandoned algorithm, both of these are really valuable. Because either you're actually achieving your goal or you are learning something so that in the future, you won't make bad decisions in the future, OK? So that is why optimism is-- we're going to see provably optimal, OK? All right. Now, of course, that means that we have to have an algorithm that leverages the information we get when we see low rewards. So we're going to have to be formal about what it means to might. We're going to formalize this as quantifying our uncertainty. So we're going to need to be precise over our confidence intervals or uncertainty bounds, and then use that to make decisions. OK. So in particular, what we're going to do is we are going to estimate an upper confidence bound for each action value such that that confidence bound-- that upper confidence bounds holds with high probability. So we're going to make sure-- we're going to be frequentist today. We're not going to be Bayesians. Don't worry if you haven't done a lot on either of those. But we're going to focus today on just high probability bounds. So we're going to need a Ut of a, where that holds with high probability, and we're going to want this to be dependent on how many times we've selected the arm. There are lots of ways to quantify uncertainty. We're going to focus today on a frequentist view and just thinking about counts. And then the way we're going to behave, the way that our agent is going to take actions, is just going to pick whichever action has the highest upper confidence bound. And there's a whole suite of algorithms that are called UCB algorithms. So there are many algorithms that are variants of this notion. There's also ones that are called Optimism in the Face of Uncertainty, OFU, OK? So it's a really simple idea. And now, the question is going to be how well does this perform and how do we quantify the uncertainty? So let's go through Hoeffding's inequality. We're going to use it in homework 3, but I'm curious who has seen it in previous classes. OK. Maybe a couple of people. But most people, I wouldn't expect you to know. So Hoeffding inequality is a really useful inequality. The idea of it is we're just going to think about, how different can our observed average be from the true mean? So let's say we have n samples that are somewhere between 0 and 1. And this is our true expectation, so is their true mean, which we don't know what it is, and this is our sample mean, just over the n samples. What Hoeffding's inequality says is that the difference between your empirical estimate and the true estimate, if they're off by U, then the probability of that happening is going down exponentially. Which essentially means that as you have more data, the chance that your empirical estimate is really different than your true mean is going down exponentially fast. If you can't have your empirical average be 30 and the real thing is 2000, if you have a lot of data, you're going to converge on the true mean. Which is, of course, what you would hope, but that this is a formal thing about what the rate is. So let's just look for a second and think a little bit about what this can imply. So let's look at this part. Let's say-- I'm going to do it for the absolute, value probability of E of X minus Xn, so this is, again, just our empirical mean. The probability this is greater than mu. So this gap between our empirical average and the true one is. And so just to back up, why are we doing all of this? We're going to want to figure out a way to get an upper bound on what the real mean is of this. And so what this equation is going to allow us to do is to try to figure out, how big do we need to set U to be in order for us to get an upper bound on what the true expected reward might be for a particular arm? OK. All right. So let's see how we can do that. All right. So we're going to say, this is less than-- I've got an absolute value here, so we're going to use this version. OK. And we're going to set this to delta. So this is going to be the confidence with which we want this confidence interval to hold. So this is going to be, want the CI to hold with this probability with 1 minus delta probability. So we're going to try to construct an upper confidence bound that holds with least probability 1 minus delta, OK? So let's just do this. And now we're going to focus on this hand, this side. So we're just going to do some algebra. Let's go to delta over 2. That means U squared is equal to 1 over n log of 2 over delta, which means U is equal to square root-- OK. So this gives us our range, and it says if we want the probability that our empirical estimate differs from the true mean by no more than U, then it is sufficient to set U equal to this, OK? So that means that we can say that Xn minus mu is less than or equal to expected value of x, which is less than or equal to Xn plus U with probability greater than or equal to 1 minus delta. So that just created our upper confidence bound. So they said with high probability, I can take my empirical estimate, I added my mu-- my mu here note, just depends on the number of samples that I have, and that gives me my upper confidence bound. So we can use this. We can use this given our data. It just requires us to count how many times we've sampled things, compute the average, and then add on this additional bonus. We often call these bonus terms in these cases. Sorry. So this is going to create the UCB1 algorithm. Which is at every time step, we're just going to compute-- this is again-- remember, the Q hat is this the empirical average. And then we add on this bonus term. OK. And this is, again, just the number of samples of a after t time steps. OK. And for those of you familiar with things like union bonds and stuff, we'll come to that shortly. So this is-- we haven't really fully made sure that all of these confidence intervals are going to hold over all time steps, so we'll be a little bit more careful about what delta needs to be soon. Yeah. It's called UCB1. Like, why is it 1? [INAUDIBLE] There's a lot of different variants of the UCB algorithm. I think this is one of the first ones. It was, I think, Auer, U-E-R 2002. I think it's the one they named first in their paper. OK? But this notion of optimism under uncertainty is certainly around before the 2000. But I think this is the paper where they first did some of these nice proofs. OK. All right. OK. So let's think about how different that algorithm would look like in our types of settings, OK? So we're going to use optimism under uncertainty. And what we're going to do in this case is, we're first going to sample each arm once, so same as before, and this is what we're going to get. And now what we do is we're going to compute those upper confidence bounds, OK? So what we want to do is compute this upper confidence bounds for each of the arms. So UCB of A1 to A3, OK? And so this would be 1 plus square root of 2 log or delta over 1, same for this one, and then 0 plus square root 2 log 1 over delta. OK. So in this case, you would pick a1 or a2 with equal probability because the upper confidence bound is identical. OK. So we select the arg max. Let's say that we pick a-- OK? And now we're going to, again, compute the upper confidence bound. So in this case, what would happen is you would still have-- you'd have the following. You would have UCB of a1 is equal to 1 plus square root 2 log 1 over delta divide by 2, UCB a2 is equal to 1 plus square root 2 log 1 over delta over 1, and UCB a3 is equal to 0 plus square root 2 log 1 over delta divided by 1. So you can see here is that we've now reduced our upper confidence bound, because we've learned something new. Now, in this case, we happen to have also gotten high reward. But either way, we learned something new. We could shrink our confidence intervals because we have additional accounts. Just to make sure I'm understanding correctly, the delta is something that we would select to figure out or to choose our confidence bounds? Yeah, great question. So, yes, we haven't talked a lot about how we set delta. Going to be a couple criteria for it. In general, we're going to need all of these confidence bounds to hold for all time steps for all arms. So we're going to need to do some union bounding to make sure all of them simultaneously hold, because we want to have it with high probability, that all of these things are valid at the same time. In the simplest setting, we know how many total decisions we're making, and so we need to use that information as well. And then you can use those two things to bound the regret, as we'll see. So you can see, this is why it's a bit different than greedy. Because we are still using our empirical averages but then these confidence intervals are going to change so that over time, these will alternate often, depending on what rewards you're getting, and you may periodically take a3. Because with that little data, there is some probability that a3 is just as good as a1 and a2, particularly after you get additional data. So we'll alternate between the arms based on these upper confidence bounds. OK. Let's go ahead-- let's skip through those here. Let's go to here. OK. So this is-- we're just asking-- it's a little bit subtle. If you have a fixed number of time steps, like, you know the total you're going to make, like, T decisions, you can set t to be roughly-- you probably want this divide by a. This is because you could use a union bound. So why are we doing this? We want these upper confidence bounds to be valid, and we need them to be valid at every single time step because we are using them to make decisions. So this is also related to false discovery and other things like that if you've heard about them in machine learning. So what we're going to use here is we're going to think about all of these as being events that these confidence bounds hold, and what we mean by that is that they really do contain the true value-- the true unknown value with high probability. So what we're going to say is the probability that all of these events hold, which means that all of our confidence intervals are valid for all of the arms, for all of the time steps, we're just going to use a union bound, which says we're just going to sum over the probability of each of them over all of those events. So that would be roughly the number of arms times T. And so that's why you can then just divide your confidence interval, your delta-- so you can just divide your delta into delta divided by t times the size of your a, and that generally is sufficient. And just to think about what that will do in terms of your bounds, so remember, we had a log 1 over delta term. So that means you would get something like this, log t a divided by delta. So generally, the union building blows up your long term. There's various approaches, including law of iterated logarithms and others to try to get this term to be smaller. So you can do tighter things on this. All right. OK. So let's think about-- I promised you that we're going to be able to use this type of idea to get sublinear regret. So let's go through a proof sketch to think about how this actually enables us to get much better performance than what we've seen before. All right. So what this statement says-- and I'll just put a pointer in. So it's in the references under the website, but there's a great book on-- I think it's just called Bandit Algorithms by Tor Lattimore and Csaba Szepesárí, which I think maybe came out in 2019 or 2000. I'm trying to remember, but they have a great book. So it came out of a series of blog posts they were doing on multi-armed bandits, and then they turned it into a book. And so this is a really nice one. And if you go there, I think approximately chapter 7, they're going to do a much more rigorous version of this proof compared to what I'm doing today. What I'm going to try to do today is just to give you a flavor of types of bounds that you might want to prove in these sorts of cases and how we end up getting sublinear regret. So what this result says is the following. If you think back, what we said before is we could bound the expected regret by how many times we make-- we choose an arm and how much gap or loss we have whenever we choose it. And so one thing that we could do is then try to just think about, well, we-- we don't know what the gaps are, but the gaps, we can just write down as the difference between the expected reward of that arm versus the true reward of that arm. That's not something we can influence. The thing that we can influence is how many times we're selecting bad arms. So what this says is that if an arm is suboptimal, the number of times that we pull it, number of times, we take that action in upper confidence bounds, scales as a constant C prime-- not going to tell you what that is. Often in the algorithms, they don't tell you what that is either. I mean, it'll be somewhere in the fine print. The point is that constant can't depend on parts of the domain. So it can't depend on the number of arms or the gaps or things like that. It could be like 37, for example. So a constant times log of 1 over delta, delta squared plus pi squared over 3 plus 1, OK? So why is this interesting, before we get into how do we prove this? This is interesting because it says if the gap is large, we're going to take it many less times. So if the gap is really small, then it means that we're going to-- we might sample that action a lot more. And if the gap is large, we're going to take it less, OK? And then we can combine that with this equation. And what happens in that case is-- I'll go through that part before we actually think about-- so what we're going to focus on doing a proof sketch of for today is to focus on this part. But let's just think, if we could prove that, why that would show the second. Well, what we would get in this case is we would say, we'd get this term plugged into here. And the main thing that would happen there is this would become delta, because we've multiplied it by a delta on top. And then here, if you assume that everything is bounded between 0 and 1, then the deltas are at most 1, 2. So you can get-- this is just the number of actions times 1 plus pi squared over 3, this term. So this just shows what your total regret would be, in this case, your total expected regret. As I said, there's quite a bit more subtleties to the formal proof, but this just gives sort of a rough idea. So we have any questions on that before we dig into how we show the first part, which is the total number of times we're going to take arms, we're going to pull a particular arm, scales with 1 over the size of the gap squared. All right. Let's go through it. So this is going to heavily rely on the Hoeffding inequality and the upper confidence bounds. So remember what we saw before is let's imagine that we've got this. So we're going to say, this was our upper confidence bound. So we had this upper confidence bound. And again, I'm going to be loose with the deltas. OK. We'd have to be a little bit more formal about it in general, but let's look at this. So this is going to be the true value, and this is our empirical estimate, OK? So what Hoeffding inequality had told us is to say, the difference between the true expected value for an arm and your empirical average is greater than this quantity, our upper confidence bound, with probability no more than 1 delta over T, OK? So now let's think about the following. Let's think about the times we pull a, which is not equal to a star, and delta a, which is not equal to 0. These are the only things we care about in terms of regret. If we're pulling a star, we have 0 regret. If we are pulling an arm that has delta a equals 0, that also means that it has zero regret, because it means it's tied with an optimal arm. So the only things that we care about bounding here is to think about for that Nt of a, how many times are we pulling arms that are not optimal, OK? All right. So what we're going to do is observed a couple. So if the confidence interval holds, so we can think of if this holds, then we have the following. We have the Q(a) minus C log over delta divided by Nt(a). So Here I'll say, if one holds, which is less than or equal to Qt hat of a, which is less than or equal to Q of a plus square root C log 1 over delta divided by Nt of a. This just says, if your confidence intervals holds, what it means for it to hold is that confidence interval is wide enough that it contains your true value. And the upper confidence part is higher than that, and the lower confidence bound is lower than your true value. So this is just holds if our confidence intervals hold, OK? Now, if we pull a instead of a star-- so under UCB algorithm, we have the following. We know that the upper confidence bound of a was greater than the upper-- because that's why you pick this alternative action. So in this case, if we pull this arm a, that means that it's upper confidence bound was different than the upper confidence bound of the optimal action, and it was more preferred. So that's the only time we ever take the wrong action, is if it's upper confidence bound is higher than the other actions. OK. So let's write down what that means in terms of its upper bounds. So the definition of upper bounds here is that Qt of a plus square root C log 1 over delta divided by Nt of a is greater than Qt of a star plus C log 1 over delta divided by Nt of a star, OK? Because that's just the definition of our two upper confidence bound. So it says, OK, I'm only going to take this other non optimal action because it's upper confidence bound was actually higher than the upper confidence bound of the optimal action, OK? And then we notice-- so let's just label them. So we're going to call this 2. I'm going to call this three. So now we're going to substitute in from 2. OK. All right. So we know that this is greater than Qt of a star from equation 2, because we know that the upper confidence bound on the optimal action also holds, so it's upper confidence bound has to be higher than its true value. OK? All right. So now what do I have? I have that Q-- and let me write one more thing here. So similarly-- let's check that I get that right. 1, 2, 3-- good. Hold on. I just to make sure I got that one right. Yes. OK. So that means that Qt of-- oh, hold on. All right. So this is going to mean that Q of a plus-- I'm confusing myself, but I'll figure it out in a second, of Nt of a times 2 is greater than Q of-- oops, I should have written 0. OK. OK. So let me just make sure I did that correctly, because I want that to end up going in this case. Let me just make sure that I did that in the right way. I feel like I'm off by a constant. All right. I'll double check the constants afterwards. I'll just write a note. And so I'll check the constants. OK. But the main formula is going to be fine. even if you drop the two here, that would-- So what are we going to have in this case-- something's bothering me. I'll see if I can figure it out in a second. So what we want to argue in this case is that the Q of a that we have plus two of the confidence intervals is going to be greater than Q of a star. And I'm confusing myself slightly now, and I'll check into it later. But what this would mean in this case is, let's assume this holds for a sec. I'll make sure I get the explanation for next week or I'll just put it on Ed. What we would have in this case is we're going to have that 2 square root C log 1 over delta over Nt of a is greater than Q of a star minus Q of a, which is equal to delta A. Let's go to the next slide. OK. So if we have in this case, what we can then argue is that in this situation, what we have here is that we can rearrange this to the other side. So let me just do the algebra for that part. So what we're going to have is we're going to say that four times C log 1 over delta divided by Nt of a is greater than or equal to delta a squared. Which means that if we rearrange this here, we have Nt of a is less than or equal to 4C log over delta divide by delta a squared, OK? And that looks really like this. OK. So what does this say intuitively? Intuitively, this is saying, if your confidence bounds hold and you use them to make decisions, then if those confidence bounds are holding, then the only time that you make a decision that is wrong is where these confidence bounds is large enough that it overwhelms the gap. And the number of times that that can occur is finite, because the gap is nonzero. And since we know from Hoeffding's inequality that the size of the confidence intervals are going down over time, eventually, they will get smaller than the gap. So we're going to take these suboptimal actions less and less often according to how quickly your confidence intervals are contracting relative to the gap in these cases. Do we have any questions about that? OK. All right. So what this means is then when we look at this, we end up getting that it achieves logarithmic asymptotic regret as a function of log t, because we had the log t here inside of the number of times we're taking these suboptimal actions. And what you can see in these cases is that over time-- so this is a previous result where we look at the amount of data that we have and what is the best performance that we have over time. If you tune epsilon greedy, it can definitely get better. But also, UCB logs definitely have this nice logarithmic shape. If you have the right-- if you set the constants correctly. Now empirically, often, it will end up being that the constants matter a lot. And so if you set the constants wrong or if you set the constants often to the theoretically prescribed value, it'll often explore for a long time. So you can often be more aggressive than that in terms of the resulting bounds. So an alternative we could have done to UCB is to always select the arm with the highest lower bound. This can yield linear regret. So I think that's a useful thing to think about. This is optional, but you can do the Check your understanding to think about, why can this lead to linear negative regret? And it's helpful to think about the upper confidence bound case and why that one works and why this wouldn't. So in particular, I guess, imagine this was on an exam, what I would be looking for in this case is for you to construct a multi-armed bandit case for which selecting based on this criteria would give you linear regret. So if you think back to the example I showed you for greedy where we considered a particular sequence of arm pools such that you would never recover and you'd get linear regret in that case, think about this sort of setting too. Where based on some confidence intervals, if you select whichever one looks like it's better in terms of its lower bound, that you would never recover and select the optimal action. So I had a question about the slides before where we were assuming that the condition was met. Then I'm assuming the other parts came from where the condition isn't met. That's right. Yeah. So in those cases, if you set the delta correctly, you can say-- so with high probability, you're going to want this to hold for all time steps, and then there's going to be this small amount of probability that it doesn't hold. And then you can argue in that case that the regret is going to be bounded from those time points. So you split the-- it's a good question. You split the expectation into a high probability event and the low probability event. So why don't we-- why don't you talk to a neighbor and see if you got the same thing? At least one person already has the right answer. [SIDE CONVERSATION] Yeah. Oh, good. OK. I'm going to interrupt you for a sec for-- interrupt you now. Where would I have to put the mean and the upper bound for a2 so that being pessimistic fails? So according to the algorithm, here, if we select the arm with the highest lower bound, we would select a1, because a2 has a lower bound. But where would I have to put its upper bound and its mean for it truly to be-- for us to have linear regret? So here, I put a reward on the y-axis. At least one person said the right thing in there, so I know one of you guys know this. The mean of a2 [INAUDIBLE] should be higher than the mean of a1. Yeah. And then the other bound should be high as well? It should be really high. Yeah, that's right. So for example, could have this. So you could be really uncertain about it. Its lower bound is lower. Once you pick a1, the lower bound here-- this is an expectation, is only going to get closer. Like, the lower bound-- these are valid confidence intervals. This lower bound really is smaller than the mean of a1. Which means on average, whenever we sample a1, this is really just going to shrink, which means we'll never pull a2. A2's upper confidence bound is higher than a1. So under UCB, we would learn this. But if you're pessimistic-- in some way, if you think about it for upper confidence bounds, if you're optimistic, either you're correct or you learned something. The problem with being pessimistic is that you may not learn anything, because you're not updating your other bounds. OK. I realized where I was being confused, so let me go back and just correct that here. OK. So how did I get these? So let me just clarify. It was this step. I was confusing myself. OK. So we had this particular equation, that the empirical average plus its upper confidence bound was bigger than the optimal arm's empirical average plus its upper bound. What I did from equation 2 is I reminded-- remind ourselves that the empirical average is always less than or equal to the true value plus the upper confidence bound. So we substitute that in for Qt to get the Q(a) plus 2 times the bound. OK. So that's why this works out. So you just substitute with this upper bound into here. So then it gets another-- it gets Q of a plus this upper bound plus this upper bound, which means this bound becomes 2. So that's where that came from. OK. So this is the first algorithm we've seen which has provably sublinear regret, which is really nice. It's also really easy to implement, certainly when you have counts. But all of this stuff can be extended to much more complicated settings. And so there's a lot of work of thinking about for function approximation and RL, and we'll see all of those, of ways to think about formalizing this optimism under uncertainty principle in order to make decisions when we don't know what the outcomes will be in order to reduce our regret over time. So what we're going to see next time is we're going to see more fast learning, but we're also going to think about it from a very different perspective called Bayesian bandits where we think of it not being these just fixed upper and lower rectangular confidence intervals, but we think of having a prior over what the distribution is going to be of the rewards for each arm. And then in that case, we can also introduce algorithms that end up being somewhat similar to optimism in certain ways as ways to use those prior informations to figure out how to quickly gather data and start to make good decisions. So we'll see that next week. Thanks.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_I_Guest_Lecture_on_DPO_Rafael_Rafailov_Archit_Sharma_Eric_Mitchell_I_Lecture_9.txt

Hi, everybody. We're going to go ahead and get started because we're going to be having a guest lecture today, which will start at 1:45. So welcome back. Just in terms of where we are, a few different quick logistics things. The midterm, as everybody probably knows, is on Wednesday. It'll be in class. You're allowed to have one side of a normal sheet of paper in terms of your sheet of notes. All the material through today is going to be eligible for the exam. That was also in the Ed post. And you can see the Ed post for any additional information around midterms and prior exams. Because Homework 2 was only due on Friday, and a lot of people used late days through yesterday, we won't be able to grade it in time for the midterm, but we will release solutions. So those will be available by the end of today. All right. So let's start with a quick refresher understanding. This is on the poles, and then I'll do a quick recap of RLHF before we dive into our guest lecture. This will be a good reminder of some of the ideas that will be relevant to today's lecture, as well. All right. We have pretty good consensus on the first one that this is true. The Bradley Terry model expresses the probability that someone will select one option over another option. So this is true. And we have pretty good consensus that the last one is false. In RLHF, we do not update the model after each PPO rollout. There's a little bit of disagreement, particularly, about these two. So why don't you turn to a neighbor and quickly see if you can resolve those? And as a hint, it's useful to think about whether things can change based on whether or not it's positive or negative. All right. I hope everyone got a chance to think about that for a second. So the second one is true. The third one is also true. Somebody want to say why the fourth one is false? It's false. This one is false. No. Yeah. If you to play by a negative constant, it's both references. Yeah, exactly. So remind me of your name. So that is exactly right. So if you multiply by negative, of course, that's exactly flipping all the rewards. And so in general, that will not preserve preferences. You can shift it by any constant. And if you go through the math, you can see that the exponentials will all cancel. So that part is true. OK, great. So what we talked about last time was maximum entropy inverse reinforcement learning, and we started talking about RLHF, including how you could use the Bradley Terry model for Markov decision processes. I'm going to do a really quick discussion of RLHF with respect to large language models before we get into our guest lecture today. And then on Wednesday is the midterm. So as we talked about last week, while you could do imitation learning, where you get sort of full trajectories, and you want to imitate those, that is less information than you might be able to get if you got pairwise preferences. And we talked about how pairwise preferences might be an interesting intermediary point between humans having to label, like they do in DAGGER at every step of what someone should do, or provide really dense rewards versus just providing demonstrations. And so this sort of has motivated a long line of work, including preference learning recently. We saw how you could learn the parameters of a Bradley Terry model. As we saw just now, these are not unique, in general. You can do translations of the rewards and you will preserve the resulting preferences. You can maximize this with cross entropy. And last time, we saw how you could do this for trajectories, as well as for bandit like problems where you only have a finite set of actions. In Homework 3, you're going to be implementing both DPO and RLHF for Markov decision processes. So you get a chance to play with this where you're using rollouts from MuJoCo like problems. But before we go on to our guest lecture, I wanted to just briefly go through how you go from doing this approach to learning reward models all the way to ChatGPT. And so for this, I'm going to draw upon some of Tatsu Hashimoto's really nice lecture notes from an NLP class. So recall from the start of the reinforcement learning course, we looked at this sort of pipeline from ChatGPT. And here, we had the demonstration data, collecting the comparison data, and then optimizing a policy. So now we've seen how those last two steps happen. So in particular, you can generate pairwise preferences, or in fact, you can generate full rankings, and then use that to learn a reward model. And so while we thought before about different ways of doing this, as a particular example involving language, you might say someone might prefer an earthquake hit San Francisco. There was minor property damage but no injuries versus a 4.2 magnitude earthquake hit San Francisco, resulting in massive damage, versus Barry has good weather, but it sometimes has wildfires and earthquakes. So you can see in this case that these are places where someone might be able to provide different rankings in response to prompts. So now you can think of the context as being a prompt, and the output as being all the actions or all the different responses you can have, and people are going to rank them. Now, sort of building on that, before you actually do PPO or something, you may want to try to check the quality of your reward model. And this is something that you'll also think about for Homework 3. So in general, depending on the amount of data you have and the complexity of your reward model, you're going to be able to do a better or worse job of being able to try to capture the underlying latent reward model of people. So in this case, this is looking at different model sizes. And these are big models. A lot of the models that people have thought about historically are things like linear models or then neural network models. But these can be extremely large models. They can be on the same order as large language models. It's not uncommon to see 7 billion parameter reward models. And what they're looking at here is validation accuracy. And so what you can see here is, when you start to get enough data, and you have a big enough model, then you can start to capture really complex reward models. And so that's a useful thing to think about when you're thinking about your projects or you're thinking about homeworks of what is the complexity we need in order to start to capture human preferences. And then once you have that, now we have everything we need to do that pipeline. So if you've gotten a lot of preferences, now, again, the question is, how many of those preferences do you need. It might be a lot. So if you look back here, this is quite a lot of preference data. Now, it's not the same amount of data that we would generally need to be using to train an LLM, but it's not like one or two either. And in fact, there is a lot of ongoing interesting work in trying to think about how do we reduce the amount of online preference data that we need in order to train these. By online, I just mean additional data compared to the historical. So in reinforcement learning from human feedback, what we can do is, once we've had that learned reward model, now you can use that with PPO. And one of the important things to note here is that, just like how we saw for PPO before, in general, we're going to need some sort of reference decision policy that maybe we've used from behavior cloning or supervised fine tuning. And we want to regularize so we don't get too far from that when we're doing PPO. And so that sort of divergence is going to be just as important as what we've seen in the previous work. And one of the things that's been noted is that, perhaps not surprisingly, given the huge success of ChatGPT, this type of approach can make a significant difference. So by leveraging rewards and doing RLHF, there really was a substantial gain over previous approaches, even when you fix for the model size. So that suggests that changing the optimization function we're using and using the reward functions really can lead to substantial gains in performance. So I think something that's important to notice here is, well, what are we doing the reinforcement learning over and how are we training the reward model. In comparison to what we've talked about mostly in this class, this is really where you're trying to do something almost like Meta reinforcement learning or multi-task reinforcement learning. So instead of training an agent to do one task, like do a backflip or solve a Gridworld, we're really trying to train a large language model here to do any possible task the user might want. And so then when we're collecting data and we're doing comparisons, you might have an enormous number of different tasks. So writing a thank you letter, to making a website, to lots of different things, all things that used to be previously considered different tasks will likely be involved in this. So another thing that I think it's useful to note is that this is a comparison from 2023, also, from Stanford. There's also been a lot of other work. This is a very important ongoing area to understand how good these approaches are. And one thing that's useful to know is that best of n is an alternative where you could, for example, use your reward model, just generate n samples from your original model, and then just use your reward model to pick the best one, according to your reward model. So that doesn't use any reinforcement learning. It doesn't use PPO. It's just using your reward model as sort of an external expert to try to pick among all of your generations. And what you can see here is that also does pretty well, relative to PPO. Now, in general, it doesn't do quite as well, but I think it's really useful to think about some of these alternative baselines, particularly depending on whether or not you have access to actually training the model again versus you might have access to being able to train a reward model. And you might have access to an off the shelf LLM, and you might be able to combine these. It's a very active, ongoing area to figure out what's the best way to train and refine these sorts of models. All right. So that was a five minute overview of how people use RLHF to train ChatGPT. And now I'm really excited to have our guest lecture on direct preference optimization. Yay. All right. OK, well, I'm super delighted to have Raphael, Archit and Eric here today to talk about direct preference optimization. I really appreciate you guys coming. I know you guys have done this rodeo before at NeurIPS in terms of bouncing between three people. So for those of you that don't know, direct preference optimization got outstanding best paper runner up at NeurIPS this year, which is the premier machine learning conference. It's also had a huge impact already, really broadly, on the LLM community as an alternative to RLHF. So I think it's extremely exciting. You guys will get to do, to my knowledge, the first homework that's incorporating RLHF and DPO, which will be really great. And what they're going to talk about today is this. And they also just had a new paper drop on Archive just a few days ago talking about some extensions. And so I think it was timely that they could all be here. Thanks so much. OK. Well, yeah, thanks so much, Emma, for having us. It's funny when you talk about the impact of the paper. You sort of want to say RL, but I guess it's LLM community. Or what even is the community anymore? It's hard to draw the boundaries between things, but I think it's so cool to see how the boundaries are kind of breaking down between these areas. So yeah, as Emma said, we're going to talk a bit about RLHF and DPO. And we have a little bit of background that I'll do to set things up for these guys to bring things home. And some of this is probably going to be review from things Emma has already covered. But just to make sure we're all on the same page, we are, in fact, talking about this setting of reinforcement learning from human feedback. And as a small piece of background or setup here, why are we talking about RLHF? Why are we doing RL on language models? Why are we talking about it now? People did not start doing RL on language models a few years ago when ChatGPT came out. People have been doing RL on language models for a long time. But this sort of ChatGPT moment, so to speak, is something that I think really brought these RL methods to language models into the forefront of people's minds, because there was sort of a sense in which things really started working for the first time in a way that maybe they didn't before. And a lot of this comes from being able to start from a really strong pre-trained model that already has a lot of interesting skills and pre learned behaviors that we can fine tune. And so we don't have to start from scratch when we're doing RL, typically, on these language models. And that makes it a lot more kind of accessible to get some benefit from these algorithms. OK, so RLHF. We have this we have this three stage pipeline that is the thing that has sort of been popularized by ChatGPT. So in this first stage, I think Emma actually showed this same figure in her slide just a minute ago. So this isn't totally new. In this first stage, there's really a step 0 here, which is do the unsupervised pre-training. This is when we just fit a big generative model of a ton of text. This is, again, where we learn, meta learn, in some sense, some skills we're going to select from. And then we're going to collect some supervised demos, so from humans. We'll have some data set of prompts, explain the moon landing to a six-year-old. And a human is going to write sort of a sensible, good demonstration response to this prompt. And we're just going to do supervised fine tuning here. And this is going to actually serve as that reference policy that Emma was talking about a few minutes ago, the thing that we're going to constrain our model to, to make learning a little easier, and also, to avoid over optimization of our approximate proxy reward function that we're learning from. In the second stage, that's when we do the learning of the reward model. So here's when we collect preference data. So we're going to sample responses, typically, from the supervised, fine tuned model that we learned in the first stage. And we're going to ask a human or opt in a collection of humans to provide ranking annotations over multiple draws from that supervised, fine tuned model. And we're going to use those preferences to learn a reward model, so a mapping from a prompt, a dialogue history, and a potential response to a scalar reward. And then in the third stage, we're going to do policy learning. So we're going to try to fine tune that supervised, fine tuned model to generate responses that receive high reward from that reward model. OK, so the first step is pretty straightforward. Supervised fine tuning. We don't really need to talk about it very much. And again, Emma already covered some of these things, so hopefully, this is mostly review. But of course, ask questions if anything seems funny. Like I said, the feedback here, the thing we're going to do is we're going to get preferences over responses from our model. OK? So we're going to end up with some data set of a prompt. And this could be single prompt, or it could be an entire dialogue history, so multiple turns and then the most recent user message. And then this is typically going to be only two responses that we're going to do give a binary preference over. You can do rankings over a more responses, but the returns can be a plateau relatively quickly. And it's typically maybe better to have more prompts and fewer responses per prompt. I think it's worth mentioning briefly, why are we talking about preferences over responses, instead of directly asking for reward annotations. You could take your prompt and your response and just ask the human, 1 to 10, how good of a response is this. And there are a couple of reasons for this. First of all, actually, in another set of slides, we have an example of this, which I think makes it quite clear. I don't think we have them in this deck. But if you take two different humans and you say, here's a prompt that says write me a recipe for making a really good cake, and you ask two different humans, you have two different responses from your model. And for human A, you say, what reward do you give to this response and that response, and another human, you ask the same question, you can end up with the same ranking over responses, but actually a lot of disagreement in the actual rewards that you're giving. So people are not really calibrated to each other in terms of the absolute rewards that they're going to be assigning. And it's also just more cognitively difficult to assign this absolute number in contrast to anchoring to one response, and then just making a decision about, is another thing better or worse. So in some sense, I think gathering preferences as opposed to asking humans to write high quality demonstrations, or asking humans to assign directly the reward itself is sort of a way to get higher return of annotation information per unit of cognitive effort of the human labeler. So we're going to get these preferences. And now, we just have this Bradley Terry model, which is a very simple model of discrete choice in humans, which relates a scoring function, or in case, a reward function, so that's this r of x and a response y here, to a probabilistic decision over two discrete choices. So here are our discrete choices. We have the thing that was labeled preferred in the data set, and the thing that was labeled this preferred in the data set. And we wanted to train a model, some probabilistic model, again, our reward model, to maximize the likelihood of this observed data. And we need to decide on some model that relates a scoring function to these choices in order to do maximum likelihood. And that is this Bradley Terry model, which we can then simply do a maximum likelihood in, or use this negative log likelihood of loss. So we're using this Bradley Terry conceptual model of choices, and this turns into a maximum likelihood of loss. So we're simply solving a binary classification problem. So we have a binary classifier here, where the logic is just our reward model, the difference in the reward we're assigning to the chosen response minus the dispreferred or the rejected response. We're treating that as a logit of a binary classifier and doing maximum likelihood. So once we do that, we get a reward model. We finished step two now. And now, we need to just find a policy that actually optimizes this reward. And really, this is the RL bit, so to speak. And here, we want to learn, again, pi theta. This is our policy that we're actually fine tuning that we're actually learning here. And the objective here is, we have prompts where we have some data set of prompts or conversation histories and expectation for responses sampled from our policy. We want to achieve high reward, but that's not the full story here. If we just optimize to maximize the reward here, what can happen? I'm not sure if you've talked about this already. OK. Perfect. Anybody have any worries about just optimizing this objective, or are we good? Because this is OK. We could tell if you haven't. You can forget the rest of the objectives. Perfect. OK, so one thing that can happen here is, you remember, this is not a true reward function. This is something we learned from a finite data set. And so there's going to be some distribution, in which this gives us accurate or meaningful rewards. And there's going to be, outside of that distribution, there's no guarantee this thing is going to generalize meaningfully. So what we typically end up doing is we actually have an additional constraint, a KL penalty from our starting model, the SFT model, or a reference model to say, I want you to maximize rewards, but I don't want you drift too far from the starting model. Because again, our reward model was trained on preferences over samples from that reference model. So if we drift far from the reference model, we're sort of out of distribution for the data that our reward model was trained on. So we basically can start getting bogus reward scores if our policy changes too much from that reference model. Yeah. Is the reference model ever changing? It depends on the algorithm. I think in the original canonical version of RLF, no, it was a fixed reference model. But there have been a lot of works since then showing ways to update the reference model and use a moving reference model over time, yeah. Yeah. The original data is coming from that reference model. You mean that both those yw and yl, they are both coming from the same model? Again, in the sort of original canonical form of RLHF, yes. Since then, people have proposed a wide variety of different sampling schemes, a way to select what pair of responses do you show the human to get a preference over. But in the original vanilla version of RLHF, yeah, you typically sample two responses from the reference model, get a preference over them, and use that to learn your word model. I've heard that in practice, but the responses come from the same model with different temperatures, or different models. Does that, in theory, mean that we're doing something wrong? Well, I'm not sure, in theory, it means you're doing something wrong. I think one way to think about this is, again, we want our reward model to perform well across the state action space. So if you think of our state space being, or our context space being the conversational history so far, and our action space being the response, you want to have good coverage over this space so that you're going to get meaningful rewards out when you actually update your policy. And so in principle, yeah, we would like to be able to cover this space, assuming we have a model that has high enough capacity to model all of it. We'd like to cover as much of this space as possible. So yeah, a more diverse preference data set is very helpful. And there's some trade offs between, we want to concentrate our preference data set on the things that are high quality, but also, make sure we do cover a wide variety so we don't overestimate rewards for bad stuff. OK, one more, and then I'll hand it off to these guys. So [INAUDIBLE] learning, we know that even if you have limited data set, but if you have a large enough network, then if you train it to near zero error on the limited train data set, it can't generalize well on the test data set, as well. Why is that not applicable here, as well for the reward model? Well, it is applicable in the sense that the same sort of phenomena, like double descent and things like this are still applicable in this case. So you will get better performance, typically, from using a larger reward model. But there are limits to this. There's only a certain amount of information content in a finite data set of preferences. And so the extent to which you can push that model to generalize to new things-- my preference data set only has questions about what types of pets someone likes. It's just not going to tell you anything about quantum field theory, no matter how big you make your model. So there are limits. But yes, you would expect some level of generalization. OK, cool. So that is a primer on RLHF. And, basically, unfortunately, what we end up with is this-- if we're doing PPO, for example, this ends up being really, really, really complicated in the policy learning stage. So there are a lot of moving pieces here, and I guess you all will have the distinct pleasure of implementing this for your homework. Congratulations. But there are a lot of moving pieces here, and that was one of the motivating reasons for why DPO came to be, basically, was that. PPO, turns out it was a little bit tricky to get it to work for a particular problem that Rafael was initiating some research on. So anyway, that's the background on RLHF. And I'm going to leave it to Archit now to give an overview of PPO. All right. Thanks, Eric. Is this working? OK, cool. All right, who's ready for the fun math stuff? So you saw the scary picture here. And really, the question we wanted to start with is, do we need to do all this just to fine tune our model according to human preferences. And unsurprisingly, the answer is going to be no, so be prepared for the ride. And yeah, we saw this objective earlier. And before we go into the math, I want to just give a high level picture of what is going to happen here. We had some reward function, which kind of told us what humans like and humans do not like. And right now, we're parameterizing that as a separate network, saying that this will give us a score for which answer is good and which answer is bad. Now, really, can we leverage the idea that our language models have these probabilities over completions? And the completions right now represent any distribution over the internet. But can we like overload it somehow to basically represent, can we only put probability on things that humans like? And that's roughly the idea we're going to try to exploit, is that there's, essentially, a mapping between the language model and the reward model itself, one-on-one mapping that you can use to directly train the policy on preferences themselves. And towards the end of this, what you're going to have is a distribution over responses that are not just arbitrary text responses on the internet, but responses that humans like. And that's where direct preference optimization will come in. How do we do that? That's where the math is going to be. So we saw the RLHF objective, which is, essentially, we want to maximize the expected reward over completions, and we have a KL constraint to the reference distribution. For now, we're just using any reward function. The math is going to hold for any reward function, but in general, it's the learned reward function. Now, I don't know if this was covered in the class or not, but it turns out that this equation or this problem has a closed form solution. OK, great. I'm not going to derive it. Maybe I'll leave it as an exercise for people. But it's a fun derivation, and it's not too hard. So hopefully, you should find the time to do it. But really, if you've ever heard of Boltzmann distribution or something of this form, this is really just it. And this is not what we're contributing. This is a known result for a while and it's very intuitive. It might look scary for a second, but really, what it's saying is that we have the reference distribution that we started with, and we had some reward function. And really, what we're doing is we're upgrading the responses by the exponentiated reward. So things which have a higher reward will have a higher probability according to the exponentiated reward. Now, if you just look at this is very simple, but this won't be a probability distribution. And the thing on the left hand side is a probability distribution. So we normalize it by this partition function, which is the Z of x. Think of is as summing over every completion for a given question x. Now, you can imagine that that's a very, very interactable quantity. If I start computing every sentence and try to measure the probability, and then multiply it by an exponential reward, that's basically not tractable. So this equation by itself is not very useful. And I went over this. This is exactly the definition of the partition function. We're summing over every response y. The pi ref is the distribution we started with and the exponentiated reward, and beta is the temperature term trading of the reward and the KL constraint. So this is intractable. We'll hold on to the partition function for a second and we'll see what happens to it. But really, this result is a relationship between pi star and the reward r. But now we can do a little bit of algebra and shuffle it around and rewrite the reward in terms of the optimal policy itself. So what does this equation say? We're writing the reward in terms of the beta log ratio, where the ratio is between the optimal policy pi star and the reference distribution we started with. And then there's this pesky partition function that just continues to stay on there. I'm going to try to develop some intuition here. This is important. What it is saying is that, if an optimal policy ways puts more probability distribution on a response than a reference distribution, the reward is higher. Does that come through? And if a probability is lower, then the reward is lower. And this is intuitively correct. This is how our reward function should also be. If a response is preferred, then it should have a higher probability and a higher reward. So you can see we are starting to develop a relationship between a reward function and the probability distribution itself. Cool. But the main problem here is that this is, by itself, not very practical because the partition function, as we said, is just completely intractable. So maybe let's go back to what we were doing in the RLHF process. The high level idea is that we have a loss function and a reward function, and we're going to use this transformation. And once we plug it all together, we're going to get a loss function on the policies themselves. And if we go back to our loss function for the reward bit, if you remember, the logit is the difference between the rewards of the preferred response and the dispreferred response. This is what Eric covered just a little bit back. Now, if you look at this, this difference is not going to depend on the input itself or the partition function itself if we look at it explicitly. So this is exactly what is going to happen here, is we're going to take the equation that we took earlier, express the reward in terms of the policy we're going to learn, and we're going to plug it into the reward modeling loss. And once you compute this difference, this partition function is going to cancel out because it only depends on the input x, and it does not depend on the output we're computing it over. And when we do this, we get our final beautiful loss function, which we called the DPO loss function. And really, is just a reward modeling loss. And let's take a second to see what it is doing. What we're trying to do is, we have a preferred response yw and a disprefereed response yl for a given question x. And we're trying to maximize this difference. That's how we would minimize this loss. And maximizing this difference means that our log probability on the preferred response should be higher than the probability that the reference distribution puts on it. And the log probability of the preferred response should be lower than the probability that the reference distribution puts on it. Does this make intuitive sense why this would change the probabilities in the right way? Cool. And yeah, the log partition function basically just cancels out. You can think of this as being a benefit to the fact that you can shift the rewards by a constant. So often, that's considered not a good thing. But here, they're leveraging it because you can just cancel out the partition function. Yeah. All right. I'll hand it to Rafael, and you can go over the results. All right. Can you guys hear me? So this is sort of like the first control experiment we ran on this project. And basically, we took this IMDb reviewed data set, which is sort of like movie reviews, and we wanted to train the model to generate positive movie reviews. So we use the pre-trained sentiment classifier as a gold reward function. In this case, we do know we have access to the underlying reward score. And then we generated a bunch of data from the base model, which was pre-trained FT, ranked it based on the sentiment classifier, and created synthetic preferences. And then, basically, we just took a bunch of baselines across that data. And we fundamentally were interested in comparing to what degree is DPO an actual good optimizer of the core objective. Essentially, there's this reward KL trade off underlying all of this. And we basically wanted to see how good of a Pareto curve can we extract from that to see, essentially, DPO's optimal trade off here in this simple rule problem. Define Pareto curve. I see. Yeah. Well, Pareto curve is a general concept in economics and sort of decision analysis and things like that where we have trade offs between several things, for example, in this case, reward versus KL. And we're interested in the optimal trade off that we can get. And we say, for example, one method Pareto dominates, another method if, essentially, we can get something, get more without giving up on something else. So in this case, for the same KL, we can get more reward using DPO than another method. And we actually played quite a bit with the baselines here. I probably spent a couple of months trying to push these PPO numbers. And essentially, it works. PPO kind of works, and you get some results there, but it can't quite catch up with the DPO objective. And what I kind of wanted to include this curve here in this talk is, essentially, I think, even now, basically, almost all of the RLHF papers that you read are actually doing evaluation potentially wrong. Because you go read these papers and you kind of get the win rate, or you get the comparisons, et cetera. But none of them really plot these curves. For none of them, you really don't know where along this trade off you are. And that number in and of itself doesn't really tell you much because it's a question of optimization. And you don't know how well that optimization worked or didn't work just by extracting one position on this curve. So I think that's quite an important point that the community is still not quite making as much. But I think when any of these new things come up, I think this is the fundamental question that should be asked. Do you think it was because the reward model is misspecified, or which part do you think it's-- where is RLHF really in this case? Do you think the core model isn't that good, or the PPO optimization isn't that good? So basically, if you look at the purple thing, that's kind of like the out-of-the-box PPO, TRL. And if you look at some of our implementation things, they do a lot better. So the core difference there, and surprising to me, people have written like numerous papers about the same thing now. And to me, it was sort of a footnote. How we got this to work better, we just sampled more answers per prompt. And essentially, that was a question of variance. And in the RLHF setting, the variance problem is even higher because of the constant shift. So we actually did some analysis around this when we writing the paper. About 60% of the reward scores are noise, essentially. Sort a signal to noise in regular PPO is about 40%. And when you mix in the whole process, the variance like completely explodes. So it's a very sparse signal to learn from there. I'm just sorry. This is just a picture question, but I'm having a hard time knowing how do we read the graph. Is it better to have higher reward, or what is this graph actually telling us about each metric? Yeah, so obviously, it's better to have higher reward. This is a core concept of reinforcement learning. You want to maximize reward. But essentially, from the RLHF setup, we maximize the reward subject to a KL constraint, subject to some KL cost. And what this graph is saying, basically, it's plotting it for a level of KL using each of these baselines, how much reward can I get. And you want that to be, basically, as I said prior, optimal in the sense that you want to get the most reward for a certain level of KL. And the other point I made is, basically, people compare only win rates or, essentially, the reward, but they don't tell you about KL. So you can compare, for example-- oops, sorry. You can compare this DPO point to this PPO point, and this PPO point will appear better because it has more reward. But fundamentally, as an optimization algorithm, that's not the case. You said our model is interpretable. Or can you maybe explain what's going on under the hood? Or it's just optimization math? What do you mean by interpret? So if you provide feedback, the behavior will change. Can you explain the whole process, or you can not explain the whole situation? If you put in noisy data, for example, can you debug it? You spend the whole process? Yeah, I think that's a more complicated question than it seems on the surface. There's whole lines of research on, basically, average of noisy feedback, average of multimodal feedback, plurality of alignment. So it's not quite like an answer I can give in one sentence. It's a lot. Yeah. Can you explain again why we have a bad signal to noise ratio in normal PPO? It's a long question. There's a whole section in the paper. It's about half a page. But essentially, by sampling more answers per response, it kind of goes away. Can you explain what the reward means for sentiment generation? It was basically the sentiment of the sentence. And 1 is very good sentiment. 0 is very bad sentiment. So like movie reviews. This one's hopefully a one sentencer. For our appreciation of the graph, about what KL divergence trade off would you choose in a real model here? Is it that you might choose something like 10 so we're really in that region? Or is it somewhere much farther? It's very much model and data dependent. OK. Yeah, this graph means absolutely nothing in a summarization set. Gotcha. Yeah. I think it's very hard to choose a specific KL. But usually, what people do is measure performance on other benchmarks they care about. And usually, if they find that, if the KL is smaller, the performance on other benchmarks is preserved. So you typically try to err on the side of lower KL. And there's no specific number. But wherever you find your MMLE performance is great, that's where you stop. OK, in the interest of time, we had a bunch of other experiments in the paper, which show basically how DPO works. But I think, really, the testament to the algorithm is it's kind of been widely adopted more in the community and larger scale. This was maybe a little outdated. I haven't looked at this recently. But couple of months ago, this was basically the open LLM leaderboard on Hugging Face, basically, the leaderboard of open language models. And I think 9 out of the top 10 models were trained with DPO. And this is the open source community. And since then, even institutions have taken this up. In particular, this is taken from the Mistral paper that basically used DPO exclusively as their RLHF algorithm. And as you know, basically, some of the strong Mistral models are somewhat competitive with GPT 4, for example. So we do definitely have evidence that this works at very large scales. And from basically last week, we w know even LLaMa3 is using DPO as part of its optimization pipeline. Interestingly enough, they're actually using it with mixed with other things. So basically, the TLDR is this kind of algorithm sort of works. And we're seeing it taken up more and being used for more and more things. So this is kind of where the paper ends. Since then, there's been like a ton of other works that we have done and other people have done. Thought a lot about what to talk about this from those works. For example, I heard you guys have learned inverse max entropy, inverse reinforcement learning. You can actually derive DPO as a inverse Q-learning algorithm in a max entropy RL setting. Sounds actually trivial, but it is possible. And that paper is called Your Language Model is Secretly a Q Function. So for example, you can do that. I heard you're going to use RLHF on control problems. I don't know. I haven't talked with the TAs. But actually, DPO does not work for control under the classical formulation, and you need formulation of preferences under regret, rather than the reward functions. So hoping they've taken that into account. That's a whole separate other work. But I guess what I decided to focus on is this sort of DPO versus PPO debate, which is going to be raging a lot on in the community, in industry, very much on Twitter. And I kind of want to give you my perspective for this. And I don't want to sound egocentric, but I think pretty much the entire debate is wrong. Let's skip that for now. But basically, there's two things. DPO fits this implicit reward function, which Archit showed. You can think about this as fitting a particular reward model. And there are two questions there. The first question is, is this implicit reward function as good as an explicitly parameterized reward function. A similar question is, for this implicit reward model the DPO fits, you can analytically extract the optimal policy. So basically, what I can do is I can get the DPO policy, or I can take the DPO implicit reward function, put it into PPO, and run that optimization loop. Under perfect optimization, absolutely perfect optimization, I'll get back the DPO policy directly if my PPO is perfect. But that is rarely the case with any sort of machine learning optimization. So we get something that's suboptimal. And this suboptimality induces some sort of regularization effect that makes my model stronger. So these are the two big questions, I think, in this debate. So they've been kind of tackled recently. There's this thing called came out, Reward Bench, which is a large scale evaluation of reward models. And it has DPO is both a generative and a reward model, discriminative model. You can evaluate DPO models as rewards. And basically, on several scores here, we have this chat, safety, reasoning, type of task. So this, for example, shows scoring reward, scoring preferences based on dialogue and chat. You can see the top four models are all DPO models and outperform, for example, proprietary models, much bigger and sort of closed source ones. And on reasoning, the top model is this proprietary Cohere model. But the next five are all DPO models. And obviously, there's always more work to be done, more research to be done. But in my mind, this sort of work solidified this, that the DPO implicit reward is about as good as the classic RLHF reward. We're not losing generality. We're not losing capability for considering this implicit model versus an explicit parameterized one. So the other big question is then, does using a weaker optimizer, such as PPO, provide a better solution, gives you some sort of regularization. And basically, started to look more into this recently. Some of the first feedback we got on DPO was, someone tried to train like a very large scale DPO model. And what they said was, it does well, and then it becomes more and more verbose, and then starts speaking more and more. And at some point, it reaches a point where it just won't stop and just kind of goes off the rails. It just can't stop talking. And we looked at this on two data sets, one on summarization, one on dialogue. And what you can see here is the distribution of lengths of answers. And the blue distribution is the preferred answer, and the red distribution is the dispreferred answer. So we can see there's a very slight bias towards longer responses. People have biases. They prefer more verbose answers. They prefer more like verbose, longer summaries, et cetera, et cetera. But once we train with DPO, under every column is a separate level of regularization. Under any level of regularization, this is blown way out of proportion. It's not only DPO is allocating probability mass within the distribution. Basically, this green histogram is the DPO length. It's pushing things way out of distribution. And you see, now we have answers which are significantly outside of the distribution that's covered in our data set. So what is happening there? And there is this concept of reward hacking. I don't know if you've covered reward hacking. But there's a very famous paper from OpenAI called Scaling Laws for Reward Model Optimization. And what they did there is essentially the sentiment experiment, but a larger scale. They got some real human preferences. They trained a reward model, a very good, very strong reward model. And then they use that reward model to annotate some synthetic data, synthetic preferences. And then they repeated the whole RLHF process on top of the synthetic preferences. And this is what they discovered. So basically, what this graph is, is the same graph I showed earlier for sentiment, except that the x-axis is a KL constraint, and the y-axis is rewards. And these things, the dashed things you see are the learned reward functions in PPO, basically, the expected reward from your model training. And the solid lines are the actual gold reward models. So what you're seeing from a reinforcement learning perspective, it looks like the models are doing really well. It's maximizing reward quite a bit. But actually, its quality is either stagnating or going down. And this concept of reward hacking has become quite prominent since then, both for practical purposes, but for example, the AI safety community is very worried about this, the whole like paper clipping thing, if you've heard about it, and the way that, basically, the model can find a way to exploit these reward functions, such that it thinks it's doing something good while it's actually doing something very bad. And basically, these things are well understood. This paper has something like 200 citations. A ton of work has been done on mitigating these things. And the thinking there is, in classical RLHF, I'm learning a reward function. I have a proxy reward, and I'm continuously querying that reward with new data, which might make it out of distribution, which might kick it off course, et cetera, et cetera. So it's not surprising that this happens. I think by and large, the community has not realized yet that this happens in direct alignment, as well, because, A, there's no proxy reward function. You're directly optimizing the model on the data. And B, there's no new data. There's no synthetic data being sampled. It's all within the data set. But what we have discovered, and essentially, this is a new result that we are currently still developing, is that, actually, reward hacking seems to be quite prominent in DPO, and actually, all of the DPO variants, things like IPO and slick, as well, do this. Have you heard of those? And actually, it might even be more prominent than PPO, because PPO is a weaker optimizer. So you have to push really hard to really hit those tail of the reward function. But DPO gives you the exactly optimal analytical function. So in a sense, it sort of almost hacks in an absolute way. So yeah, this is currently, I think, part of the dialogue and the research that the community is not quite figuring out yet. And that's my goal to put these things out that this same reward hacking phenomena, very surprisingly, because it sort of goes against all the intuition we've had from before, happens in these sort of algorithms, as well. Right. So it's kind of the same type of plot you see on the left, the x-axis, the KL divergence. And y-axis here is GPT 4 win rate, so basically, judgments by GPT 4. And each checkpoint, each data is like a different checkpoint evaluated to train with DPO. And kind of similar to before, you see that, basically, it's different model sizes. And these are different data, but what I'm pointing out here is the pattern, this comb shaped pattern. You kind of see, the more you train, sort of like the higher KL you go. Actually, your performance doesn't improve. It goes down. So it's the same reward hacking phenomenon. The theory tells you that this thing should be monotone. You give up some KL, you get some reward. But that's not the case. And kind of the point here is this seems to be more prevalent in this-- Technically, the DPO reward function is just as good as any other reward function. But if you're optimizing it too much, we might be in this reward hacking phenomenon. And this is where, potentially, a PPO optimization could be more stable or could be more beneficial because it's a weaker optimizer, essentially, from a [INAUDIBLE]. So yeah, I think this is sort of where we are with these type of algorithms right now. And I think there's kind of exciting work to be done again. In conclusion, yeah, we saw of these things. But I think it's kind of interesting what the next steps are. A ton of work has gone into making RLHF robust. Basically, now we're showing that these alignment algorithms are very prone to reward hacking, as well. So I think a lot of work will need to be done to make direct alignment algorithms robust, as well. There's a lot more interest, as Professor Brusco mentioned, on the online fine tuning algorithms. How do we elicit preferences? How do we actually fine tune these things efficiently? There's been explosion of RLHF across modalities, not just language models. We've done vision language models. We've done diffusion models. In particular, Stable Diffusion 3, for example, is also trained with DPO. We've done text to image. There's text to video work being done. Potentially, speech and music is our next frontier to be tackled. In a couple of weeks, we'll be releasing a paper on protein synthesis with feedback and actively working on things like robot safety for things like large scale robotics foundation models. We're trying to do multi-turn interactions, which classical RLHF cannot do and things like agents to use. And all those things are, basically, things that are in the pipeline and we're looking into. So I think there's kind of a lot of exciting things that are happening in this field, and still, it's been on for a while. But I think only now, we're just starting to get deeper into understand a lot of the finer points of these alignment algorithms. I'm sorry if we run a little bit over time. Yeah. That's great. You got some time for questions. So I want to see if you haven any. Isn't the reward hacking implicitly used by the Bradley Terry model itself? It's a sort of work in and of itself. It's a finite data type of issue. If you have uniform data coverage over everything, reward hacking will go away. But it is fundamentally a finite data thing. Because you have ratios or exponentiated ratios in the reward formulation, and you're using that everywhere, because your model will try to maximize that, it will essentially try to skew that ratio. I'm wondering, if you had some other reward function, then maybe-- Yeah it could still happens. So if you use hinge objective, it still happens. If you use a square type objective, it still happens. And basically think about why this happens. So if you think about it as, you see that you have cheetah running along. And basically, imagine you have cheetah, and your target's running at a target speed of 10. And you run a target speed of 8, that's better than running a target speed of 7. Learning speed of 9 is running better than a target speed of 8, et cetera. Then you think about, well, probably running a target speed of 11 is better than target speed of 10. But you've never seen anything running a target speed of 11, so you're just extrapolating in a way that's just wrong. It's basically like this picture. Right? We think long things are better, so longer thing is always better, too. Is there a question there? Yeah, it's kind of a niche question. But I'm kind of wondering, so what if for a particular prompt, all of the samples aren't that great, but obviously, whoever's ranking them has to rank all of them and doesn't have any way of indicating that even the best sample isn't that great? I was wondering if there is a way to account for that, any sort of weighting that could be applied to the rankings that would indicate that the rankings are more or less confident overall. No, I don't have the means. Feel free to interject, but that's a great question. I think general problem around-- this is almost like the exploration problem in RL, is if you do not like ever see good trajectories, what are you going to learn without it. I don't have any easy answers, frankly. But I think some things that work is, there's other forms of feedback, as well. So this is like comparative feedback, where you're comparing two things. But you can give thumbs up, thumbs down. And then if all of them are just bad, you can indicate, optimize in a different way, such that you're down weighting most of the responses. But yeah, this is a good open problem to look at. I think the exclamation pointed to it, in that one thing that people ask a lot is, how can PPO work. Because with PPO, you get to sample from your policy during training so you can explore, and that has to be helpful. Right? DPO is just from your fixed preference data set, and you're never sampling during training. But I think your question actually points out the fact that, in some sense, because we have this issue of we're optimizing only a proxy reward, we don't get to optimize the real reward. The important exploration is actually the exploration we do when we gather the data that we're getting preferences over that we're going to learn our reward function from. Because if we do good exploration at policy training time, but we sample some great trajectory that our reward model doesn't correctly label as good, it doesn't help us. So yeah, in that sense, it's basically an exploration problem, and it's very important. That's why I think these sort of multi-term or an iterative process could be really helpful. Let's go to the middle. Yeah. Do you think a similar idea could be applied for, say, multi-step kind of reward, in which you get a reward after multiple steps, and you have a preference at the final step? But the reward function was explicitly comparing two preferences between exactly two. Can you repeat that? I just didn't quite catch the question. I think if you have a multi-step kind of reasoning process and a reward, which comes at the end of that, would this idea apply? Yeah, it does work. As I said, you can think of this as a Q-learning problem, actually. That is, however, not trivial to show. But it does work. If you have a problem where, basically, you have a sparse reward at the end, the model does end up doing some sort of credit assignment on the intermediate tokens. If you think of this as a per token MDP, you will end up with something that does something interesting for those intermediate steps. It's not doing explicit bootstrapping, obviously, but you do end up with some sort of credit assignment. And there are either several results now showing that, if you have sequence level rewards, you can end up doing something interesting, even though you don't have this intermediate rewards. Do you have a question next? Yeah. Can you go a few slides back, when you talk about the synthetic data? Yeah, there. Can you just explain, again, what the difference is between the real and synthetic, what they're doing in both cases? Yeah, it's part of the same sort of sentiment problem I was talking about before. They had real human data and train a reward function on this. So they want to be able to measure the real reward function. So they get this goal reward model, which is trained on this real human comparisons, and they generate data from their base model, and rank that data using this code reward function. So essentially, they have access. They can query the goal reward function and know what the actual score of these synthetic generations is. So basically, they can essentially create these graphs. So the reason we're getting reward hacking here is because we're not using the actual reward function? We're using this synthetic reward function? If you train on any reward function with a finite amount of data, and if you do, in the limit of infinite data, you would probably not see this phenomena. But because you're training on finite data, there will be errors outside the distribution that it's trained on. And some errors would like skew positive or overestimate the reward. Some would skew, underestimate the reward. But because you're optimizing against that, you'll end up giving responses where the errors are skewing positive. So that's why you start seeing phenomena where your learned reward is increasing, but your true reward is actually decreasing. If you guys think back to DAGGER, where they saw this propagating herds and supervised learning, it's the same thing. Here's also another interesting tidbit of information. All these checkpoints, even the very high KL ones that have quite low success rates, actually have very low losses and very high accuracies as reward functions. So basically, the quality of the reward function is not necessarily connected to the performance of the downstream policy, which is quite a surprising result, I would say. You had a question? Yeah. So back to the pairwise comparison you did, so if your objects that you are comparing cannot perfectly do this kind of pairwise comparison, so for example, say, the game rock, paper, scissors. So rock is bigger, much preferred against scissors. But it's not a perfect partially ordered set, then what can we do? If the reward function is not transitive, it's a great question. Yeah so there's an interesting outcropping of work that is basically trying to get away from the reward maximization framework and think of this as a game, where, instead of saying I want to generate responses that are the highest reward responses, we should think of this at the policy optimization level, and I should search for a policy where the average win rate-- if I take the expectation of the win rate of sampling an action from the policy that I'm optimizing, and then I have some comparison or adversary policy, I'm going to sample an action from that adversary policy, what is the expected win rate of the action sampled from my policy compared to the action sampled from the adversary? And so now we have to pick like an adversary policy class, which kind of makes sense in your rock, paper, scissors example. Because yeah, there's not like an optimal action to take here. It depends on what the policy of your adversary is here to know what's good and what's bad. So in this case, it exactly does address this issue of, if you have only a partial ordering, you can't necessarily compare all pairs of responses. We can still use that kind of data. We don't have to be bottlenecked by fitting a reward function first. So there are methods like a-- Nash policy? Yeah, like direct Nash optimization, or Nash learning from human feedback are this other kind of newer, I guess, family of algorithms are really interesting. Well interestingly enough, rock, paper, scissors doesn't actually have a deterministic Nash fixed point. It has a stochastic one, but stochastic ones are just like equal probability over everything. That's related to some deeper results that actually say examples like this of plurality of preferences are actually unsatisfiable. So you cannot actually theoretically train, and in practice, train a model that will satisfy that set of preferences. I think it's good. That's one way that is motivated, too, is if you have different distributions of populations with different preferences, even if each of them are internally consistent with transitivity. They may not be across. Yeah. So assuming that reward hacking is not happening, what in DPO prevents it from taking large steps in the accusation? What do you mean by that? Assuming we're hacking is not happening, is there something that DPO is doing that's preventing it from taking too large of a step? Yeah, I think the KL regularization, if you look at the beta term, if the beta term is higher, the sigmoid essentially saturates after a point. Right? And if the beta term is higher, you have to increase the differences less to satisfy the loss. So roughly, the beta controls how quickly you change the loss function. But there are other parameters, as well, like learning rates and so on, which also change affect this. So yeah. Yeah. For this reward hacking problem, one of the methods that people usually try to use to address this issue is using ensemble models. Right? Is that something that could be done with direct methods like PPO, an ensemble of DPOs or something like that? You could. The problem with that is then you have to keep all the models around. But there's smarter assembling things you can do. You don't have to have complete copies of your entire model to have an ensemble, for example. So you can ensemble sub pieces of your model, or even represent your reward model as a distribution, instead of a single scalar. And this starts tying back into these situations where we have a variety of preferences in our data that aren't always consistent with each other. One way of modeling this data better is to say, I have a sort of a non deterministic, or I have a multi-modal reward function instead. And if you have a way of representing this with this generative model architecture, then you can still just stick this into a DPO looking loss. Yeah, your answer already kind of answered my question. But I just wanted to ask, in general, what were the promising directions for addressing reward hacking in DPO? Well, there's a number of reward hacking works on classical RLHF. There was a huge number of those. Some of those transfer pretty straightforwardly. Here's something I'm kind of excited about. And interestingly enough, that came from the open source community in a way that they didn't actually understand what they were doing. They kind of like stumbled across this very randomly by a very questionable group of researchers on Twitter. What they discovered is, basically, if you just take a bunch of random RLHF models and you just, literally, weight average them, they just become better. Take the weights, take the average, and it just becomes better. And it turns out there's a ton of work on this from 2018 around the optimization landscape of these things. And they very randomly stumbled across it. Now it seems to work now, but there's a paper called WARM, Weight Averaging Reward Models, which kind of makes that point for reward models. So if you train an ensemble of reward models, you don't keep the ensemble, but you average them, weight average the ensemble. That significantly improves your robustness as a reward function. And the same seems to be actually happening with DPO. If you train ensemble of DPO models, or already pre-trained DPO models, and you weight average them, that seems to actually significantly improve the robustness, as well. And Twitter randomly stumbled across this without really understanding it. But it seemed to work for them. And it turns out there's a really sort of deep reasons behind this. So that's one thing I'm kind of sad about. And actually, after we get this paper out, we have right now something on the order of 400 checkpoints or something. The next thing we're probably going to do is try to see how much robustness we can squeeze out from some sort of-- and people do smart things now, like evolutionary merge and things like that, how much robustness we can squeeze from some sort of evoluntionary merging strategy. I'll give you one thing that's sort of interesting, also, is that we're starting with this KL penalized reward maximization objective. And that was the original policy learning objective is maximize rewards subject to this KL penalty. And the intuition is that, yeah, we want to keep the KL small so we don't over optimize our reward function. But this is kind of a crude way of encoding this desideratum, basically. And something that might be closer to what we really want is to say, well, the places where my reward model has high uncertainty, those are the places where I want to be conservative. But if I have something that's out of distribution, but my reward model is really confident, or I have low uncertainty over what the reward should be, or basically, the lower percentiles of reward are still quite high, then it's OK to change my model a lot in these places. And so I think one direction that I think is also interesting here is getting away from the KL regularized policy optimization objective, which is nice because it gives us this one-to-oneness from policies to reward models. But also, I think it's possible this is a bit too crude and it leaves some performance on the table, because we're over constraining our policy. Another quick question, a quick point. As you said, you can think about all these algorithms as Q-functions, essentially, an interesting framework. I think that's kind of interesting. What we're pursuing is, initially, DQNs were really hard to get working. Right? And there's, after a couple of years, they would work great because a lot of tricks were used to make them stable, to make them perform, and make them not bootstrap, and make them not overfit, et cetera. And I think a lot of these things could potentially transfer from that to the LLMs. And particularly, the weight averaging thing is very much, I think, also inspired by results in DQNs, where you have a target function. I don't know if you guys did the DQN homework already. We have a target Q function, which is actually weight average, so some sort of Polyak averaging. So it's kind of like staggered. And that seems, for example, to improve stability a lot. I think a similar result holds for LLM cues, so to speak. But again, it's still sort of in the pipeline of experiments to do. Yeah. I'm not sure if this was already touched upon, but are there any risks with overfitting? And is there certain domains, like medical domains, where there's very, very small data sets? Is there a scope for this kind of work on those? This is essentially an overfitting problem. You have limited data coverage extrapolated in the wrong way. It's a little bit more trickier, though. People have actually found that, in DPO and other settings, this overfitting is somewhat beneficial. So you can do multiple epochs on small data sets. And for some of our experiments, you can get that very tiny preference data sets, as well, and it still sort of works. But people do multiple epochs, and they're very clearly overfitting, but the performance still keeps on improving. But again, a lot depends upon how you evaluate these models. And you're probably losing somewhere else. So it really depends upon how you're going to use this. One thing to keep in mind is that we've been talking a lot about reward over optimization, or reward hacking, which is this discrepancy between the proxy reward that we're actually optimizing against, the thing we learned from feedback, and the true reward that we don't actually get to observe. But there's another discrepancy that we haven't really talked about that Archit just mentioned, which is that, when we evaluate these models, in practice, we're actually typically not evaluating average reward. We're typically evaluating something more like a win rate, which is comparing to some baseline policy or something. So the setup is more like almost a satisficing rather than a maximizing kind of situation. And so that's another layer of disconnect between the thing that we're using as our training objective and the thing that is actually providing utility for the human, or the person who's actually building this thing. So there's another layer of where we can get overfitting, in a sense, to the objective. Yeah. So my understanding is it's kind of two stages. The first one is the normal supervised training of the language model. And then the second stage is the DPO training. And then you use the KL divergence as a means to make sure you're not moving too far away from your original supervised model. It seems like two stages. Is it possible to combine this preference learning during the normal supervised training, so as you're training the model from the start, you're also digging into these? Because it seems like you're using KL as kind of a proxy for making sure doesn't move too far away. But if you do them at the same time, maybe it'll help address that. There are a few works that have tried. So you're talking about merging the supervised instruction tuning and the preference tuning part? Yeah, because they're one after another. Yes. Yeah, so there's a few works that have tried to do that. I think it's still an active area of research. But the general idea why, so maybe it's useful to understand why we even do instruction tuning before doing RLHF. It's that when you start with a pre-trained model, it will give you gibberish responses, which are not even aligned with the instruction you're giving. So the instruction tuning sort of helps us generate the right preference data set, where you're starting to follow the question being asked. So typically, in a very typical RLHF pipeline, when you don't even have a preference data set to begin with, that's why you do the instruction tuning bit. But if you have preference data sets already, people are coming up with methods where you can both combine instruction tuning and preference learning bit into the same optimization algorithm. They're not very different. They're usually some elements of the loss functions you already see. But it's still somewhat of an active area of research. Yeah. Can you do maybe a DAGGER esque kind of thing where you train the model and then you do fine tuning? There's methods which do that, as well. Yeah. In my personal experience, they didn't work very well. But there are papers that claim that works really well. You've also have problems trying to get DAGGER [INAUDIBLE] to work. It doesn't mean it's impossible. Yeah, it doesn't mean you can't. There's a lot of details that go into these. Yeah. And also, personally, I'm somewhat suspicious of these things because the optimization landscape is so-- basically, as you're seeing from this optimization thing, the optimization landscape is so complicated and can form so many different pitfalls, that I think trying to combine this and navigate that in a single shot optimization direction is pretty hard. Probably not impossible, but pretty hard. To me, it's not really clear what the benefits from that are. But again, I also think it kind of goes back to the exploration question, which I think how Archit framed it there is-- on the first day, OpenAI said, let there be NSFT model. And there was no preference data you said yet. And so in order to actually get the preferences, you needed a source of exploration to get the trajectories to get preferences over. So you had to do one and then the other. And so I think that's the original way to think about that. But now, if we're trying to do this in a single offline stage, well, now we're sort of stuck. We're just stuck with whatever data we have in terms of the exploration. And there's only going to be so much can do when you have purely offline data, so if you're doing this iteratively. But being able to sample and then get new preferences over those samples is useful to do. So you mentioned the discrepancy where we're training to maximize the reward function, but then during evaluation, we're evaluating based on win rate. So could we just use a different objective function to optimize directly for win rate? Is that possible? Yeah. These NASH algorithms basically are doing that. So, instead of deriving this as we have some reward function and we're maximizing reward, it's like, literally, I have some either baseline policy and I want to-- if I can only evaluate the preference function, not the reward function, so a function that takes two responses and says which one is better, one objective I could come up with is, the expectation under my policy of that preference function computed on one response from my policy, and one response from this baseline policy, or one response from an adversarial best, worst case, best adversary policy. And so now, you're basically explicitly optimizing for either average case or worst case win rate against some reference or comparison. So how does that compare to PPO? It depends on who you ask. The papers introducing these methods show improvements. I think one of the ways that it's helpful is that you're not, again, going through a reward function. And so you're not requiring-- you're not explicitly training to have some complete total ordering over all of your responses. And so this can be helpful. It's not as constraining of a sort of framework to think about. At the same time, any policy we end up with, compared to some reference model, we can interpret as a reward function. So I'm not exactly sure how to think about the advantages there. But yes, if you look at the experiments in the papers, they will say, yeah, we have improvements in win rate, which kind of makes sense. Right? You're evalling with win rate, and now we're training for win rate instead of training for reward maximization. It's not that surprising. You can see improvements. There's also another point that, if you do consider the Bradley Terry model to be true that this is your preference model, that this is the data generation model, maximizing reward and maximizing probability of win rate are actually identical. So as I said, this reward maximization thing, because of the Fried parameter, has very high variance. So what OpenAI does in other papers, but it's always like a footnote in a 100 page paper is, they actually normalize the reward functions. So they subtract some human baseline. So the reward of the human completion or the human data is zero. And what this gives you is, essentially, actually, the log probability. Then the reward function they optimize with PPO is the log probability. The generation is preferred over the human generation under Bradley Terry. So these things are very tightly coupled. And the normalization part, from our perspective, actually doesn't change the optimal policy. In things I've seen in experiments, it actually significantly reduces the variance, which is the intuition there. But actually, there's a very direct way to tie that with, essentially, maximizing probability of winning, essentially. It's like a baseline. Yeah, it's exactly a baseline, essentially. And this baseline actually works. The variance actually significantly goes down. Why don't we do one more from somebody who hasn't asked a question yet? So are those DPO applicable to multi-objective correlation? There's a paper called MODPO, which stands for multi-objective DPO. And yeah, you can basically-- yes, you can do DPO in this setting where you basically condition on a scalarization of your multiple objectives, like a particular weighting. You don't have to learn any reward function, or you have to learn n minus 1 reward? No, on of you guys, correct me if I'm wrong. I think you're still learning, basically, a weighting conditioned policy, where you can pick the mixture. You have all of your different objectives, and you can pick like what weighting over these objectives you want to use, which policy do you actually want to end up with. How do you trade these off? And you don't have to retrain for every single different scalarization. There are others that do this with uncertainty over the reward model, as well. Now let's thank our speakers again. Thank you very much for having us. Thank you. Thanks so much for coming. All right. Good luck with the midterm, everybody. See you Wednesday.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Offline_RL_1_I_2024_I_Lecture_8.txt

All right, while we work on getting these started, I'm just going to write a couple things up about general logistics and continue to work on this for a sec. All right. OK, great. Well, why don't we dive into this. Let's see. So I think everybody agreed from the first one, which is great. So this is true. So this is true. There was a bit of disagreement about B and C So why don't you talk to your neighbor for a second and see if that changes your mind or resolves the confusion. [SIDE CONVERSATIONS] Yes, that's correct. Because one of the downsides is constantly evaluation. And I think that's what I thought where DAGGER needed a little bit more than just images because it's able to put together-- All right, so the first one is-- this is false, and this is false. And so, DAGGER, if you think back, unfortunately required the human to keep around forever. They would constantly be getting asked, hey, for the policy that the agent followed, was this an optimal action or not? And behavior cloning does not require knowing the dynamics model. It allows us to reduce reinforcement learning to supervised learning. And the idea is that we take the expert demonstrations, and we just try to learn state-to-action mappings. And so we can just treat it as a standard supervised learning problem. Great. all right, so I think as we go further in the course, we get to go to more and more exciting topics. And today, we're really going to start to see how-- skipping all the NLP side, but how do we actually get to reinforcement learning that can do some of the amazing things that we see large language models doing? So for example, when I was preparing this lecture, I was like, please write me a program to demonstrate how our life works. Be brief in your explanations, and then show me the code. And within about five seconds, it generated me code that used Q-learning and other things to generate an actual example of how RLHF, which stands for Reinforcement Learning From Human Feedback, which is how they trained ChatGPT, amongst a whole bunch of other things to do. So it could generate me a small example of how to do that in code that you can run. So that's pretty extraordinary. This was not possible two years ago. I started offering this class in 2017. So when I first started offering this class, this was definitely not possible. And this only really became possible with ChatGPT. So it's pretty phenomenal that we now have AI that can do this. And the question is, how do we get there? And what sort of RL techniques are being used to help accomplish this? So that's what we're going to start digging into now. So today, what we're going to do is we are going to continue on from imitation learning and talk a bit about Reinforcement Learning From Human Feedback. And then next time, we're going to have a guest lecture from one of the authors of the direct preference optimization work, which received best paper runner up at neural information processing systems, which is kind of the premier machine learning conference. So he's going to come talk. He's one of the graduate students here at Stanford. And this have become-- I guess, maybe like it's starting to replace or exceed performance on RLHF on a lot of benchmarks, but super exciting. And it'll be great to have him. And in fact, because everybody here always is innovating, which is awesome, he was like, oh, well, we actually have a new paper coming out on archive like next week that shows how we can extend this to all in all these different ways. So I asked him if he had time to cover that a little bit. So there's a lot of work to be done in this space to think about how do we better use RL in combination with these incredible function approximators of large language models to create the amazing performance that we could see of a system that could do something like this. So that's where we're going. What we're going to focus on today is to continue talking about imitation learning. And I think imitation learning is a nice way to build into this, because imitation learning is one form of using human feedback to try to train reinforcement learning agents. And then when we get into RLHF, well, that'll be sort of a different way to leverage human expertise. So to start, we're going to go back to imitation learning and to talk a lot today about Max entropy inverse reinforcement learning. So let's just remember where we were last time. What were talking about when we talked about imitation learning was the idea of taking demonstrations from people. And these either could be explicit demonstrations. Like, I show the robot how to pick up a cup, and it records all my movements, and then you can use that for later training; or it could just be natural trajectories. So you take electronic medical record systems, and you just look at the decisions that are made from doctors. And we use that to try to either equal doctor performance or exceed doctor performance. So often, we just have observation data, which may either just be done in normal sort of business as usual, or that is explicitly being given as a demonstration trajectory. And this is just going to be the sequence of states and actions. We're not going to have rewards in general. And so the idea was, well, it might be easier in some cases either because it's just sort of natural data traces that are being generated as part of their normal work, like, electronic medical record systems are; or because it's hard for people to write down a reward function that kind of captures all the complexity of what they're trying to do in their objective. So that was one of the motivations for this. And we saw a few different ways to try to think about this setting last time, including behavior cloning, where we just map things back to supervised learning. And we try to learn a policy directly to match the expert. We saw DAGGER. I'll put that on here, too. So another thing that we saw kind of in between these two was DAGGER, which tried to address a challenge of behavior cloning, which is that when you make mistakes in your supervised learning system, you may end up in parts of the state and action distribution that you don't know. You don't have good coverage. So we talked about this kind of race car track example where once you go off, you've got a distribution mismatch. And we'll hear more about distribution mismatches in RL later in the course. And there we wouldn't necessarily know what to do. And so what DAGGER said is we have to keep an expert around, and then they will always tell us what you should have done. So they're kind of a coach go back. They replay how you did in that hockey game; what you should have done at each moment. And there's a lot of really interesting questions of thinking about those counterfactuals. And then we thought about this broad question of, well, could we recover the reward from looking at these demonstrations? And this could be useful within its own right to try to understand the objectives that people are using when they're making their decisions for different areas, as well as potentially for learning a better policy or learning the policy. And then can we also-- one we have that r-- generate a good policy or generate a good policy directly? So one of the ideas that we talked about in this case is, well, what is sufficient to be able to accomplish mimicking? So in particular, we said, well, if we want to get a policy that matches the expert, that is equivalent to generating trajectories, where that distribution over those trajectories is the same as what the expert would have done. So we think of this strong relationship between policies to trajectories, which also is to states and actions because we can think of there being a policy that induces a distribution over states and actions. And two policies that induce the same distribution over states and actions will have the same reward because we're assuming that the rewards are only a function of the states and actions. And so we talked about how people had sort of leveraged this assumption to think about different ways to try to learn reward features. So, for example, if you have a set of features to buy your policy-- so this might be mu, which could be things like, how quickly a call service agent responds to a calls; how many times they use positive sentiment, things like that. And of course, in the case of a robot, it might be how many times they hit a wall, how far it went, others, any of these sorts of features. You could imagine that your reward function is just a linear combination of those features. And so we saw-- So these features are just things that people can come up with for every problem? Great question. So [INAUDIBLE] asked, are these features like people are writing down per problem? Historically, yes, I think one of the big things with deep learning has been like, let's at least go as close to the sensors as possible. So can we use just images instead of features on images? But in the case of something, like, say, online marketing, a lot of them would be potentially predefined. So purchases and what web pages you looked at, and what things you-- search queries you did. So you would have to still enumerate a set of features in this case that you're defining your reward over. But ideally, it's sort of as close to the sensor level of the data you're collecting as possible, or at least that often has a big advantage. So what we saw here is that essentially, because we assume if things are linear, and we assume there's just this unknown weight vector-- so this is a vector. This is a vector-- we could say if you could have, make sure that your distribution over features is really close. And if you bound the norm of the weight vector, then being really close in features is the same as being really close in reward, which means if your policy can induce the same features, you can get the same reward. This is a recap from last time, but it's useful to think about as we go forward. So one of the big challenges we talked about last time is that there is not a unique reward function that is compatible with the observed data, even if you assume your observed data is optimal. So we talked about how even the zero reward is compatible with any policy you might see. And so in general, it's not can be identifiable. We can't just say, if we observe these trajectories, and we know the policy is optimal, this is what the reward is. There's too many rewards that are compatible. And so what we're going to spend a lot of time on now is to think about one choice for how to break that ambiguity. And this is where we left off last time. And what we're going to focus on now is Maximum Entropy IRL. GAIL is also-- the second one is known as GAIL. This is also a popular approach. This was developed by Stefano Ermon's group here at Stanford. But we're going to start with max entropy because also, there's a lot of other follow-up things that could be useful from this idea. OK, so we're going to talk about Max Entropy Inverse RL. This came out in 2008. And it goes first with the principle of maximum entropy. Raise your hand if you've heard of this before in the context of probability distributions. OK, a few people-- more than I would have expected. Cool. All right, so remember that the entropy of a distribution p-- so think of this is a probability distribution. So remember, we've got this. This is something. So we'd have sum over all [? s ?] if you have a discrete state space. This is just a probability distribution. OK. So the entropy of a probability distribution is minus the sum over all of the states, the probability of that state, times the log of the probability of that state. It helps capture how distributed our distribution is. And what the principle of max entropy says is that the probability distribution, which best represents the current state of knowledge-- what do we mean by current state of knowledge is if we have some previous data, the one that we should pick is the-- so the probability distribution we should write down is the one with the largest entropy, given the constraints of the precisely-stated prior data. So you can imagine you have your expert data. And what this says is that-- and we haven't talked about what these probability distributions will be yet. But what this says is that we're going to try to write down distributions over your-- well, we're going to look at trajectories in particular-- that are compatible with our observed trajectories, but otherwise have the highest entropy. And so, intuitively, you could think of if you have some data, you want to find probability distributions that are consistent with that, but have the highest entropy, given that they're consistent. So we're going to end up with something where you have constraints. Yeah. I don't understand the motivation of imitation learning. I'm trying to not deploy the expert model because it's expensive. I would try to distill the model. What are we trying to do with imitation learning? Yeah, what's the motivation for this? Because we already have access to an expert model while learning another model. Well, the idea is that you have access to trajectories from the expert. So you don't have access to the expert at all point. You don't have their policy. You just have observations. So you can imagine something like, if I'm an expert doctor, you could look at all of the ways that I do surgery. And you could look at all of my movements and stuff. And then what I want to do is have a robot that can imitate that. And so I need to distill it and make it sort of intuitive, an explicit, parameterized policy. Doe that answer your question? Yeah. OK, cool. All right, so this is an interesting idea. This is sort of saying this is one way to break ties. There's a whole bunch of different reward functions, a whole bunch of different ways you could maybe be compatible with the observed data. Let's pick ones which have the maximum entropy, OK? So this is just a choice you could make. What does it mean for a probability distribution to be consistent? Great question. So hold on to that for a second. I'm going to say-- yeah, so the question was, what does this mean? How do we actually make this mathematically formal and algorithmic? And we'll see that in the next slide. It's a good question. We're going to write this down in a formal way. But this is the principle. And so this is what Brian Ziebart and his colleagues thought about in terms of this method. And I'll just say a little bit about the motivation. So Brian was a grad student at the time at Carnegie Mellon, and they were interested in trying to understand taxi driver behavior. And so what they wanted to do is-- when you're driving, there's lots of different constraints. Particularly, if you're a taxi driver, you want to think about distance and potential traffic and tolls and all these things. And so what they wanted to do is just to take trajectories of people driving through the streets of Pittsburgh and then try to infer what the reward function was that taxi drivers were using, as well as be able to have a policy that did as well as good taxi drivers. So this was sort of part of the motivation. And again, they had to deal with this question of how do you-- you can't just learn a unique reward, so let's just try to find something that's got maximum entropy. And let's see what this means in this case. All right, so in the linear reward case, what we're going to be interested in, or how we're going to think about where max entropy applies is to say, we're going to have distributions over trajectories. So we're going have distributions over trajectories. And we want to find a distribution of trajectories that matches our observed distribution over trajectories from the expert, but otherwise has really high entropy. So what you could be learning in this case is a probability distribution over trajectories that has the maximum entropy subject to the fact that it is a true probability distribution. So that's one constraint. So this is just subject to-- using a subject to or such that, but the other ones are constraints. And the other is that, in this case, we're going to say we're going to want to match the features. And we saw before that matching the features was equivalent to being able to match the rewards in the case where you have a linear function. So in the linear reward case, what we want to do is we want to say, I've got my distribution of trajectories. Let's say mu is a function that just takes the trajectory and outputs a set of features. And we'll talk about some of the choices for that soon. And we just want that to match what the features where we observed from the trajectories from D, where D is a data set from our experts. This is from our experts. OK, so this is how we would write that down. Now, I haven't told you yet how we're going to learn the reward function. I haven't even told you how we're going to learn this, but this is where the maximum entropy assumption is being applied. It's saying what we mean by maximum entropy is we want to think about getting a distribution over trajectories that is compatible with our expert data, but otherwise has the maximum entropy. Yeah. Remind me your name one more time. When you say distribution over trajectories, does that mean distribution over policies that create that trajectory? Or is that something else? Great question. That sort of isomorphic. So you can just think of it directly as being distributions over state action, state action, et cetera. Or you can think of it as it's implicitly going through a policy that is generating this. Yeah. And we'll become clearer, too, about where the policies come in. Great question. So this is what this would say, but we haven't got into rewards yet. And we need to think about how do we go from this to thinking about learning a reward model and learning policies. So in general, we don't have rewards, but if we did have rewards, what we would like to do is to get a policy that induces trajectories that match the same reward as our expert. So we would like to get a policy that has as high reward as our expert. If we knew what those were, like, if we had a way like this r phi, then we would say we want r distribution. So let's say we're going to learn a distribution over trajectories. We want this to be the same as what the expert is. And I'll just highlight that here. I'm using this p hat for expert. OK. So this is expert. So this looks almost the same as above, except for I've said, well, let's imagine that we don't necessarily have to have a linear reward function. In general, we just want to say we would really like that whatever our distribution of trajectories is that it matches the reward of the experts because we know the experts is optimal. So if we achieve this, we're good. So we would like to be able to solve this problem. We don't know what r is still, so we can't do this. But we're just going to look at what would be the solution to this problem. And where we're going to go from this is that we're ultimately going to end up with an algorithm that does something like the following. We are going to assume we have a reward function, or compute one. Once we have a reward function, we're going to learn an optimal policy, and then we're going to update our state or trajectory features to update our reward function, and we're going to do this many times. So we're going to be thinking really a lot about the relationship between reward functions to optimal policies, optimal policies to distributions over states and actions, distributions over states and actions to how can we update our reward function. And we're going to step through all of those steps. And in the original paper there, assume the dynamics reward model is known. All right, so let's step through the first part because I think it's really helpful to see-- often, when people talk about max entropy, then they introduce this sort of exponential family, and it may or may not be clear where that comes from. So remember that we have this constrained objective. So we have this thing here. All right. So what we would like to understand in this case is given constrained objective, if we knew the costs, what would be the form of the distribution. Over tau? OK. Because remember, what we've got here is we have a max. So what this thing is this is an objective. This is an optimization problem that says the right distribution over trajectories you want is the one that maximizes that expression there. And what we're going to do now is we're going to see what would it be like? If we knew all of those things, what would the sort of structural form look like? And then we're going to use that to make some other steps. So now this is just to get intuition over this functional form. What we're going to do is we're going to rewrite this as using Lagrange multipliers. OK, so we've got p here. We're going to introduce lambda. And I'm just going to write this as follows. And I think this is illustrative because it'll make it really clear where the structural forms come in that we're going to use. OK. all right, so I'm just writing down our first Lagrange multiplier. And I suspect most of you have seen this, but if you haven't, feel free to come up to me afterwards. OK, we're just rewriting the constraint optimization problem. So we rewrote our constraint optimization problem as a single equation, and now we're going to take the derivative with respect to this because remember, we want to optimize this. So we're going to do D. We're going to do it with respect to our trajectories. So we're just going to get log of p of tau plus P of tau times the derivative of log, which is just 1 over p of tau plus-- and in the third case, the third one doesn't have any p of tau. This term does here. So you'll get lambda 1 r phi of tau. Yeah. Is this something that shouldn't go away because we're taking the derivative with respect to one specific trajectory? Yeah, exactly. So we're going to assume we're taking-- we're trying to get what is the derivative with respect to 1-- this probability of this particular trajectory tau. That's why everything goes away. And the important thing to notice here is this goes away because there's only a p hat there. That was from the expert. So that term just disappears. It's not a function of p of tau at all. all right, so now we want to set this equal to 0. Set this to 0 because we want to find the max, and then we're going to just do some algebra. OK. Now, we're just going to exponentiate. OK, why did we do this? Because I wanted to illustrate that what this means is that the probability distribution over trajectories, which maximize the entropy subject to some constraints, is exactly proportional to-- this is the proportional side-- the exponential of the reward for that trajectory, which means that, in general, if you observe this, you would put sort of exponential more weight on things that have higher reward, subject to a constraint that you have a probability distribution. All right. So what this shows here is that if we want to take this principle of max entropy, then what we end up getting is that the functional form over our trajectories is this exponential. It's proportional to an exponential. And that's an exponential family for those of you who have seen this before or seen exponential families. So this is like the structural form. This is the distribution that maximizes the entropy. And so once we know that, we can leverage that to now start to try to learn a reward function. So let's see how we do this. Because remember, we just did this, assuming that our r phi was known, like, for a particular r phi. We're not taking a derivative with respect to phi here at all. It's just a derivative with respect to p of tau. All right, so what this means is that we can think of maximizing the entropy over the probability distribution with respect to tau is equal to maximizing the likelihood of the observed data under this particular max entropy distribution. So I'm going to just write out what that would be here. So remember, that's what we saw here. We saw that if we max entropy, the functional form we get, looks like this normalized exponential. So in particular, we'll just write that out again here. So what we get is we say the probability trajectory, probability of a particular directory tau I, given some reward model phi, is equal to 1 over z of phi. And I'll define that in just a second. e r phi tau I, where z of phi is our normalizing constant because we have to have a well-formed probability distribution. So let's say this is structurally like what it looks like. And notice that we can also write this in terms of states. So this is also equal to e to the sum over all the states inside of your trajectory. where I'm sort of abusing notation a little bit to both use r phi of tau or r phi of state just to mean the reward you get from a particular state, or the reward you get from a whole trajectory. So notice we can use each of these, and this is our thing. So why is this helpful? So we don't know what the reward function is. We don't actually have that, right? Yes. But what this means is that since we know what the functional form is of the probability of tau under the max entropy principle, we can now say, OK, I'm not going to worry about this part. I'm going to assume this is the structural form. Now my unknown is just phi. Now I'm going to try to maximize the likelihood of my observed data by changing the parameterization of phi. So this observation. And when I say this observation, I mean that the probability of tau that maximizes entropy, constrained entropy, looks like normalized exponential means we can now estimate or learn r phi by maximizing the probability of our observed data. So we're going to treat this as a maximum likelihood problem. All right, and I'll just note here. This is a really elegant observation. This came all the way back from Jaynes in 1957. So when people were thinking about what does it mean to maximize the entropy of something-- subject to some constraints-- they realized that you could make this could convert it to this exponential family. And then once you have that, now you have something where your uncertainty is only with respect to this phi. And in fact, this type of insight at a very high level will be related to what you'll see next week in terms of direct preference optimization, where sometimes we might be able to reparametrize our objective function to be able to get rid of-- you might call them almost nuisance parameters-- things that you might not care about directly-- where you have one parameter you really want to learn. OK, so let's see how we can do the maximum likelihood. So now we're going to try to do is we're going to try to actually learn that reward function, and we're going to leverage the fact that we the structural form of this probability distribution over the trajectories. So what we're going to do is we're going to say we're going to maximize phi of log, the probability of all of our data. This is our expert of the probability of each of those trajectories. So we're just saying we're going to try to maximize the probability that we observed the data that we did under our reward function. And because of our structural form, we can rewrite this as follows. This is going to be a sum. So I'm just going to say log of product is the same as sum over the logs. Then I'm going to plug in what my form told me that my probability distribution has to look like for my trajectories. All right, so this is just me plugging in that sort of max entropy form of the trajectories. And now I'm just going to split that apart. So I'm going to rewrite it. The log and the exponential cancel, and then I have log of the normalizing term. Now notice that in terms of this part-- so notice that this is independent of tau star. So this is-- independent of all those, we end up with two things. We have max over phi sum over tau star and our data set of r minus the size of our data set times log sum over tau. And the reason that happened there is this was all inside the sum. This sum, this was completely independent of tau star. So I could bring it out. The number of trajectories that I have in D is just the cardinality of D. All right, so now what we can do is take a derivative. So I'm going to call this whole thing J of phi because it's all parametrized by my particular reward function. I'm going to take the derivative of that because, in general, we're going to do everything with gradient descent as usual. OK, so this is going to look like the sum for all my trajectories of my expert data set, the derivative with respect to my reward function, minus-- let me see if I can make this nice and big for this part. OK. So we're going to have two things. We have this log. So we're going to have to take the derivative of that. This all goes on the bottom. And we're going to play a small-- we're going to observe something in just a second. So then we're going to still have our sum over tau e to the r 5 tau times the derivative. So I'm just taking the derivative of that whole term. All right, the important thing to notice here-- so I just took the derivative of both of these parts. This thing should look a little bit familiar. This is, in fact, just the exact expression we got for what is the probability of a particular trajectory. let me just put that in. So note this is just equal to the probability of tau, given phi. Let me make sure I put given phi because that's just this normalized exponential divided by that. So we're going to have this. When you put that-- I shall be careful. Let me just make sure I write that carefully. So this is going to go to this because this is equal to e to the r phi of tau divided by the normalizing constant. So we could move this-- we can move this outside part into here. And then that expression in there, which is this, is just equal to probability of tau given phi. all right, so this-- move back to here. OK, so we just end up with the following-- we get the derivative with respect to the reward function for every trajectory inside of our expert data minus the number of different trajectories we have times the sum over all trajectories, the probability of that trajectory given by times the derivative with respect to phi at that point. And that's our gradient step. So what this would say here is if you want to take a step towards optimizing, then what we would do in this case is you could compute the derivative with respect to your reward function. All right, we have a few more steps to go. So the next is-- this is all in terms of trajectories we'd like to get in terms of states. So for that, we can just observe the fact that, as before, the probability of a trajectory can be broken down into its components. So this is just equal to t equals 1 to length of your trajectory. The probability of your a given S is like your policy and the probability of St plus 1, given St and At. This is just the probability of a trajectory. And we've seen this before. If we have that the probability of a trajectory is proportional as we've seen to e to the minus r phi of a trajectory, and we know that we can write that also is equal to e to the minus sum over s inside the trajectory of r phi of that state, so then we can think of plugging that in for our derivative. And what we get is the following. So it's just the same derivative now, but in terms of states instead of trajectories. So this is for all the states inside of your expert demonstrations. You take the derivative with respect to that state minus d sum over the states probability of the state, given that in the trajectory, and then the derivative with respect to that state. And why is this interesting? This is interesting because basically what we're getting in this case is we're getting stuff that looks like us trying to match the distribution over states that we see in the data set. Now, when we think of doing this, one other thing to note is where do these sort of state densities come from? So essentially you could think of it as I have some observed states and actions, and I'm going to think about it under a different policy what states and actions that will induce? If you know the dynamics model, and if it's tabular-- so if it's tabular. So if you think back for a few weeks ago, tabular, and the dynamics are known-- then you can actually compute the state action distribution directly using dynamic programming. And pi is given. Then you can actually just directly compute the states and actions. So let's see that. So you could say U1 of S is equal to P of Ss. And then for t equals 1 dot, dot, dot T. So this is time indexed. OK, so this is like-- again, remember, the high level what we're trying to do here is we're learning that we're trying to match the state action frequencies between our observed policy from our experts and what we induce under our reward function. What this is going to say is that you're going to try to estimate a reward function. You're going to try to compute an optimal policy given that reward function, and then you're going to try to count and see what your state action distribution looks like under that resulting policy. If it matches your experts, you're done. Otherwise, you need to keep changing your reward function, your policy, and the resulting state action distribution until they match. All right, so I'm just going to go through briefly so you can see how this is computable. All right, so what this would say is that your distribution of states on the next time step depends on your distribution of states on the previous time step, the probability under your policy-- this is your policy. I shall be a little careful there-- The probability of the action, given the state, and the probability of the state, given the state and action. And you can use this then to sum up over all time steps. what your average density is for a particular state. So what this means is that when you're trying to actually compute the derivative of your objective function with respect to the reward, then what you end up getting is that you can plug these in. And if you have-- so you can write down this. I'm going to get-- so you can see that it's fairly involved, but it is definitely possible. Sum over all your states, probability of the states, given phi t, and your reward function. And this will simplify a bit if you have-- so let me just write out. If your r is equal to just this times some features, then when you take the derivative with respect to the phi, you just get the features. So this would mean that dr phi of S. So if linear, just equal to your features. OK, so I know this is a lot of algebra. But what this is saying is that your derivative about how you want to change your reward will just end up being a sum over all the features you have inside of your data minus this additional term. So you compute all of this with respect to your observed features and the features you have in your data set. All right. So how does this all work when we put this fully together? What we have in this case is you give as input some expert demonstrations. You initialize your phi. And then what you do is the following-- you first compute an optimal policy, given that r phi, e.g. Was something like value iteration. You compute the state visitation frequencies. You compute the gradient on the reward model, and then you update your reward model phi, and you repeat over and over again. All right, I'll just write out what that equation is here. So your derivative here would be your sum over all your trajectories inside of your data set, your features for each of those trajectories minus the sum over the states, the probability of the state, given your current parameterizations and the features for those states. So this is under a linear reward, which is what they derived it for. And so this is what you would do over and over again. All right, so let's pop up a level and check our understanding. Given all of this, what steps in the above algorithm rely on knowing the dynamics model? Is it computing the optimal policy? Is it computing the state visitation frequencies? Is it computing the gradient or nothing required it? I told you that they did say they assumed access to the dynamics model. And I'll just write out that gradient again right here. So let's just take a second to review the algorithm, check in, any questions we might have about it. OK. All right. And all the slides are on the web, so you're welcome to go back, though, I guess it might be helpful to-- most of them I've just been writing out here. But you can also just think back about value iteration and how I was just starting to show that we could use dynamic programming to compute the state visitation frequencies. So remember, you probably all remember value iteration. And then this was the type of equations I was writing out. So to compute the distribution over the next time step of the states, we were doing things like summing over the distribution for the previous time step, as well as the probability of action given a state, as well as the dynamics model. So that's the type of dynamic programming algorithm they were proposing there to be able to do this. And then you can also just think back to what we need and value iteration to be able to do this. All right, why don't you talk to your neighbor and see what you got? There's a lot of variance. [SIDE CONVERSATIONS] All right, good. So this is a nice sort of reminder of algorithms that we've seen from long ago. So the answer in this case is 1 and 2. So to compute the optimal policy generally with value iteration, you need to have access to both the reward model and the dynamics model. So this one is true. This is true. And then the dynamics programming algorithm we looked at also required access to the dynamics model in order to back up and say, if you're in this state now, what's the distribution over states you're going to be in the next time step? Once you have those, don't need this for the gradient. So I'm just bringing that up there. Once you have all the state-- once you've computed all the frequencies, and you have this don't need it again. So assuming that you've done these two things, you don't need it again for the gradient. But it is being heavily used. And as you might imagine, that's also a pretty strong assumption. So do we actually know what the dynamics is? The dynamics model is for, say, like, a human physician making decisions or a surgeon, it seems quite unlikely. You might know it for, like, MuJoCo or something, but probably not in general. So let me just summarize where these things are. This approach has been incredibly influential. As we said, the initial one use linear rewards, and it assumed the dynamics model is known. But there is a lot of follow-up work to this, including some really nice work by Chelsea Finn, who is faculty here now and have been for a while. But as part of her PhD, she showed that you could use general reward and cost functions, like, things like deep neural networks and others. So you could use much more complicated functions and state spaces where you're not going to be able to use dynamic programming to be able to enumerate the distribution over states. And then also she removed the need to know the dynamics model. So they had a really nice paper in 2016 showing how to do this with really sort of very general rich, complex state spaces, which has also been highly influential. But I think this idea of saying like how-- at a high level the challenge was, what do we do about the fact that there are many reward models that are compatible with people's behavior? One thing you could do is you say, well, the one we're going to learn is the one that has maximum entropy. And this provides a recipe or an approach to trying to tackle that problem. And it turns out that can be very effective in many cases. And in Brian Siebert's approach, they ended up using it for trying to model taxi drive cars-- taxi car drivers, et cetera. But it's been used in many cases since. So let's pop up a level. We're just finishing sort of our introduction to imitation learning. What we've seen is that imitation learning is this nice approach where if you have access to existing demonstrations-- and it might be hard to write down the reward function-- you could try to learn from those what optimal behavior is to try to match the behavior of access to. In some cases, it can greatly reduce the amount of data needed to learn a good policy. We haven't talked a lot about that precisely, but there's some really nice work on the theory of imitation learning in RL and thinking about some ideas we'll see later in this course around sample complexity of is it provably harder to learn from optimal demonstrations versus in the RL setting. So there's a lot of really nice aspects for imitation learning. The things that I think you should know in terms of going forward is you should certainly be very familiar with behavior cloning because it is a technique that is used very frequently. So you can just reduced RL to supervised learning when you have demonstrations. But it's also good to understand what this principle of maximum entropy is doing, how that relates to distribution over trajectories, and how that is then formed into a maximum likelihood optimization problem to learn the reward model. And I think one thing to notice in this case is when they did this, they are not claiming that is actually the reward model used by people. It is the reward model that is compatible with people's demonstrations that maximizes-- and a distribution that maximizes entropy. So it is not necessarily claiming that it is exactly mapping human preferences. Awesome. So now we're going to get into-- this is one example of human feedback or human input to trying to use that to make good sequences of decisions under uncertainty, but there's actually a huge number of different ways to do this. And so now we're going to-- this class and next class, we're going to talk some about human feedback and reinforcement learning from human preferences. And I think you can think about this from many different levels. You can think about it in terms of how could humans actively try to help reinforcement learning agents that they are trained to do something? Like, maybe they want to train the robot how to clean up their counter in their kitchen, and they have a particular way they want to do it. And so they might be actively trying to help an agent do a particular task; or we might be just trying to align, say, large language models with our values, our intents. And so then could we provide information that's going to shape their behavior across many tasks? So it is relevant to both of these different types of objectives. And I'm going to go through some different ways that people could be using human input in terms of these sort of training. So one thing to note is that people have been thinking about this for quite a long time. I like this work by Andrea Thomaz and Cynthia Breazeal from MIT. And they had this thing-- it looks pretty primitive now, but this thing called Sophie's kitchen. And the idea in this case is that you would be trying to teach an autonomous agent how to make a recipe or do some basic different tasks in the kitchen. And of course, as you can see with this, we've come a long way in the last 20 years, which is wonderful. But the kind of key insight here was, well, maybe we could learn much faster if you have a human. Like, instead of having an agent that's trying out things like Epsilon-Greedy and sort of exploring in the world by itself, that's not how humans do it most of the time. Most of the time, we have things like schools or guardians or friends that are giving us lots of feedback and help when we're trying to learn tasks. And so their insight was to say, well, let's try to do more effective and efficient robot learning by leveraging the fact that you can have a human in the loop that's providing feedback to the robot. And in this case, I think one thing that's important to note is that the robot is getting two different forms of input that they're trying to maximize. They're both getting input from the human, and they're getting input from the environment. So, for example-- I don't remember if this exactly was in that particular domain-- but you could imagine something like maybe there's intrinsic reward if you drop something like a big cost. But then maybe the human also says that's good when you make the right recipe. So there's two forms of signals that are being used to train. So this is an example where it's more like DAGGER. You have a human in the loop, and they are trying actively to help the robot all the time. Another version of this is the TAMER framework from Brad Knox and Peter Stone over at UT Austin. And what they were, again, looking at is like, well, maybe we can train agents to do things much better and much quicker if we're willing to be in the loop. And these are all different approaches than the DAGGER approach. In this approach, so what are we looking at? Again, this was older. So this is looking at Tetris, a video game. We're trying to stack blocks and clear lines. And what you could see in this case is a lot of the previous work like of policy iteration-- and it doesn't matter exactly what these algorithms are, but there are these sort of competitive algorithms at the time. We're, at game three, getting nothing. Like, they just weren't clearing any lines. But after a while, they could start to learn much better. They could get many more later. And what they found here is that by using human feedback, they were taking human feedback, and they were learning an explicit reward model. So one thing you could imagine you could be doing is doing something like model-free RL where you're getting signals from the human, and you're using that to update the agent's policy, but then you drop it. You're not doing any parametric modeling of the reward model. In this case, they are trying to explicitly build a reward model from the human feedback. And you could see that they could get much better performance very quickly. But just kind of maybe the problem with DAGGER, people aren't going to stay around for thousands of games. And so you may not be able to exceed performance, at least in this case, if you allowed the agent to train for much longer. But I think this is another example of where-- so this is sort of a place where they're starting to do model-based approaches, where you were actually explicitly training a reward model from human feedback. And I think it's nice to think about there being at least one sort of continuum of the type of support that humans could provide-- really, probably is multi-dimensional, but at least one-- which is, you could think of-- if humans are willing to provide data at all to train RL agents, one might be, I'm only going to give demonstrations that I'm going to do anyway as part of my normal behavior, or maybe that I'll do once. And then in another extreme is something like this DAGGER or this constant teaching, where I'm willing to be a coach for my agent, and I'm just going to sit there the whole time. And one of the things you might wonder is, like, well, what's in between. This is clearly a spectrum. And one thing that a lot of people have thought about quite a bit over the last 15 years is where their preference is. Pair comparisons is that sweet spot. So the idea in this case is that you're not going to ask people to do constant teaching. You're not going to-- but you are going to ask them to do a bit of work. And in particular, you're going to ask them to be able to compare different types of behaviors and which do they like better. This is kind of in between on the level of human experts. So one of the first places this was discussed a lot was recommendation ranking systems. So Yisong Yue, who's a professor now at Caltech, together with his PhD advisor, Thorsten Joachims and others at Cornell, did some really nice early work on thinking of if you have recommendation ranking systems. So imagine you have two different retrieval functions, and you put in some query. And this gives you this-- the retrieval function A gives you this series of outputs. And retrieval function B gives you the other. And you'd like to learn because you are Google or Bing or things like that. You like to learn which of these two is better. And so the idea that they came up with in this case is, well, we can ask people which one is better. And in particular, you could ask people to compare, say, maybe the first item returned or the second or the complete ranking, which one is better. And that's something that might be much easier for people to do than to specify a scalar reward function for how good it is that CS159 is returned to your query. Is that 17.3? Or is that 2,006? Or is it minus 7 billion? It seems very hard to ask humans to do that, but they probably can do the comparison. And they can say, well, it seems a little bit better. So that's one area. And that was sort of one of the early areas of people thinking about how could you get feedback on recommendation systems so that we could make them better? But there are lots of other applications. And as you can see, robotics and driving is one that people have thought lots about. So this is work by Dorsa Sadigh, another one of my great colleagues here. And what they were doing here is to think about if you're training a car to have different behaviors on the road, how do you get input from humans about which types of behaviors are going to be acceptable? So, for example, most of us would probably prefer the thing on the left than the thing on the right, because the thing on the left is not involve a car accident. But it is hard to write down an objective function for all the things you want to do when you're driving, including, like, if it's hailing, or if a car suddenly stops in front of you, or it's pretty subtle. And so what Dorsa and her colleagues showed is that people can do this sort of preference pairs. And in fact, she has done-- she and her lab here has done lots of really interesting work on thinking of which preference pairs do you show to people so that you can quickly try to get a sense of what their preferences are to try to respect this human effort aspect? So these are just two of the examples of the places that people have thought about this. And of course, ChatGPT is another, and we'll see more about that in a second. So in general, pairwise comparisons might be in this sweet spot, because it's often easier than writing down a reward function. And it's much less work than sort of DAGGER and having to constantly say-- you could imagine in recommendation systems. What is the perfect response to this query of, I don't know, which courses involve this? That might be really hard for people to write down, but it's easy for them to do the comparisons. Now, how do we think about this? Well, one way we could think about how do we mathematically model this is the Bradley-Terry model. And as we've seen with trying to understand modeling scalar rewards, when we start to think about having people compare preferences, often we will still be really interested in understanding a latent reward model. So the idea will often be that we assume that people have some reward function in their head that maybe is hard for them to articulate. And what they can do is produce preference pairs that are compatible with that underlying reward function. Now, they might be noisy. A lot of us all make mistakes. We all make mistakes. And so we're not going to assume that necessarily what I say is perfectly corresponding to my latent reward function, but it's going to be noisly related to that. And so Bradley-Terry model is one of these types of models that tries to relate kind of internal preferences over items to how we might compare them. All right, so let's first start off with a simpler setting before we get to RL of K-armed bandits. We're going to see K-armed bandits shortly in the course. But for right now, you don't really need to know what it is except for there's only actions. So there's no states right now, just like you have K different actions. That's all. And they all have different rewards. And what we're going to assume is that a human makes some noisy pairwise comparisons where the probability she preserves prefers Bi, so item B, compared to Bj, is like the exponential model that we saw before. Exponential models come up a lot. So it's going to be the exponential of the reward for Bi divided by the exponential of reward for Bi plus the exponential reward for Bj. OK, so what will be the probability that I prefer Bi to Bj if actually their reward is identical to me? Actually, I'm like, I don't mind whether I have, I don't know, like deep dish pizza versus flat. I actually do have preferences. But imagine that I don't. So what would the probability be in that case, according to this model? So my internal reward for both of them is like plus 20 because I really like pizza. So what would that be for this probability if the two items are identical? Fisher said. Yeah, so this is automatically normalized. So it's 50% at most. If I like one thing much more-- I do like deep dish pizza a lot more. So it's probably more like that would be, say, 100 versus 10. And in that case, my probability would be more like, say, 0.9 or 0.95, something like that. So this was just a particular model. It is noisy. If you read the-- if I put a link later to the reinforcement learning from human feedback paper, they make some additional assumptions of how people make preference pairs on top of this model. But this is the basic model that a lot of people have been looking at recently to understand how internal latent rewards relate to external preferences. One of the important things to note here is that this model is transitive, which means that if I want to know what this sort of probability is between i and k-- so those are two particular items-- I can deduce it from my probability for i to j and my probability from j to k. So you can kind of chain things. So this is a transitive probability model. So this was introduced roughly 70 years ago. It's a very popular model, and it came up early on in terms of recommendation systems and others. So another thing that's useful to think about is, OK, in this setting where I just have k different actions I can take, and I want to understand-- I want to learn what the reward is for somebody, for all of them-- If I want to think about finding a maximum one, say, like what's the best arm or what is the best action, I might want to try to understand how, under these different preference models, what it means for something to be good. So in the class so far, we've often talked about just maximizing the value function, and we want to find a policy that's good. Now let's just think about if I have no k arms, which of them is best? So a Condorcet winner is one where, for every other item, you prefer item i to all the other items. So like, of all the types of pizza, if I like deep dish most, that's the Condorcet winner. And it doesn't mean that it has to-- that those probabilities have to be probably one. It just has to be greater than 0.5. It means I have to beat all of the other options. And I'm bringing these up right now because there's also been some later discussion of how does all the recent RLHF work relate to ideas from social choice and computational economics about what are we computing? What is the sort of underlying objective we're computing, and how are we distinguishing between different sorts of responses LLMs could give us? So the second thing-- so this is a pretty high bar. This means that there has to be one thing that beats everything else. A Copelan winner is a little bit less. It just says it's the winner if it has the highest number of pairwise victories against everything else. So that doesn't mean that you have to prefer it to everything else. It just that means, on average, it beats the others. And a Borda winner-- an item is a border winner, if it maximizes the expected score, where the score for item Bj is 1. If you prefer Bi to Bj, it's 0.5 if they're equal, then it's 0 otherwise. So it's sort of like this discretization of the wins and loses. And typically, algorithms for K-armed or dueling bandits-- again, we'll go into what bandits are more later-- what they focus on doing is trying to find this. I don't necessarily need to find an item, say, like, a ranking system that is always better than everything else. I want to find one that on average beats everything else, and they often would construct these kind of pairwise matrices where you can think do these different actions, beat these other actions. All right, so how would we learn these? So the question, is we have all of these noisy pairwise comparisons. And what we want to do now is to see if we can extract these underlying reward functions. Why would we want to do that? Well, once we have these underlying reward functions, we can figure out which arm is best or which action is best. And in the reinforcement learning case, we could try to optimize for that reward function. So how do we do that? What we're going to do is we're going to assume we have N tuples of the following form. We have item 1, item 2, So item i, item j, and mu, where mu is 1 if you prefer the first thing. Mu is 0 if you prefer the other thing. And it's 0.5 if you don't care. So this is just like a classification task. You can just think back to your supervised learning where you just have-- it should look very much like a logistic regression task. And you can maximize the likelihood with cross-entropy. So we had to map it back to a standard logistic loss where we say these reward models-- and in general, we're going to parameterize these as like deep neural networks or some other complicated function. It could be linear. It just depends. But once we have that, then we can try to find the set of parameters that maximize the likelihood. So that's how we could fit a reward model when we are given preference pairs and observed preferences. Now, you might wonder, how do we do this in RL? Because in RL, we have states. We have multiple actions. We have trajectories. The idea is pretty similar in some ways to what we were seeing with max entropy. And that what we're going to do is to say, well, we have a trajectory. If we have a trajectory, we can think of there being a series of rewards. So the reward of that trajectory is just the sum. So I plug in all of those sums. And I prefer a trajectory if I can get higher reward for that trajectory than the other, according to the same model. So I essentially just map it back to as if it was kind of like a bandit, just now that I have two different trajectories. And we'll see an example of this in just a minute. OK, so what do we do? We now are going to ask people to compare trajectories. We'll use that, then we're going to learn our reward model. And then once we have our learned reward model, we can do PPO or something with it. So this gives us a reward model for our domain, and now we can try to optimize our policy with respect to it. So let's look at an example. So in the reinforcement learning from human feedback, more precisely called deep RL from human preferences-- this came out in 2017. And they wanted to train something to do a backflip. And what they noticed here is they needed about 900 bits of human feedback in order to learn to do this. So let's see what it looks like. All right. OK, so what someone's going to be doing in this case-- so remember, they're trying to train this sort of little MuJoCo-like thing to do a backflip. So what they're going to show people is they're going to show them little clips. And they're going to say, is the thing on the left doing a better job of trying to do a backflip, or is the thing on the right? And they're just getting people to click left, right, left, right, left, right, left, right. And so they're not having to say what is the reward function for doing a backflip? They're just saying, I don't know, this one looks closer to a backflip or better. And so what you can see here is that some of them are going to be much better at getting close to doing a back flip. So some of those is actually pretty good. And what they are saying is that they only needed about 900 examples in order to train it so that it could actually learn to do a backflip, which is pretty good, particularly if you think back to deep Q-learning and the enormous amount of data and training that they're often doing for, say, trying to learn Atari, et cetera, which is literally millions. So this is really cool. This is possible to do. And this is something that you're going to be doing in your homework. So in homework 3, you're going to be doing both RLHF and DPO. So I'm really excited about this assignment. This is the first time we're doing it. So you can actually see how this works. So you can see how we can actually learn from human preferences. We're not making you do the human preferences. We're going to give you a data set for how we can actually train these agents. Now, I know we're almost out of time, but I'll say just a little bit about this. I'll probably have a bit of time on Monday before we have our guest lecture. But just I want to give you at least a little taste. So this was in 2017, that paper was. And there was attention to it. But I feel like in many ways, there wasn't a huge amount of work on that until much more recently. So I just want to share a couple of slides from Tatsu Hashimoto's NLP class. So if we just think back-- I think I showed this slide on the very first day of class-- how is RLHF being used for ChatGPT? What they're doing there is they're getting demonstration data, and they're doing supervised learning. This is basically what we would call behavior cloning. Then they're going to get this comparison data and train a reward model. Now, in their case, they might not just use two. You could actually have people rank between, say, four or something like that. And you can extend these models to do that. So you get labelers to do that, then you train the reward model. And then use PPO to actually update the large language model. Now, one thing that I think is important to note in this case is that this is all really an instance of meta reinforcement learning in the sense that what they're going to be trying to do here-- unlike where we've seen like you want to train something to do one task, like, being able to do a backflip-- they're trying to learn in general a reward function that covers all the tasks that people might want to do with large language models. And so it's this multitask problem. And so when they do this, they're going to give you a new prompt, like, write a story about frogs. And then they will want the agent to do well on that, which is likely a task that has maybe never seen before in its data. So I think that's also another important thing to note here is that the reward models that are being trained now are things that probably would have been considered multitask settings before. But now we're sort of lifting them and saying your task is just to do whatever humans want to do with this ChatGPT in terms of answering questions. And so how do you train a reward model that will be good for any of those? So we'll continue talking about this next week. We'll talk probably either before or after the guest lecture a bit about how we actually do this. But it basically just follows exactly along the framework that we've just seen there. And I'll see you then. Thanks.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Policy_Evaluation_I_2024_I_Lecture_3.txt

Hey, everybody. And welcome back. We're going to get started with a refresh your understanding poll. You can go to Ed and to see all of the polls for today. Just remember to log in first so that we can log it for participation points. The two questions ask you to think about what we talked about last time in terms of Markov decision processes and what sort of guarantees or type of properties they have. Yeah. Could you read the extend model of the tabular MVP again? Yeah, great question. Tabular MVP is where we can write down what the value is of a state as a table. So like can just have one entry for what the value is for each state. This is in contrast to neural networks or things like that where you don't have one parameter per state. OK, we'll take like another one or two minutes. We have a good amount of controversy on these questions, so we'll see how this converges. I remember seeing this A to the power S number in a previous context. Was it in the context of policy iteration, or what was it? Yeah, does someone want to remember why is A to the S important? I remember why it is. Yeah, remind me of your name. Is it the number of total possible policies? Exactly right. Exactly right. So there's at most A to the S potential deterministic policies. All right. So most people selected the correct answer for the first one, which this is true. So asymptotically, value iteration and policy iteration are correct in the tabular discrete Markov decision process case. And they will asymptotically both converge and compute the right value function. The second one looks like it's pretty evenly split, so I'd like you to turn to someone near you and argue for what you said for the second one. [INDISTINCT SPEECH] All right, so maybe we vote if your answer changed. So the answer is true. And about half of you said that. Does someone want to tell me why this is true? Yeah. I'm not sure if this is correct, but I think that value iteration might not be guaranteed to converge. So if it's not guaranteed to converge, it could just be unbounded. So that certainly would be the case. But fortunately, it is guaranteed to converge if gamma is less than 1. But you're correct that it can require more iterations. And remind me your name. So my thinking was that in making policy iteration, we know that in each step, we're going to improve to a new policy. But in each value step, you might not reach a new policy. You might take multiple steps to reach a new policy. That's right. So in policy iteration, and I talked to some people about this, too, there can only be A to the S. So in policy iteration, there is at most A to the S because you only go through each policy once. But for value iteration, it can be more. And I'll give you an example. So one way I would think, in general, if you see this sort of question is to think about, well, can I come up with a counter example to say where this would be different. So consider a really silly Markov decision process where there's just one state and one action. So if there's one state and one action, there is literally one policy. You can only do one thing, and there's only one state to do it in. So that means that policy iteration is going to take one round. But for what value iteration is going to do, is it's going to keep going until the value function stops changing or stops changing within a very small amount. And so what would happen in value iteration, and feel free to go back to your notes from last time, is we would start off. And let's say the reward is 1, and the gamma is 0.9, and we initialize the value function to 0. So then if you use a geometric series, and if you haven't seen that before, just come chat with me about it. I'm happy to say. The V star of this date is 1 over 1 minus gamma, because you get 1 plus gamma times 1 plus gamma squared times 1 dot, dot dot. Because you're always staying in that state, and you're always taking that action. And day in and day out, that's what you get forever. So the actual value function that you get eventually is 1 over 1 minus gamma, or about 10. But after the first iteration, value iteration V1 of s is just 1. So that means 1 is not close to 10, or not that close to 10. So we haven't converged yet. So we'll have to continue to do a bunch of iterations of value iteration, whereas at that point, policy iteration would stop because it would just evaluate the value of the one policy we have, which is to take that one action, and you'd be done. And I bring this up just to illustrate that, even though both of those algorithms are guaranteed to converge to the right thing, eventually, they can have quite different behavior in the short term. You got a question? I was just going to ask, is it only converging in the limit? All of them are only converging limit. So all of them are converging asymptotically as you do this over and over again. Good question. Great well, welcome back. If you just came in, feel free to go through the questions later. What we're going to be doing today is to continue in this more simple setting where we don't do any function approximation. But we are now going to think about the fact where we don't have models of the world. And what I mean by that is that we're not given a dynamics model, and we're not given a reward model. Our agent just has to try things out in the world to learn how good they are. And we're going to start with model free policy evaluation, in the case where we still have a small enough number of contexts or states that we could write down a value for every single one of them separately. So that's why we call it tabular. I'll just say one thing in terms of logistics. So office hours are on the website. We'll try to keep that calendar updated. It's just a Google Calendar. It'll include the location. And if you go to Q status for the one-on-one office hours, we'll make sure that the Zoom is either on that or on the website. I'll have office hours starting this week on Thursdays. And mine are for project and conceptual questions. So feel free to come ask me about anything in class. I won't be going through code, but you can go talk to the TAs about that. Or you can come in and brainstorm about projects or general questions about reinforcement learning. I'm sure some of you are starting to think about projects. Just in general, some people are asking, so what's kind of in scope. I put something on Ed about that. But also, in general, it can be a new idea for a reinforcement learning, a new application. It can be something you're doing if you're already doing research in reinforcement learning. It can also be replicating part of an existing paper. And this is really helpful, actually. It's helpful for the whole community because there is a lot of work going on, and people are making different choices about hyperparams, and seeds and things like that. So it really is very valuable to see what things we can replicate. Does anybody have any questions about that or other logistics of the class before we get going? All right. So let's get into policy evaluation. So as I said, what we're going to be doing today is to think about how do we learn through direct experience how good decisions are. And we're going to assume that we have a fixed policy. So again, like our boss says, how good is this way of advertising to customers. Or maybe you're in a setting where you're trying to see how good is the patient outcomes from the current protocol. And the idea is that we're only going to be using data from the environment. And today, this experience is going to come from executing that policy. Let me just move this up so you can see a bit better. OK. Today, we're going to assume that when we get this data, it's from directly executing a particular policy. Later, we'll talk about other relaxations to this. And so I'm going to try to motivate today why this is a useful thing to do and what sort of properties we would want to try to compare different algorithms. So it will turn out that this type of thing comes up all the time. It comes up when we actually want to make decisions and learn better policies. And it's going to be an important part of much more complicated algorithms, like deep Q-learning, and policy gradient and other things, where we want to be able to see how good our current things so that then we can take gradient steps or improve our policy. So this is what we're going to try to cover today. We're going to cover Monte Carlo policy evaluation, temporal difference learning, certainty equivalence, and batch policy evaluation. And maybe raise your hand if you've seen temporal difference learning before. OK, raise your hand if you've seen Q-learning. OK, great. Q-learning is the control version, basically, of temporal difference learning. So you'll see a lot of similarities there. All right. Before we dive into this, I just wanted to recall a couple of definitions we'll have. We're going to use G for the return, which means that, from this particular state, what is the discounted sum of rewards we get for a particular episode. The state value function is saying, on average, what is that reward we get. And the state action value says, if I start in this state, take this action and then follow this policy, what is the expected discounted sum of rewards. So we saw last week that we might want to do dynamic programming for policy evaluation when we do have access to the models. So again, what I mean by that is that where someone gives you a function for the reward and a function for the dynamics model. And so we saw we could do this sort of Bellman like backup for a particular policy. So it's different than the Bellman equation because there's no max. I'm not trying to take a max over different actions. We're just taking whatever action is specified by the policy. And in this equation here, this is for like a deterministic policy. Otherwise, we'd need some additional averaging over all the actions that could be taken by that policy. And just to remind ourselves here, before we converge, this policy, this V pi k, is an estimate of the value of the policy. It's not the actual value yet. It's just an estimate, and it's, hopefully, improving over as we do more iterations. And another good thing to remind ourselves of is that this is here, this sort of expected discounted sum of rewards, what we're doing in this equation is we are plugging in this term as an estimate of the expected discounted rewards for the future. So we're saying, we've got this estimate of the value function. We're going to plug-in and say, my reward is my immediate reward plus my discounted sum of future rewards. And this is what I'm using for my discounted sum of future rewards. And so this is known as bootstrapping because we're plugging in one estimate in order to help us do another estimate. And we'll see a picture of this graphically later. All right. Monte Carlo policy evaluation is a really simple idea, but it's very useful and it is commonly done. So essentially, the idea with Monte Carlo policy evaluation is, we are just going to simulate or act in the real world. So this is just saying you've got a policy, which means what action to take in every state. Today, we'll mostly focus on deterministic policies just to make it easier. So I'll just say that, for most of today, assume pi is deterministic. But all of these ideas can easily be extended. Just easy to write that down without having to do an expectation over actions everywhere. OK, so what's the idea in this case? Well, the value function is just an average over the returns. It's an expectation over the trajectories or the returns you could get by following the policy. And therefore, the value is just the mean of returns. And we know how to approximate means. We just do things a bunch of times and we average. And so as an example of this, it might be something that someone says, OK, I want to know if we, say, give a particular set of patient treatments, and maybe those treatments take a year, for example. And someone wants to on average, how good is that. Well, what you could do is you could have 100 patients. I'll go through that particular protocol for a year and then average their outcomes. And that would be an example of Monte Carlo policy evaluation, because you just execute the policy for many different episodes, and then you average. And one thing just to note here is that you can have cases here where not all the trajectories are the same length. So imagine, in the patient case I just gave, you might have that some people drop out of a trial during the year. Or maybe they finish their treatment successfully, and so then they're also done. So all the trajectories may not be the same length, but essentially, you can just think of it as you just have many, many trajectories. Maybe this one has a G equals 10. This had a G equals 5. This has a G equals 10. And you just average over all of them. And that is your value function. Now, one of the benefits of this is that, when we do this, we're not actually-- benefits or drawbacks. And we'll talk about this. This is making no assumption that the system is a Markov decision process. It's just averaging. So your system might not be Markov. What I mean by that is that you, in general, will have a finite set of features to describe the state. So if we think about that patient example I just had, maybe you have different vitals of the patient. Maybe you have static demographic variables. But we do not have all the features that are probably going to describe how someone's going to react to a set of treatments. And so because of that, you may or may not think that, in the features you have access to, the system is Markov. But this doesn't require the state to be Markov. It's just averaging. You just roll out your policy many times and you average. Now, a really important thing here is that it can only be applied to episodic MDPs. So what do I mean by that? I mean your episode has to end in order for you to see what the total return was. So if you have horizon lengths or episodes that last for a year, that's OK. It's a little bit slow, but you could do that. But if you want to just think of how good a policy is, if you just are going to act forever and never stop, this wouldn't work, or not without some additional approximations. Somebody have any questions about either of these two things, about it not assuming the state is Markov? Yeah. You were talking about medical treatments. So does this only work if the treatment only lasts the same amount of time for every patient, like six months? Because they have different lengths. How can that be episodic? Great question. Yeah, so what [INAUDIBLE] is like, well, could it be episodic if the episodes are different lengths? It could be. So it could be that you have a fixed policy. And maybe that policy says, if someone doesn't respond to this type of treatment, we do this additional type of treatment. In a clinic, that's very common. As long as the episode is guaranteed to end, you know the treatment could only last, say for a year total, you can still average over all those outcomes. You just sum over the return for those different ones. Yeah. For each of these trajectories, are we supposed to begin with this different state? Or we can actually start with the same state? Great question. So if we want to get a value function for all states, we need to see all the states inside of these trajectories. And we'll talk about how we estimate these in a second. So I think this will be answered. So for example, how might we compute this? So we would like to get the value for all the states that are reachable inside of your policy. So what you could do is, we can initialize two different variables. One, N of s is just going to be the counts, the number of times that we've updated our estimate for state s. G of s here is going to start with 0, which we've never seen any returns from this state. So what every visit Monte Carlo does is it samples an episode. So this goes up to some time step T, TI. And this has to be finite, but it could be different on different episodes. And then we compute the discounted sum of rewards for that episode. OK, so starting at time step T, how much reward do we get till the end of the episode? And then for every time step until the end, we see, is this is the first time that state's been visited. Then we update the total number of times we visited this state for the first time per in each episode. We increment the total return and we average. So it steps along, and it just says, OK, maybe the first time I reached-- if you think of the Mars Rover example, I have it in the next one. No, I think I put it-- so for a lot of today I've moved a lot of the worked examples till the end of the slides. But if you want to go through them later, I encourage you to. So for example, in the case of the Mars Rover, you might imagine you start in state S3. And then on that particular one, you get a reward of 1 for that episode. And so then you would average in 1, starting in that state S3 till the end. So this is first visit Monte Carlo evaluation, which means you only update a state at most once in each episode. So if you had something like this, S1 went to S2, went to S3, went to S2, went to S3. So in this case, dot, dot, dot, you would update S1 once in this trajectory, and you would update S2 once, and S3 once, even though you, in fact, visit those states multiple times. You only update them for the first time you visit. Yeah. So this have the problem where the state is really rare or uncommon, then we get a really bad [INAUDIBLE] or we just never visit it at all? So if you don't ever visit a state under the policy, that's OK, because then you don't have a value for it. But then it's kind of undefined because you never would reach it. In this case, as [INAUDIBLE] says, if there's a state that's really rare for you to reach inside of your policy, it might take a lot of trajectories in order to get a good estimate of it. So maybe there's some rare side effect of a treatment plan, and it's going to take a lot of trajectories to observe that. And that was one of the challenges with the COVID vaccine, is that, of course, it was a finite number of people. It was a pretty large number, but pretty finite. And some side effects don't show up until you get many, many more. It's true, generally, for treatments, even if on average they're totally fine. But you won't see some of those rare side effects until you get an enormous number of trajectories. Now, COVID vaccine certainly had-- the benefits there way outweigh side effects. But my point is just to highlight that, depending on how frequently you see the states, it will take you more or less a number of total episodes in order to observe. Yeah. Is it fine if it doesn't matter that I saw S2 again or S3 again in this trajectory? For this algorithm, no. It probably affects the reward still. It's just that you don't use that data. So an alternative is called every visit, where every time you see the state in that trajectory, you update it. And as you might imagine there, let's say you see it's a really long trajectory, and you see S too many times. Then you would update for all of those. So I'm just going to show you three common different ones. So this is a worked example you can go through later if you want, which is, if you imagine this is the Mars Rover, the rewards are on either side. This is the particular trajectory. You can compute the first visit and the every visit Monte Carlo estimates. So both of those are totally reasonable things to do. Perhaps more common is what's known as incremental Monte Carlo. And this does kind of what you would expect it to do, which is you maintain a running estimate for what is the value under a policy for a particular state, and you smoothly update that as you get more data. So what we would do in this case is you keep track of the number of times you visited that state, and then you weigh your old estimate by the number of times you've visit the state minus 1 divided by N s, plus your new return you just observed divided by N s. So that's just sort of your way you're kind of constantly updating your value function for this state as you get more data. And for those of you who have done machine learning, which is probably most of you, this should look pretty familiar. This is kind of like a learning rate. This is your updated value, and this is your old value. And in fact, that's what we're going to see here. OK? So you can think of this in general. It doesn't have to be 1 over N. It can just be any alpha here. So any sort of alpha here is just a learning rate. And we're just smoothly updating our estimate of what is the value function for a particular state. And we'll see lots, lots and lots of algorithms like this, similar to probably what you saw in machine learning. The key thing here is the estimates that we're using, which we also might use the reward targets often. We have this estimate here, and then we have our old estimate. So this is one sample of what is the return, starting in this state till the end of the episode. So I think it's helpful to think a little bit about what this looks like pictorially. And this also relates a lot to-- we'll see this when we talk about things like AlphaGo. So I think it's helpful today. And it'll look somewhat familiar to you if you've seen things like minimax trees or expectimax trees. Who here has seen expectimax trees before? OK, so maybe one person. So most people, not. So this might be a useful representation. So I think one way to think about this is, what we're trying to do is we're trying to think of what the value is starting in a certain state. And we know what the action is we're going to take because we've got to fix policy. So that says we start in this state s. We take this action a. This is prescribed by our policy, so this is pi of s is going to equal. And then after we do that, because the world might be stochastic, we're going to have a bunch of next states we could reach. So we have probability of s prime, given s and a. And what we're trying to do when we do policy evaluation is we're trying to get an expectation over all the potential futures we might end up in by following this policy. Maybe in some cases, there's really good patient outcomes, hopefully, most of the time. And maybe sometimes, there's less good patient outcomes. And we want to do an expectation over all of this. So we can think of that as a tree, as just, we start in a state, we take an action, we look at the branching factor of all possible next states, and then we repeat. Excuse me. So if we think of what the policy evaluation diagram is doing, for each state, we know what the next action is we take. And then we branch again in terms of states. So this is like s prime, and this is s double prime. And so we can just think of this tree of possibilities going out. But we don't have any there's no branching on the actions because the actions are fixed by our policy. So then if we go all the way out, and then what we want to do is we want to figure out what the value function is, you can think of this as an expensive way to do dynamic programming. What you would do is you would take an expectation over the states, and then you would propagate those values back up to the root. And if you don't find this a useful conceptual way to think about it, it's fine. I think it can be helpful to then think about what these different algorithms are doing in terms of approximations. So this is what we would like to do in order to get v pi of s. But we want to do this in a much more computationally efficient way, and also, sample efficient way. So what Monte Carlo policy evaluation is going to do is it's going to look like that particular equation here. And what it is doing here is it is going to approximate these full expectations via sample. So in particular, what it's doing here is it's updating the value estimate by using a sample of the return to approximate an expectation. And we do this many times. We average over many such returns. So it's kind of like saying you have this enormous branching tree. You could do an expectation over all of that explicitly up from the roots, or you could just sample many times, and that's also going to approximate the tree. And the more samples you get, the better that's going to be as an approximation of the tree. And this type of idea has been used in many different types of algorithms. There's some really nice work in the mid 2000s by Michael Kearns and others. And then similar ideas were really the foundation that then led to some of the advances of Monte Carlo tree search, and then that went into AlphaGo. So this is what Monte Carlo tree search is doing. So notice, it's not doing any form of bootstrapping. There's no dynamic programming that's going on here. It's just rolling out. And then what we have here is, it is using this here, this sample as an approximation. OK? All right. So that's how Monte Carlo policy evaluation works. One natural question in this case is, how good is that estimate. So we're going to see lots of different ways and lots of algorithms for trying to do policy evaluation. And so you might now ask, well, how do I pick among them. What are the properties I should think about? So one pretty basic property that you might want is consistency, which means that, as you get more and more data, does your estimate actually converge to the true value of the policy for all the states. And this is something you probably want in many cases, at least, because otherwise, it means that, even if you had infinite data, your estimate is still going to be wrong. Now, as we start to think about more complicated settings, we might have to be satisfied with this less good objective. But here, for right now, we're hoping we can just write down the value of every state as an entry in a table that we should be able to get consistency. A second thing we might want is computational efficiency. We'd like this not to be too expensive for us to compute. We'd like us not to require too much memory. And we'd like it to have statistical efficiency, which is, essentially, how does the accuracy of the estimate change with the amount of data. And what that means here is, more formally, we'd like to know how quickly do these things converge as you get more and more data. And then in reality, we often care about empirical accuracy, just what is our mean squared error for our types of our estimators. So how good is Monte Carlo? Well, let's just first quickly remind ourselves that the bias of an estimator is this. So if we have an estimator theta, which we're going to be thinking of as our value function approximation, it's going to be the difference between, on average, what our estimator is versus the true value. That's our bias. And the variance of an estimator is the difference between this and its expectation squared, the expectation of that. And the mean squared error is going to be variance plus bias squared. So generally, you would like an estimator that has low mean squared error, which means we want it to have low or zero bias and low variance. Something to think about if you're less familiar with these, is whether or not if an estimator is unbiased, is it consistent. It is not necessarily consistent, just so you know. So what we would like here is that, asymptotically, the probability that our estimator-- so N here is the amount of data we're using to construct that estimator-- the probability that, as we get an infinite amount of data, that our estimate is different than the true value by more than epsilon. It has to go to 0. OK, so we would like it to be consistent. So how does Monte Carlo fare on these sort of properties? Well, first visit is unbiased. So it's an unbiased estimate of the true policy. And by the law of large numbers, as the amount of data you have goes to infinity per state, so if you have really rare states, you're still going to need a number of samples to estimate them. But as the amount of data you have goes to infinity, you'll converge. So it's consistent and it's unbiased. Every visit Monte Carlo is biased. One way to think about that is, in the first case, all your data is IID, independent and identically distributed, in that every visit case-- imagine that you visit state s2, and then four steps later, you visit s2. Well, their returns are going to be correlated because they're both in the same trajectory, so they're not IID anymore. So that's just some intuition for why it might be biased. But it's also consistent and it often has better mean squared error because you get to use more of your data inside of a single trajectory to do more updates. And then incremental Monte Carlo methods depend on the learning rate, as you might expect. So see that here? So let's imagine that we are going to have our alpha parameter, which is our learning rate, which is trading off between our new estimate and our old estimate. It can actually change per time step. So just like how you can generally decay your learning rate, you can change your learning rate here. And if your learning rate is such that if you sum up all of its values for a particular state, it goes to infinity, but the square is less than infinity, then you will converge to the true value. And again, these are pretty common types of criteria we'll see for some of the algorithms we have that, under some sort of smoothness, guarantees for the learning rates we'll have some decent properties. Yeah, remind me of your name. If those conditions aren't met, do you definitely not have a guarantee, or are there other conditions that can give you a guarantee, and those are just some other queries? Great question. So he's asking, is it required to have these conditions. These are sufficient. They aren't necessary always. A lot of that will depend on the particular problem domain, too, and what the dynamics and the reward is. To my knowledge, I'm not sure if there are other really general conditions like that, but there might be for specific problem classes. It's a good question. Now, one of the problems with this is that, in general, it's a pretty high variance estimator. So you're kind of getting, certainly, with every visit, or certainly, for first visit Monte Carlo, you're only updating the state at most once per episode. So it can take a long time. So you can imagine that, if you have very different outcomes from the same starting state, so maybe most of the time, you have pretty average outcomes, but maybe one in 100 times you have a really bad outcome. It's going to take a long time for that estimator to converge. So in general, this is a pretty high variance estimator, even though it is often unbiased and it is consistent. And then the other big requirement is that it requires episodic settings. So you have to wait till the end of the episode to update your estimate. And for here right now, that might not seem that bad. But when we start getting into control and decision making, you might want to use the data you have already in that episode to change the behavior of the agent. So you can imagine something like, if you're doing self driving cars or something, you're already getting some evidence that the car is not working as expected within a single episode, that might be really long. You might want to use that information to change how you're steering, for example. All right. So just to summarize here, what it does is it's not using the Markov process. It's updating your value function estimate, using a sample of the return to approximate the expectation. And under some pretty mild conditions, it converges to the true value of the state. And in some cases, it will turn out that, even if you actually know the true dynamics model and reward, you might still want to do this. And I think one thing that's useful to think about here is systems which you think the Markov property might be violated, at least with the features that you'd be using to represent the state. All right. Now let's go on to temporal difference learning. And this is, again, sort of related to Q-learning, which we'll get to in the next lecture. So Sutton and Barto, which is a textbook that is an optional one for-- yeah. I had a quick question. So if we don't the rewards model, how do we calculate the rewards for the trajectory? Great question. So the assumption here is that it's kind of like you either are in a real setting where you can sample these from an oracle, or something in the real world is giving you these. So you may not have an explicit representation for the reward model, but you can get them, so if your customer buys something or they don't, or you have a side effect. So you don't necessarily have a parametric model, but you are getting real rewards. That's a good question. Anybody else have any other questions about Monte Carlo before we go on to temporal difference learning? And I'm going to call it just temporal difference learning now, and then I'll specify that it's actually TD0 for most of what I'm going to talk about. So I'll just specify, mostly discuss TD0. And I'll specify what I mean by the 0 shortly. So Sutton and Barto, which is one of the optional textbooks for the class, says, if one had to identify one idea as central and novel to RL, it would undoubtedly be temporal difference learning. And what their point is, is that it really is sort of a way still to construct estimators, both for control and for policy evaluation. And the idea is, if we think back to that tree I showed you, and I'll show you some more, there's going to be a way to combine between the idea of sampling to approximate expectations and bootstrapping to approximate future returns. And we'll see that in a second. It is model free, meaning you don't need to have a parametric representation of the reward function or the dynamics model. And the nice thing is you can use it in episodic settings, or in infinite discounted horizon settings. You just set off your robot, and then it's just going to have to learn to act forever. And one of the key ideas is that we're going to update our estimates of the value of a state immediately. So I'll put pi here because we're still talking about a policy after every single tuple of state action reward next state. So let's see how that works. So again, remember, our goal is just to compute the expected discounted sum of rewards for a particular policy. Now, let's think back to the Bellman operator. So if we know the MDP models, and we have a particular policy, we could write the Bellman operator like that. And what we were doing in incremental every visit Monte Carlo is we were updating the estimate using one sample of the return. And the idea now is to say, well, this was one sample. But maybe we could just maintain-- we have access to a value function. Why couldn't we look up and instead of having what the rewards were starting the state till the end of the trajectory, we observed a particular reward. We got to a particular next state. Why don't we use the value function for that state? So what we're doing in this case is, instead of using G, we're plugging in the immediate reward plus gamma times the discounted sum of future rewards, using our current estimate of the value function for that next state we reached. And here, the reason, one of the things is that we don't have to wait. We can do this immediately, as soon as we reach s prime. So as soon as we reach s prime, as soon as we see the next state, we can immediately update the value of our current state. So we'll have to wait till the end of the episode. We can use this for infinite horizon problems. So this is what that looks like. And we're also going to call that the TD target. And again, that should look like machine learning, and it should look like what we just did with Monte Carlo that what we're plugging in here is we're saying, we're taking our old estimate. And we are moving it. We are shifting it a little bit by our learning rate towards our target, which is our reward, plus our discount sum of future rewards, using that plug-in estimate. And when we think of how much our estimate is changing, we often call that the TD0 error, which looks at how different is my current estimate of the value of a state versus the estimate that I'm plugging in. And again, if you've seen Q-learning before, this is going to look really similar to what we had, but there's no max or things like that. You'll see those soon. So the TD0 learning algorithm just looks like the following. You sample a tuple of a state action rewards next state. You update the value for that starting state, and you repeat. And so your t goes to t plus 1, and then you get the next tuple. You just do this over, and over, and over again. So in our Mars Rover example, you have state action reward, next state, you update, and then you just shift along. Let's see what that might look like here. So in this case, let's imagine we have a policy where we always take action a1. We're going to make our discount factor 1 to make the math easy, and we're going to assume that any action from state 1 or s7 terminates the episode. And then what we see in this case is, we have the following trajectory. We start in state s3, we take action a1. We get a reward of 0. So this is the reward. We transition to state s2, and so forth till the end of the episode. So what we would have in this case is that we would make it so that the first update we would do be V of s3. And what we would say is that's my old estimate of V of s3 times 1 minus alpha. I've just rewritten the above equation here because this was basically 1, and this is alpha times minus V. Plus alpha times the immediate reward plus gamma times V of s2. So that's what that would look like. And here, imagine that I've initialized all of them to be 0 to start. So this would still just look like 0. And, in fact, what would be the only state I would update to not be 0 in this episode? For it to be updated, not to be 0, either its immediate reward has to be 1, or it has to be transitioning to a state whose value is not 0. Yeah. So what we're seeing here is that, in this case, we have state action reward next state. And so this is the TD update. And what I was saying here is that we've initialized all of them to be 0, which means that, in order for their value to change from being 0, either their immediate reward has to be non-zero, or we have to transition to a state whose value is not 0, because all of them, their current value is 0. Were you going to guess which state is up? Which one? Well, when you're in state 1, you have a reward 1. That's right. Yes, so you don't see any reward here until you get to state s1. I'll just highlight it here. So at that point is when you update your value function. That's the first time that you get to anything that any reward becomes non-zero. So in that case, what you get is s1 is equal to V of s1, 1 minus alpha, plus alpha times 1 plus, gamma V of s terminal. s terminal is always 0. So it just becomes alpha times 1. So why am I making you guys do a lot of algebra here? I want to do it because, if you work this out, and I won't go through it here, but I think it's a useful exercise, the TD episode, TD estimate you would get for your whole value function at the end of this episode is quite different than what you get with Monte Carlo. So TD updates after every single tuple, every single state action reward next state tuple. And so that means, when you reach the end of the episode, if you look at what your value function would be, and I've written the value function here just as a vector, but this is the value of s1. This is the value of s7. So I've just written it as a vector. This is what your value function would be. It would say, my current estimate for s1 is 1, and everything else is 0. But if you look at first visit Monte Carlo, it's quite different. And if we make gamma equal to 1 here, which I said it would be, it would be 1,1, 1, 0, 0, 0, 0. Why is this? Because Monte Carlo waits till the end of the episode, and then it uses the returns to update any state that was visited once in that episode. And the reason that's important is that, now, actually, we filled in a lot more things because we knew. We observed in that case that, not just did we get a reward here, but then we saw what s2 got, which was also a reward of 1 and what s3 got, which was also a reward of 1. And the reason I bring this up is that there's going to be different choices about how these behave, particularly when you don't have a lot of data at the beginning, which may be more or less data efficient or sample efficient. And ideas of sample efficiency will come up a lot. We'll see that a lot later on, but we'll see it on Thursday, as well. All right. So what does this look like in terms of the tree? So if we go back to our tree, which is like expanding out potential futures, what we can see here is that TD is updating the value estimate using a sample of st plus 1 to approximate an expectation. So in reality, if you're doing dynamic programming, you would want to do a weighted expectation over all the next states you could reach, weighted by the probability of getting there. What TD is doing is it's just sampling one of those. And that sample is an approximation of that expectation. So we're going from this to sampling the next state. But similar to dynamic programming, it is then bootstrapping. So unlike Monte Carlo, which goes all the way out to get a sample of that value function, here, we're just plugging in V. So this part looks the same. So TD does both sampling to approximate expectations, and it bootstraps by using your existing estimate of the value function. All right, so let's just do a Check Your Understanding. So this is a poll. So what I'd like you to think about is how this learning rate might affect things. So whether different choices of this is going to weigh the TD target or more or less than the past V estimate, and what might happen when your state space is stochastic, meaning that when you start in one state, you might end up in multiple next states. What does that mean about convergence and the implication for learning rates, as well as thinking about deterministic Markov decision processes? Deterministic Markov decision processes, what I mean by that is that p of s prime, given s, a, is equal to 1 for exactly 1 s prime, meaning that there's no stochasticity. When you're in a state in action, you always go to one particular next state. So that's a deterministic Markov decision process. So just take a few minutes now and look into this. And you should be able to select all that are true. But if you can't, let me know. You cannot? OK. All right. Well, then, again, these are only for your thoughts. So just try to write down for yourself which of these you think are true, and then we'll talk about in a second. I'll check into these for next time. I don't think we can select multiple answers. Yes, that's what I just heard. Sorry about that. So I'll try to fix that for next time. Just try to have in your head of which ones you think are correct, and I'll ask you to compare with someone in a second. Thanks for letting me know. All right, turn to your neighbor and check, and particularly, focus on the last two and see if you agree on your answers for this. [INDISTINCT SPEECH] All right, great. I had some great discussions. OK, so for the first one, this is going to be false because, if we have alpha equals 0, then we don't care about the TD target at all. It just totally drops out, so we never update. In the second case, this is true because this means, if alpha is equal to 1, then this part and this part cancels out, and we just have this. So that means whenever we see an update, we always update. We totally change our estimate, potentially. The third one is a little bit subtle. This is true. Does somebody want to give me an example where this might occur? Yeah. If you had two states where they just keep pointing at each other, is that the case for this one? Yes. And in particular, if you could go to either of those states with some probability. Yeah. So I sometimes think of it as like a coin flip. So imagine that you have one state where, after this, you either go to a state. Maybe it's 50% probability you get plus 1, and 50% probability you get minus 1. And then your problem just resets. So imagine, it's like a really short problem. You start off, you get 0 reward, you transition to a state, and then your episode resets. So in this case, either on that round, you're going to get plus 1, or you're going to get minus 1. So you either get plus 1 or minus 1 here, and so you'll just flip back and forth between the plus 1, minus 1, plus 1, minus 1, plus 1, minus 1. So that's just to highlight that, if you do have systems which are stochastic, the fact that, in your target, you are using a single sample of that stochasticity to approximate the expectation can be bad. But that does not mean-- and I guess this gets to, I think, that was asking before that, in many of these cases, there's the cases where it might be possible that this would happen, but it won't always. So in this case, there do exist deterministic systems where, even if alpha is equal to 1, you can converge. So again, think of something. I like often to think about really small MDPs to get some intuition for this. If you have a case where there's just a terminal state and there's no more transitions, so you get to some point where you always go to some terminal state and it's plus 10 there, and there's no more updates, then it's just plus 10. There's no more expectation. And in general, any case where, if there's no stochasticity, and you're near the end, and there's no more stochasticity in that episode, those can be cases where you'll still converge. OK, great. So I encourage you to go through some of the worked examples, if you want to, just to see some more comparisons over the difference between Monte Carlo and TD methods in this case. Just to summarize what we're doing in TD learning, we are bootstrapping and sampling. We're sampling to approximate our expectation over all the stochasticity. We're bootstrapping because we don't want to use a full return. We are, instead, taking V to approximate that. It can be used in episodic or infinite horizon settings. It is generally lower variance for doing lots, and lots, and lots of updates. It is a consistent estimator if your learning rate alpha satisfies the same conditions specified for incremental Monte Carlo policy evaluation. I here today only introduced TD0. What TD0 refers to is, you take the immediate reward, and then you immediately bootstrap and plug in the value of the next state. So we did r plus-- we did r plus gamma V of s prime versus summing up all your discounted rewards for the whole episode. In general, you could have something kind of in between. So you could have plus rt plus 1 plus gamma squared. So you could have something like this. So in general, you could do some sort of combination of using partial returns, and then bootstrapping. There's a lot of different TD methods that interpolate between taking one step and then plugging in the value versus only plugging, not using any value function approximation. And if you want to think about this graphically, it's kind of thinking about do you plug-in v of s prime here, or do you plug it in-- no, way lower. Yeah. Is there an empirical estimate of what a good trade off for TD for computational complexity versus performance? Have they found a good number for it? Good question. Unfortunately, in many cases, it will be depending on the domain. One thing, I think, to think about here, too, is that you can think of this part doesn't require the Markov assumption. So if you have a system where you're not confident, but maybe you're like, well, maybe I'm willing to say that I'll plug-in a Markov assumption eventually, because it's going to be lower variance, but I want to preserve the fact that maybe it's not Markov, then I sort of have a short horizon. Often people do use something in between the two. So they often do consider this between for multiple reasons, but it gives you some of this flexibility. It often is a lower bias. That's a great question. All right. What we're going to do now is think about also how some of these ideas relate to dynamic programming, which is what we saw in an earlier lecture, because we could use this also for policy evaluation. We know how to use it for policy evaluation if we are given the models. But some of you guys might have been thinking, well, we have data now. If we have data, because we're taking the policy in the environment, couldn't we use that to estimate a reward model, or couldn't we use that to estimate the dynamics model? And that's what's known as certainty equivalence approaches. So the idea here is that you're going to be getting data as you execute this policy, and you can compute a dynamics model from that data. So you could use a maximum likelihood MDP model. Remember, right now, we're in the tabular setting. So we can have a parameter for every single state in action. So we can just count. We can just say, how many times was I in this state, took this action, and transition to this next state, divided by the number of times I was in that state in action. So this just gives you a maximum likelihood estimate of the dynamics model, and you can do the same thing for the reward model. And of course, as you might imagine, you can do this with much more complicated function approximators, like deep neural networks, too. But the idea is that, once you have this model, and it's called a certainty equivalence model because we're now going to ignore any error in these models. So we have finite data. These models will definitely be wrong, but let's ignore that for now. So once you have this maximum likelihood MDP model, you can just compute the value of a policy, using the same methods we saw last week, because you have a dynamics model now and a reward model. And you can see some examples about this at the end of the lecture slides. So one of the benefits of this, and this gets to the question, is this is really data efficient. So I showed you an example for the Mars Rover before, where we only updated one of the states with TD learning. We updated three of the states with Monte Carlo. What this does here is it tries to update. It computes a dynamic small reward model for all states and actions, and then it tries to update all of them. OK, so it's going to compute a value for every single state. Now, the downside of that is that now we're doing policy evaluation with a full model, which is either going to be something like s squared a for iterative methods, or maybe even worse. So it's computationally expensive, but it's really data efficient. Because as soon as you reach any state for which you get, say, a positive reward, you can kind of propagate that to any other state that is possible to reach from there. It's still consistent. It's going to converge to the right thing for Markov models, and it can easily generally be used for all policy evaluation, which we're going to get into. Yeah. Sorry. What is Nsa in this equation. Sorry. Great question. Nsa here is the number of times we've been in that state and taken that action. Yeah, so this is counts. Yeah. It seems pretty similar to Monte Carlo. So is just the difference that you're generating probability, as opposed to calculating G? Great question. We're going to hold that thought. He's asking how similar is this to Monte Carlo. It's actually going to be pretty different. We're going to see this in a second. We are using our data similar to Monte Carlo. We're going to use the data here to compute models and then propagate information. But they're going to end up making some interesting different decisions. So let's see that now. It's a great precursor. OK. So let's get into batch policy evaluation. So I've said there are these different methods. They might have different computational complexity. They might be more or less data efficient. So one thing that you might imagine doing, and we'll see this a lot shortly, a lot more next time, is, well, if I have some data, how could I best use the data that I have. And this comes a lot up a lot in the research that me and my lab do, because we're often dealing with patient data, or student data, or legal data, or others, where it's really expensive to get the data, or it's costly, or it could be harmful. And we want to get as much information as we can out of the data we have. So when I say batch, what I mean is, imagine that you have a set of k episodes. And now what you want to do is you want to do policy evaluation just with that data. So what we're going to do is repeatedly sample one of the episodes that we have of those k, and we're going to apply Monte Carlo or TD0 to that episode. We're just going to do that over, and over, and over, and over again. And we'll see this a lot more next time, as well. And so the idea is to just understand, if we do this, given that finite amount of data, what will Monte Carlo and TD0 converge to, in terms of the evaluation of the policy. So let's go through that. There's this really nice example from Sutton and Barto to illustrate this. We have a really small domain. We have two states. And we're going to say gamma is equal to 1. There's no discounting. And we have eight episodes of experience. So in one episode, we started in state A, we got a reward of 0, we transitioned to B, and we got another reward of 0. OK? So in this case, you can think of it as this. You get a trajectory like that. In some episodes, we started in state B, and we just got an immediate reward of 1, and we observed that six times. So in six trajectories we just happened to start in state B, and we got a reward of 1. And then in one trajectory, we started in state B, and we got our reward of 0. So first, imagine if you ran TD updates over this data an infinite amount of time. What do you think the estimate of VB would be? Try to remember what it works for there, as we have 1 minus alpha times our old estimate. So that's VB plus alpha times our immediate reward, plus gamma, times the next state. But here, it's just terminal because we always terminate. After B, we always terminate. So you never get any future discounted rewards in this case. So what the updates look like for TD learning is that you would have 1 minus alpha times your old estimate, plus alpha times whatever reward you get in B. And imagine you just iterate over these over, and over, and over again. Somebody have any guesses of what the reward would be for V of B? Would it be 0.75? Yeah. Somebody else want to explain now why it's 0.75? The season mods? Yeah, because in this case, we had eight episodes. In two of them, when we started in B, we got 0. And in six of them, we got 1. So we just average those rewards. And imagine, we're just doing this many, many, many, many, many, many times. So eventually, you would just converge to this estimate being 0.75. What about for Monte Carlo? So let's do Monte Carlo for V of B. What would that look like? So remember, for Monte Carlo, it would be 1 minus alpha times V of B, plus alpha times G, where you start in state B. Is it going to be the same thing? Is it going to be different? So Monte Carlo, we're averaging over all the returns we get starting in that state. Yeah. So when we start at B, so then wouldn't it be 6 over 7? 6 over 8. 6 over 8, yeah. Yeah. But don't we start at A in the first one? In one episode, we start in A. But we're just trying to compute the value of B right now. We'll get to A in a second, yeah. You're thinking ahead. Yeah. So the Monte Carlo estimate. So we're just trying to contrast. So just to recap, what we're trying to do here is we're trying to see, will these two algorithms converge to the same thing or not. We're going to start off and look at what the value would be of state B. In TD, we would converge to 0.75 because the immediate reward is either 1 or 0, and the discounted sum of future woods is 0 because we terminate. And in Monte Carlo we just average over all the returns we get when we started in B, and that is also 6 divided by 8. All right, so now this is the hard one. OK. What about of V of A? What will we converge to in these two cases? So let's do Check Your Understanding, and you can respond in the poll. And feel free to talk to someone next to you. And again, the intent of this is to think about, are these actually computing the same thing or not in this case. And remember, this is a different setting than what I told you before, that both of these things can be consistent. But that was if you get infinite data. What this is looking at is, if you only have a finite amount of data, and you just go over it over, and over again, either with Monte Carlo updates or with TD, will you converge to the same thing. And if you're not sure or you're confused, feel free to put that in the poll, too. There's lots of different answers here. So this is a great one to talk to somebody nearby you. So why don't you see if you're getting the same things? And we can use our collective intelligence. [INDISTINCT SPEECH] OK, I'm hearing a lot of good discussion, so I'm sorry to interrupt. But this is kind of a fun one. So let's start with TD. So someone want to explain why it's 0.75 for TD? There are multiple people that got that. Yeah, would you explain what you and your partner were saying? Yeah. So I think if you just-- so we're just looking at this. And remind me your name. Sorry. Yeah. There's this one episode where we're at A. So in that episode, the immediate reward is 0, but then we have to do plus gamma times the reward of the next state, B. And then V pi of B is 0.75. We got it in the previous part. So the value is 0.75. That's right. So Monte Carlo is not that estimate. So TD gives you 0.75. What does Monte Carlo give you? It's not 0.75. And again, multiple of you guys got it correct. I just have a question. Is gamma that same as alpha here? Good question. No. Here, I'm assuming that gamma is 1. And it's a great question. So someone else was asking this, too. So I'm assuming that alpha is set correctly for these to converge. Yeah, it's a good question. Sorry, someone else had that, too. So I'm assuming that we're going over our data an infinite amount of time, but we're decaying alpha correctly as we do that. It's a good question. Do you want to explain what Monte Carlo is? It's not 0.75. Going to be 0? Yes, it is. That's great. Someone want to explain why it's 0? So Monte Carlo is 0. Yeah. So when we see one trajectory where A shows up, and [INAUDIBLE]. That's right. Remind me of your name. Yeah, so what [MUTED] said is exactly right. So we've only seen one trajectory, and I know some other people made the same observation. So we've only seen one trajectory where there was A at all. We, for Monte Carlo, we just average over all the returns we've seen where that. So that's only 0. So I bring this up because, even though, asymptotically, all of these things converge to the right thing under some mild assumptions, with finite data, which is what we're almost always going to have in reality, even if you go over it multiple times, they're converging to, sometimes, totally different things. And here is what they are converging to, in general. Monte Carlo is converging to the minimum mean squared error with respect to the error observed returns. So it's just going to set it so it minimizes the error between the observed returns it's seeing and its value. So in this case, that would be V of A equals 0. So that is the minimum mean squared error. TD0 converges to the dynamic programming policy for the MDP with a maximum likelihood model estimates. So you guys remember how we just talked about certainty equivalence? What we were doing here is, we're taking all our data. The answer you get from TD0 if you do this batch process is the same as if you had computed your maximum likelihood Markov decision process from the data you have, and then you did dynamic programming with it. OK? So that will be exactly the same as this. And so in particular, it is leveraging and using the Markov assumption. And that's why it could actually chain these things together. So you could see here, for Monte Carlo, it doesn't know that the value of A has to be related to the value of B in terms of this bootstrapping relationship. But TD is making that explicit. It's using the Markov decision process to say, the value of A has to exactly be equal to the immediate reward you get in A, plus gamma times the states that I could get into, which is always B, so the value of B. So TD learning is explicitly baking that into the solution you get, whereas Monte Carlo is not. Monte Carlo is just trying to minimize the mean squared error for the returns you see. So they can end up giving you very different solutions. And depending on whether your Markov property is really satisfied or not, you might want one or the other. Awesome. So this just summarizes quickly some of the different properties and approaches and just highlights here that temporal difference really does exploit this Markov structure. And that could be really helpful if you want to leverage that to get better estimates of earlier states, like in the case we just saw. So just to summarize, we finished going through policy evaluation with tabular settings. And then on Wednesday, what we're going to do is talk about control, and we'll start to talk about function approximation, as well. All right. Thanks. See you then.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Policy_Search_1_I_2024_I_Lecture_5.txt

Hey, everybody. Welcome back. We're going to go ahead and get started with our refresh your understanding. [SIDE CONVERSATIONS] OK. Hopefully, everyone had a chance to think about this a little bit more. So let's go through the answers. The first one is true. So if you are trying to evaluate the value of-- this is in the tabular case. So this is where we're assuming we're going to sample each tuple at random, and then we do a Q-learning update. And we do this an infinite amount of times. We know for a standard tabular learning, we can converge to the true value of a policy under-- as long as our learning rate schedule is such. So if there's an existing learning rate schedule under-- if you're decaying your learning rate at the right level, then you will converge to the true Q value in the tabular case, because there's no function approximation that's happening there. In the second case, this is also true. So we talked a bit about how we could think about doing these things in a batch way, where we do it over, and over, and over again. We take our existing data, and we run it through our either TD-learning update, or our [INAUDIBLE] update, or other things. And we said that the TD-learning updates, if you do it in a batch way, are equivalent to just taking a certainty equivalent model, which means you estimate the dynamics model and you estimate the reward model-- excuse me-- from your existing data, and then you do dynamic programming. So that's what we saw-- I think we saw that in Lecture 3. This one is false. Does somebody want to say why it's false? This one is not true. There's a number of reasons why it could be false. Anybody want to share why? Why is DQN not guaranteed to necessarily converge to the optimal Q function? Yeah. Remind me your name. [MUTED] would you need to enforce a certain number of iterations for it to have any chance of converging at all? So good point [? related to that. ?] So certainly, if you don't do enough iterations, but even if you do an infinite number of iterations, it also might not be guaranteed to converge. Can anybody tell me why even with [? infinite-- ?] [INAUDIBLE] you certainly need a lot of iterations. But even if you had a lot of iterations, you still might not be guaranteed to converge. I think here it helps to think about what we often call realizability, which is we don't know what the functional form is of Q. And so you could think of the fact that-- I'm going to draw it in-- as if the state space was one dimensional. But in general, of course, the state space is like this vector or it's images, and so it's really high dimensional. But imagine that it was one dimensional. Even here, you don't know what your V function or your Q function might look like. And so if you are using the wrong approximator, if you are using, say, a line instead of a multi-degree polynomial, then no matter how much data you have, you're not going to converge to the optimal Q function. [INAUDIBLE] Because you just can't even realize it. So in general-- and there's all sorts of additional instability things that mean we can't be guaranteed it's going to converge. So we're not guaranteed it'll converge. But empirically, it often does pretty well. So we'll see-- [INAUDIBLE] If you look at the empirical results, it often does really quite well. Great. So far in the class, we've talked a lot about these value function based methods, where we thought about having an explicit representation of the expected sum of discounted rewards starting in a state or starting in a particular state and action. And so we talked a lot about value functions and Q functions. And now, we're going to talk a lot about policy search. And so we're going to still think about there being this policy, which is a mapping of states to actions or a mapping from states and actions to a number between 0 and 1, such that it sums to 1, because we always have to do at least one action in every state. But we don't necessarily have to have an explicit representation of the value function anymore. So these have been very popular and important. And if we think back to what our RL algorithms involve, they involve optimization, delayed consequences, exploration, and generalization. And we've seen examples of all of these ideas so far. And we'll go a lot more into some of them as we go through the course. But one thing you might be wondering about is, could we play the trick that's often done in computer science and try to reduce reinforcement learning to another problem? So could we do something like just like online optimization? So we know that we don't know how the world works. And we're trying to find a good control policy. But could we do something sort of online optimization where we're trying to search for a good policy? And in this way, you can think of policy gradient being related at a high level to this type of idea. It's not a reduction-based approach, but it's thinking about, well, can we just directly search to find a good policy. And policy gradient methods have been extremely influential, particularly over the last 5 to 10 years. So they're used for lots of areas. They're used for things like sequence level training with recurrent neural networks. That was based on REINFORCE, which is an algorithm we're going to go through today, that has been used for things of end to end training of deep visuomotor policies. So this was really influential work in the robotics community about a decade ago. And I'm just going to show you a quick video of it. So this is work that was done by Professor Chelsea Finn as part of her PhD thesis, along with Sergey Levine and others at Berkeley. And let's see if this will work with audio. [VIDEO PLAYBACK] - [INAUDIBLE] [END PLAYBACK] So what you can see there that what they're doing is-- what they're going to be trying to do is learn from-- so they showed you a really-- they showed a big network. And what they're trying to do is go directly from pixels to learn what the robot should do. And this is one of the first examples of people trying to do this directly from images. Let's just go back to some of the tasks that they're using. And so that was part of the motivation, too, is that they want to be able to learn these tasks in a way that will generalize. [INAUDIBLE] So this is another example of trying to do direct policy gradient methods in order to go from really large complex state spaces into direct decisions. Now, in Homework 2, and we haven't covered PPO yet, but you're going to be implementing Proximal Policy Optimization, which is one of the methods that build on the methods that we're going to talk about today. And that was used-- PPO was used as part of training ChatGPT. So as you can see, all of these algorithms have become incredibly influential, in part, because they can scale really well to extremely complex inputs, whether it be images, or high-dimensional robotic tasks, or even things like natural language. And so they're very powerful. They're often used in conjunction with things like state action values, as we'll talk about later. But you don't have to use them with them. So they're really useful sort of class of things to know about. So in particular, just like how last time we saw that you could approximate a state action value or a value function with a set of parameters-- so we can do function approximation. In those cases, we thought of directly learning a value function or a state action value function, and then it generate a policy from the state action value. So something like e-greedy, where we either take what the Q-value suggests is the best action or we act randomly. And what we're going to do today instead-- and I'll try to be careful about not using the same-- we used w before to parameterize our state action values. And I'm going to try to be careful about using theta just to make it clear. We're going to directly parameterize the policy. And we're going to try to learn parameterized policies. So we can think of these as like deep convolutional neural networks, which at the end will output either an action or, if we have an action as input, will output a probability. And the goal in this case, as is normal, is that we want to find a way to act in the world that will give us high reward. So we want to find a policy with the highest value function of V pi. And we're, again, not going to be focusing on model-based learning. So we're still going to try to directly learn from experience. And we're not going to assume we have access or that we're explicitly building a model. And I think one of the things that's helpful to think about is there's these sort of different views or lenses into reinforcement learning. So this is a nice picture from David Silver, who's an amazing person in reinforcement learning. He was one of the main leads on AlphaGo and a number of other incredible papers. So you can think of it as you have some methods, which are value-based. We're explicitly building a value function. We have other ones that are policy-based. And the ones that are in the intersection are often known as actor-critic methods. Who here has either implemented or heard of actor-critic methods before? OK, so some people, but not everyone. They're extremely popular. And in actor-critic methods, you will often combine between the benefits of value-based and policy. And so, for example, AlphaGo is an actor-critic method in the sense that it is often having an explicit representation of the policy and of a value function. So we'll get to actor-critic methods later today. We're going to focus on policy-based. So now that we're going to-- most of the time, we've thought about policy so far, we thought about deterministic policies or e-greedy policies. And now, we're going to think much more generally about stochastic policies. And that's going to be important, because as we saw last time, if you only have a deterministic policy, it's much harder to learn about actions you don't try. Whereas, now we're going to think about having stochastic policies, where you're going to be getting information about lots of different actions. So let's think about a particular example, also to illustrate some of the things that policy gradient methods are going to help us handle. So who here has played rock, paper, scissors? Most people, I think. It's called "roshambo" in Chinese. It's a very popular game throughout the world. It's a stochastic game where each side can pick a particular strategy and the state-- you can think of there being a state, you could keep track of what your opponent has done over time. So think for a second about whether a deterministic policy can be optimal if you're playing this game repeatedly. So raise your hand if a deterministic policy can be optimal. Raise your hand if you think a stochastic policy is optimal. OK, someone who said stochastic explain why. Yes. [INAUDIBLE] is like circular. There's no [? 1 ?] [? plus ?] [? 1 ?] [INAUDIBLE] That's right. Yeah, so there's no best-- there's nothing that strictly dominates all the other strategies. And also, if you're deterministic, what can your opponent do? Like, if I say I'm always going to pick paper, what does my opponent do? Yeah, they're always going to pick the other one, like rock [? to ?] [? beat. ?] So anything you do that's deterministic can be exploited by your opponent if you are playing repeatedly. And so the optimal thing to do here is to be stochastic. So the optimal policy has to be stochastic here. Otherwise, all deterministic policies are strictly dominated by good stochastic policy. And now, you might think, all right, well, that sounds different than what we've seen so far. But one of the challenges here is the system is not Markov. So it's not stochastic what your adversary will play next. They're not random-- or it might be if they're playing a stochastic policy. [CLEARS THROAT] Excuse me. But in general, [? they ?] can react to what you've seen so far. And it's not just like a random environment, like a coin flip on the next time. And in this case, actually, a uniform random policy is optimal. It's a Nash equilibrium. So that's one case where having a stochastic policy would be really helpful. So you could just have a fixed stochastic policy, and it would be optimal, but you couldn't necessarily write this down easily as a Q-function and just take the argmax. There is not a deterministic policy for this environment that is optimal. And so it's less clear how you would write that down directly in terms of a Q-function, in part, because the system is not Markov. So here's another example where we might want to have stochastic policies. And it's where we have aliasing or partial observability. So imagine this case where you have a robot that's walking along. And maybe they have sensors so they can tell how far they are from the walls. But under those sensors, these two gray boxes look identical. Because like from the agent's point of view, if they have only immediate sensors, both of those places will look identical. And so they can't distinguish those gray states. And imagine that you just-- because you have a feature representation that just tells you about what you're-- whether you have a wall to the north, to the east, to the south, or to the west, so those two gray states would look identical if that was your feature representation. So you could have a value-based reinforcement learning representation where you use an approximate value function, where you take in this as the state representation, or you could have a policy-based one that takes in those. So the challenge here is that if you're value-based, you have to do the same thing in those two gray states, because you can't distinguish them. So from your perspective, it's like you're in the same place no matter which of those two places you're in. So if you're going to do a value function based, and then extract a deterministic policy, you would either always have to go say to the left in those cases or always go to the right. And neither of those would always be good. So under aliasing, meaning that we don't know whether which of the two gray states we're in when we're in one of them, an optimal deterministic policy will always move west in both states or east in both states. And either way, it might get stuck and never be able to reach the money. And that's what's going to happen if we do a value-based reinforcement learning approach. So that's not great. You're going to traverse this for a long time. You're not going to be getting high reward. What could you do if you wanted to have a stochastic policy? So that allows you to act randomly or stochastically in any state. What do you think would be the right thing to do in the gray states if you could have a stochastic policy? With just some probability, you go either east or west. Yeah, exactly. So you could just randomize it. So an optimal stochastic policy will randomly move east or west in the gray state, because it doesn't know which one it's in. And half the time, that'll be the right thing to do. So now that means much more of the time, it'll go into here. And it generally will reach the goal state pretty quickly. So this is another case where the system is not Markov. This is not [INAUDIBLE] the state features. So because we have aliasing, meaning the system is partially observable, it is not a Markov system. One way to handle that is to treat it as a partially observable Markov decision process. [INAUDIBLE] talks a lot about those in his classes. But an alternative is to use a stochastic policy. And you can also [? act ?] very well here. So those are two examples of the type of thing that might be able to be easy to handle with policy gradient methods or stochastic policies that might be hard to tackle with the type of methods we've seen so far. So now, we have to think about if we have policies, and, in general, we're going to want them to be stochastic, how are we going to learn what are good policies? Like, we have this-- now, we have a function space over policies. And we want to learn which of them have good values. So if we're in an episodic environment, we can use the policy value at the start state. So we can just say I'm going to similar to the Monte Carlo methods, if I start in this state, I run this policy, what would be my expected reward be until the end of the episode? We're going to mostly focus on the episodic case today, but you can extend these to more of an infinite horizon case. All right, so once we think of it in this way, we can really think, OK, this sounds like an optimization problem. So we really just want to find the parameters that maximize the value. So you could say-- here, you can think of this as being your thetas. So I'm just drawing it in one dimensional. But in general, this could be all the parameters in a deep neural network. And then, this is V of theta of a particular starting state. It might look like this. And what your goal would be is to find the parameters of your policy that maximize the value function. And so this is an optimization problem, but it's a hard optimization problem, because we don't have that function. You can only estimate it through data. And so you can imagine, like you start off, you have no idea how theta maps to V. And then you have to learn that over time. So once we think of it as an optimization problem, where we don't know what the function is, there are a lot of different methods we could think about to try to solve this problem. And what we're going to focus on today, mostly, is ones that are going to exploit something about the structure of sequential decision processes. But there are methods that completely ignore all of this. And in particular, you can even use things that completely ignore gradients. So you can do things like hill climbing, or genetic algorithms, or cross entropy methods, where you may not think of any of the type of structure of the parameter space. And in some cases, that can work really well. So there's a really nice example by my colleague, Steven Collins, who's over in the mechanical engineering department. He does some really interesting work on-- oh, yeah. [INAUDIBLE] Yes, but you can make it a distribution over actions. So you can output something between 0 and 1. And you can have an input action. And so [INAUDIBLE] [? have ?] a stochastic policy. [INAUDIBLE] You would then compute it for all of the actions, then you would have to have some-- yeah, you'd have to pick a random number, and then use that to select. Yeah. Yeah, good question. So my colleague Steven Collins over at mechanical engineering does a lot of work on exoskeletons. And there are lots of reasons exoskeletons could be really helpful, particularly for people that have motor impairments. But one of the challenges of them is that you have to have them actually help people walk. So if you clamp something on to, say, your leg, then the way my physical configuration is may not be the same as your physical configuration. I'm pretty tall. And so you'd really like to make sure that this helps each individual in the best way possible. But you don't want it to have to learn over the course of two years how to best optimize to someone, because they're not going to wait that long to use it. So what they did is they use policy methods, policy search methods to quickly personalize the parameters of an exoskeleton. And they called this human-in-the-loop exoskeleton optimization. So the idea in this case is what they're trying to figure out is what is the parameters of their exoskeleton. And what they're looking at is, essentially, how much it helps you walk. So how much it reduces the effort needed to walk. And so what they could do in this case, they're not using a gradient-based method. They're using just CMA-ES, which is-- [? so the treatment is ?] continuous optimization, is they'd have people walk under a few different control parameters. They would see which of those seem to be most effective, and then they would move the policies they try in that direction with some stochasticity. And I think it was within maybe two or three hours using this, they could find substantially better policies. I think it increased metabolic efficiency like maybe by 20% or 30%. It was pretty remarkable. And so this was published in Science about seven or eight years ago. But that's another example of a place where you can do this sort of online optimization, but you don't necessarily have to think about the temporal structure of the policy. Can I ask a question? Yeah. [INAUDIBLE] [? --with ?] a default policy, and then try to improve that default policies? Great question. Yeah, so in all of these cases, we're going to have to assume that we initialize our policy parameterization in some way. Just like how we initialized our value function to 0 to start, now we're going to-- or if you had it for the deep Q-network, it would be whatever your neural network parameters were. Yeah, great question. Now, it's just useful to know about these, because they often work pretty well. So I think sometimes we like to leverage the structure specific to, say, our Markov decision process. But in some cases, just leveraging these ones, which may not use very much structure at all, can actually do really well. So it's just good to keep in mind that there are a lot of ways to do online optimization. All right, so this is often a great baseline to try. The great thing about this is it can work with any policy parameterizations, even if it's not differentiable, because it's not using gradients. So it doesn't need to be differentiable. And it's also often very easy to parallelize. So CMA-ES, for those of you who haven't seen it before, you'll have a number of different policies you kind of try in parallel, and then you'll use that to update and shift to another set of policies. And that's what they did, Professor Collins did. And in a lot of cases, the problems that we think about are places where you'll have many customers. You'll have many robot arms, and so you can parallelize things. One of the limitations is that it's often less data efficient, because it's ignoring the temporal structure. So if you have temporal structure or if you have a [? gradient ?] information, it may be more effective to use [? that. ?] So what we're going to focus on in this class is differentiable methods. So we're going to focus on places where we can do stochastic gradient descent, including on the policy parametrization. So if we have our policy parameterized by a deep neural network, we can propagate through that and update those parameters. So we're going to focus here mostly on methods that do use gradient descent and that often leverage the sequential structure that we're making a sequence of decisions and we want to optimize to make those sequence of decisions. So to do that, we're going to explicitly define the gradient. And we're going to write down the value function in terms-- as a function of the policy parameters, so that we can be clear that this value function relies on those policies. And we're going to focus today on episodic Markov decision processes, where we go for a single episode, stop, reset, and keep going. So now, what we're going to do is we're only going to be trying to get, in general, to a local maximum. Now, it's possible you're lucky and you're sort of convex in the space of the value-- in space of the policy parameters. But in general, we're not going to assume convexity. So at best, we're going to hope to just get to some sort of local maxima in our space. So if we have-- again, if we only had one parameter, and we have something like this, we might get to here. We might get to here. In general, we're not going to make global optimality guarantees. This is in big contrast to the tabular cases we saw before. We were guaranteed to get to the optimal Q-function, optimal value function. Now, we're just going to hope to-- given our policy parameterization, let's try to get to what's a local optima in that policy parameterization. So it's sort of a policy specific-- policy class specific guarantee. And it's only a local optima. And what we'll be doing is we're just going to be trying to take the gradient of the policy with respect to the parameters. And as usual, we're going to have a step size parameter. So we're going to take the gradient of the value function with respect to the parameters and take a small step. And the key thing is going to be thinking about places where we can do this all directly using smooth functions. Now, one way you could do this-- of course, when you see this now, you probably immediately think of autodiff methods and think about we can just back propagate, et cetera. But it's worth noting that when these methods began to start to get popular, they didn't necessarily have autodiff yet. I know. There was research then still. And one of the things people started thinking about for this is how you could use this for robotics. So this is a nice paper from 2004, so 20 years ago, by Peter Stone's group. It was right around then-- I think maybe RoboCop was maybe 60 years old then or something. I think they started it back in 1998 or so. So there are these little quadruped robots. And the goal was to think about getting robotics to the stage where you could have robots play human players. I think that was the goal by either 2030 or 2050. I forget. But what they were just going to start with-- it was quadrupeds. So one of the big challenges at the beginning, because everywhere, you start with the beginning challenges, and you go from there, it was just getting them to walk fast enough. So if they're going to score goals, and they're going to compete, you need them to walk quickly. And so there's this question of just how do you learn fast walks, so that they can-- sort of trying to teach robots to run. And what they found here is that they could use policy methods and policy search methods just to learn a faster way for it to walk. And so they parameterize the curve of how the foot moves as a set of parameters. And that defined the policy for moving those joints. And then, what they did is they just had these walk back and forth many, many times. And what they would do is they'd have them walk with some particular policy parameters. They would see how fast they walked. They would do finite different methods. So they weren't trying to explicitly do autodiff or anything there, and then they would slightly change the policy parameters and repeat. And they learned to substantially faster walk during that time. And I think it took maybe around four hours or so, but they just had to replace the batteries a couple of times. So just an example to say like, it's lovely to have autodiff. You can do really complicated things now. But these methods can work even in really basic settings, particularly where you think you have pretty bad models of how the world works. And so now, you can just be directly data driven. And why is this hard problem, for those of you who haven't done robotics? It involves a whole bunch of contact forces. The ground may be-- well, they have to learn on this particular ground. You may not know, because it's commercial hardware. You may not know exactly all the parameters that the designers put in. So you can just be data driven. OK, as opposed to maybe having a physics simulator. All right, so just to summarize so far, the benefits of policy-based RL is that we're going to have often better convergence properties, because we're often going to be able to guarantee that we get to a local optima. Whereas, we didn't have that for deep Q-learning. They're often really effective in high dimensional or continuous action spaces. And you can learn stochastic policies. But the methods we've seen so far might be more inefficient and higher variance. And we often only get to something in the local optima. And we'll see some things to help with of the inefficiency in a second. All right, so now, what we're going to dive into is how do we do this when we are willing to have differentiable policies. So the hope is that we can actually compute the policy gradient analytically, so we don't have to do it with finite differences. And we're going to focus on policies where it's differentiable as long as it's non-0. So we're going to assume that we can always compute the gradient of the policy parameters themselves. And there are a number of different classes we can do this for. And there are many popular classes, including, of course, deep neural networks. So popular ones are often softmax. Softmax is used all the time. I'll explain what is in a second. Gaussian and neural networks. And again, just to be clear here, what I mean by a policy class is what is the functional form we are using to give us a probability of an action given a state. So are we having something like-- well, I guess we can just see on the next slide what this will look like. So we're going to assume I'm going to give you some examples of those of what softmax, and Gaussian, and neural networks look like in a second in terms of how we differentiate them. But these are just different ways for us to parameterize what is the probability of an action given a state. Actually, I guess I'll give a quick example of Gaussian. For a Gaussian, you could imagine-- let's imagine I have a robot. And I'm trying to figure out, say, how much speed to apply. Then, you might have a policy class that says the action I take is equal to a Gaussian centered around 0.5 with some standard deviation. So it would be a stochastic policy. And it would say the average amount of speed you're going to apply is 0.5, but you're going to have some variability around that. That would give you some stochastic behavior. So sometimes, your robot would go really slowly. Sometimes, it would go fast. Sometimes, it would go in a negative direction. OK, so let's keep assuming that the policy is differentiable. Whenever it's non-0, we know the gradient. That still doesn't tell us how to solve policy gradient methods yet, because what we want to do is take derivatives of the value function. So we want to say I want to find the maximum, the policy that has the best value function, which means I'm going to need to take the derivative of the value function with respect to the policy parameters. So remember that the policy value, the value of the initial starting state under a policy, is going to be the expected sum of rewards. We don't have to use discounting for most of today if we assume it's finite. So I'll just say we're going to assume we're in the episodic case. So this is finite. So no discount [? counting ?] for now. So we don't need discounting for now, because it's always a finite length, so we're never going to have infinite reward. So the policy value is just the expected sum of discounted rewards when we follow the policy parameterized by theta till the end of the episode, starting from the state [? s0. ?] And there are lots of different ways for us to write this down. So one way is for us to write down [INAUDIBLE] r is equal to-- well, it's equal to the state action value averaged over the probability of us taking each of those actions under our policy. So this here just says, what is the probability of me taking this action? Starting state s0. If I have policy parameterized by theta times what is my Q value. Starting in that state, taking that particular action, and then following that policy for the rest of it. So this is one way to write it, but we can also think of a quite different way, which is let's think about trajectories. All right. Don't want these [INAUDIBLE] in a second. So this is a trajectory. What's a trajectory? That's going to be s0, and then it's going to be an action, and then s1, dot, dot, dot-- sampled from pi theta. So another way we can think of the value-- and then, this is going to be the reward for that trajectory, that whole trajectory. And we've called [? R G ?] before. So sometimes-- I'll just write down that in case. [? Sometimes-- ?] [INAUDIBLE] So another way we can think of the value is we say, well, let's just sum over all possible trajectories we could reach under this policy. And what would be the reward for each of those trajectories? And I'm just going to take a weighted sum. Now, of course, you might be thinking that's totally intractable. And yes, in general, if you have a really long trajectory, then it's going to be [INAUDIBLE]. And you have a really large state space. And you could reach many states. In general, it's not going to be possible to actually enumerate this. But this is mathematically well defined. This is just an expectation over the reward of trajectories. And we know whenever we see expectations that we can approximate those with finite samples. You can think of just taking n samples, just like what we saw with Monte Carlo methods, and using that to approximate a trajectory. So in general, this is intractable. In general, intractable, but we can approximate by sampling. So this is one way we could write down. So this is another also a valid way to write down what is the value of starting this state and following the policy. So I've written that down more neatly here p of [? tau ?] theta is the probability of our trajectories. [INAUDIBLE] [? when ?] you execute that policy starting state s0, and that is the sum of the rewards for trajectory. In this class, we're going to focus on this latter definition. But instead of setting [? bar, ?] they have a nice way to think about policy gradient methods that starts from the other definition. So you can always look at that. But both are totally valid definitions. So now, we're going to focus on thinking about likelihood ratio policies. So we're going to be thinking about this case where we have a distribution over trajectories, and then what is the sum of rewards for each of those trajectories. So we have our value function. And now, what we want to do is find the argmax, so that we maximize [INAUDIBLE] having probability of getting trajectories with high reward. So that's nice. So instead of just thinking about the value function, we now can think of it as, OK, I want to have policies that induce trajectories through the state space, through the state and action space that give me high reward. So what we're going to need to be able to do is to take a gradient through the right hand side. So that's what we're going to do now. [INAUDIBLE] OK, so we're going to take the gradient of this. Because once we have the gradient of the value function with respect to the policy parameters, we can update our policy parameters to increase, hopefully, the value of the policy that we're at. So what we're going to do is we're going to say we're going to take the gradient with respect to the right hand side. We can rewrite this by pushing in the gradient. Now, R of tau doesn't depend on the policy parameters. That's just what is the reward once you've told me what a trajectory is. So we can put that on the other side. So the only part that depends on the policy parameters. And now, I'm going to play a trick. I'm going to note that this is going to be equal to-- well, I'm going to do something that's going to seem not very helpful for a second, and then we'll see why it's helpful. I'm just going to multiply and divide by the probability of a trajectory. I haven't done anything. I've just multiplied by 1, and I've happened to multiply by the top and bottom by the probability of that trajectory. But then, I'm going to note that the derivative with respect to log of the trajectory and theta is just equal to 1 over the probability of the trajectory of theta times the derivative with respect to the trajectory and theta. Because the derivative of log is just equal to 1 over the value times the derivative of the thing inside the log. So that looks exactly like this. So that's the trick that we're playing here. And so we can rewrite this then as the probability. And I'll tell you why we did this in a second. All right, let me just rewrite it one more time, so it's more easy to see. Why did we do this? The reason we did this is that, in general, it's going to be hard for us to think about-- or it might be tricky for us to think about how do we propagate our derivative through something that's an expectation. We had an expectation over all the trajectories weighted by the reward of those trajectories. We now want to take a gradient with respect to it. We want to end up with something that is computable from samples, because it's easy for us to get samples. We can actually run our policy in the environment. So by playing this trick, what we now have is something that we can also sample, because this is now an expectation over trajectories of the reward of the trajectory weighted by the gradient of the log of the probability of that trajectory. And we'll talk soon about how you compute this part, but this expectation can be sampled. Because this is just a probability over trajectories. And so we could sample, say, hundreds of them and approximate that outer expectation. So that's one of the right reasons why this is-- I'm just writing this out more neatly here on the next slide. This is called the likelihood ratio, this term here. And so that's one of the benefits to doing this, is that we want to end up with something that is computable. We want to be able to get this gradient with respect to the value function for the policy parameters. And so this is going to give us something that we can approximate with samples and we can compute. All right, now, you still might be a little bit concerned, because-- all right, maybe you think, yeah, I can maybe compute this by writing things out in the environment, but I'm still going to have to take this derivative. And how am I going to do that? And what does it end up depending on? So let's do a next step. So as I said, what we're going to do here, this is an expectation, so an expectation. We're going to approximate that expectation with an empirical estimate. So we're just going to-- instead of actually taking all possible trajectories, particularly in the case of vision input, you could imagine that would be completely insane. So we're just going to approximate it by taking m samples. But we still have to handle this. So that's what we're going to do next. So this first part should all seem clear. The second part should, at least certainly, for most of us, would not be clear yet about how we do that second part. OK, so what do we do with that? What we're going to do now is we're going to decompose that latter part into states and actions. So remember that what this means here is this is going to be a particular trajectory we get by following a policy for t steps or until the end of the episode. OK. So let me just remind ourselves what t is going to look like here. So this is going to be like time step here. I'm using the subscript as time step. OK, so let's just write out what a trajectory is and what those probabilities are. Are we assuming or approximating the probability of the trajectory just to be 1/n? No, good question or sorry. Yes, for this part? Yes. Yes. We're assuming that we are [INAUDIBLE] for each of the trajectories, we're using a Monte Carlo estimate. We're just using 1/m. But if some trajectories are more likely than others, they'll appear more in that set of m. Yeah, good question. OK, so let's now try to express what the probability is of a trajectory. OK, so the probability of a trajectory we can write out as follows. So we're going to still have that outside log. We're going to do the following. OK. I'm going to say mu of s0 is equal to the probability of s0. That's just like what is our probability distribution over our starting state. OK, so that's mu. And then what we're going to have is the following. t equals 0 to T minus 1. We're going to have our policy. So this is going to say, what is the probability that I pick action I picked, given the current state I'm in times the probability of st plus 1, given s0 to t, a0 to t. So what I've done is I've just written out. What is happening in my trajectory here, as I start, I have some distribution over C in this initial state. Under my policy, I have some probability of picking a0, and that's here. Then I'm going to assume for a second that rewards are deterministic. But you could add in a reward term here. And then I'm going to say, well, what's the chance that I get to state s1, given my history, given the previous states and the actions? So I've just written this out as a joint probability. And now what I can do is I can use the fact that log of a times b is equal to log of a plus log of b. So I'm just going to decompose all these terms. So I'm not applying my gradient yet, but I'm just going to have log of mu of s0 plus sum over t equals 0 to T minus 1 just plus sum over t equals 0 to T minus 1. Open the log. Sorry, it's a bit messy. I'll make sure to add a clean version. So what I've done is I've just decomposed my log. But now this is really nice because this term is not a function of theta. This is just my initial starting state distribution. It has nothing to do with my policy. So this drops out. Does this part depend on my policy? Yes. Does this part depend on my policy? No. No. So when we take the derivative of it, it disappears. So that is beautiful because now it means we don't have to know about our dynamics model. So the only term that is still around after this is this thing. All right. So this is great because now we don't depend on our dynamics model. We have written down what this term is as a function of-- so we're just doing this term right now as just the sum of the derivative of the log of the policy at that particular point. So we're summing up for each of the different actions we took along the way, what was the log of their probability and taking the derivative of that whole term. All right. So we don't need any dynamics model, which is great. And I'm just going to say here, I'm going to make sure that something is consistent here. Oh, yeah. I had a question on the slide with [? all the math. ?] With all the math? Yeah. Uh-huh. So in the dynamics model, the ps at t plus 1, for the given part, why did we look at the entire history of not just the past state and action? Great question. So what I've written about-- so this question is a good one. I wrote down here the dynamics in a really general form. I am writing them down and I'm not making the Markov assumption. We could make the Markov assumption. But what I wanted to point out here is that you don't have to make the Markov assumption. It does not matter. So because the dynamics model are independent of your policy, when you take the derivative, they completely drop out, whether they are Markov, whether they are non-Markov, et cetera. And so that's really nice. It shows that in this case, it's not making the Markov assumption. Now, I did make the Markov assumption somewhere. I made it here because I assumed that I made the Markov assumption in the sense I assumed my policy was Markov. My policy is only depending on the current state. But your policy also could depend on a history of states. You could have a recurrent neural network or any of the other representations you might want to choose there, and then this would just depend on your history. Good question. All right. So I just want to go, and I want to make sure that I wrote it down neatly in terms of the most general form. That's why I'm skipping this right now. One of the things to note here in terms of just notation is that people often call this thing here a score function. So this derivative with respect to log of the policy itself, we often call a score function. So in general, the nice thing is that it's generally not very hard to compute the score function. So if you have a differentiable function, we can compute the score function pretty easily in many cases. Let me just make this a bit smaller. OK. So let's see what that might look like for a couple of different policy classes. So one thing we could do, which is a pretty popular thing to do, is to do a softmax policy. So the idea in this case is that let's take a linear combination of features, so phi s, a dot product with theta. And then you could say the probability of your action is proportional to the exponentiated weight. So you take the exponent of that dot product between the features, and then you normalize it. And that gives you generally a stochastic policy. You can also have a temperature parameter in there if you want. And the nice thing about this is that we can write it. We can take the derivative of this very easily. So we can just do that quickly here just to illustrate. So what this is just to illustrate that it is often very feasible to take the derivative with respect to the policy parameterization. So this is just going to be the derivative of the log of e to phi of s, a t theta divided by phi s, a OK. So we can do this here, and we can rewrite this here as-- and so this is just going to be equal to phi s, a minus [INAUDIBLE]. So I'm just taking the derivative of this for a particular theta. And so we can just rewrite that as phi s, a minus sum a theta. OK. So what I've done here is I've taken the derivative with respect to this function for a particular theta. And then what I've said here is, well, you could notice that this here is exactly just equal to my pi theta of s, a. So it's like I'm getting this weighting over the features. OK, put this on the next slide neatly. OK, so the score function for the softmax policy is just going to be equal to the feature s, a phi s, a minus the expected value of the policy of the features. Yeah. I'm sorry. What does phi usually mean? Great question. Well, if phi could be, for example, you could think of it as like if you have a large neural network that's doing some representation, it could be the last layer, like the second to last layer. And then you could just do like a linear dot product of that. Yeah, that's a good question. Or in case of customers, it could be a whole bunch of different features. And then you have different groups over there. All right. So this is also possible to do for other functions. So for Gaussians, we often want to think about that for continuous action spaces which are really useful for robotics, where you might have continuous torques or continuous accelerations, et cetera. You can think of there being a mean, which is a linear combination of some state features. Your variance might be fixed or it could also be parameterized, and then your policy is a Gaussian. So maybe you're sampling some particular action dependent on your state along with some variance. And then you can again, just directly compute what the score function would be in this case in closed form. But in general, you're often probably going to be using this with deep neural networks. And then you can just use autodiff to do this just to illustrate that there's a number of different functional forms where you can compute this analytically. OK. All right. So just to recap this, what we've shown so far is that we can have policy methods where we have a direct parameterization of the policy. We can write down the value function as being a weighted sum over the trajectories generated by that policy times the reward. It turns out that when we want to take the derivative of that, we can re-express it so that we just think of we don't need the dynamics model, and we're weighing these score functions. So now let's just do a small check your understanding about likelihood ratio, score function policy gradients. And so I'd like you to do is say, does it require that your reward function is differentiable? Can you only use it with Markov decision process? Is it useful mostly for infinite horizon tasks? a and b; a, b, and c; none of the above. or not sure? Let's just take a second to do that. All right. We have a good split of opinions. Nobody is not sure, but there is a lot of spread. So why don't you talk to your neighbor and see if we can come to more consensus. [SIDE CONVERSATION] I'm sorry to interrupt some good discussions, but I want to make sure we get through reinforced today. So there's a little bit of a tricky one. In fact, when I was giving it to one of my TAs, I forgot to put a none of the above, and he was like, wait, what the hell. So it's none of the above. And so the first one's part of the actual elegant aspect of policy gradients. So as you can see here, you need the policy function to be differentiable, but the reward function does not have to be. The reward function is not a function of the policy in the way that we've written it here. So that's pretty elegant. So that has motivated people in a really wide range of areas where you really might have very complicated reward functions to be interested in using what we're going to see soon, which is reinforce, which is based on this idea because you just need the policy parameterization to be differentiable. So that's really cool. B doesn't have to be Markov because as we saw, the dynamics model drops out. And so what you're saying in that case, it doesn't appear at all. So it doesn't need to be Markov. You don't need differentiability. And we are assuming that it's finite horizon so that we can actually-- episodic so we can get m more than one. If it was infinite horizon, we'd only get m equals 1. So all three of these are false. So let me just make sure I circle that. OK. So just to give brief intuitions because to make sure that we get to reinforce, you can think of if this is a generic way of writing this down, we have some function times the derivative of log of some other probability function. And you can think of this first part as measuring how good a sample is. And what the idea is that when you have the derivative, you're trying to move up the log probability of samples that have high reward. Because you generally want policies that visit parts of the state and action space where you get high reward. So that's the intuition. And the nice thing is that f doesn't have to be differentiable. It could be discontinuous. It could be unknown as long as you can get samples from it. So it can be extremely flexible to what is that reward or objective function. So I put a couple of slides here. I believe it was John [? Schulman ?] who originally had these ones. I put some credits at the front. But you can think of taking a combination between what the probability is of your input of your x as well as your function. So in our case, that's going to be the reward function. This is generally going to be the reward function over trajectories, and this is going to be our policy. It gives us probabilities of the trajectories. And so you can think of combining between these two to actually change your parameter space. So just to give a little bit of intuition over what this sort of gradient estimation is doing. So in general, we can also write down a policy gradient theorem, which says, we could either use something like episodic reward. Or we could be trying to look at average reward per time step. Or we could be trying to look at average value. And in all of these cases, we can end up writing something that looks really similar to the equation I showed you before, which is the derivative with respect to these value functions or something like a value function is going to look something like the derivative with the score function, the expected value of the trajectories you're going to get, of the log of the parameters times the Q function or the return for that particular state action pair following the policy. And there's a nice derivation in Sutton and Barto about that. At a high level, I think the useful thing to know here is just that can extend it beyond just thinking of like the sample of return. And we can think of there being Q functions. All right. Now what I've shown you so far is something that is correct, and we can turn it into an algorithm, but it does not leverage much of the temporal structure. So what do I mean by that? So what we've written down here is a valid gradient. It's unbiased, but it can be very noisy. So we're estimating this by Monte Carlo method because we have these m samples. And as we know from Monte Carlo methods before, they are unbiased, but they can be very high variance. And so some of the ways to make this more practical, and what I mean by that is a better estimate of the gradient and hopefully with less data, because ultimately, we're going to have to be using this information to update our weights to try to get to a good policy. So we want this to be data efficient-- is we can try to leverage the temporal structure, and we can also include baselines. All right. So let's first see the temporal structure. So what we've done before is we've summed up all the rewards from a whole trajectory, and we've multiplied it by the sum of the score function for the whole trajectory. That's what we've done so far. We can instead think of it as, what if we have the gradient estimator for a single reward term? So this is just for one time step. We can think of it for there, which is we have that single time step times the score function for the remaining time steps or for the time steps up to that point. So it's like we just think of the partial trajectory until we got that reward. So we want to think about the derivative of this. This is the reward we got at this time point. So instead of having this whole sum, we just think of, well, what is the trajectory that we got up to that time point and all of their score functions? Does that make sense? I remember having questions about that part. OK, so this is like for a single time step t prime. And so now what we can do is we can sum this over all time steps. So instead of having the sum of all rewards times this, we can say, well, we know that for one time step, it is equal to the expected value of the reward for that time step times the score functions up to that point. So let's just rewrite it like that. Now we're just going to sum over the rewards we got for all time steps. All right. So now what we can do is we can do slight rearrangement. So what we can notice is that for each of the points, so you can think of it as, I have-- so this is t, 0, 1, 2, 3. And you can think of all of these score functions I have at each time point. So the score function at time step 0 is going to appear for r0, r1, r2, r3, dot, dot dot. So that's what I've just done here. I've said this first term is going to appear with a reward function of all of the subsequent time points. Because that decision happened, and then we got a reward, and then we got a whole bunch of rewards later. On the first time step, it can affect the reward you get at time step 1 and all the feature after that. So this term here, this score function can influence-- the one that we get on time step 1 can influence time step 1 all the way out to the end of the episode. The one we get on time step 2 can influence time step 2 all the way out to the end of the episode. So essentially this is like saying, my reward on time step 3 cannot be impacted by decisions I make on time step 4. Time only flows one way. So if we think about what those score functions were and like we think of the trajectories that were generated, they're a temporal structure. And so it means that we cannot have-- if we change the policy parameters such that decisions in the future change, that can't affect my reward on earlier time steps. So this is leveraging the temporal structure. So this just allows us to rewrite the equation so that now we have for each of the different score functions essentially which of the rewards they influence. And the reason this is important is because here you could see that we're multiplying each of the score functions by all of the rewards, and now we're only going to multiply them by the rewards they influence. And so in general, that's going to be way less than having the full set of rewards. So this is going to reduce the variance of our estimator without causing any bias, just leveraging the fact that decisions in the future can't affect your rewards in the past. All right. So that is one of the first things that we're going to do in this case. So we're going to write-- so remember in this case that if we sum up all the rewards from the current time step to the end, we just called that the return. We've seen that before from Monte Carlo. So we can just rewrite this expression like that. And that gives us the reinforce algorithm. So this is the reinforce algorithm that has been incredibly influential in NLP and robotics and many, many areas. And so what this says here is that the way we change our parameter is just our learning rate times our score function times the return we got from that time step till the end of the episode. So we still have to wait till the end of the episode to update anything. But what happens is we run a full episode with our current policy. And then for each time step, we slightly change our policy parameters by using a learning rate, the score function for that time step plus the return we got from that time step till the end of the episode. And then we just step through that for the whole episode. And that's given us T different updates to our policy parameterization. And then we just repeat over and over and over again. And what that guarantees to us is that eventually we will land in a local optima of the value function for the policy parameterization. So this is called Monte Carlo policy gradient or known as the reinforce. I believe this was in roughly 1992, so about 30 years ago. And it's been many, many, many policy gradient algorithms are built on this idea. Now, when you're looking at this, you might still be concerned, from remembering back from the Monte Carlo methods we've covered, that this estimate G can often be pretty high variance. So in general, if you're just directly averaging over sample returns, that might be high variance. So one of the next fixes we can do, and we'll get to this more on Wednesday, is to introduce a baseline. And I'll just say the goals here is that we're going to hopefully try to converge as quickly as possible to local optima. So we want to reduce the variance over our gradient estimate. And so the baseline is going to allow us to hopefully reduce, well, in general, yes, reduce the variance over this estimation process. And we'll see two ideas next, which is introducing a baseline and then thinking about an alternative to the Monte Carlo returns. So those are the ideas that we're going to go through next. I guess I'll just do one more thing, and we will go through the proof of it next time. So I'll just introduce the concept of baseline and then we'll prove it next time. So the idea in this case is that we're just going to subtract something off. And we're going to subtract something off that only depends on the state. This only depends on the state. OK, so this is not a function of your policy, only depends on the state. And it will turn out, and we'll prove this next time-- it's pretty elegant-- that for any choice of something that only depends on your state, the gradient estimator is still unbiased. So you couldn't subtract off anything there. That is only a function of your state, and you didn't change the bias of your estimator, which is wild. And we'll prove that next time. But the goal is that we can hopefully reduce the variance of our estimated gradient by subtracting off the right thing. And just intuitively, the way to think about the baseline is that you don't necessarily just care about whether or not the gradient is positive or negative and whether the returns were good or bad. You might care about, well, how much better or worse are these returns compared to something else I could have done? Like, I want to know whether this policy A is better than policy B. And maybe both of them give you positive returns. One of them gives you 100 and of them gives you 90, but you'd really like the one with 100. So you'd really like to move your policy parameters in the direction of stuff that is better than other alternatives. And that's the idea of a baseline, is to say like, well, maybe I know that I could probably always get like 90 for this particular state. How much better is this policy for this state compared to something I could do on average? And so we're going to intuitively increase the log probability of an action proportionally to how much its returns were better than expected, where the baseline is giving you that expected value. And we'll see formally on Wednesday how by doing this with the baseline, it doesn't introduce any bias. So it's going to be one of the ways that we're going to get better gradients. The other thing that we're going to do on Wednesday is we're at least going to start talking about PPO, which is part of your homework 2, bless you, which is going to involve more ways to be more efficient and effective in the policies that we do. I'll see you then. Thanks

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_MultiAgent_Game_Playing_I_2024_I_Lecture_14.txt

All right. They should be up now. All right, just take a second and then compare your answers to someone near you. The reason I'm asking you about these particular algorithms is because some of the ideas today, even though we're going to be talking about AlphaGo and Monte Carlo Tree Search, will be related to some of the things that helped make those advances possible. So just check. Good chance to refresh your understanding of how upper confidence bound algorithms work. And the one I thought might be somewhat controversial in particular is the third one of whether or not if you have a reward model and it's known, whether there's still any benefit to using an upper confidence bound algorithm. [INAUDIBLE] All right, let's come back together. So, it looks like there was good agreement on the first couple. So the first one is true, which is, you can think of upper confidence bounds as being a way to balance between our uncertainty over outcomes when we have limited amounts of data and yet use that information to still try to acquire high reward. And these algorithms can be used both in bandits and Markov decision processes. The third one is a little bit tricky. Actually either answer would be fine depending on which setting you're looking at. So does somebody want to argue why, if the reward model is known, there is no benefit to using upper confidence bound algorithms. So in some settings there would not be. Someone tell me a setting where if you knew the reward model, you should not use an upper confidence bound algorithm, something that we saw over the last couple of weeks that was different than the reinforcement learning framework. The multi-armed bandits. That's right. Yeah, so in the multi-armed bandit case, where there's no state and there's no dynamics, the decisions that you make don't influence the next state at all. Then exactly what you said. If you knew what the reward model is, you'd know how to act. Like if I knew whether a customer liked ad A or ad B better, I would just show them ad A So in a multi-armed bandit setting/ so in a MAB setting, this is true. In general, it's not true in Rl that in generally false. Somebody want to tell me why in general, it's false? Even if the reward model in reinforcement learning, you might still want to use an upper confidence bound based algorithm. Because we want to know is the value function rather than just the [INAUDIBLE] reward. That's right. So assuming that you don't know what [MUTED] which is, I'm assuming you don't know the dynamics model, so you don't know how to compute what your optimal value function is. It's still often helpful to use upper confidence bounds. And in fact, in many cases, you might know the reward function for when you reach a state, like you know when a customer clicks on something that's good. But the hard thing is maybe to drive them into states where they are going to click on something or they are going to make a purchase. So in Rl this is generally false and we'll see some other examples today where it's helpful to use an upper confidence bound algorithm, even though we know quite a lot about the world. So what we're going to be talking about today is Monte Carlo Tree Search and AlphaGo. And before we get into that, I'll just remind us a little bit about where we are in the course. So we have just a few more weeks left. You should have all gotten feedback on your projects. I encourage you to come talk to me or anybody else about any questions you have. I have two office hours this week because I was traveling late last week for a conference, so you're welcome to come to my office hours today, which are right after this class or on Thursday. There's also a lot of other office hours. In addition to that, a week from this Wednesday, we're going to have a quiz. The quiz will send more details out about On Ed. But the main idea is that it's going to be multiple choice. It is designed to be easier than the midterm, but we'll give you the full amount of time. People generally take the full amount of time just to check their answers. And it will cover the entire course, so everything up through the day before. Does anybody have any logistics questions before we get going? All right. So we're going to talk about Monte Carlo Tree Search and AlphaZero. So as many of you may know there was this amazing series of results from DeepMind and kind of like the 2016 to 2019 time period around showing how you could use reinforcement learning and AI to conquer the board game Go. And this happened about a decade earlier than people expected. And this was really considered one of the huge achievements in AI. So people really thought it was going to take a lot longer to do this. Chess had already been mastered, checkers longer before that. But there was a lot of different innovations that came out of a long history of work that DeepMind used to make this possible. And I also think that it incorporates a lot of interesting different ideas that one might think could be helpful to try to solve other problems. The other thing that I think is interesting about this is it's quite a different form of reinforcement learning than we've seen before. It's really reinforcement learning for computation. And we'll see a lot more about that. So, what we're going to start with is thinking about simulation-based search. And simulation-based search is going to sound quite familiar because we've been seeing ideas around this with Monte Carlo search and Monte Carlo methods. But then we're going to think about combining these with using different parts of the stochastic decision making process. All right. So in particular, one of the major ideas that we're going to be looking at today is the idea that we're going to be mostly focusing on how to make-- figure out what we should do in the current state only. So in general, in class, whenever we've been computing a policy or a value function, we've been computing it for the entire state space. So we might have a policy, and if anybody gave us a state, we could immediately tell you what action or action distribution we should use or we compute a Q function for the whole space. One of the key ideas today is to say, well, maybe, particularly if we've got an enormous space, that we don't really care about trying to compute an optimal policy for everything in the space. Maybe we just want to use our computation to really focus on a good decision for right now or for whatever state that you might end up in. And there are lots of reasons to think that that might be important, particularly in really large domains. So you could imagine, if you're the Fed and you're trying to make some sort of federal monetary policy, you probably don't care about doing this for all the scenarios which the US is not in. You really want to figure it out for the current scenario. In the case of the board game, Go as we'll see, there's just an enormous space of potential states you could end up in. And it may not be important to have a perfect way of acting in all of those. So one big idea here is that we're going to be mostly focusing on computation to figure out what's the right thing to do in the current space. So one thing you might do in this case, given all the ideas we've seen in class, is you might simulate. So imagine that someone gives you a policy and what you want to do is try to do at least as good as that policy and maybe a little bit better. So one thing you could do is say, well, I'm in a current state, like a current real state, say, ST, and what I'm going to do is I'm going to say, think about all the different actions I could take next. And then I'm going to roll out using my default policy from those states. So this is just like for K episodes from the current real state. I'm going to roll out in my brain what might happen. Now this means that I need some access to a dynamics model. So I can only do this if I have access to a model. And what I'm going to mean here by model is dynamics and reward model. So you might imagine that you actually know how the world works or that someone that you've learned some sort of model from the past, this could be estimated or true. And so then what we can do is we can just do sort of Monte Carlo evaluation. And what we're getting here is an estimate of the Q function that says, if I start in this state, and I take this action and I roll out under my simulation policy, what is my expected return? So that just gives me an estimate of the Q function. We've seen this before with Monte Carlo methods. And then you can just pick whatever the real action is, like this is what you're going to take in the real world with the maximum value. And what you can think of what this is doing, and I'm just going to augment this with pi to make that even more clear, is what I'm essentially computing is from the current state, what is the Q value of my simulation policy. And so I could roll out under those policies. And then I'm going to do one step of policy improvement, given that. And so if someone gave you a budget of computation, this would be one reasonable thing you could do with it. And then we're going to see a lot of things that you can do that are much better than this. But this is one thing you could do that would be viewed sort of as simulation-based search, which would allow you to do better than the current policy you have. All right. I think it's helpful to think about this in terms of what the tree structure is. So the idea in this setting is that we start in some state and we're going to roll out under and we're going to take a certain action. So that's going to be our action A here. That's our action A and then after that, we're going to get to a next state, S prime. And then from then onwards, we're going to follow our policy pi. So this here is policy of S prime. We're going to sample an action according to that. So at the root for my current state, I'm going to consider all possible actions. But then after I take that action and transition to a next state, I'm just going to roll out by following my policy pi. And then I do that all the way out till I hit a terminal. So T here equals a terminal state. And so one thing you could do is just do the simulation and then average over the root nodes. But you also now have a whole bunch of data. So you could do other forms of reinforcement learning given that data. Does the pi have to be fairly optimal for this [INAUDIBLE]? Good question. So it depends how we think of what this is computing. So, if this is just computing Q pi of SA, it's just doing one step of policy evaluation. So this will work, whether pi is a good policy or a bad policy. But exactly, I think, to your point, if pi isn't very good, then you probably aren't going to get a very good-- you're just doing one step of policy improvement here. You're not necessarily going to get Q star unless pi is really close to optimal. OK, so this is one thing you could do, but I think the nice thing of visualizing the tree in this case is it starts to make it really obvious that you could do other things that could be better than just following whatever your current policy is or whatever policy you might have access to. And I'll just make this clear with the model and a default So we might instead, if we have limited amounts of computation, instead of just doing rollouts, we might want to try to get something that's closer to Q star. And one way we could try to do this is by trying to construct an expectimax tree. So raise your hand if you've seen either minimax trees or expectimax trees before. OK, like a few people, but not everybody. So this will be a quick introduction to those. But the idea is if we think about what this forward search is, really what we are doing when we construct this tree-- so this is the action. So this could be like, say, A2 and this is a A1m and this is a next state. Imagine that you just have a few states in these examples. So the black nodes are all actions and the white nodes are all the states. You can think of this, and we've seen similar graphs to this a while ago, think of this as just rolling out your Bellman backups. So you could think of what happens in the world is, I take an action and I transition to some state, and then I take another action and I transition to some states and sometimes I terminate. And what I would do normally in this case is, then I would back up along this tree. So whenever I have states, I would take like an average or an expectation. And this is really just representing the probability of S prime given SA, So it's representing that sum. And then every time I have actions, I would take a max. And this is just representing inside of the Bellman backup that I take the max over the actions. So you could think of this as just approximating the max over, max over ARSA plus gamma sum over S prime, probability of S prime given SA B of S prime, except for instead of having B of S prime, then you would just expand this out all the way until you hit the terminal state. And this would require us also to keep track of the rewards we obtain as we go down this tree. So, for example, here you might get reward of S prime A. So does that make sense? So if you have access to a Markov decision process and its dynamics model and its reward model, one way you could use that to figure out what's the optimal thing to do in your current state is you build this tree. Build this tree until at a leaf you reach a terminal node or for a fixed horizon H, and then you back up by doing wherever you see a branching according to states, you take an average weighted by the probability of each state. And whenever you get to a set of action nodes, you take the max. Anybody have any questions about that? We might get to this later. But if we're considering complex games, like go, the state space is like massive, right? It's very unlikely that you're going to run into the same composition of the board twice. Like, how do you deal with that? Great question. Hold on to that. We'll get to it. Yes, yeah, absolutely. So right now, well, and in fact, on the next slide, we'll talk about how big this tree is. But this, at least conceptually, should be something that you think, yeah, we could do this. I could imagine doing this. So why might this be better than before? Well, this might be better than before because you're not actually solving the whole MDP. You're only doing sort of Bellman backups starting from the current state you're in. And so you might imagine that if the space is enormous, even though you're sort of rolling this out in terms of this kind of exponentially growing tree, it still might be smaller than your whole state space. But as I was saying, this is huge in general. So if you want to actually expand the whole tree in general, the size of the tree is going to scale by the size of your state space times the size of your action space to the H. H here would be her horizon. And so as you could imagine, this is going to be terrible really quickly. We don't want to-- if you think about Go or you think about Mountain Car or other games where you might be-- or other environments where you might be having sort of 100 or to 1,000 steps, this is going to be completely intractable. But as you might notice when we're looking at this, here, when we wrote it out, we thought about all the next states we could reach. But if that's a really large set, we know that we don't necessarily actually have to sample all of them and compute that exactly in order to get a good estimate of the expectation. We know that, in fact, we can just sample. So if you sample, what's the next state 100 times and average over all of their values. That's a pretty good approximation of what the average value is, even if there are 10 billion states. Because you can approximate an expectation by an average and that tends to concentrate really quickly. So that's going to be one of the really big ideas of using Monte Carlo Tree Search is that we're not going to have to expand all the next states. We're just going to sample them. So let's see how that might work. So this is where we get into Monte Carlo tree search. And note I highlighted a tree here because we're not doing Monte Carlo search anymore. We're not just rolling out with a policy. We're essentially going to try to sample parts of that tree. But we're not going to just do single pi rollouts. So we're going to build a search tree rooted at the current state. We're going to sample actions the next states, and we're going to explore different parts of that tree. We're not going to always follow the same simulation policy pi. OK. And then after the search is finished, we're going to take an action in the real world by whatever has the highest value, as we estimate at the root. At least that's one way we could do things. We'll see some other ways to do it. And, well, let me just give a little bit of intuition of why does this work. This works because what we're doing in this case is we are approximating. expectations with averages. OK. So we're not actually trying to expand all the next state. We're just going to approximate it with averages. And that will turn out to concentrate pretty quickly. And that's going to be really helpful. So let's do a quick check your understanding. So oops. Well, there you go. That's OK. You can think about whether or not you agree with this. Monte Carlo Tree Search involves deciding on an action to take by doing tree search. So think about whether it's a good choice for short horizon problems and why, long horizon, and large state and action space. And actually the middle of this is slightly debatable. So take a second and think about this. Uncover this so that when I upload it later, people can-- So why might the first part be false? Why would we not want to do this further? Well, first of all, does anybody have any questions on what Monte Carlo Tree Search is doing in terms of how it's different than the other things that we could do. So then tell me why it's not probably a good choice for a short horizon problems with small state and action spaces. What would you do instead in those cases? Yeah, [INAUDIBLE] We better do what? Monte Carlo [INAUDIBLE]. So maybe. I guess what I was thinking more is in that case, maybe you should just do dynamic programming. Yeah if the state space and the action space is really small, you can just do value iteration. Yeah. Monte Carlo search could work too. But in particular, if things are really small, if you think back, it's been a long time I know, but in Monte in standard dynamic programming, it's only like S squared times A for each backup. And then you're just doing that. If you're only just doing the H times, that's nice. You don't have any exponential dependence in that case. So if it's really small. Just do Bellman backups. And the order of that is roughly A squared A times the horizon H, roughly. So at least it avoids the exponential. It will be a good choice for long horizon problems with a large state space and action, a small action space. Because what we're doing in this case is we're approximating that expectation by samples. So we approximate-- so this is true and this is false. Approximating an expectation by samples. And so that means instead of us having to get that like enormous state space that we're multiplying by, whether S squared or such, we're just sampling from that. And so we can have something that's more like a constant with respect to how much we're sampling. Now the middle one is actually a little bit controversial. And we're going to see different ways to tackle this. Why should this be somewhat controversial? Well, in Monte Carlo Tree Search, the initial way we're getting the big gain is we're sampling next states instead of enumerating them. But it shouldn't be obvious that for actions we want to maximize. For actions, we want to take the best over all the actions. And so Monte Carlo Tree Search a priori still has to just sample the whole action space. And so it's not clear yet that unless we do something special, that Monte Carlo tree search is necessarily going to help us when we've got really big action spaces. Because in general we've replaced the expectation by a set of samples. But it hasn't told us yet how to do anything smart in terms of the action space. So this one is sort of debatable, may be false. Depends how you think about it. But of course, there are a lot of algorithms that combine with Monte Carlo Tree Search to show us how we might be able to tackle this problem. So what we really want to be able to do is solve long horizon problems with enormous action spaces and enormous state spaces. So we're going to need ideas beyond Monte Carlo Tree Search to tackle that. An Upper Confidence Tree Search is one idea for how to do this. And I think UCT came out in around maybe like 2007, 2008. People started using it for Go around then. And the idea in this case is that in addition to doing the sampling over next states, let's be strategic over what action we take when we're expanding in our tree. So when we decide to sample, the next action doesn't have to be from a default policy pi. Let's think carefully about essentially where do we want to fill in our search tree. And this is one of those other really big ideas. Because this is really where we're going to start to think about ideas from reinforcement learning essentially to optimize computation. Because right now we're still assuming that we know the MDP, that we know what the dynamics model is and we know what the reward model is. So in theory, if computation was no issue, we could just do value backups. The challenge is this is going to be completely enormous and thus totally intractable. So the idea here is to say, well, maybe if we have access to those, we can still think of trying to approximate sort of like Bellman backups or approximate maxes. But we don't actually have to want to enumerate all the actions as much, and we want to really focus where we're using our computation. And DeepMind has been really a pioneer in thinking about using reinforcement learning to prioritize computation, to solve a lot of really important problems. And I'll try to come back to that at the end. OK, so how does UCT work? The idea is, and this is why I asked you guys about this and refresh your understanding is, we are going to treat each node, where each node that was sort of like a state node inside of our tree search as a bandit. And so it's like we have many, many, many, many bandit problems inside of our search tree. And we're going to then maintain an upper confidence bound of the reward of each arm inside of a node. So the first node, you would have is your root node. All right. And so it would have, say, A1, A2, A3, and we would think of that as a MAB, as a multi-armed bandit. And then when you get further down in the tree, so let's say we this goes to next S prime. This would be another. A1, A2, A3. And this would be another multi-armed bandit. And you would have you'd have to store in memory lots and lots and lots of different multi-armed bandits. So you're maintaining huge numbers of multi-armed bandits. And just like what we normally do in upper confidence bound, we're going to maintain an upper confidence bound over each arm. But we're going to be thinking of that as essentially what would happen if I take this action and then act optimally till the end. Now, one big challenge is, of course, we don't know what the reward would be of acting optimally. So there's going to be a lot of different policies that are moving at once. But let's see what that might look like. So here's the idea. So let's say what we're going to call. We're going to say we have a node I. So this could be our root node or it could be any other node. The way we are going to I'm just going to call this AI. We're going to try to maintain an upper confidence bound over what is the potential expected discounted sum of rewards we'd get starting in this node and taking this action as the following. Let's say that we've been in that particular node and we have rolled out from it using some strategy that we haven't really talked about yet, NIA times. So this is the number of times we've been to this node before and we've happened to expand the A action. What we do is we look at all the returns we've gotten under those cases. So what's a return again? So a return would be, you go back to here. So let's say you've done this, OK. What you would do, what your return in this case would be is it would be a sum of all of these rewards you've gotten along the way. So G we're going to use to denote the return. So this would be, this reward from starting in state A, taking A1, and getting the rewards out to the terminal state. And maybe next time you go down this action, you actually get to here and you get a different return. So those are just like your Monte Carlo returns from before. And it's just for all the other times you went through that action. So that's part of it. So that's just an average and it's kind of a weird average because it might be that your which nodes you visited and which actions you took have changed. So we're not committing it to it to be a particular policy. It's just like we've taken some action and we've followed-- we've made a series of decisions until we got to a terminal state. We added up the rewards and we keep track of that here. So that's one thing and that's sort of a we probably look at it and think this is a very loose approximation of what the optimal Q value is for that state in action. The second term looks like upper confidence bound, which is, you have some constant C/ you have some log term, which depends on the number of times you visited this node divided by NIA, number of times we've been in that node and taken that particular action. So this just looks like a bandit term. It's an upper confidence bound over the reward that we can get. And so what Upper Confidence Tree does is that the way it picks the action, the next action to take from the current node is it picks whichever one of these has this higher upper confidence bound. Now, it should seem slightly suspicious that this works because in bandits, when we took an action, we knew that this really was an unbiased estimate of the reward of that action because we just would see that one action. And then we knew from Hoeffding that this really was an upper confidence bound on the true value of that arm. But now we're in a much more weird case, where we are thinking of this for a sequence of actions we're going to take. We're trying to do expectations over states, and the actual actions we're taking from this node onwards may not be optimal. One time we might go through-- I'll draw it on the board. Like one time we might go through this zig zag. Another time we might go through this zig zag. Another time we might come back here and then we take a different action. So it's not like we're not doing one step of policy improvement here. We just have lots of different things that we're trying and we're averaging over them. So you should be slightly suspicious whether or not this is going to be doing a reasonable thing. But it's certainly something you could do, something you could imagine coding. And then we'll do this many, many times. And then at the very end-- so this will expand essentially different parts of your tree. And when you're following this, in particular, you're going to start to expand parts of the tree which look promising more. So if this one happens to have been getting-- this one gets like plus 100 and this gets plus 100 and this gets plus 90. Whereas let's say one other time when you took this action, you went down here and you got minus 10, well, then when the next time you get to your root node, you're probably going to be more likely to keep going down this path. So it's going to selectively expand parts of your tree. It's not going to hit and you'll have these unbalanced trees where you'll often see parts of things are getting filled in. And then maybe if something else becomes more promising, you'll switch to another part of the tree and fill in things there. So it's sort of this unbalanced construction of your forward search tree. And the way that it's unbalanced is that using the Monte Carlo aspect to approximate all the expectations. And you're using this upper confidence bound to of selectively prioritize across your actions. And that's what's going to help with our enormous action space. Now, you still might be concerned that when there's a really enormous number of actions, like we're going to see in Go in other cases, that this still isn't going to be enough, because if the number of actions you have is a million, these things generally-- you'll have 0 for this part right before you have taken any actions and your counts will all be the same. So, it should still be concerning because what should you do if you have these 1,000 different actions, and like you might not be able to do anything essentially until you've visited everything once. Because before then, as long as you've defined something that's a reasonable upper confidence bound, everything is going to look awesome. It's like action 99 will be awesome/ action 100 will be awesome. And so you'll have to sample all of them at least once, and that generally will be completely intractable. So we'll see ways to further reduce this. But what you can think of this part is doing is saying, well, if you can at least sample every action once, you can at least mean that you're not going to have to focus on unpromising actions later because you're going to quickly use this upper confidence bound. So these sort of Monte Carlo Tree Searches are starting to look really promising. A lot of people have used tree search based algorithms, as some of you might have seen in other machine learning algorithms or other AI classes probably in particular. But what this Monte Carlo and UCT based approaches is, there's this highly selective best first search, but with simulations as well. And they're using sampling to break the curse of dimensionality. On UCT and UCT to help with large action spaces. And the other really nice benefit of these ones is they're parallelisable So when you're sampling things, you could certainly imagine trying to expand, do these sort of rollouts many, many times and then collect the results. So you can start to parallelize these methods as well. And that's going to be really helpful. So that's the background between Monte Carlo Tree Search. But now, of course, the really big breakthrough that this allowed or people built on these ideas is to achieve AlphaGo, AlphaGo and then AlphaZero and then MuZero. So there's a whole sequence of them. And let's just get up a movie for that for a second. And who here has played Go. OK a few people. I was thinking it could be fun to have us all play it so we could see that it's really quite hard. But another time. Go is the world's oldest continuously played board game. It is one of the simplest and also most abstract. Beating a professional player at Go is a long standing challenge of artificial intelligence. Everything we've ever tried in AI just falls over when you try the game of Go. The number of possible configurations of the board is more than the number of atoms in the universe. AlphaGo found a way to learn how to play Go. Building suspense. We'll see how the network goes. This is a documentary of DeepMind's efforts to try to beat the world class people in Go. Let me see if I can make it work. I think it's probably decided it doesn't like the internet right now. Let's double check if I can get that to work. So what they ended up doing is they are going to use reinforcement learning to help solve this problem. We'll see whether or not the technical difficulties resolve. And then what they did is they started playing against grandmasters and they tried to-- then they played against Lee Sedol, who was one of the best people in the world at Go. And I think one of the really interesting things about this is that it really shows that A, it's now possible to use AI to beat the best people in the world at Go, but also the types of strategies that it built were very different than what people were doing before. And so I think this is a pretty important aspect for AI because we've often thought of AI as sort of automating things that people already know how to do. And I think this illustrated that they're really starting to be places where computers go beyond even the best humans and what we know how to do. And since then, there's been a recent paper, I think maybe a year or two ago by Been Kim trying to look whether or not you can teach grandmasters using the strategies that AlphaGo and its descendants invented. And so then there's this really interesting opportunity and question to think about, can we actually learn from computers in these new ways and try to exceed both human level performance and computer level performance. So I will post this later. You guys can look at it. Let's go back to there. OK. All right. So how does Go work? Well, it's a really, really old game. It's considered one of the classic, hardest board games. And it was considered a grand challenge for AI for many, many decades. This sort of game tree search that we saw before, something like a Ford search. Now, it couldn't be expectimax in this case because it's a two player game. Go is what's considered a zero sum game, meaning that someone either wins and loses. And in this case, whenever we think of a next state, rather than it being an expectation, it's really a minimax problem because each opponent is playing to win. So in this case, it's good to think about what is actually uncertain in this case. When we're playing Go, the rules of the game are known. They actually have another descendant now where you didn't have to know the rules of the game. But certainly for the first few, the rules of the game were known. So what's unknown in Go, if we wanted to think about building a tree or trying to learn in this case, if we know the rules-- Yeah. Well, you might expect your adversary to play the best move. That might not always be true. They might be seeking different strategies, so you wouldn't know that. It's a good point. So it might be as [MUTED] saying, that you might not know exactly what the best strategy is, or you might not know whether someone's going to play the best strategy. I think the other thing that I think of is that we don't always know what the best strategy is. It's just incredibly hard to compute this in this case. And so in this case, that's sort of next state, if that next state is really from an adversary, it's not clear you've got stochasticity in that because you don't know what the optimal game is. Now, of course, once someone picks a move, everything is deterministic. So in some ways it's all deterministic, it's all known. The key thing is that because it's this adversarial game, it's not clear what the optimal strategy is. So that's kind of one of the really hard parts. All right. Just a couple basics of the rules of Go. So normally it's played on a 19 by 19 board. But when people first started to-- well, kids and also when researchers started tackling this game in earnest, starting like the late 2000s or 2000, David Silva, who's one of the authors of this and an amazing researcher, I think as part of his PhD in the late 2008, 2009, he was doing things on a 9 by 9 board. And just as a couple of basics, there's two different players, either someone playing the black stones or the white stones, and you're trying to surround stones that are captured and then you win. It's as I said, it's a zero-one game, zero-one game, which means a winner takes all. One of the interesting things about Go is that in general, there's no intermediate reward. So you have to play till the end of the game to see who's actually winning. And so there's just a single reward at the end, which also makes it very hard for credit assignment and to understand what moves caused the resulting game. Yeah, so AlphaGo and AlphaZero, AlphaGo was the first one that was used. Then they developed a number of variants. They then played against Lee Sedol, and then there was AlphaZero. And what they exhibit in this case is a number of different really interesting features. So they have self-play, strategic computation, highly selective best first search. They use the power of averaging. They leverage local computation. And then they learn and update heuristics. For those of you that have seen tree search based methods before, you've often probably seen ideas around heuristics, which are other ways to think about how do you expand the tree. One of the interesting ideas in these papers is that they're going to learn those heuristics and update them over time. It's another important aspect. So let's see how it works. So how does self-play work? So the key idea in this case is that we're going to have the agent play itself. So there's going to-- you can think of it as there being two copies of this same agent right now. And what will happen when they're playing a game is they compute the best move of the current state and then the opponent does the same. And they have access essentially to the same policy or the same sort of algorithm, but they're both just using it in an adversarial way. And so that means the only bottleneck in this case is computation. We have no humans involved. And self-play also provides a well-matched player. So take a second and think about, what are the benefits that will happen with self-play and what's going to be like the reward density. Are there going to be lots of rewards when you do self-play? Are there going to be very little rewards? Let's just take a second and I'll check and see whether I can make the networks work. Maybe talk to a neighbor and see if you guys both have the same idea of whether self-play will be helpful or not. All right. What does this do to policy training? What happens when you do self-play? Do you have higher reward density. Do you have low reward density? What happens? Here, raise your hand if you think you have high reward density. If you think you have low reward density. So, all right. Was somebody who think we have high reward density you want to explain why. That's right. We do a pretty high reward density. Why do we get that when we do self-play? What happens? Or I think it's easy, maybe easiest to think of like if you play against someone that's much, much better than you, what happens? Just kind of lose-lose. Lose all the time, right? I mean, everyone's probably done this before. You play against like a friend of yours that maybe much better at a board game than you or something like that. Or you're a better friend of yours is much better than you at tennis or something, and you go and play with them. And it's not normally that fun because you just lose all the time. And when you lose all the time, you may not get very much signal about what things are even doing better or worse at because you always lose. And so that would be a case where the reward density is very low because the players are really mismatched. And it means that most of the time the agent is not winning. Now the same thing is true if the agent is much better than the other agent. But self-play means you're sort of matched cheaters, like matched at the same level as someone who plays tennis the same level as you, or you're matched with someone who has the same Helo score as you and in chess or Go, which is sort of a way to quantify the player's skill. And the nice thing about that is that you would expect that roughly, if you play someone that's exactly the same level-- now here you're an Rl agent, so you're going to play someone that's actually just on the other side. So they're exactly the same level as you, and that means you'd expect you'd win about half the time. I think, on average. So that's really good density for something that is a zero-one game because you're not just like every 3,000 games getting a zero or a one here. About half the time you'd expect to get a one and half the time you'd expect to get a zero. And the reason that might be beneficial is hopefully that's going to give you a lot more signal of how you should change your policy in order to figure out how to get better. So I think that self-play is a really interesting one because you could think of it in some ways as kind of providing an automatic curriculum. Go to the next one. The rewards are going to be pretty dense. And for those of you that have seen curriculum learning before in other machine learning stuff, just like in classes where you often build up with math over time and you don't start with calculus. You start with addition or what a number is, and then you slowly build up. So you're always trying to be on roughly the right level. Similarly here, the agent should do that automatically because they're going to start off and they're going to both be terrible at Go, but they're still going to get pretty high density of reward because they're both terrible at Go. And then over time, the agents are going to get better and then now they're automatically always playing an agent that's roughly the same level as them. Now, we'll have to see why the algorithm will help them get better. But intuitively, as we saw, even with the Monte Carlo simulation, not even tree search, there, it was doing like one step of policy improvement. So you can imagine that even if each round were just doing of one step of policy improvement, over time, we would hope that we're going to get better and better. So this idea of self-play I think is a really interesting one. It works really well in games and it's been exploited a lot. I've often thought like it would be really interesting to see are there other places you can set up to essentially be like a game. Because what you could think of here is what self-play is leveraging is that sort for the dynamics part of your environment, you now have a simulator you can plug in, which is the agent itself. Now, in general cases, you can't do that. Like if I'm going to simulate patient dynamics, I can't just plug in, like, I can't do self-play for that. An action is how the patient responds to some treatment. And I can't like play against two patients. That doesn't make sense. But I think in some cases here, it's really a very reasonable thing to do to use self-play and it can be really efficient. Because you can think of it in a way as like it's changing the strategies in a way that sort of iteratively updating the complexity of the environment you're trying to solve. OK, so [MUTED]? I think I have a good idea. But what is the exact definition of reward density. What is it with respect to? A good question. So here I just mean about lots of things here. What I mean by reward density is how often you're going to get a-- you're going to win. And here rewards only happen at the end. So it would just be, of the games you play, are you going to get a lot of reward? Are you going to get zero? If the agents are really mismatched, in general, the reward density is going to be either saturated, which means you always win or near zero because you're never going to win. And neither of those are very informative. And the idea is that if you're getting reward about half the time, that might be really informative, because you get lots of signal of like, that thing worked. That didn't work. That thing worked. That didn't work. And so you're going to have a lot of stuff to estimate kind of a gradient or an improvement for your decision policy. Yeah. With self-play, can we say that if you play against someone who has a completely new strategy, might not be able to generalize well enough, because I was always playing against myself and always using the same kind of strategies. Great question. So yeah, so that's a really great question, which is, OK, well, so self-play might be good, but then what if you suddenly play against someone really different. So, what we're going to have to see in this case is whether or not over time, you do get to something that's essentially like a minimax policy. So if you get to the optimal policy, you could hope that you really are at like grandmaster or beyond. And one of the exciting things here is that they will get to that. So as this ratchets up and ratchets up after lots and lots of trading and after using very, very complicated networks, you can get to that level. Does that work-- well, Does that work for games where moves are not deterministic, like, I don't know, like gambling games, like poker or something where there is some sort of probability associated? Yeah, interesting. So there's also been a lot of work. There are really, really good AI agents for poker now. I think it was 2019 that Noam Brown, had a paper in Science showing that you could beat-- I don't know if you could beat, but it was certainly sort of competitive with top humans, I believe. So. Thomas Sandholm and Noam Brown, who did his PhD at CMU, had have got an agent to do well at poker. The algorithms are slightly different, but Yes, you can. It's a good question here. We're assuming that it's also going to leverage the deterministic nature. Yeah, all right. So how does this work? Let's go through what it's doing because it relates to Upper Confidence Tree Search. But there are many changes. So there are many improvements that were needed for it to get much better. But it is going to be similar in the sense that it's going to try to compute. First, we're going to start with, it's going to simulate many, many, many games, and it's going to iteratively try to learn better strategies. One of the things that is going to be different compared to naive Upper Confidence Tree is that we're going to actually maintain a neural network. So let me just get back to there. So what we're going to do in this case is we're going to have a neural network that, given a state can produce both an estimate of V of S and a policy distribution for that state over actions. So we're going to maintain a single neural network. This is what AlphaZero does. It maintains a single neural network that, given an input state, will output both an estimate of the value of that state and a policy for that state, a distribution over actions. And we're going to talk about how we train that shortly. But for now, just assume to start that we've already trained that or that we have access to that and we're going to use that now, when we're going to do a number, we're going to play a number of games. And in particular, let's first think about how we're going to compute the first move in a single game. So we're going to do some self-- we're going to do self-play in this case between two agents. Value's the same. What we're going to do is we're going to do an Upper Confidence Bound based thing. OK, what does these upper confidence bounds going to be based on? And then we're and then we're going to decide the max between them. So this is going to look like UCT, but slightly different. So what U is going to be equal to, in this case, so U of IA, so let's say this is node I. It's going to be proportional to the following. It's going to be proportional to PSA. It is divided by one plus FSA. This is from our policy. Look, I'll write it as this. So, this is from our policy network. This means that our upper confidence bound is going to include in it a bias towards some actions versus others. So our policy network is going to say, if you give me a state, I will give you a distribution over actions. And that's this. And that is going to help us with the fact that we have an enormous number of actions. And so this is going to prioritize some actions that we think in general might be better for these types of states. So this will be a deep neural-- like that neural network up. There is going to be a huge crazy deep neural network and it's going to try to leverage similar types of states to suggest which actions might be useful in this particular state. The other thing you can see here is that this is going to decay as we visit a state in action more. Here in this case-- so I'll just be a little careful. this is all going to be operating, I believe this part, is really I. So this is, I think at the node level. I'll double check that. But the PSA has to be at the state level. So remember, you'll be in some state at this point and you could feed this into a convolutional neural network. It's an image of the board or some other deep neural network. So that part has to generalize. But I'm pretty sure-- I'll double check this, that the count here is actually specific to this particular node. Now, why is this U interesting? It's interesting both because it incorporates sort of a priority function over actions. You might say some actions are better or worse and that's going to change which ones we expand. The other is that we are decaying faster than normal upper confidence bounds. So recall the UCT, U was proportional to one over square root. Yeah. So this is going to decay a lot faster. This means we're being a lot more aggressive in our upper confidence bound. We're shrinking fast. And so what that means is that we're going to do a lot less exploration of things that we think are not so good. So that's one really important part of how we're going to pick what to expand. The other part is this notion of Q. So how are we defining Q for this node? Q is going to be equal to one over NIA. And again, I'll double check this is nodes rather than states. Sum over S prime, V of S prime. So what this means here is that this is going to be an empirical estimate of what the value is over the states that we've reached by following this particular action in this node. And we're going to see where that comes from shortly. It's going to be a little bit different than what we saw before. But these are the two components that we're going to use to decide which action to expand. So yeah. By referring to the node here, we're talking about the identity of the node in the matrix or the state of the node in the state? The state. So you can think of what a node here is, in his case is it is a particular board game configuration. So it's like saying the white pieces are here and the black pieces are here. So it's like you could think of it as just like an image, an image of the board. And the earlier work, in fact, was using convolutional neural networks to take in essentially images and features. Yeah. Is there a meaningful difference between nodes and states? Yes, so it's a great question. So in general, there may be a difference between nodes and states because, well, this is I'm not a Go expert, so I don't know. But in general for these type of algorithms, you could reach the same state at different parts of the tree. And if you can do that, then you would have different bonuses there. Now, I don't know enough about Go to know whether that's always possible and it's certainly possible in some cases that it would be isomorphic, that the nodes and states would be identical. But in general, these sorts of algorithms can work in cases where you can certainly imagine for checkers or chess and stuff, you could end up in the same board game state later on, but it would be a different part of the tree. OK. All right. So this is just the start. This is just starting at the root, trying to figure out which action we're going to take from the root. And then what we do is we repeatedly expand. So in this case, we would follow the right hand side. Now, what we would do at this case, which is pretty interesting, is so this would deterministically-- I would put down, say, a piece on the board. In this case, I decided to put down this piece. And then what I would do is I would flip over and pretend to be the opponent and it would do the same thing using its Q and U. Now it's Q and U are going to use the same neural network approximation. So this is just self-play. But it's just useful to in this case that they are going to be optimizing for the opposite. One is trying to optimize that the Black pieces are going to dominate. The other one is going to try to optimize so the white pieces dominate. So now we're going to have that the opponent selects the max Q plus U. So it's just useful to think of you're sort of repeatedly flipping back and forth between these two. But you're using exactly the same neural network parameters when you do that. So this is going to continue going all the way down until we hit a leaf node. So this is again, we haven't even selected a single action to take. All of this is going to help us finally take a real action in one game. So right now we're just going to do a whole bunch of computation to figure out what that action is. And just to note again here, so we're assuming that we have access to this parameterized deep neural network. And whenever we do this expansion, we are using our P function because that's what was going into our upper confidence bound. So our U was a function of P. So it's a function of these probabilities. And so we could weight different actions more. So we keep going all the way down until we hit a leaf node. And at that point we plug in V of S, so when we hit a leaf node. So if this is terminal. We're going to do V of S. We're going to use our neural network to plug in V of S. So this is different than what we saw before. Because before we were thinking we could actually get the rewards along our trajectory until we get to the final end. Or if we didn't have any rewards, we just get whether we sort of thought we were in a winning or losing state at that point. We're not doing that anymore. We are plugging in an estimate of the value of the final state according to our value network. And that means also that we can either go all the way out till we win or lose a game, or we can terminate. We can say after 700 steps, plug in our V of S, which would give us an estimate of how likely we were to win the game at that point. So once you have that, we're going to propagate all of this stuff back up. So if we're going to select that, once we go all the way down and we get to some V, this is going to go back up. And remember, what this is going to do is we're going to update our Q function. So our Q was equal to one over NIA, sum over all of our times V of S prime. So we're going to update our value all the way back up. So we used our P function when we were expanding out to figure out which actions to take as well as our upper confidence bound. And then we use our V prediction to do the backups. So the way that it would work is we go all the way out to a leaf node, and then we go all the way back up along the ancestors to the root node. And then we do the whole thing again. We do that many, many, many, many times. I'd have to remind myself, I think it's like, say, for example, it might be 160,000 times, for example, just to give you a sense of the scale. So it could be something like 160,000 times, and that means you're going to fill in parts of the tree. And then after all of that, we have to decide what actually to do. So that's just to compute a tree to decide the current move. So we do this many, many, many times. So we do this many times and then at the end, we are going to decide what to do with our root node by the following. And this, again, is a little bit different than what we've seen before. We are going to compute a policy for the root node by figuring out which actions did we mostly visit underneath it. So we're going to look at NSA. So sort of which actions, how many times do we take each of the actions, from the root node to one over tau. I think this should be minus. Let me just double check. Yeah, I guess it just depends how you set tau. So tau is just going to be a temperature parameter. So if tau for example, was minus one in this case, then it would be one over NSA. Would be proportional to that. Or if N was one, you would be sort of proportional. You would take things according to that divided by the total. So if n is one, sorry, if tau is one, tau is equal to one, then it would be NSA divided by N of S. And as you increase or decrease this, then you get things closer to taking a max or just averaging. So this allows you to have a stochastic policy at the root node instead of necessarily just taking the argmax. So this is quite interesting. So this is what they're going to end up doing. After you do all of this, you're going to actually take an action and then you're going to-- so that gives you a policy and then you are going to sample from that policy to actually make a decision. OK, so this is how a game works. You do an enormous amount of computation. At the end, you get this policy according to the number of times you've taken each action from the root node. And then you sample from that policy. You reach a new state. So like, let's say you put down that thing. Then the opponent does exactly the same thing and they put down something. And you repeat this all the way out until the game ends. Now, even if your DeepMind, you care about computation. And so in some cases, they will truncate games if they think there's definitely going to be one outcome or the other. But in general, you would just keep going this all this way and Z here would be who won or lost the game. Yeah. You said they will truncate games, but do they actually sit behind the computer and watch the games being played or like-- No, no, it's absolutely all automated. So this is going on billions of times. And what they will do is, if I think it's after 700 moves, if they're not-- if they think either it's going to end in a draw or it's definitely going to be a lose, and then they try to bound like false positives and stuff like that. But it was interesting to me that they included that. Just indicates that it probably saved them a substantial amount of computation time. Yeah, no, everything is totally automated. So what they do is now at this point, so this is like a single game-- as you could imagine, this is an enormous amount of computation. After a single game, but a lot of this could be parallelized. You're now going to train our neural networks. So remember that we used a neural network to both give us an estimate of the probabilities, like give us a policy for each state as well as a value. And what we do is now we have-- so from that game, that one game, we have one observation. So this is our Z, you know, who won, who won the game. And we had, from each step we had these policies that we computed. And we're going to use those as targets to train our neural network. So what we do is we go back and we say, OK, well, in that time when you were in state S and you computed a policy, and eventually you got a value of Z, you either won or lost the game, we are now going to train our crazy big, deep neural network to predict for this state. This is the policy. And for this state that is the value. And this is just a supervised learning problem. And then they do the same thing for every state that was reached in that particular game, all using the same final state, which is either you won or lost. And so this is just an enormous network. I can't remember. I think it's maybe like, let's say, 40 layers. And they try, and we'll see shortly, the influence of architecture too. The architecture matters. And so again, just this neural network goes directly from states to both predict. It's got two output heads, both predict policies and values. In their earlier work, they had separate neural networks, but one for policy, one for values. Here they just combined it. All right. So that is how it works in a nutshell in terms of what they're doing. And then they do this for an absolutely enormous amount of time. The final thing I think was trained for 40 days over, like many TPUs, et cetera. Yeah. Does this mean like, if you think about it, is this kind of like a loss function with respect to the value? The policy is not actually a component of that loss function? Is that what-- Yeah, that's a great point. So it is a really good point. So these are just two different heads. And you can think of it as what they're sort of assuming in this case is that the representation you're learning is going to be helpful for both, but this value may or may not relate to this policy. And this is just saying, we think that the features we're going to learn about this, like the sort of way that we're encoding the game states. And also just to note here, it's not just the current board that they're using. The states they use tend to use history as well, because again I'm not an expert in Go, but there are various rules in Go which mean like I think you can't repeat a move and stuff. So because of that, they have to maintain a short history of the previous game states. So you can think of S really as being like multiple game board states of the past. And I think the intuition for this is that you're going to learn feature representations from that. They're going to be helpful for predicting both of these. Now ultimately you would hope that this sort of, there is some relationship between these two, but they're not constraining it. So just to recap, what are the key features that they're using? So I guess also to specify in this case, they're going to do this across many TPUs, over many, many, many days. And what they're doing when they do this is that they're constantly retraining these neural networks. And at the end of all of this, when the actual play kind of test games, say, against other human players or against other AI agents, is they're going to still do the Monte Carlo Tree Search. So they're going to take their final neural networks and then they're still going to do the Monte Carlo Tree Search method that we've just seen before they make decisions. And so what we'll see in a second whether that's important or not. So in particular, some of the important questions that they consider in this paper is what is the influence of architecture? Does it matter which architecture you use in these cases? What is the impact of using MCTS? Obviously, they're still learning a policy and they're learning a value function. And the question is how much additional gain do you get even after 40 days of training this by doing Monte Carlo Tree Search and how does it compare to human play or using human players? So the first way that they did this is they, instead of having this neural network that was predicting a policy and a value, is they actually did supervised learning on human play. And that gave you a way to prioritize actions. So that's what they've done when they did AlphaGo to start. And I think that's what they did also for when they won against Lee Sedol. And then what they've been trying to do in this paper and others is to remove some of those assumptions to see if you could learn without even human knowledge. Now here, I'll just specify that they still know the game rules. And then they have later paper where they want to not even need that. But here the algorithm moves. So the first thing to note is that higher is better. This is talking about the performance of the resulting approach under different architectures. So what they do is they actually have the same training data that they use and they just use different architectures. They use data in this case from some of the runs of AlphaZero, which is the algorithm we've been talking about. So all of these have the same data and then they look at what the performance is if you train the neural networks with that data. So same data, just differences architecture. And there is a huge difference. This is like from 3,000 to 4,500. So their current one-- so this is a convolutional neural network, which is separate, meaning that you have a different policy network from a value network. Whereas this is a ResNet and they're using a dual representation. So you can see that you get a significant benefit by leveraging representational strength across both of these targets. And also that this is better than using convolutional neural networks. So I think this is a good reminder that like when we're doing reinforcement learning or we're doing decision making, we still want to build on all the amazing advances that are happening in deep learning in general. And the complexity of the neural networks that we use and the functions they can represent really matters. So that's the take home from this part. This is a huge difference in performance. The second is the impact of Monte Carlo Tree Search. So I think this is important to know. This is if you use the raw network. So you take the network. This is after those 40 days of these crazy numbers of TPUs and you don't do Monte Carlo Tree Search on top in your evaluation games. And again, this is much, much, much worse. So this is AlphaGo Zero, the algorithm we've been talking about, AlphaGo Master was another one they developed shortly before this. This is the one that beat Lee Sedol. AlphaGo, what they call Fan is the first big AlphaGo paper. And these are some of the other approaches that happen before their methods. And again, you can see that even though they now have all this beautiful, different architecture, et cetera, if you don't do Monte Carlo Tree Search on top of that, you miss a lot. So it really is important to do this last mile of additional computation even after you have these really, really good neural networks, this kind of local computation matters. This gives you a sense of the training times involved. So this is the Lee Sedol paper or Lee Sedol Method. I don't think they published this before. This is one of their master methods they had. And this is showing, for a particular size approach, how long it took of training before you got something that exceeded all of those. So it gets there, but it also just highlights the enormous amount of computation needed and the importance of the architecture. So I know we're almost out of time, but I just want to highlight two things. So again, in this case, it didn't need any human data, no supervised learning. And they noted, though, that it was less good at predicting human play than some of the other prior methods. So that, again, just highlights that these methods really are helping agents to discover strategies that are not necessarily the ones that are used by humans. They're discovering very different ways of solving these sort of incredibly complex optimization tasks. And I think that's really interesting in terms of the future of human AI collaboration. We're almost out of time for today. I'll just highlight as well that these sorts of ideas of how to use Rl to optimize computation and solve really, really big search problems have also been used by DeepMind to solve things like Alphatensor and these other ways of trying to start automatically searching for new algorithms, which I think is really, really exciting because you can think of the space of algorithms or the space of different search algorithms, et cetera. And those are enormous. And so you could think of using these types of strategies to prioritize which things are most effective. All right. So I'll leave this here because we're out of time. You're welcome to look at this, to think a little bit more about the aspects of UCT search. And then on Wednesday, we're going to think more about rewards in Rl and what are the implications of which ones we're choosing. I'll see you then.

Stanford_CS234_I_Reinforcement_Learning_I_Spring_2024_I_Emma_Brunskill

Stanford_CS234_Reinforcement_Learning_I_Tabular_MDP_Planning_I_2024_I_Lecture_2.txt

Hi, everybody. Welcome back. This is lecture 2 from Reinforcement Learning. We're going to start with a Refresh Your Understanding. Again, these are just a sort of a quick way to check your conceptual understanding from the most recent lectures, or occasionally we'll go back a little bit. To do this, you just need to log into Ed. Everybody should be added to Ed. If you're not, just send us an email to our mailing list. So if you go to Ed, please follow the steps given to log in first before you click the links. So if you follow those steps and then you're logged in with your SUN ID, then when you click on the poll links, it should just take you right there, and it will just log all your responses. If you're curious about how we use these for participation points, you can just go to the website to see how we calculate it. I think we use just a percentage of these. If you do a sufficient percentage, then you get full participation points. It's optional. All right. So we're going to start with this today. The question is, in Markov decision processes, a large discount factor gamma means that short-term rewards are much more influential than long-term rewards? And then a second question to start thinking about is, in general-- so last time we started talking about sequential decision making under uncertainty, and one of the things we often would like in real-world systems is monotonic improvement, meaning that if we get more data or we get more computation, we know that the system is going to be better, make, in our case, better decisions than it could if it had less computation or less data. And so the question that I'm posing to you now and that we're going to discuss today is, is it possible to construct algorithms for computing decision policies so that we can guarantee with additional computation-- we can also think of often as iteration-- that we're going to monotonically improve the decision policy? And you can start to think about if you're already aware of any algorithms that might have that property, if you think it's impossible, or if you think-- if it's true, do you think that all algorithms would satisfy that? That's not for the poll. That's just to start thinking about, and we'll come back to it later. All right. So I'll just give you another minute or two to do this Refresh Your Understanding. It's just a quick one, and then we'll go. And again, these are not assessment questions, so you're welcome to look back on lecture slides from last time. You're also welcome to talk to anybody right next to you. All right. It looks like we actually have maybe a 2/3-1/3 split on this question. The correct answer is false. Does somebody want to say why it's false? Yeah? And remind me your name. Yeah, I think because you multiply the longer-term rewards by the gamma. So a large gamma means that the long-term rewards are weighted decently [INAUDIBLE]. That's right. So if you have-- exactly what [AUDIO OUT] said. So if gamma was 1, you would care about short-term rewards exactly the same as long-term rewards. In general, if gamma was 0, you would not care about long term rewards at all. You'd be entirely myopic. But as gamma gets closer to 1, it's sort of a relatively weighting more of longer rewards than you would otherwise. Great. All right. And yes, as I said, we'll get more into the conceptual question later. The other thing that I wanted to clarify-- I saw there was some questions on this last time as well as after class as well as on Ed-- is I had mentioned when I was making distinguishments between reinforcement learning and other forms of AI machine learning this notion of optimization. But I think that that was a little bit-- I think it was more confusing than it was helpful, and depending on how you think of it, in machine learning or AI, we always have some form of metric or optimization. So you can think of a loss as also being-- we're trying to minimize the loss, and so that also sounds like an optimization problem. So you can just ignore that distinction for now. I do think in general, when we're thinking about decision-making, it's going to be very important what we think of as that metric. And so it won't necessarily just be loss functions. We can have lots of different scalar values or even multiple objectives. But the distinction of whether or not supervised learning is using optimization is perhaps not so helpful. OK, great. So let's go ahead and get started. So I do also just want to highlight that for some of you-- and I got a question about this. We've also got a couple of questions about this. This first week or two will overlap a little bit with some of the other classes you might have taken. So particularly if you've taken 238 with Mykel Kochenderfer, the beginning may overlap. The things that will probably still be different in the first couple of weeks is I expect there's going to be a higher level of theory in the first week or two about the properties of some of these algorithms and what sort of guarantees we have. And then afterwards, I suspect after that, most of the content in the rest of the class will be quite different. If you have any questions about how this compares to a lot of the other decision-making classes that are offered at Stanford, don't hesitate to reach out to me in office hours on Ed or after class. All right. Now, why do we do this? Because also, you might be thinking, we want to get to AlphaGo, or I want to get to controlling robots, or I want to get to optimizing LLMs. Why are we starting with systems like the seven-state Mars rover that we're going to look at. And the reason is because actually a lot of the ideas that enabled people to solve AlphaGo and do things like RLHF, or reinforcement learning from human feedback, really builds on this fundamental notion of decision processes. And I think it's much easier to really cleanly see how these ideas come up when you can actually see these in the world as tabular. You can just write down all the states. So that's why I think it's helpful, but even today we're going to start to see where those ideas might be applied. So we're going to start to do things like policy search, which is the foundations towards things like policy gradients, which are extremely widely used. So you can think of all of these as just being building blocks that we're going to use to build up to get to the point where we're later going to be-- and very soon, within a couple of weeks-- tackling things that are state-of-the-art algorithms. All right. So what we're going to be doing today is really focusing on making good decisions given a Markov decision process, and so that means both being able to understand how good a particular decision policy is as well as what is an optimal decision policy. And when I say we're given a model of the world, what I mean is that we are given that dynamics model, which tells us how the world evolves when we make decisions, and we are given a reward model, which tells us how good decisions are. And last time we talked about Markov processes, and we were starting to talk about Markov reward processes because they can end up being really useful when we're trying to evaluate how good a particular decision policy is. And we'll see a lot of the same ideas from Markov reward processes to MDPs. All right. So let's just refresh our memory. So this is the question that we had before of, how do we think of the influence of discount factors? As was said, what happens is we multiply the next reward by the discount factor two rewards away by the discount factor squared, et cetera. And so as you can see there, if the horizon is really long or as it goes to infinity, rewards will have 0 value eventually because gamma is less than 1. So the idea of the value function was to say-- remember, this is a Markov reward process. We don't have decisions yet. It just says, how much is the expected discounted sum of rewards we will get starting in this state and acting, most of the time today, forever? So most of the time we'll think of today of just getting to act forever and how much reward would you get. And because this gamma-- as long as the gamma is less than 1 here, that will be a finite number. So we're just starting to talk about how could we compute this. So again, remember, the return is going to be a particular series of rewards you might get if you start in this state and act forever, and V is going to be, on average, how much reward would you get if you start in this state and act forever? All right. So one of the key ideas here is that computing the value of an infinite horizon Markov reward process leverages the Markov property, which was this idea that the future is independent of the past given the present. So given your current state, you don't have to think more about the history. So what that implies when we try to compute what the expected reward is, future reward is from a state is we can think of, well, what is the immediate reward we get in that state plus all the different states we could get to next under our dynamics model and then the value of their reward? How much do we weigh each of those? Well, we weigh each of those just according to what is the probability I could get to each of those next states. And if you're familiar with things like tree search, you can think of it as just, I'm in my starting state. I think of all the next states I could go to. Each of them have some weight, depending on the probability I get there, and then I sum all of those up according to their values. And this is going to be the basis of the Bellman equation, which we're going to see lots about. So if we wanted to think about how we could solve this, one way we could think of it as if we have a tabular world, meaning that we can maintain a scalar value for every single state separately-- so this is like our Mars rover case-- then we could just express the value function in a matrix equation. So we say the value of each of the states is exactly equal to the immediate reward plus gamma times the transition probability to all the next states. And so that's nice because now we can just directly solve for what the value is. So we know that this has to halt, so now what we're going to do is just invert that to solve for V. So what we would say in this case is we would say v minus gamma times P of V-- this is P. And again, I'll apologize that in the different things you see online or the textbook, et cetera people sometimes use T for transition matrix. They sometimes use P for probabilities, going to the next state. If it's ever confusing what notation is being used, don't hesitate to reach out. OK, so we just rewrite it like this, as equal to R, and then we move this. So we have V of I minus gamma P is equal to RI equals the identity matrix, which means V is equal to I minus gamma P inverse times R. So why do I show this? I show this because if you know how the world works, you have the dynamics model, you know what the reward function is, and the world is small enough, you can just directly solve for this. This isn't for decision yet. This is just showing us what the value would be of each of the states. So this is one way to solve it. We would call this the analytic solution. And one thing to note here is this requires a matrix inverse, and so there are faster algorithms than N cubed, N being the number of states. But in general, matrix inverses are fairly expensive. So this is being done once, but this is a fairly-- if your state space, the number of states you have N, is large, this can be expensive. And it also requires that the identity matrix minus gamma times the dynamics, the dynamics model is invertible. OK, so this is one way we could solve this. Yeah? And remind me your name. In practice, what usually happens? Do people just go ahead and take the matrix inverse? Let me reword the question. In practice, do you usually find that these kinds of matrices are invertible? And if yes, do people just go ahead and take the matrix inverse, or do they [INAUDIBLE] something? It's a good question. So in practice, is it invertible, and what do people do? In practice, normally, we're dealing with state spaces that are far too large, so we can't do this. Yeah, good question. There might be cases where it's small enough, but in general, no. So that's a great motivation for a second approach, which is instead of doing it directly analytically, we're going to use dynamic programming, and we're going to design an iterative algorithm. And this is going to be very, very similar to what we're going to see for decision processes. So the idea in this case is we're not going to do this in one step, but we're going to avoid that matrix inverse, which might be pretty expensive. So we're going to initialize the value of a state to 0 for all s, and you can think about whether or not it actually matters what we initialize to. But just imagine we do that. And then for a series of iterations, k is our iteration variable. For all the states in s, what we do is we say-- we're going to make a new copy of our value function, and we say Vk of s is equal to R of s plus gamma sum over s prime probability of going to s prime given s times the value that we already have for k minus 1 of s prime. And we just do this over and over and over again until our value function stops changing, and we'll talk soon about whether it will stop changing. The nice thing about this is that it's only s squared for each iteration, so this would be an iteration, instead of a matrix inverse. All right. So this is how you could compute the value of an MRP. Now we're going to see how we could do that for an MDP. So a Markov decision process is very similar to a Markov reward process, but now we get to add in actions. So now we're actually going to be starting to make decisions. And the idea now is that the dynamics transition model will probably depend on the action you take. So you're going to get to different distributions of next states. And so it could be something like you think of, depending on the ad you show a customer, they might do different things. Depending on the controls of your robot, it's going to move or manipulate its hand in a different way. Generally, these dynamics are going to be a function of the action, and we are going to, for right now, assume the reward is a function of the state and the action you take. So you often say that an MDP is defined by a tuple-- S, A, dynamics model, reward model, and gamma. So we could think of that for here. So now we have our same little Mars rover, but now we actually have two different dynamics models, one for if we take a1 and one if we take a2. This is just an example. In these cases, these are deterministic. In general, we can have them be stochastic. And we would also need to specify what the reward is, so maybe we have 0 reward in all of these states and plus 1 here and plus 10 at the end. And this would just define. So once you've defined the state space, the action space, the reward function, the dynamics model, and the gamma, then you've defined your MDP. All right. So now we actually get to start to think about policies, which is what we'll be talking about throughout the course, which is, how do we make decisions depending on the state we're in? And the policy is going to specify the action to take, which can be deterministic or stochastic, and often we're going to think of it as being stochastic. And we'll talk about the properties of stochastic versus deterministic ones and why you might want one or the other quite a bit in the class, but we can generally do everything we're doing in each case. All right. So an MDP plus a policy is just a Markov reward process. Why is that? Because once you specify how you're going to act, you've removed the policy part, and so if you want to know how good that policy is-- so let's say someone says-- again, your boss says, hey, how good is this thing at advertising to customers, for example? Then once you've decided what the policy is, we can think of the reward as just being a weighted sum over the probability that's taking that action in that state times the reward for that state in action and then your dynamics model, which is a little more subtle, which is now you're taking a weighted sum over all of the transition dynamics according to the action you take weighted by the probability you take that action. So it just defines a Markov process because now you just have this transformed dynamics model where you've merged in the policy. So why is this helpful? And this is something that you may or may not have seen in previous classes. One of the reasons why this is helpful is because now we can just say, oh, any techniques we have for Markov reward processes we could also apply to evaluating the value of a particular policy in a Markov decision process because we've just reduced an MDP and policy evaluation back to an MRP. All right, so if we think about doing policy evaluation with an MDP, we can just plug in the actual policy that we would be using. So what we have in this case is that instead of-- now we actually get to make decisions, and so then we get to say, what is the probability of picking the action in this state times the expected discounted sum of rewards at that point? So this looks very similar to an MRP, except for we're saying, based on the probability for each action, what would we get next? And we call this a Bellman backup for a particular policy because this is going to specify what is our expected discounted sum of future rewards if we start in this state and follow the policy? And just notice that if the policy is actually deterministic, we can reduce it back to a case where we've sort of averaged over these rewards. So remember, this was just going to be if you have a particular action, then you're just going to index into what the reward is for that particular action. So we can see that here. And just raise your hand if you've seen this before, if you've seen the [INAUDIBLE]. OK, good, so probably at least 2/3 of people. All right. OK. So if you want to check your answers, if some of this is new for you, then one thing to do is to try to check that you can do this sort of value iteration or this policy evaluation for the Mars rover example. We won't go through it in class, but you can check the answers. I'll release them at the end of the slide just to check that you know how to apply this. All right. So of course, shortly we're going to be interested in not just evaluating the value of a single policy but finding an optimal policy. So one question is, how many policies are there? And is the optimal policy value unique? So we'll just take a second. You can go to the polls and enter in your answer. OK, great. So it looks like most people got-- the vast majority of people got the right answer for the first one, which is it's 2 to the 7. In general, the number of policies we have is going to be A to the S because for every single state we could choose any of the actions. And also, most people got the next one right, which is great, which is the optimal policy, the one with the highest value, is not always unique. It can be unique-- it depends on the problem-- but it's not going to be unique whenever more than one action has the same identical value, so when you have ties. Yeah? How do we generally deal with invalid actions? Because for example, if we're in S1 and we choose left, I would imagine-- to me, that's an invalid action. I'm not sure how we really deal with that. Yeah, so the question was if we have invalid actions. So in general, you can have a different action space be possible in every state. That's also very common in recommendation engines that you'd only-- it's only a subset of articles you might show to some people based on their state. In this particular example, we're going to assume that it's not actually go left. It's try left. And so if you try to go left and there's nothing in the rest of the world, you just fail, and you stay in the same place. But in general, most of the time in the class, we're going to assume the action space is the same for all states, but in some cases, it might be different. Good question. OK. So in MDP control, we're going to want to not just have the policy-- evaluate a particular policy, but we're going to compute the optimal policy. So we want to take the arg max over the policy space, which in general is that A to the S space, and there is going to exist a unique optimal value function. And the optimal policy inside of a tabular MDP in an infinite horizon is unique and deterministic. So those are two properties that are good to be familiar with. So now we're going to think about how do we actually compute this and what its other properties are. So one is that it's stationary. What I mean by that here is that in infinite horizon problem, you always have an infinite number of additional time steps, and so the optimal thing to do just depends on your state. It doesn't depend on the time step. We'll think more about what happens when you only have a finite number, like where H is finite and what might happen there, but for most of today, we're just going to focus on the infinite horizon problem. And as I said-- and most of you guys already knew that this, in general, is not unique. So one option is policy search, and this is where we are going to get into-- oh, yeah, and remind me your name. Is the optimality conditioned on the initial state? It's the optimality conditional on the-- what do you mean by that? The state of the [INAUDIBLE]. Yes, the optimality, yes, it will be per state. Yeah, so the optimal policy will be defined per state. The idea is that you can take a different action in every state, and you want to know what the optimal thing is to do to maximize your expected discounted sum of rewards from every state individually, like pointwise. Good question. Yeah? Yeah, two interconnected questions-- why is there a unique optimal value function? And second is, can you remind me again of what was the reason why it may not necessarily be unique? You mentioned a specific case related to this. So the optimal policy is not necessarily unique because there could be more than one action with the same value, and the optimal value function is unique for reasons we'll see later in this class, like later today. We'll prove it. OK. So one of the things-- and this is going to go back to the conceptual question I put at the beginning of class-- is we would like to ideally have methods and algorithms that have monotonic improvement capabilities, and so policy search is going to be one of those. So what we're going to do here is we're going to try to search to compute the optimal policy. There's A to the S deterministic policies. In general, you could imagine just enumerating all of them and evaluating them all, but we can often do better than that. And when I say, "better," what I mean here is we can reduce the computation needed to try to identify the optimal policy, so we shouldn't have to iterate through all eight of the S policies. So how does policy iteration work? The idea is that we're going to alternate between having a candidate decision policy that might be optimal. We're going to evaluate it, and then we're going to see if we can improve it. And then if we can improve it, we will, and otherwise, we're going to halt. So what we do-- how we do this is we're just going to initialize it randomly, which just means we're going to start off, and we're going to say, for every single state, we're going to pick an action. And then while our policy is still changing-- so this is the L1 norm. It measures if the policy changed for any state just as a refresher. What we're going to first do is we're going to evaluate the policy, and then we're going to try to improve it. So in order to do that sort of policy improvement step, it's going to be helpful to define the Q-function. Again, I know for many of you, this is probably a review. The Q-function of a particular policy is just, what is the reward of the immediate state and action plus the discounted sum of future rewards if we were to, after that action, act according to the policy? So it's sort of like saying, OK, first, when you're in this state, you're going to take this action, and then from then on, you're going to follow whatever your policy tells you to do. And for any of you who've seen Q-learning, you've seen this sort of idea lot. So what we're going to try to do in this case-- why would we want a Q-function? It turns out it's going to make the policy improvement step really easy. So what we're going to first do is we're going to say, I'm going to take my particular policy. I'm going to compute the Q-value for that particular policy, pi i, because we're going to be iterating. And then after that, we're going to compute a new policy, pi i plus 1, by just taking the arg max of Q. So for our Q-function, we're just going to say, according to this Q-function, which says, What is the expected discounted sum of rewards? if I start in this state, take this action that follows pi, which of those actions is the best? And we can define that per state. Yeah? Is there any relationship between the Q-function and the value function? Because it kind of looks similar. Yeah, yeah. So we often call it the Q-function the state-action value function. All right. So this is sort of just what we do now. Now we're going to have this Q-function. We're generally going to do this by having this Q. And then we will do pi i plus 1 of s is equal to arg max over of a of Q of s, a per state. And then we just repeat this over and over again. OK, so there's a number of questions you might have about this. You might say, OK, this seems like a vaguely reasonable thing to do, but does it have any formal properties? Are we guaranteed to improve? What can we say about this? So to do that, I think it's useful to delve into what the policy improvement step is actually doing. So what the policy improvement-- when we compute the Q-function, this is the equation for the Q-function. So we take our old policy pi i, and then we compute the Q-function of this. And we can do this iteratively. And now what we want to do in this case is think about what is the performance going to be of the new policy we extract. All right. So what the Q-function says is we're going to be able to show that the Q-function-- the best thing of the Q-function is better than the value of the old policy. So what does this say? So the first thing is just how we've compute it. This is just the policy evaluation step. And we know that if we have a Q-function over s and a, for a particular s max over a of Q pi of s, a, has to be at least as good as the Q-function for any of the actions. So we know that this has to be-- this thing is always greater than or equal to Q pi i of s, a for all a. And then this is just that equation. This is just what exactly is Q pi i of s, a, just the definition, almost, except for it's particularly for the actions that-- this is for specifically if we were to follow the previous policy. So remember, this is the equation for Q pi i of s, a. Think about one of those actions that you could have done is exactly what the old policy would have told you to do. That is what this equation is. You just take a here, and you plug in pi i of a. So that's just exactly what this is, and that is just the definition of V pi i of s. OK, so what is this saying? What this is saying is if you were to take your new policy, pi i plus 1-- so remember, pi i plus 1 is defined as the arg max of this Q-function. It's whatever maximizes your new Q-function. So what this says is if you were to take pi i plus 1 of s for one action and then follow pi i forever-- so that's what the Q-function represents-- then our expected sum of rewards is at least as good as if we'd always followed pi i. So that's what this equation is telling us. It's like, if I get to make one decision differently and then from then on, I follow my old policy, the value I can expect is at least as good as the value I could expect if I had always followed the old policy. Does anybody have any questions about that? Because then the next step is going to build on that. Yeah? Can you go back to the algorithm [INAUDIBLE]? Sure. For the policy improvement? Yeah, yeah. The next slide, actually, on [INAUDIBLE]. So the step that we are talking about is this one, right, the policy improvements? Yeah. We're trying to see, when we do the policy improvement step and we extract out, instead of max here, arg max to get out the new policy, how does the value of that relate to the value of the thing you could have done before in that state? And so this is just trying to say, what really is Q pi i of s, a? It is the value you get if you first take a and then you follow pi i from then onwards? So it's saying if you were to do that, then this new action you've computed, this arg max policy, is actually better than what you would have gotten before or at least as good. But the thing that should seem slightly strange to you is I am not creating this sort of hybrid policy where I take one new action and then I follow pi i forever. I'm creating an entirely new policy where I'm not just going to follow pi i plus 1 for one step. I'm going to follow it for all remaining steps. So this should not yet convince you that doing that is actually any better than my old policy. This would say, if you take one new action and then follow your old policy, it's going to be better than your old policy. So that's why we have to do additional work to show that we're actually going to get a monotonic improvement if we just follow this new policy. always. All right. So let's go through that. So what we're going to prove is we're going to say, actually, that's true. The new policy we construct through this policy improvement step is somewhat remarkably going to be strictly a monotonic improvement compared to the old policy unless it's identical. So that means at every step of policy improvement we're going to get a better and better policy for every state. And the only time we're not is if we've already converged. So let's go through that. So this is going to prove to us that the value of the old policy is less than or equal to the value of the new policy, meaning we're going to get this monotonic improvement. So what we're going to do in this case is we are going to first write out-- so this is just the definition. This is the definition of max over a over Q pi i. All right. So let's just write out what this is. This is going to be equal to-- and it will be written out more neatly on the next page too. OK, so what did I do here? I noticed that the definition of pi i plus 1 is exactly the arg max of this expression instead of max. So when we did the policy improvement, the way we did the policy improvement was we took the arg max of the Q-function. So instead of having this max out here, I'm just going to plug in pi i plus 1 because that's going to give me something that's exactly equal to the max a for that whole expression. All right. And so this is exactly equal to that, but what we can show next or what we can do next is that we can just add the same terms and notice that this is the same. This is less than or equal to Q pi i of s prime, a prime because the value of pi i for s prime, so following a particular policy, always has to be less than or equal to taking the max over the Q-function for that policy. Why is that true? Because either the max is the same as the pi i action or there's a better action. All right, so that's the less than or equals. And then we can just expand this expression out, and this is going to start to get a little bit messy, which is why it'll be nice to have it on the next slide too. But what will happen here is you can see how the expansion works. And why is this important? This is important because this is going to allow us to think about not if we just take this new action on the first step but on all future steps. So what we had here is we had this thing which was max a over Q pi i of s prime, a prime. We're going to expand out what that expression is. Because notice, this thing here is exactly equal to this thing, which we know is here. So we're just going to substitute it. So we can put that in here. So this is R of s prime, pi i plus 1 of s prime plus gamma sum over s double prime, meaning s double prime here I'm just using to be two time steps away. Why was that useful? Well, what we've just said is that the value of pi of s is less than or equal to taking this new, better action for one time step and then following the old policy. I've now done that recursively, so I've said, well, now that's also less than or equal to if I take that new action once and then I take it again and then I follow the old policy. And then you just repeat this. So you just keep nesting this. And what you can see here is that you have these less than or equal that happen when instead of plugging in the value of the old policy you allow yourself to take a max over that Q-function. And what happens if you do this all the way out? This will exactly become equal to V of pi i plus 1 of s, so dot, dot, dot, dot. So I have that here. So what this has shown here is that the value you had under the old policy of the state is less than or equal to the value of that state under the new policy, so this proves the monotonic improvement, which is super cool. So this now says if we do policy evaluation where you just keep computing the Q-function and taking a max, you will always monotonically improve unless you stay the same. All right, so now let's do our next Check Your Understanding, which is, given everything I've just said, if the policy doesn't change, can it ever change again? And is there a maximum number of iterations of policy iteration? Yeah? On the previous slide, is the dot, dot, dot supposed to represent just algebraic manipulation or-- Yeah, you just keep expanding this all the way out, yeah. Good question. All right, let's take a second and do the poll. Yeah? What's your name? At what point did we show that this is actually leading to an improvement? Can we just like stay in the same value level? Because the inequality was greater than or equal to, so is it possible that you're always equal to where you started? Yeah, it's a great question, and, in fact, that really-- so I've just shown less than or equal to. What we can-- well, I guess we can discuss this in a second, but it will be a monotonic improvement unless you're the optimal policy. So if there's any state at which you can improve, you will, and if you stay the same-- we'll, actually, we'll talk about this now because it's nicely split between the answers for both of these questions. So maybe everybody turn to somebody nearby you, and discuss whether you think the policy can ever change if it didn't change initially. And is there a maximum number of iterations? Because it's pretty evenly split amongst people who voted. [INTERPOSING VOICES] I think that's the maximum-- [INTERPOSING VOICES] I guess in this example, there's only one option-- [INTERPOSING VOICES] Yeah, I agree, totally. [INTERPOSING VOICES] I don't see how people saying that-- I want to make sure to clarify something because that came up in a good conversation, which is let's assume for the moment there are no ties. I know I said that, in general, the optimistic policies can have ties, and that's true. But for the point of view of this question, it is easiest to think about if there is only a single unique optimal policy. So why don't we do that? Again, none of these are for assessment. They're only for your learning. But just in terms of what you're thinking through, my intention was to think about the simpler case where there is a single optimal policy and then under that case, whether the policy can ever change once it hasn't changed once. What I mean by the policy doesn't change-- meaning when we have had a policy and we do policy improvement and our new, improved policy is the same as the old policy. So under the case that I just said, which is that it's deterministic and that there is a single optimal policy, raise your hand if you said the policy, once it doesn't change, it can never change again? That's the correct answer. Does somebody want to explain why? You're all correct. Yeah? Remind me your name. It kind of intuitively made sense in the sense of you're doing the expected value. So you're summing over all-- or you're summing over all of the actions. Even if there's stochasticity on the system, you're still taking the average value. So like if it didn't change before, it won't change change. Yeah, you are taking those and so definitely along those lines. So if we look at what was the definition of the policy improvement step-- let me just go a couple of slides back. So what we said is we computed the Q-function, and then we extracted a new policy. If pi i plus 1 is the same as pi i, is Q of pi i plus 1 equal to Q of pi i plus 1? All right. I probably said that wrong. There's too many i's. Let me just write it out. So the question is, if pi i is equal to pi i plus 1, is Q pi i equal to Q pi i plus 1? So if it's the same policy, do they have the same Q-function? Yeah, I'm seeing a bunch of nods. OK, so if your policy hasn't changed, meaning that your old policy is the same as your new policy, then Q pi i is equal to Q pi i plus 1, which means that when you do this for Q pi i plus 1 and then try to extract a policy, it'll be exactly the same. So once you're stuck there, it'll be stuck forever. Now, if you have ties, it's more complicated. So if you have multiple actions that can achieve the same Q-function, it depends how you break them. If you break them deterministically, you'll stay in the same place. If not, you may oscillate between all the policies which are optimal, otherwise known as all the policies for which they have the same Q pi i. But in the simpler case that I mentioned, once you've got to that single policy, you won't ever change. And what that means is, given that we also only have a finite number of policies if it's deterministic-- so we're assuming if we stick to-- so I'll just say, no if pi star is unique for all s. So that means for every state there's a unique optimal action. Is there a maximum number of iterations for policy iteration? If you have deterministic policies, there's only A to the S policies, as everyone was saying before, which is great, and so since the policy improvement step either improves the value of your policy or halts, that means you only go through each policy once, at most once. There might be some never bother to go through. And so that means that policy iteration will halt, and it will take, at most, A to the S policies. If it takes A to the S, that means that you evaluated every single policy. In general, you won't. So this is what shows that we actually do get this monotonic improvement. This is really nice. With every single-- because you could imagine in cases where there's a-- oh, question? Yeah, sure. [INAUDIBLE] that we're going to do better than random, right? We haven't guaranteed that whatever we converge to is better than random. We've proven that we're going to get to the optimal policy. And the optimal policy may be just random, right? Because depending on the environment, you might just-- there is no-- you can't do better than random. You mean in terms of how you design actions? Yeah, so for example, if it is the case that all of your actions have exactly the same reward, it doesn't matter whether you act randomly or you follow a policy. The value you would get would be exactly the same as random. Whether or not you can do better than random will depend on the domain. The hope is, in general, we can do a lot better. OK, so we've shown now that here is an algorithm where, as we do more and more computation, we get better and better policies. And this is great because you may not actually want to go-- particularly if the state space is very large, you may not want to go until where your policy entirely stops changing. So if you have an energy time requirement, you can still guarantee that, hey, I'm getting better and better, and maybe I stop after 100 iterations or 1,000 iterations and just use that policy. So this is one which has that nice monotonicity guarantee. Sorry. Can you say what that is, for example? Oh, sure, yes. And what's your name? Yeah, so A here is the number of actions, and S here is the number of states. So the decision policy space is for every state, you could pick one of the actions, so you multiply all of those. So this also shows here about how, yeah, exactly what I said on the previous one, that if your policy doesn't change, it'll never change again, again, assuming pi star is unique per S. OK, so that's one way to go. One way is that we can do policy iteration. And the interesting thing about policy iteration is that every time point you have an explicit policy, and what that policy tells you is how to act forever using that policy. And when you compute the Q-value, it says, how much reward do you get if you take this action in this state? And then follow that other policy forever. So again, remember, today we're in the infinite horizon case unless I specify otherwise. But along the way, a lot of those actions and the decisions we make may not be very good. So your early policies might be pretty bad. We know we're monotonically improving, but the early policies might be bad. Value iteration is different. The idea is that at every iteration, we're going to maintain the optimal value of starting in a state but as if we only get to make a finite number of decisions. So remember, in policy iteration, we always have a policy, and we have the value of acting in it forever. It just might not be very good. Value iteration is, what is the optimal thing for me to do if I can just make one decision, I can take one step? OK, I'm going to figure out what the optimal thing is to do for one step. Now I'm going to imagine I get to take two steps, and I'm going to build on what I know I can do for one step. And so now I'll build the optimal thing to do for two steps. So the interesting thing with value iteration is you always have an optimal value but for the wrong horizon. So one has a value for the infinite horizon. It might be a bad policy. The other one has the optimal value and thing to do but for the wrong horizon. And the idea in value iteration is you just keep going to longer and longer and longer episodes, thinking of getting to do H plus 1 steps or H plus 2 steps, and then you build upon your previous solutions using dynamic programming. So let's see how to do that. OK, so this is where we get into the Bellman equation. This is the seminal work of Richard Bellman. And the idea here is, as we've said, is that for a particular policy, we satisfy the Bellman equation, and we can turn that into an algorithm. So in particular, there's a thing called the Bellman backup operator, and what it says is if you give me a value function, which right now we can think of as just being a vector-- later we'll get into function approximation-- and we do a Bellman backup, essentially, it's like saying, I had a value function, and I want to think about, what should I do if I get to do the best thing that maximizes my immediate reward plus my expected future reward given that value function? So it says, I'm going to figure out, if I take a max over all the actions, what's the reward of that state in the action plus the discounted sum of rewards using the value function you've given me? And what that does is it yields a new vector of values over all your states. So this is being done per state, and this is called the Bellman operator. It comes up all the time. All right. So how do we do value iteration? Well, we're just going to do this recursively, so we're just going to loop until we hit convergence. Just as a refresher, this is the L-infinity norm, and what that means is that it is equal to the max over S V of s. Maybe I'll write it out more carefully, equal to max over S Vk plus 1 of s minus Vk of s. What that just means is that if you have two vectors, you look for every single entry, and you find the entry in which those two vectors are most different. And that's the L-infinity norm just as a refresher for some of you who might not have seen it or not seen it recently. So what value iteration does is we're just going to have a loop. It's going to look very similar to what we saw for the Markov reward process. We're going to initialize our value function, and then for each state, we're just going to do this Bellman backup. And so it's like we took our previous value function. We do our Bellman backup, and we get a new value function. And we do this over and over and over again until our value function stops changing. So for policy iteration, we kept going until our policy stops changing. Here we keep going until our value function stops changing. And what that condition means is it says, I keep going until the difference between my old value of estate and my new value of estate is really small for all the states. Yeah? And remind me your name. I have a question on how to connect this value iteration to-- you just said it works with finite horizon and why policy iteration works if it's infinite. Yeah, good question. So what you could think of this as-- so great question. At the beginning, you don't get to make any decisions. The expected discounted sum of rewards you get from many states is 0. You don't get to make any decisions. You never got any reward. The first round of this, it's like you're saying, OK, I get 0 reward if I don't make any decisions. k would be 1 here for the next round, so we'd say, if I get to make one decision, then I would take a max over a, my reward, times discount factor times 0. So it's now saying, what is the best thing I can get to do if I get to make one decision? So what this will be is on the first round, this will just be equal to-- so if V is equal to 0 for all s, then what we would get when we do this backup-- we'd get Vk plus 1 is just equal to-- let me put it over s-- is equal to max over a r of s, a because this part will be 0 if your value is 0. So now this is like saying, OK, before I get no reward because I make no decisions. Now what's the best thing I should do if I get to make one decision? The next round, I'll say, what if I get to make one decision now and then plus this will get plugged in as your value? Yeah? So the expression that we're plugging into the max function, is that the same as Q of s, a? How does that [INAUDIBLE]? Good question. So in general, that's going to be max over a Q of s, a, yeah, because here we're requiring ourselves to take the max over the actions we're taking. Yeah, great question. Yeah? In policy iteration, we were also initializing the values of V. In the policy iteration, we were just randomly initializing our policy. So we're saying, in this state, you go left, and in this state, you go right, et cetera. And then we were evaluating the value of that. We could when we do that evaluation-- yes, in that part, we were setting V equal to 0 and then doing this iteratively. If we [INAUDIBLE] policy iteration, then we would be able to detect cycles. If we had done this in-- well, we can do-- we can do this as part of the policy evaluation for policy iteration, but what do you mean we would be able to detect cycles? So states having [INAUDIBLE]. So there we are comparing successive policies, and so for that, we are comparing successive value functions. [INAUDIBLE] That's right. So it's a good point to say, inside of the policy iteration one, instead of just halting when your policy has stopped changing, you could also halt if your value function has stopped changing, very few functions. Yeah, [INAUDIBLE]. Yeah? And what's your name? Do you have any guarantees on [INAUDIBLE], based on value iteration and policy iteration, which one converges faster [INAUDIBLE]? Yeah, it's a great question. I'll look it up for next week. To my knowledge, there isn't over that in one that would not be instance-dependent. In practice, policy search is very, very popular. Clearly there's a good reason why we [INAUDIBLE] policy search [INAUDIBLE]. I think part of it may be-- I think part of it is probably that it also often has this nice monotonic improvement, so value iteration does not necessarily have a monotonic improvement requirement here. So it is always the optimal thing to do for the wrong horizon, whereas the other one says, it may not be optimal for ages, but it will always be monotonically improving. Great questions. OK, let's see what the properties are for value iteration because these are really useful, great questions, and we'll see why this whole thing ends up working. So just I want to highlight here, you could think of policy iteration also as Bellman operations, and I think this gets to what your question was about too. So the Bellman backup operator for a particular policy pi is defined as follows. You see you don't see the max anymore. You just are committing to doing a particular policy. And then policy evaluations amounts to computing the fixed point, and I'll define that more formally in a second. And so to do policy evaluation, you just repeatedly apply this operator until v stops changing. That was the iterative algorithm we saw before, just with different notation. All right. Let's start to talk soon in a second about fixed point. And then what we'd say here is when we do policy-- another way to do policy improvement is to explicitly do another backup but take the arg max instead of the max. So that's the only difference. This is the same as what we're doing for the Q-function, so this is Q pi k of s, a. I'm just showing you different notations for the same thing and also how people sometimes talk about the Bellman backup for a particular policy. But normally, when people say, "Bellman backups," they mean for the optimal. All right. So let's just go back to value iteration because while I've told you how to compute a value function, I haven't told you how to get out a policy from it. So the standard way to do this would be you would go through this process, and then you would do it, say, one more time and extract the arg max instead of the max to actually get your policy. So normally in this case, you don't bother to compute a policy along the way. You just do value iteration a bunch of times, and then at some point you extract a policy. All right, let's see about some properties of this. So why this is a good thing-- I've already told-- we've already seen that policy iteration is guaranteed to converge because there's only a finite number of policies. You never repeat a policy, and so either you're at the optimal policy already or you keep improving. For value iteration, it may not be clear yet that this should converge. So I'm first going to define a contraction operator. So let's let O be an operator, like the Bellman backup, so you can just think of it as an algebraic equation if this is something-- if you haven't seen operators before, which is totally fine, and then this is just going to denote any norm, so like the L-infinity norm or others. If when you apply the operator to two different, say, value functions-- we can just think of these here as vectors-- and that distance after you apply the operator is smaller than the distance before, then it's a contraction. So just give it a bit of intuition for this in case the contraction operator isn't something you've seen before. If you think about having two value functions and then there's some states on which they really differ in their value, what this says is that if you then apply an operator to them-- and we're going to prove that the Bellman operator is one of them-- that afterwards they get closer together. So that max difference between the states is smaller afterwards. Yeah? Yeah, is this like a iff or just if? Can there be contraction operators that don't satisfy this-- We're going to look at this specifically-- you mean are there other-- so I'd have to check. I'm not an expert in all of contraction operators, so I'll hesitate. What I will show is that the Bellman operator satisfies this statement, and therefore, then we can show that we are going to converge to a fixed point. All right. So particularly under a couple minor assumptions, your discount factor being less than 1, or you end up in a terminal state with probability 1, essentially, both of these make sure that your expected sum of discounted rewards is bounded. Then the Bellman backup where you do this max is a contraction, which means that your distance is going to shrink and shrink between two value functions. So your Vk plus 1 versus its difference to Vk, that distance in terms of the max difference in any of the states is going to get smaller. And we'll go through that now. So this is proving that we end up getting the-- this is a contraction, so the Bellman backup is a contraction operator on V for gamma less than 1. Let me just make this large. So we're going to use the infinity norm, which, again, is just saying where is the max difference in the values for any two states, and what I'm defining here is two different value functions. So this could be anything. And what I'm going to try to show is that after you do the Bellman operator, that can be no larger than the max difference before you did the Bellman backup. All right. So what we have here, this is the first inequality, so this is important. What I'm going to say is right now, so this is just the definition of the Bellman backup operator. What you can see here is I have two different maxes because I'm going to do the max over a for the first value function and a max over a prime for the other value function. What I'm going to say now is if you do that, instead, this has to be less than or equal to if you pulled the max a out and you required both of them to use the same action. And why is this true? Because essentially, what we're allowing here is we're allowing us-- before, we could pick different actions for both Bellman backups, and now we can pick one. So that means that instead of getting to maximize the second thing separately, we're just going to try to maximize the difference. So that's the first place this less than or equal is going to come in. Once we do that, now everything's taking the same action on the same time step, so we can get rid of these because they're identical. So we can just say this is just exactly equal to max a. I'm going to pull out the discount factor of sum over s prime probability of s prime given s, a of Vk of s prime minus Vj of s prime. Now, again, what I'm going to do is I'm going to bound this, and I'm going to say, the difference between the two value functions that any state is always less than or equal to their max difference across all the states. So this is less than or equal to max over a gamma sum over s prime probability of s prime given s, a Vk minus Vj because the difference between any particular states is always less than the max difference between any of the states, so I upper-bounded it using this expression. But now this term now does not depend on states, and so I can take it out of the sum. This is just some constant. It's, like, 7. So this is equal to max over a gamma. But this is just a transition model, and the probability that we go to some state has to sum up to 1 if we sum over all next states because for any state in action you're in, you always have to go to some next state. So this is just equal to 1. So we get that this is just equal to max over a. And now there's no more dependence on a, so it just disappears. So what that said is that the distance, the max difference between the Bellman backup value functions we get from starting with two different value functions, has to be no larger than the max difference between the value functions before times gamma. And if your gamma is less than 1, that means you're strictly contracting because it means that that max difference has to be smaller than it was before. It would be, like, 0.9 times whatever the distance-- at most, 0.9 times whatever the distance was before. So that's really cool because that means now that if we apply value iteration where we're repeatedly doing the Bellman backup, we're shrinking the distance-- so if you think of having a series of value functions, so you've got, like, V0 and V1 and V2 and V3 and V4, dot, dot, dot, you can think of what this distance is. And what this is saying is that these distances are going to be shrinking over time. And I've told you before that the value function is unique, so that means as you shrink and shrink and shrink and shrink, this is eventually going to become a unique value function because if there were-- you can think about it too. If there were two different value functions, you can think about what would happen after you do a Bellman backup operator. They are different. So this proves it's a contraction, and just to note this here, even if all the inequalities are equalities, this is still a contraction if gamma is less than 1. It's still making progress. All right. So here's some thoughts in case you want to think about this more. To prove the value iteration converges to a unique solution for discrete state action spaces, whether initialization matters at all, and is the value of the policy extracted from value iteration at each round guaranteed to monotonically improve? So these are all great things to think about. So let's go back to more practically. This is then value iteration for finite-- well, actually, I'll pause here in case anybody has a question about the proof. Yes? And remind me your name. Can you go back to the proof? Yeah. I understand all the steps except the first one, where we take the max out of the norm and max over as an action. Why is that greater than [INAUDIBLE] having a separate max on the inside? Yeah, good question. So what this is saying here is we have-- you can think of this as a Q-function. So we get to pick a max over that Q-function and then we subtract off the max over another Q-function. And you could imagine that you could think of there being lots of different pairs of actions in this case, and either this max is the same as this one, so it's actually-- so let's say in particular, concretely, that this gives you a1. That's this one. So this one is either a1 or another a. If it's another a, that's just because this is actually larger than what Q s of-- so let me just write this out just in case. So we can think of there as either being Q s, a1 under this or Qj of s, a where a is not equal to a1. So either this max is exactly the same as this one, in which case this is equal, or this is different, and the only time it would be different is if that value was actually larger than the value of Q s, a1. And if this was larger, this difference would be smaller because you'd be subtracting off a larger value. So that's why we can turn this into an inequality. It's either the same if they happen to have both picked the same action, or it would have picked another action, for which that whole difference would have been smaller. Good question. Yeah? Can you go back to the questions you were posing? Yeah. Is the value of-- the third question out there that, is the value of the policy extracted from value iteration [INAUDIBLE]? Isn't that implicit within value iteration that with each-- each new value function is better than the previous one and therefore the policy will also be better? That's a good question, yeah, the question of whether it has guaranteed that. We have not proven anything about that yet. We prove that for policy iteration, but this is just to think about it in this case. OK, so let's go now-- anybody else have a question on the proof? All right. So one thing I just want to mention briefly and it'll come up on the homework is thinking about this for finite horizons. So most of today, almost all of today, we've assumed that we get to act forever, so we have an infinite horizon. But there will certainly be cases where there's a finite horizon. And in this case, this goes back to the thinking of value iteration as just computing the optimal value for each horizon. So if we have k equals 1 to H, H being the max horizon you want to compute for, then for each state at each round you would have a value function, k plus 1, which tells you how many decisions you make, what your horizon is, and we would just do this backup. So this looks exactly the same as what we saw before, but now you could also get a policy here-- so this would be the policy associated with that value function-- of what is the arg max action. So it would compute a series of policies, one for each step. One other thing I want to mention here-- and we'll talk about this on the homework too. One of the other thing I want to mention here is that you can also just simulate the value of a particular policy. So this is also really popular once we start to get into really big state spaces. So if we think of the fact that in a lot of these algorithms we're doing some sort of policy evaluation, one thing you could do is you just take your policy. And you know what your dynamics model is, and you know what your reward is. And you just roll it out. So if you're in a state, you simulate what the next state might be. Then you get a reward. So you can just generate a really large number of episodes and then just average them. So I'm just like, oh, how good is this policy? If your boss asks you, you just run it on 100 people. You average their rewards, and then you're done. And this is something that becomes really popular when it starts to be, say, hard to write down what that dynamics model is explicitly or do that sum over s prime. But it's really easy to sample. And I'll note for that you can use concentration inequalities, like Hoeffding inequality and Bernstein, for those of you familiar with them, to bound how much data do you need for this estimate to be close to the true one. And the great thing is that it's not that many, so if you have an enormous state space, like, I don't know, your Amazon or something like that, or you've got patient data, and it's incredibly high dimensional, you don't have to do that huge sum over S prime. You can just sample, and your accuracy generally improves by 1 over square root n, the number of samples you're doing. And the nice thing is that this also requires no assumption about the Markov structure. So you might have a partially observable scenario, which also comes up a lot in things like healthcare, and then you can just roll out your policy and just see how well it works. Well, in healthcare, you probably wouldn't just randomly roll out any policy, but pricing might be. Yeah? Remind me your name. [INAUDIBLE] That's right. So here this is just the policy evaluation stage, exactly, and you could either do it to compute the value of a policy or, as you just suggested, do it for the Q-value. So you start off in a state for each of the different actions. Then you roll out the policy until the end. And this is just a really popular other technique, and it'll come up in other places. So I wanted to start saying, we can do that here too. So you can also think about doing all of these in the case of the Mars rover. And I won't go through it now, but you can use these as examples to step through these different algorithms and think about how you would compute these type of policies. All right. So I will-- maybe I'll get to the end of this, but I'll leave you with two things. One is a thought question, which is, is the optimal policy stationary? What that means is independent of time steps in the finite horizon tasks. And we'll explore this issue, too, on the homework. And I also just want to refresh some terminology. So in the context of Markov decision processes and reinforcement learning, when we say, "models," what we normally mean is a mathematical model of the dynamics and reward. Policy is a function mapping from states to actions that can be deterministic or stochastic, and the value function is this expected discounted sum of rewards from starting in a state and following a policy. The things that should be clear to you is you should be able to understand what a Markov process is-- a Markov reward process is an MDP-- the Bellman operator, contraction, model, Q-value, and policy. And you should be able to implement both value iteration and policy iteration, and you'll have practice doing that on the homework. You should also understand some of the strengths and weaknesses in terms of what we've discussed in terms of the computational complexity of some of these different operations and be able to prove contraction properties about these as well as understand which of these are really leveraging the Markov assumption versus which of them don't require that. So next week we'll continue to talk about this, and we'll start to talk about function approximation and how we can also learn when we don't know what these models are. I'll see you then. Thanks

Introduction_to_Political_Philosophy_with_Steven_B_Smith

19_Democracy_and_Participation_Rousseaus_Discourse.txt

Professor Steven Smith: Good morning. My name is Borat. Anyone see the movie yet? Yeah, I saw it over the weekend. Had to cheer myself up a little bit after Saturday afternoon but there's still another week to go. Still time. Good morning. I want to talk today about my favorite part of the Second Discourse, a book that never grows old, that never fails to produce. Last time, in talking about Rousseau's account of the origins of inequality, I focused on a famous passage in which Rousseau claims it was the establishment of private property that was the true formation of civil society and the beginnings of inequality and all of the subsequent miseries of the human race that he wants to describe. But in fact, that's not really true. Rousseau himself knows it's not quite true. If Rousseau were only interested in issues of class and economic inequality, there would be very little difference between him and materialist theorists of society like Karl Marx although Marx was in fact a very appreciative reader of Rousseau and got most of his best lines against capitalist society from him. Nevertheless, Rousseau understands that even for institutions like property and civil society to be possible there must be huge and important developments that go on or take place even prior to this, moral and psychological transformations of human beings. And it is for Rousseau far more what we might call "the moral and psychological injuries of inequality" than the material aspects of the phenomenon that is of concern to him. Rousseau very much takes the side of the poor and the dispossessed but it isn't property, or it isn't poverty rather, that really rouses Rousseau's anger as it is the attitudes and beliefs shaped by inequalities and of wealth and power. It is Rousseau the moral psychologist where his voice truly comes out. In many ways, Rousseau like Plato finds his voice when discussing the various complexities of the human soul. So what is the chief villain in Rousseau's Second Discourse and his account of the beginnings in development of inequality? Real inequality begins in a faculty or a disposition that is in fact in most editions of the book rendered simply by the French term because it is really untranslatable into English. It is amour-propre, the first term I put on the board, which is the first and most durable cause of inequality for Rousseau. Amour-propre, again, is an untranslatable word but in many ways is related to a range of psychological characteristics such as pride, vanity, conceit. In the translation that you have, I believe, the translator refers to it as egocentrism, a kind of ugly modern psychologistic term I think but better and more accurately, evocatively translated by terms like vanity and conceit or pride. Amour-propre for Rousseau only arises in society and is the true cause, he believes, for our discontents. And in a lengthy footnote that I hope you checked--in a lengthy footnote, he distinguishes amour-propre from another disposition that he calls amour de soi-meme, a sort of self-love. How are these distinguished? He says in that note: "We must not confuse amour-propre with love of oneself. These are two passions very different by virtue of their nature and their effects." Love of oneself, amour de soi-meme, "Love of oneself is a natural sentiment," he writes, "which moves every animal to be vigilant in its own preservation and which directed in many by reason and modified by pity produces humanity and virtue." So there is a kind of self-love, he says, that is at the root of our desire to preserve ourself, to be strong in our self-preservation, and to resist the invasion or encroachment by others. But then, he goes on to say amour-propre is an entirely different kind of passion or sentiment. "Amour-propre is merely a sentiment that is relative," he says, "artificial and born in society which moves each individual to value himself more than anyone else, which inspired in men all the evils they cause one another and which is the true source of honor." Listen to that last expression. "Amour-propre," he says, "is what moves every individual to value" him--or herself--"more than any other, which inspires all of the evils in society and," he says, "is the true source of honor, both evil and honor, the desire to be recognized and esteemed by others." How can this passion of amour-propre be responsible for these two very different sort of competing effects? How did this sentiment arise first of all? How did it come about and I suppose fundamentally and more importantly, what can or should be done about it? For Hobbes, recall, and this idea of pride, vanity, what Hobbes called vainglory, you remember, a very important part of Hobbes' political and moral psychology in Leviathan, pride is seen as something natural to us, Hobbes writes, you remember, it is part of our natural--pride is part of our natural desire to dominate over others, but for Rousseau by contrast amour-propre is something that could only come about after the state of nature, a state that Hobbes, you remember, had called solitary, poor, nasty, brutish, and short, after the state of nature had already begun to give way to society. Hobbes' account for Rousseau is incoherent. If the natural state is truly solitary, poor, nasty, brutish, and short, what would it mean in such a state to feel pride or vanity that requires human sociability and requires the esteem of others and somehow the gaze or the look of others? How could pride have arisen in a state of nature which on Hobbes' own account is solitary? Rousseau uses Hobbes in a way to prove his own point, that amour-propre, vanity, is not a natural sentiment but, as he says in that passage I just read, a sentiment that is relative and artificial, could only have come into being once we enter society in some ways. But how did that happen? Rousseau speculates about this and, again, this is part of his hypothetical or conjectural history. He speculates that amour-propre began to arise and develop as soon as people began to gather around a hut or a tree and to look at one another, as soon as we became conscious of the gaze of another, and it is from that gaze, from the look or gaze of another, that the passion of vanity was born. Listen to the way in which he speculates how this arose. "Each one," he says, "began to look at the others and to want to be looked at himself and public esteem had a value. The one who sang or danced the best, the handsomest, the strongest, the most adroit or the most eloquent became the most highly regarded and this was the first step toward inequality and at the same time toward vice. From these first preferences were born vanity and contempt on the one hand and shame and envy on the other and the fermentation caused by this new leavens eventually produced compounds fatal to happiness and innocence." So the rise of this passion to be seen, to be seen to be best at something, produced in many--for many people again, as he puts it, pride and vanity from some shame and envy on the part of others and from this fatal compound grew tendencies that were, as he says, fatal to our happiness and innocence, and Rousseau, I think is very much onto something here. Amour-propre is presented in the passage I just read and throughout much of the Second Discourse in largely negative terms but it is also related to something positive, in many ways, for the development of humanity in society, the desire felt by all people once we enter society, to be accorded some kind of recognition or respect by those around us. That too is a part of amour-propre, the desire to be seen and recognized and respected. The desire for recognition, he says, is at the root of our sense of justice and underlying this, I think, is the intuition powerful and in many ways I think deeply true, that our feelings, beliefs, opinions and attitudes be acknowledged and respected by others around us, that we matter in some way. When we feel that our opinions are slighted, when others do not recognize our worth, we feel angry about this and this need for recognition, which is part of this passion of amour-propre, is for Rousseau also a cornerstone of justice but it is also, as he says, at the same time the demand for recognition can easily become cruel and violent as we demand this from others. Consider again just the following. I want to read one other passage from the same part of the text. He writes: "As soon as men had begun mutually to value one another and the idea of esteem was formed in their minds, each one claimed to have a right to it, each one claimed to have a right to esteem or recognition, and it no longer possible," he writes, "for anyone to be lacking it with impunity. From this came the first duties of civility even among savages and from this every voluntary wrong became an outrage. Every time someone was harmed or injured, it became an outrage because along with the harm that resulted from the injury," he says, "the offended party saw in it contempt for his person, which often was more insufferable than the harm itself." Think about the psychology, the moral psychology that Rousseau is invoking here in his talk about harm and injury. It's not the physical aspect of the harm that bothers him. It is the sort of contempt that is implied or entailed in the act of injury. Hence, he goes on to say, "each man punished the contempt shown him in a matter proportionate to the esteem in which he held himself. Acts of revenge became terrible and men became bloodthirsty and cruel." That is to say, amour-propre and society gave rise to the state of war. Does this sound familiar? I think it should. I was trying to think of some example that might fit this and one I came up with when I was thinking about this earlier--consider a story that was much in the news. I forget if it was last spring or last summer sometime. The Danish cartoon controversy. Do you remember that, about the cartoons of the prophet Muhammad and the outrage and the protests, often violent, that occurred about that? To some degree, Rousseau might argue, the protests were about disrespectful cartoons of the prophet but he would argue, I suspect, that the deeper cause seemed to be that the protesters believed was disrespect being shown to them, to their beliefs, to what it is they held sacred in some sense. It is their beliefs that were being disrespected and were the cause of the protests. Amour-propre, as Rousseau I think himself recognizes, is this very volatile passion. It contains the desire to be respected again and acknowledged that is at the root of justice and virtue and yet at the same time this passion, as we know, is easily manipulable by those who wish to convince others that their basic entitlements or views are not being respected. To some degree, I think, Rousseau would believe the protesters over those cartoons had a point. Their views were not being respected and to which you might say a Lockean or a liberal formulation of the problem or response would be, "Well, so what?" The task of government, according to Locke or the liberal view, is to ensure the security of person and property, to protect you from harm and of course to provide you the freedom to practice what religion you like, consistent with the freedom of others to do so too. It is not the business of government to ensure that your beliefs are being respected. This was clearly the view, for example, of the Danish newspaper editors that published the cartoon as well as the Danish prime minister who refused to apologize for this on the ground again that the government's job is not to impose a gag order on what can and cannot be said on the grounds that some people might find it offensive. This is a respectable, sort of liberal line of thought going from Locke to John Stuart Mill, and yet, while I am inclined to agree very much with that point of view, there is something powerful and true about what Rousseau has to say about it, about this kind of issue. Lockean liberal thought was addressed in many ways to people who had experienced the crucible of civil war, a century of religious conflict and were looking for a way to settle their religious and political differences. Toleration in many ways is a liberal virtue because it requires us to distinguish between beliefs that we may take with the utmost seriousness in private life and yet nevertheless bracket them in some way once we enter the public world. This, in many ways, is the peculiar liberal virtue of self-restraint or self-denial, that we refuse to allow our own moral point of view to, in many ways, dominate in the public space. But it is one thing, you might say, to tolerate other views and another thing to accord them respect and esteem. That seems to be something very different from what Locke talked about. To tolerate simply means not to persecute, to leave alone, while respect for something requires that we esteem it. You might ask yourself, "Must we esteem and respect values and points of view that we do not share?" This seems very different, again, from the sort of liberal understanding of toleration that means only extending acceptance to views again that are very different from our own. It doesn't require us to, as it were, censor, self-censor, our own views on the ground that they may be--our views may be in some ways disrespectful or hurtful to others. This is a vast topic. I've sort of used the opportunity to sort of move away from Rousseau a little bit but his point is I think that amour-propre, the desire to be esteemed, recognized, and to have your values and points of view esteemed by those around you is in fact a violent and uncontrollable passion. It is the passion very much like Plato's thumos, spiritedness, back in the Republic. It is a passion that makes us burn with anger over perceived slights and makes us also risk our lives and endanger the lives of others to rectify what we believe to be acts of injustice. Like Plato, in many ways, Rousseau wants to know whether amour-propre is purely a negative passion or disposition or whether, like thumos, whether it can be redirected, in some way, to achieve social goods and social benefits. All of this is entailed in that short discussion of amour-propre in the Second Discourse. So much of Rousseau's subsequent account of civilization and its discontents grows out of this peculiar psychological disposition and passion. So let's talk a little bit more about civilization and its discontents. In Woody Allen's movie, Annie Hall, you might recall a scene in which he says there are two kinds of people. They're the horrible and the miserable. The horrible are those who have suffered some kind of personal tragedy, a disfigurement of some kind, who are facing a terminal illness. The miserable is everybody else. Rousseau wants us to be miserable. He wants us to feel just how bad things are, how bad we are, how bad off we are. The only exception to this general human misery is, as he tells us at one point, kind of early primitive society. These societies described by him, not quite the state of nature to be sure, maintained a kind of middling position between the pure state of nature and the development of modern conditions. He says these were the happiest and most durable societies and the best for man. It was primitive man, not the pure savage of the state of nature, where Rousseau finds a happy equilibrium between our powers and our needs that he says is the recipe for happiness, bringing our powers and our needs into equilibrium, but the end of that happy state came with two inventions, two discoveries: agriculture and metallurgy. With agriculture came, here we see the division of land, the division of property, and the subsequent inequalities that came with it. With metallurgy came the art of war and conquest. With these two developments, he tells us, humanity entered a new stage, one where laws and political institutions became necessary to adjudicate conflicts over rights, and the establishment of governments that this entailed rather than bringing peace, as it would for Hobbes or Locke, the establishment of governments had the effect simply of sanctioning the existing inequalities that had begun to develop. For Rousseau, there is something deeply shocking and deeply troubling about the assertion that men who were once free and equal are so easily, as it were, led to consent to the inequalities of property and to rule by the stronger, which government brings into being. The social contract, as he presents it in the Second Discourse, is really a kind of swindle. The establishment of government is a kind of swindle that the rich and the powerful use to control the poor and the dispossessed. Again, rather than instituting justice, this compact merely legitimizes past usurpations. Government is a con game that the rich play upon the poor. Political power simply helps to legitimize economic inequalities. Governments, he tells us, may operate by consent but the consent they are granted is based on falsehoods and lies. How else can one explain why the rich live lives that are so much freer, so much easier, so much more open to enjoyment, than the poor? That is Rousseau's real critique and real question. And it is the establishment of government that is the last link in the chain of Rousseau's conjectural history, the last and most painful, in many ways, legitimation of the inequalities that have been created after our emergence from the natural condition. But what, again, is most painful to Rousseau is the emergence of a new kind of human being that this stage of civilization has been brought into--that this state of civilization has brought into being. And Rousseau is the first, I think, to use that term so powerfully, which became used very much in the next two centuries, the bourgeoisie. The bourgeoisie is Rousseau's invention and most striking about this human type for him is the necessity to appear to be one thing where actually being something else. Go back again to think of the way in which Plato or Socrates uses that distinction between seeming and being when he talks about the just man in Book II of the Republic, someone who seems to be and someone who is just. It is this tension between the two that is so central to Rousseau's account of what he calls the bourgeoisie. "Being something and appearing to be something," he says, "become two different things and from this distinction there arose grand ostentation, deceptive cunning, and all the vices that follow in their wake." And in the penultimate paragraph of the Second Discourse Rousseau describes the dilemma of the bourgeoisie in the following way. He says, "The savage lives within himself. The man accustomed to the ways of society, the bourgeoisie, is always outside of himself and knows only how to live in the opinions of others and it is, as it were, from their judgment alone that he draws the sentiment of his own existence." Think of that sentence. It comes from the next to the last paragraph of the book, that in society we only live through the opinions of others, through the gaze of others, through what others think of us. We are constantly our own sentiment of existence, he says. Our own sentiment of self and existent comes entirely from the judgment, as he puts it, of those around us. The bourgeoisie, in other words, is someone who lives in and through the opinions, the good opinions, of others, who thinks only of himself when he is with other people and only of other people when he is by himself. Such a person is duplicitous, hypocritical, and false. This is why this is the true, you might say, discontent of civilization. This is what our perpetual restlessness and reflectiveness have made of us. Goaded on perpetually by amour-propre, this is the particular misery that civilization has bequeathed us. So the question at the end of the book is what to do about this and here, in many ways, one has to say the Second Discourse falls short. The book ends on a note of utmost despair. It offers no positive answer to cure the problem of civilization but only hints at best at two possible solutions. One is suggested, you will recall, by the letter to the City of Geneva which, in a sense, prefaces the book. Perhaps the closest approximation to the early state of primitive society lauded by Rousseau are the small, isolated rural republics like Geneva in its own way where a kind of simple patriotism and love of country have not been completely overwhelmed by the agitations of amour-propre. Only, he says, in a well-tempered democracy like Geneva is it possible for citizens still to enjoy some of the equality of the natural man. Democracy for him, this kind of simple rural democracy like that of Geneva, is the social condition that most closely approximates the equality of the state of nature and that of course is a theme that Rousseau will take up powerfully in his book the Social Contract. But Rousseau offers another hint to the solution of the problem of civilization, what to do about it. How can we restore happiness in the midst of society? The Second Discourse leaves us to believe that all society is a state of bondage and alienation from nature, from our true being. We have lost our true humanity that he describes in the state of nature, our state of--our capacities for pity and compassion and the like, and the answer to the problem of society is, in many ways, to return to the root of society and this root of society is not just the need for self-preservation but a kind of primordial, as he calls it in that passage I read a minute ago, sentiment of existence, the sentiment of our own existence. By giving oneself over to this feeling of existence without a thought for the future, without care or fear, the individual somehow psychologically returns to the natural state. Only a very few people, Rousseau writes, he being one of them of course--only a very few people are capable of finding their way back to nature. The type of human being who can find their way back to the sort of pure sentiment of existence is not going to be a philosopher, is not going to be a person of high order reflection like Socrates, but will more likely be an artist or a poet. He is one of those rare aristocrats of nature, you might say. His claim to superiority is not based on a higher understanding but a superior sensitivity, less on wisdom than on compassion. Rousseau believed himself to be one of these people. Maybe you also are one of them. Yes? But it requires you, in some way, to distance yourself severely and psychologically from all of the possibilities of society, to return inward, and it was that inward journey that Rousseau took and that he writes about so powerfully in his Confessions and his final book, The Reveries, where you find the Rousseau, founder of the romantic disposition that you get again in writers in America like Thoreau and others who look inward and return to nature in some way, their natural self as opposed to society. But the Second Discourse leaves us, to be sure, with a paradox. The progress of civilization is responsible for all of our miseries. Yes, it is society's fault. It's not your fault. It's society's, he wants to tell us, and yet he also leaves us with no real apparent way out. He denies that we can, as a practical solution, return to simpler, more natural forms of political association but how then do we resolve the problem that he leaves us with? And his answer to it, his political answer to it, his most famous political answer to it is contained in his book, yes, called the Social Contract, Du Contrat Social, published in 1762, seven years after the Second Discourse. Here he attempts to give one such answer, and I mentioned one such answer because it is not his only or final answer, but one such answer to the problems of inequality and, again, the injuries of amour-propre. The Social Contract begins with one of the most famous sentences in all of the history of political philosophy, "man is born free and is everywhere in chains." Always begin your essays with a good, strong sentence like that. Rousseau knew this. He knew something about how to write. The phrase seems to be perfectly in keeping with the Second Discourse. In the state of nature, we are born free, equal and independent. Only in society do we become weak, dependent, and enslaved. It is what follows after that sentence in a way that is the shocker. How did this take--how did this change take place, Rousseau asks. I do not know. What can render it legitimate? I believe I can answer this question. What can render it legitimate and by the "it" I take it he means the chains as in--that states man is born free and is everywhere in chains. In the Second Discourse, he had attempted to completely delegitimize the bonds of society, saying how the Social Contract and the creation of government was nothing but, in many ways, a sophisticated swindle. Now in the Social Contract, he asks the question, "What can give these chains or bonds moral legitimacy?" He says I believe I can answer that. Has Rousseau simply undergone a massive change of heart in the seven years between these two books? I don't think so but I think these are--this is part of his--one of his answers to this fundamental question. But before going into the details of this, let's consider some of the differences between these two very powerful books. Right? The Second Discourse, the discourse on inequality, presents itself as a hypothetical or conjectural history of human development from the state of nature to the civil condition. It is written in a vivid language, which is why it is always- it is often considered one of Rousseau's most powerful pieces of writing, a vivid language drawing on in many ways the biological sciences of his day and newly discovered knowledge of animal species like orangutans and other kinds of anthropological investigations of the Caribs and North American peoples, a very vivid work. The Social Contract, by contrast, is written in a dry, even a kind of bloodless language of a lawyer. It is very much written in the genre of a legal document. Its subtitle is The Principles of Political Right. It is a work of considerable philosophical abstraction whose leading concepts are abstractions like the social contract, the general will, and so on. The book, he tells us in short preface, was originally part of a longer investigation of politics which has since been--which he says has since been lost. Also, the Social Contract presents itself in many ways as a utopia, an ideal city, in some respects an answer to the Calipolis of Plato's Republic and yet this is also--this seems to be not quite true. The work begins, even before the famous sentence about man being born free, the work is prefaced with a statement that could have come directly out of Machiavelli's Prince. "Taking men as they are and laws as they might be," Rousseau says, "I will try in this inquiry to bring together what right permits with what interest prescribes." Taking men as they are… You remember the fifteenth chapter of The Prince. Let us look at the effectual truth of things, not what is imagined to be but the way people actually are. Let us take men as they are, Rousseau says, following Machiavelli. He will not begin, he tells us, by making any heroic assumptions about human nature, no metaphysical flights of fancy, but rather stay on the low but solid ground of recognized fact. What does he mean by this and what are these facts of human nature, men as they are, he says, that Rousseau claims to describe in the Social Contract? And here we get to the basic premise of the book. The basic premise, I think, from which the entire Social Contract unfolds is the claim that man is born free. All subsequent relations of hierarchy, obligation and authority are the result not of nature but of agreement or convention. Society and the moral ties that constitute it are conventional, you might say, by agreement, all the way down. There is nothing natural about any of the social contract. And from this basis of man as a free agent, that we are born free, Rousseau attempts to work out a system of justice. The Principles of Political Right, again is the subtitle, suggests that are appropriate to human beings conceived as free agents responsible to themselves alone. But how do you do that? How can you do that? Rousseau's political philosophy begins, at least he believes I think, with the realistic or even empirical assumption that each individual has a deep rooted interest in securing the conditions of their own liberty. The state of nature and the social contract presuppose individuals who are in competition with others and each attempting, as it were, to secure the conditions for their own liberty. He does not presuppose altruism on the part of any human being or any other kind of self-other regarding characteristics, what I called a moment ago heroic assumptions. He doesn't make the assumption that we act for the interests of others. We are selfishly concerned with our own freedom and the best means of preserving it and protecting it. Each of us has a desire to preserve his or her own freedom and that social order will be rational or just, that allows us to preserve that freedom. The problem, of course, is that in the state of nature the desire to preserve my freedom comes into conflict with the selfish desire of everybody else to preserve their freedom. The state of nature quickly becomes a state of war based on conflicting desires and conflicting again means of liberty preservation. So how do we preserve our liberty without lapsing into anarchy, that is the state of war? This is the question that the Social Contract sets out to answer and to which his formulation, his famous formulation of what he calls the general will, is the solution. I'm going to end on that note today and Wednesday I want to talk about the general will and how Rousseau sees it as a sort of collective answer to the problem of the securing of individual liberty. So meditate on that if you like for the next day.

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17_Constitutional_Government_Lockes_Second_Treatise_1319.txt

Professor Steven Smith: I want to look at two sets of issues. I want to in, a way, conclude my interpretation, my reading of the Second Treatise, by focusing on the role of executive power in Locke's theory of government, Locke's theory of the constitutional state, particularly focusing on the role of the executive, vis-a-vis the legislative branch of government, and then I want to turn a little more speculatively to thinking about Locke and the American regime and the current state of political philosophy, modern contemporary American political philosophy. But let me start first by going back and sticking with the Second Treatise by talking a little bit about the role of legislative and executive power. The last time, I think, I was concluding by arguing that Locke doesn't endorse necessarily one particular form of government from any other. He is an advocate of what we have come to call limited government, of constitutional government. There is that important passage where he ridicules the Hobbesian sovereign as a lion and tells us we did not enter into the social compact to be devoured by lions. He says, the form of government must be limited although he's relatively open or at least non-committal, agnostic you might say, as to what particular form that government may take. One feature of this form of government that he thinks is very important, is that it must in some sense embody a separation of powers, powers must be made to check one another, what he calls in the book the subordination of powers. This is Locke's doctrine and you will see it there. We often associate it with Montesquieu or sometimes with the federalist authors but, in fact, Locke himself is a strong advocate of what he calls the subordination or separation of powers, not exactly the same as we'll see between our understanding of executive legislative and judicial, but nevertheless a separation nonetheless. However, in the first instance, Locke emphasizes and in fact he continually affirms nevertheless the primacy of legislative authority. In England, in the England at his time and even today, that means a doctrine of what is called parliamentary supremacy but he says that the first and fundamental positive law of all constitutions is in establishing that of the legislative power. The first act, after the completion of the social contract, he says, is establishing the legislative power. It is the lawmaking authority of government that is supreme, he wishes to emphasize. This seems to push Locke, you might, say more in the small ‘d' democratic direction. It is not so much executive power, the power of a prince, but rather the legislature, the parliament that is supreme. There is nothing more important, in Locke's theory of constitutional government, than the existence of what he continually refers to as settled or known laws, settled laws that serve against arbitrary rule. In many ways, the purpose of government for Locke is much less to offset the dangers of returning to an anarchic state of nature as it was for Hobbes than to prevent the possibility of the emergence of tyrannical or despotic power, tyrannical or despotic sovereign, and of course, Locke's writing is very much bound up with the big and major constitutional crisis of his time leading to the overthrow and expulsion of a king, James II. Yet in many ways, even though Locke is the great advocate of legislative supremacy, he obviously cannot and does not wish to dispense altogether with the role of executive power. He often treats the executive, whether that be in the form of a prince, a monarch or perhaps even a body in a cabinet of chief officers as it were. He treats them often simply as if they were an agent of the legislative or of the legislature. The purpose of the executive, he sometimes seems to write, is merely that of carrying out the will of the legislature. In Locke's language, "the executive power is ministerial and subordinate to the legislature," section 153, I believe. The executive, again, on some aspects of Locke's writing seems to be little more than a cipher in comparison to the doctrine of legislative supremacy. And yet, Locke here is not altogether consistent, one has to say, because he understands in every community there is a need for a distinctive branch of government dealing with matters of war and peace. Locke calls this the federative power. Every community, he says, like Hobbes, is to every other community what every individual is to every other individual in the state of nature and a distinctive federative or war-making power within the government is necessary for dealing with matters of international conflict, conflict between states. And in a remarkable passage, Locke notes that this power, he says, cannot be bound by antecedent standing positive laws but it must be left to, quote, "the prudence and wisdom of those whose hands it is in to be managed for the public good." In other words, Locke seems to suggest that this particular kind, this branch of government, this federative branch which falls to some degree under the executive, must have a certain latitude even apart from the law that relies, he says, on the prudence and wisdom of those whose hands it is in to manage it for the public good. In other words, matters of war and peace cannot be left to the legislature or to standing laws, as he calls them, alone but requires the intervention of strong leaders, what he calls in an absolutely stunning passage god-like princes, section--if you don't believe me, section 166. Locke's reference here to god-like princes seems to recall Machiavelli in many ways, Machiavelli's talk of armed prophets. It is necessary, in extreme situations, for such princes to call on their prerogative power. It is impossible, Locke writes, to foresee and so by laws to provide for all the accidents and necessities that may concern the public and that during, in other words, contingencies or emergency situations the executive must be empowered with this prerogative power to act for the good of the community. For this reason it seems, the executive is not simply a tool or an agent of the legislature but he says, again, must have the power to act according to discretion, that is to say, according to his own discretion for the public good without the prescription of law, those are Locke's own words. How to balance his argument for constitutional government and legislative supremacy with this doctrine of prerogative power and what seems to be a kind of power of what he calls in no uncertain term the god-like princes and their need to exercise this power? Locke's prerogative is, in many ways, the result of simply the inability of law to foresee all possible circumstances, all possible contingencies. That's an argument that goes as far back as Aristotle, we've seen. Our inability to make rules that can apply to all possible events, makes it necessary to leave some discretionary power in the hands of the executive to act for the public safety. One of the examples that Locke gives of the use of this power is in fact a domestic, not an international issue, which is to say, in the case of a fire in a city it is sometimes necessary, he says, in his day for the fire department to tear down the house of an innocent person to prevent the fire from spreading to other houses. This is acting for the public good of the community, even while in some ways it's clearly a violation of rights of property and so on. He understands this as a piece of prerogative power acting for the public good. In fact, the example is not so far fetched. Think today for example about arguments we have today. Even in Connecticut, there's a big argument going on about the right of what's called "eminent domain," the right of the government to absorb or to take over private properties whenever, usually for things like schools or airports but also for general improvement when it is thought it will enhance the public good. There's a big debate going on right now out in New London and in Brooklyn also with the argument about the creation of some civic center, some sports arena that will require the demolition of certain neighborhood houses. And there's a big debate about this eminent domain. What is that, but in a way Locke's example of prerogative power, acting, doing something that is somehow said to be for the public good but that represents some kind of extra constitutional power? But the question for Locke, as for any constitutional lawyer, is what are the limits of this prerogative power? What check, if any, is there on this power to prevent their abuse? Well, Locke doesn't exactly say. Yes. Right. He doesn't exactly say. He raises this question to be sure, of fundamental importance for constitutional government. Does executive authority, he asked us, extend to all things even or especially in times of war? Think about the debates that are going on now about detainees at Guantanamo or the issues of domestic spying when it comes to issues of the war on terror. Are these examples of prerogative power, that is to say, the executive acting outside the limits or the bounds of constitutional authority for the sake of protecting the public good or are these examples of kind of political absolutism? Is the invocation of this power, in some ways, going down the slippery slope to despotism and absolutism? I will leave it to you or your sections to try to discuss these matters but Locke himself praises those who he calls the wisest and best princes of England as being those who have exercised the largest prerogative on behalf of the public good. This is beginning to sound more and more in respects like Machiavelli than the advocate of, again, limited government. This power comes into play, he says, especially during times of national crisis or emergency when it is necessary to act for the public safety in some ways. And again, this seems to have special resonance for us today as we face issues like states of emergency and states of exception. There are in fact political theorists, one name comes to mind, a twentieth-century German legal philosopher by the name of Carl Schmitt who argued that the state of emergency or the exceptional situation is the essence of politics and that the person or body who has the power to declare the exception is none other than the sovereign. So from Schmitt's point of view you might say this idea of prerogative is a kind of extra constitutional power that the statesman must of necessity utilize when ordinary constitutional operations, like the rule of law, prove to be inadequate. But consider another example if you like, that prerogative power, about prerogative powers that maybe granted by the Constitution. Consider Lincoln's famous suspension of habeas corpus during the Civil War. Lincoln, interestingly, did not take this extraordinary step by appealing to an extra-constitutional power that obtains in times of crisis. Rather, Lincoln argued quite forcefully that this sort of prerogative power is already deeply embedded within the structure of constitutional government. He cites the Constitution when it came to the suspension of habeas corpus. The Constitution writes, "The privilege of the writ of habeas corpus shall not be suspended unless when in cases of rebellion or invasion the public safety requires it." In other words, the Constitution itself seems to allow for this extraordinary kind of action at least in cases of rebellion or invasion when it says the public safety requires it. The Constitution seems to embody within itself, our constitution that is, this Lockean power of prerogative that comes into effect or can be legitimately exercised in times of rebellion or invasion. Are we living in that kind of age now, not rebellion perhaps but invasion? Well, think about that again. Are these arguments applicable to our situation today, in some sense, when it comes to debates about the extent of executive power to embark on these extraordinary measures? And yet at the same time, Locke is aware clearly of the potential abuse of this kind of prerogative. He asks, who will judge, who can judge whether the discretion of the executive is being used for the public safety or the public good or whether it is simply a kind of usurpation of power? In these moments of high constitutional crisis between conflicting powers of government, in such cases, Locke says there shall be no judge on earth. He says the people have no other remedy in this but to appeal to Heaven. This is in section 168. How much is contained in that term "appeal to Heaven?" What does Locke mean in terms of high constitutional crisis when he says there is no judge on earth, the people must appeal to Heaven? Does that mean they should fall down on their knees and begin to pray, what they should do? Unlikely. By an appeal to Heaven, Locke means the people's right to dissolve their government. He raises this question at the very end of the book. When a conflict between the people or their representatives and the executive becomes so great that the very conditions of social trust have been dissolved, who will be judge? And he answers emphatically: the people will be judge. Locke affirms here a right of revolution. An appeal to Heaven, or what he calls an appeal to Heaven really refers to an appeal to arms, to rebellion, and the need to create a new social covenant. Locke, you can see, is attempting to hold together a belief in the sanctity of law and the necessity for prerogative that may sometimes have to circumvent the rules of law. Are these two doctrines incompatible? I think in many respects or at least in some respects they are. Can the prerogative power of the executive be in a way constitutionalized so that it does not threaten the liberty of its own citizens? Locke alerts us to this timeless as well as this very timely problem. One of the best sources for thinking about many of these constitutional issues today, regarding privacy rights and other kinds of citizen rights, can be found in, I would say the last five chapters or so of Locke's Second Treatise. I can't think of a better source. So in the end Locke's appeal to Heaven or Locke says the people have an appeal to Heaven, that is to say an appeal to arms, an appeal to revolution, suggests that at the end of the day Locke was a revolutionary but I would say also a sort of cautious and moderate one, if this is not a complete contradiction in terms. I won't go through chapter 19, the famous chapter on revolution in full, to talk about the conditions under which he believed the people can rightfully appeal to Heaven, as it were, but Locke's doctrine of consent and legislative supremacy, this should make him in many ways a hero to Democrats, to radical Democrats. His beliefs about limited government, the rights of property should make him a hero to in some ways constitutional conservatives and even libertarians. In the end, I think Locke was neither or both. Like all of the great thinkers in some ways, he defines--he defies, excuse me, simple classification but there is no doubt that Locke gave the modern constitutional state its definitive form of expression. And the problems of our state, the problems, the legal, the constitutional and political problems that we experience are very much problems rooted in the philosophy of John Locke and are unthinkable without the influence of Locke. So that takes me to a theme that I want to talk about for a little while, which is Locke's America, John Locke's America. No one who reads Locke, even superficially, and I would not accuse anyone here of being a superficial reader, after all, but no one can fail to be impressed by the harmony, in many ways, between Locke's writings and those of the American Republic that he helped to found. His conception of natural law, rights, government by consent, the right to revolution and all are all part of the cornerstone of our founding documents. To some degree, as I've just been suggesting, a judgment on America is very much a judgment on the philosophy of Locke and vice versa. In many ways, if anyone is, I think Locke has the title to be considered America's philosopher-king. So how should we think of Locke after more or less three centuries of consistent Lockean rule? How should we think of Locke? For many years and for many people, even today, the affinity, the affiliation between Locke and America has been regarded in a largely although not wholly, largely positive light. For many historians and political theorists, our stability, our system of limited government, our market economy has been the result of a sort of broad consensus over Lockean principles, over Lockean first principles. But for many other readers of American history, this relationship has been seen as more problematic. In the 1950s, a book written by a famous political theorist and historian, named Louis Hartz, a book called The Liberal Tradition in America, complained of America's, what he called "irrational Lockeanism." That was Hartz's line, that was Hartz's quote, "irrational Lockeanism," by which he meant a kind of closed commitment to Lockean principles and ideals that shut off all other political alternatives and possibilities. Hartz was someone very much interested in the question, as many political theorists have been since, why has there been no socialism in America, why did America not evolve or develop along European lines with social democratic parties and socialist parties like the English Labor Party and other kinds of labor movements. And Hartz's argument was that we were sort of arrested in this Lockean phase of development, what he called our irrational Lockeanism that closed off in many ways other principles. And for still other thinkers, more or less on the left, Locke legitimized what was called an ethic of what was called "possessive individualism," particularly Locke's focus on property and the rights of private property that focuses entirely on market relations or puts the market values ahead of all other things. And for still others, in many ways more recently, thinkers of a more sort of communitarian direction or bent, Locke's emphasis upon rights and the protection, that government should protect natural or certain unalienable rights, suggests a purely or overly legalistic conception of politics that has no language for talking about the common good, the public good or other sort of collective goods or benefit. So my point is that Locke's influence has not been altogether accepted by everyone. There has been much ground for criticism of this peculiar affinity between Lockeanism and America. But today, I would say that Locke's theory of liberalism or Locke's theory of limited government, constitutional government, is confronted by another alternative that, in many ways, has deep roots in the very tradition which Locke himself---the very liberal tradition in many ways of which Locke himself is the founder. And I am referring, in particular, to a book that many of you will read at some point in your Yale experience, a book, widely read and widely acclaimed book by a recently deceased political philosopher by the name of John Rawls who wrote a book in 1973 called A Theory of Justice. In many ways, Rawls' book was an attempt to update the liberal theory of the state. He invokes the idea of a state of nature, an original condition, as he calls it, a theory of rights although he does so in many ways through the techniques of contemporary philosophy and game theory and Rawls' book is probably the single most important contribution to Anglo-American political philosophy in the last generation. It is a book that situates itself within the liberal tradition beginning with Locke, developed by people like Immanuel Kant and John Stuart Mill in which Rawls himself hoped to, in many ways, bring to completion in his book. A theory of justice, as he calls it, stands or falls on its theory of rights from which all else is derived. And what I want to do for a few minutes is to contrast Rawls' general theory, so powerful and influential today, from that of John Locke's, the original founder of the liberal theory of the state, and see how they have diverged. Consider the following propositions, if you will. Here is John Locke, section 27 of the Second Treatise. "Every man has property in his own person. This nobody has any right to but himself and where there is property," he writes, "there can be justice and injustice." Here is John Rawls, one of the opening pages of his Theory of Justice. "Each person," Rawls writes, "possesses an inviolability founded on justice that even the welfare of society as a whole cannot override. For this reason," he continues, "justice denies that the loss of freedom for some is made right by a greater good shared to others." Okay. So far, so good, in other words. Both of them present their theories of justice as justified in terms of the liberal principles of equality, freedom and the sanctity of the individual and individual rights. Both regard the purpose of government, in many ways, as securing the conditions of justice as deriving from the consent, or the informed consent, of the governed but both it seems to me go on to differ profoundly about the source of rights and therefore the role that government has in securing the conditions of justice. Let me explain a little bit more what I mean. For Locke, going back to chapter 5 of the Second Treatise, rights derived from a theory of self-ownership. According to his view, you will remember, everybody has a property in his or her own person. That is to say, no one has a claim on our bodies other than ourselves. It is on the rock of self-ownership, the fact that we have property in ourselves, it is on the rock of self-ownership that Locke builds his edifice of natural rights, justice, and limited government. To put it in a slightly different way perhaps, a person has an identity, what we might call today a moral personality or an identity by the fact that we alone are responsible for making ourselves. He uses this metaphor of the work of the body and the labor of our hands but we are literally the products of our own making. We create ourselves through our activity and our most characteristic activity is our work. Locke's fundamental doctrine is that the world is the product of our own free creativity, not nature but the self, the individual is the source of all value for Locke. It is this self, the I, the me, the ego that is the unique source of rights and the task of government is to secure the conditions of our property in the broadest sense of the term, namely, everything that is proper to us. Now, using that as a sort of shorthand, contrast this to Rawls' idea. Rawls adds to his idea of justice something that he calls the "difference principle," the DP as it's sometimes referred to in the literature on Rawls. What is the difference principle? This principle maintains that our natural endowments, our talents, our abilities, our family backgrounds, our history, our unique histories, our place, so to speak, in the social hierarchy, all of these things are from a moral point of view something completely arbitrary. None of these are ours in any strong sense of the term. They do not belong to us but are the result of a more or less kind of random or arbitrary genetic lottery or social lottery of which I or you happen to be the unique beneficiaries. The result of this, in other words, is that no longer can I be regarded as the sole proprietor of my assets or the unique recipient of the advantages or disadvantages I may accrue from them. Fortune, luck, Machiavellian fortuna, in that way, is utterly arbitrary and therefore, Rawls concludes, I should not be regarded as the possessor but merely the recipient of what talents, capacities and abilities that I may, again, purely arbitrarily happen to possess. So what does that mean in terms of social policy or theory of government? The result of Rawls' difference principle and its fundamental difference with that of John Locke could not be more striking from this point of view. The Lockean theory of justice, broadly speaking, supports a meritocracy sometimes referred to as "equality of opportunity," that is, what a person does with his or her natural assets belongs exclusively to them, the right to rise or fall belongs exclusively to them. No one has the moral right to interfere with the products of our labor, the products of--which may also include not just in a primitive sense what we do with our hands and bodies but what we do with our intelligence and our natural endowments. For Rawls, again, on the other hand, our endowments are never really our own to begin with. They are part of a common or collective possession to be shared by society as a whole, the capacities of hard work, ambition, intelligence and just good luck that, for example, got you to a place like Yale, on Rawls' account, do not really belong to you or at least the fruits of those ambitions and intelligence and good luck do not belong to you. Again they are somewhat arbitrary as a result of upbringing and genetics. They're not yours or mine, in any strong sense of the term, but rather, a collective possession that can be or should be the fruits of which distributed to society as a whole. Consider the following passage from Rawls. "The difference principle," he writes, "represents in effect an agreement to regard the distribution of natural talents as a common asset and to share in the benefits of this distribution whatever it turns out to be." Your intelligence or your drive or your endowments are, again, what he calls a collective asset. Think about that. And it is this conception of common assets that underwrites Rawls' theory of distributive justice and the welfare state, just as Locke's theory of self-ownership justifies his conception of limited government in the constitutional state. According to Rawls, again, justice requires that social arrangements be structured for the benefits of the least advantaged in the genetic lottery of society. His thought experiment that he calls "the original condition" specifies that nobody would know in advance in this condition what their particular endowment intellectually, in many other ways, would be. Therefore, every individual would, in contracting with the whole, would agree to share equally in the benefits of this, as it were, genetic lottery. So redistributing our common assets does not violate, on Rawls' account, the sanctity of the individual because again the fruits of our labor were never really ours to begin with. Unlike Locke, whose theory of self-ownership provides a moral justification for the individual, for the self, for our moral personality, Rawls' difference principle maintains that we never again belong to ourselves at all. We never really have ownership in ourselves but are always part of a larger social "we," a social collective, a collective consciousness whose common assets can be redistributed for the benefit of the whole. Locke and Rawls, the point I'm trying to make is, they represent two radically different visions of the liberal state, one broadly libertarian, the other broadly welfarist, one emphasizing liberty, the other emphasizing equality. Interestingly, again, this transition, this evolution represents a change which has gone on within in many ways the liberal tradition itself. Unlike some of these other critics, Rawls does not come to be claiming from a tradition outside of liberalism but to be developing certain arguments from within the liberal tradition and yet has moved in a way clearly very different from its Lockean formulation. Both of these views, again, they begin from common premises but move in very different directions. Locke's theory of self-ownership regards the political community in largely negative terms as protecting our antecedent individual selves and individual rights. Rawls' theory of common assets regards the community in a far more positive sense as taking an active role in reshaping and redistributing the products of our individual endeavors for the common interests. The question for you, just like the question for any of us, is which of these two views is more valid or which of the two strikes you as more powerful or plausible? My own view, and I loathe to editorialize, but my own view is far closer to American theory, to Locke's theory, which I think--than Rawls'. The Declaration of Independence, the charter of American liberty, states that each individual is endowed with unalienable rights among which are life, liberty, the pursuit of happiness. The very indeterminacy of the last phrase, the pursuit of happiness, with its emphasis upon the individual's right to determine happiness for themselves, suggests a form of government that allows for ample diversity for our natural talents and abilities and although the Declaration certainly intends that the establishment of justice is one of the first tasks of government, nowhere does it imply that this requires the wholesale redistribution of our individual goods and assets. And second, although Rawls is clearly attractive, excuse me, Rawls is clearly attentive to the moral ills of inequality and we will turn to that problem emphatically on Wednesday when we look at Jean-Jacques Rousseau's Essay on Inequality. There has never been a more powerful, passionate and persuasive critic of the ills of inequality than Jean-Jacques Rousseau but while Rawls is certainly attentive to the moral ills of inequality, he seems very naive about the mechanisms, the actual political mechanisms, by which inequalities will be rectified. Rawls wants government to work for the benefit of the least advantaged but this will require the extensive and often arbitrary use of judicial power to determine who has a right to what, far in excess of the powers of the court. The result would be, I think if we follow Rawls' teachings to their letter, the result would be not a class of philosopher-kings, but rather a class of chief justices endowed with the power to rearrange and redistribute our collective assets for the sake of achieving the maximum degree of social equality. It is no surprise that the warmest reception that Rawls' writing gets today is in the schools of law, is in the law schools where he has had an enormous influence on shaping the education of the current and the next generation of lawyers, judges and possibly chief justices who may be looking to again, looking not to the Constitution but to Rawls' theory of justice as a litmus or a tool for bringing about social redistribution. So, I leave you on that sobering note but a return to Locke such as it is, even if such a return were possible, is by no means a panacea to what ails us. I am not suggesting for a moment that Locke is some kind of cure all. Some historians, let me just mention again, Louis Hartz was but the most famous, treat America as a nation uniquely built upon Lockean foundations. America, he believed, remained something of a Lockean remnant--a Lockean, yeah, remnant, fossil in some ways, in a world increasingly governed by more radical forms of modernity. In fact, it has been our sort of stubborn Lockeanism that has, in many ways, prevented the kinds of extreme ideological polarization and conflict that one sees throughout much of the nineteenth and twentieth centuries. But Locke's effort to build a kind of modern republican government on the low but solid foundations of self-interest and self-ownership and the desire for comfortable preservation could not help but generate its own forms of dissatisfaction. Can a regime, dedicated to the pursuit of happiness or to the protection of property ever satisfy the deepest longings of the human soul? Can a regime, devoted to the rational accumulation of property answer those higher order needs or higher order virtues, like honor, nobility and sacrifice? Can a regime, devoted to the avoidance of pain, discomfort and anxiety, produce anything more than contemporary forms of Epicureanism and Nihilism? In any case, I'm suggesting no more than any other land could America insulate itself from the great heights as well as the great depths of later forms of modernity. America, as a former teacher of mine once said, is the land where the many facets, the many faces of modernity are working themselves out. We are but a moment in the kind of comprehensive self-dissatisfaction that is modernity so that a return to Lockeanism, in many ways, is not so much a cure for the pathologies of modernity. I would suggest that those pathologies are themselves already rooted in the pathologies of Locke. I will end on that sober note and encourage you to take Rousseau's advice about loving one's country seriously on Tuesday.

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15_Constitutional_Government_Lockes_Second_Treatise_15.txt

Professor Steven Smith: It's so nice to see you again on this gorgeous autumn day. And we had a wonderful, wonderful weekend, didn't we? Yes, we did. Okay, today, I want us to begin… we move ahead. We're moving ahead. Today we begin with Mr. John Locke. For the next three classes, Mr. Locke. It is hard to believe that a little book like this, in this not terribly distinguished edition, mind you, but nevertheless, in this edition of just over a hundred pages, that a book of this length could have such world shaping effects. If anyone would ever doubt the importance of ideas, political ideas, in history, I would only say to you to consult the history and the influence of John Locke. Remarkable. I want to talk today a little bit about Mr. Locke. John Locke is, for our purposes today, I mean, there are many reasons why one would read him in different kinds of classes, but for our purposes, John Locke gives the modern state, the expression that is most familiar to us. His writings seem to have been so completely absorbed and adopted by Thomas Jefferson when he wrote the Declaration of Independence that Locke seems to have become virtually a kind of honorary founding father, as it were, of America. Among other things, John Locke advocates the natural liberty and equality of human beings, our natural rights to such things as life, liberty, and what he calls "estate" or property, the idea that government, at least legitimate government, is government by consent, that legitimate government is necessarily limited and limited government constituted by a separation of powers, and that when governments become repressive or that when governments become abusive of natural rights, that the people have a right to revolution. In addition to this, John Locke was a famous advocate of religious toleration. His name is forever linked with our ideas today of what we might call liberal or constitutional democracy. He gives the modern constitutional state, again, its definitive and, in many ways, most familiar expression. Yet, Locke did not arise ex nihilo, nor did anyone. Locke's writings come from somewhere and from some source. They were prepared, in many ways, in part by Machiavelli, who had died approximately a century before Locke's birth. But more importantly, by another English writer, or by an English writer with whom we have spent some time, namely Mr. Hobbes, Thomas Hobbes. Hobbes took Machiavelli's idea of The Prince and, in effect, turned it into a theory or doctrine of sovereignty. The Hobbesian sovereign is at the basis of our ideas of impersonal, or what we might call representative, government. He transforms princely rule, Hobbes does, into an office called the sovereign. And this office is, for Hobbes, the creation of a social contract, or covenant, as he calls it, responsible to the agents or persons who have created the contract. Hobbes had taught that the sovereign is representative of the people who create his office in order to ensure peace, justice, and order. Without the power of the sovereign, we would find ourselves in a condition of nature, a state of nature, a term coined by Hobbes to indicate a world without civil authority or at least with only weak civil authority, unable to enforce common rules and laws. Hobbes gave voice to the doctrine of secular absolutism, one that invests the sovereign with absolute power to do whatever is necessary to ensure, again, the rule of law, justice, and political stability. And out of these rather harsh and formidable premises, Locke created a different, what we would think of as a more liberal constitutional theory of the state, while being still, in many ways, very dependent on the premises that Hobbes, again, modifying Machiavelli, had undertaken. Locke set out a process of domestication. He set out to tame or to domesticate Hobbes's fierce or harsh theory of absolute government, which had found few defenders in his own day. Locke's most important work of political theory, of political philosophy, is his Two Treatises of Civil Government, of which we are only reading the second. The book we have before us is often simply referred to as the Second Treatise, but you will probably have suspected, I think, that the Second Treatise was preceded by a first treatise. The First Treatise is much longer than the Second Treatise, and it was an elaborate and painstaking, one could almost say deconstruction, of the theory of the divine right of kings, which in his era, had received expression by a man named Robert Filmer, whose name appears, I think occasionally, in the Second Treatise. Filmer had written a book called Patriarcha, and the Patriarcha had argued that all political authority derives from the grant of authority that God had given to Adam, and therefore, that all legitimate authority has divine right behind it. Locke's First Treatise is a very important, but also, I have to say, extremely tedious work, and you should be grateful that I am not assigning it to you. But it's a very interesting book, in its own right, of in many ways, biblical criticism and exposition. But it's only in the Second Treatise that Locke set out to set out his own positive theory of government, as it were. This book was written, we now believe, shortly before the famous Whig Revolution of 1688, and in it, Locke sets out his theory of parliamentary supremacy, rule of law, and constitutional government. To put it maybe slightly oddly, Locke was in his day, to some degree, what Aristotle was to his. The Second Treatise is intended as a practical book. It was a book addressed not so much to the philosophers of his age, but to Englishmen, written to them in the everyday language of their time. He wrote to capture, in a way, the common sense of his time, although this is not to say Locke was not, at the same time, a deeply controversial figure. Locke had the ability, and it's a very desirable ability, to take, in many ways, radical or even revolutionary ideas and express them in a kind of language that makes people believe that this is what they had thought all along. And that is, to some degree, the genius of the Second Treatise. In many ways, that is easier for us, because Locke's language has become, for us, the kind of I would almost say common sense, or shorthand language, for the way we think about politics. And it was, again, a mark of his genius to be able to create that language and give it a stamp that seemed to make people believe that this is what they had simply been thinking all along. Locke was himself a deeply political man, but he was also, at the same time, as I've just been hinting, perhaps, a very reticent one. He lived in a period of intense religious and political conflict. He was just a boy in school when a king, Charles I, was executed and he was an adult when another king, James II, was overthrown and forced into exile. He was a younger contemporary of Hobbes, but he lived in a period of immense civil conflict and war. Locke spent many years at Oxford, where he was both a student and a fellow, and he was suspected, throughout much of his time there, of harboring radical political sympathies, but he was so cautious and careful in expressing them that after many years, even those closest to him were unclear as to what his opinions were. The master of Locke's own residential college at Oxford, Balliol College, described Mr. Locke as the "master of taciturnity," a master of taciturnity because he could not discover, through questioning and so on, Locke's opinions on religious and political matters. Just think of it. There used to be a very wonderful bust of Locke in the lobby of the British Arts Centre and I used to recommend to students, when they were down in that part of campus, to stop in and look at his face, because as with Machiavelli and others, the face is very revealing. And I used to ask people to see do you detect in here the sense of the master of taciturnity that his college master had discussed? Locke was a private secretary and a physician to a man named Anthony Ashley Cooper, later known as Lord Shaftesbury. Shaftesbury had a circle, the Shaftesbury Circle, of political followers who were opponents of the monarchy and who were forced into exile in 1683. Locke followed them into exile. He spent several years in Holland in 1683 before returning to England, again, shortly before the Whig Revolution, where his book, the Second Treatise, was published and where he lived until his death in 1704. Just two years ago, in fact, Yale, at the Beinecke Library, held a major conference in commemoration of the 300th anniversary of the death of Mr. Locke. So those are a few things about his contributions and his context. I want to begin today the substantive part of this talk by focusing on the theme that, in many ways, forms the central core of Locke's political doctrine, his Theory of Natural Law. This is a term that has come up from time to time. There is no modern thinker that I'm aware of who makes natural law as important to his doctrine as does Locke. The best way to observe the working, or to reconstruct the working, of natural law is to follow a procedure that we have seen before; to think about what is the condition of nature, the state of nature, where we can see the natural law in its operative form. The state of nature, for Locke, in many ways, as for Hobbes, is not a condition of ruling and being ruled, as it is for Aristotle. The state of nature is not a political condition. Locke describes the state of nature as a condition of perfect freedom. While Aristotle said that we were, by nature, members of a family, a polis, a moral community of some kind, bound by ties of civic or family obligation, Locke understands, by the state of nature, a condition without civil authority or civil obligations. The state of nature is not, for him, an historical condition, although he does occasionally refer to the vast tracts of North America as suggesting a condition of nature, but the state of nature is a kind of thought experiment. What does human nature like in the absence of authority? The state of nature, Locke suggests to us, is not an amoral condition, as it was for Hobbes. It is not simply a condition of war, of all against all. The state of nature, he tells us, is in fact a moral condition. It is governed by a moral law, or a natural law, that dictates peace and sociability. There is a moral law of nature that determines that no one should harm another person in their life, liberty, or possessions. This natural law, Locke affirms, "willeth the peace and preservation of all mankind." So the natural condition, for Locke, is a moral state, one in which a natural law, again, dictates the peace and preservation. It is not a war of all against all. Locke's natural law, in some ways, seems like a very traditional form of moral law, familiar to readers of his time; readers who would have been familiar with the natural law tradition, going back to Cicero, the Roman Stoics, St. Thomas Aquinas, and in Locke's own day, an important Anglican divine by the name of Richard Hooker. Locke's theory of moral law, or natural law, sounds comforting and traditional, and to some degree, it is. All civil authority has its foundation in a law of reason that is knowable, by virtue of our rational capacities alone. The law of nature declares, according to Locke, that we are, in his famous term, the "workmanship of one omnipotent and infinitely wise maker." And as products of divine workmanship, we ought never to harm anyone in their lives, liberties, or possessions. Locke, again, seems to effortlessly weave together the Stoic tradition of natural law with these Christian ideas of divine workmanship into one seamless whole. You can see the way in which Locke's rhetoric here, in his writing, brings together different strands of the philosophical and theological tradition, weaving them together in a kind of effortless whole almost. Do not be simply seduced by this. Why do I say that? Because even within the same paragraphs, Locke's natural law, the law that, again, mandates or dictates "peace and preservation of all mankind," turns into a right of self-preservation. From the beginning, you have to say, it is not altogether clear even whether the natural law is a theory of moral duty, duties that we have to preserve other's duties and obligations, or whether it is a theory of natural rights that mandates that the highest priority be given to individual self-preservation and whatever is necessary to achieve the preservation of the individual. The state of nature is a condition without civil authority. The law of nature, in other words, has no person or office to oversee its enforcement or its application. So this state of nature that he once describes, or early describes in the book is a condition of peace and mutual distrust, quickly degenerates into a condition of civil war, or of war, where every individual serves as the judge, jury, and executioner of the natural law. The state of nature quickly becomes a Hobbesian condition of essentially every man for himself. Consider the following passage in section 11 of the Second Treatise. "The damnified person," Locke writes--someone who has been mistreated in the condition of nature--the "damnified person," who has been injured or mistreated, "has this power of appropriating to himself the goods or services of the offender by the right of self-preservation, as every man has a power to punish the crime to prevent it being committed again, by the right he has of preserving all mankind, and doing all reasonable things he can in order to that end." In other words, if you have been wronged, or feel you have been wronged, in the state of nature, you have, according to the natural law, for Locke, the right to, as he puts it, appropriate to yourself the goods or services of the offender. And you have that--to take from them their goods, their property, their services in some way, whatever you feel appropriate as, again, the person who has suffered some kind of wrong. Every person becomes, as it were, judge and executioner of the law of nature. The fundamental law of nature, Locke says here, is the right of self-preservation. And this states that each person is empowered to do, again, whatever is in his power, to preserve him or herself. Again, consider the following in section 16: "And one may destroy a man who makes war upon him." May destroy another who makes war upon you. "Or has discovered an enmity to his being, for the same reason that he may kill a wolf or a lion," because such men "are not under the ties of the common law of reason." They "have no other rule but that of force and violence," so also, they may be treated as beasts of prey, "those dangerous or noxious creatures that will be sure to destroy him whenever he falls into their power." Listen to that language. From an original moral condition, where we are under a natural law not to harm others, a law to preserve and protect the well-being of others of our kind, we have become like lions and wolves to each other, beasts of prey and other noxious creatures. What is the state of nature, but, in the words of Dorothy Gale, "lions, tigers, and bears, oh my!" This is what we are to one another. This is what I've come to think of as Locke's bestiary, and in fact, the Second Treatise is rife with language of comparing human beings and our behavior to animals. He speaks about lions and wolves. Elsewhere he speaks about polecats and skunks and foxes. If, in fact, we are all beings, as he says, created under a natural law, we seem to quickly degenerate into almost bestial behavior. Beasts of prey, far from being cooperative and peace seeking creatures. The very freedom that such beings as ourselves enjoy in a state of nature leads us to abuse that freedom and, in turn, requires or is at the basis of the need for civil government. However, in the meantime, the question that any reader of the Second Treatise has to ask of themselves--and I hope you've put this forward in your sections to one another--is whether the natural condition, as Locke understands it, is one overseen by a moral law of justifying or sanctifying peace and security, or whether Locke's state of nature is simply a thinly veiled description, a thinly papered-over description, of the Hobbesian war of all against all. Was Locke simply Hobbes, in some way, in sheep's clothing? Remember his famous taciturnity. Locke seems to be speaking two very different languages, in other words, one of traditional natural law that holds out duties to others as primary and the other, in some ways, a modern Hobbesian conception of natural rights that maintains the priority of right and each individual's right to self-preservation. Is Locke, in other words--and this is perhaps more of an historical than a theoretical question--is Locke a member of the ancient, in some ways, Ciceronian and Thomistic tradition of natural law or a modern Hobbesian? Do his politics derive from a theological conception of divine workmanship or an ultimately, you might say, naturalistic conception or account of the human passions and the struggle for survival? Do his priorities go to duties or to rights? Or is Locke simply confused? Is he confusing two different languages or is he being intentionally ambiguous in his account? A recent book, by a well-known scholar of Locke has argued, I think quite powerfully in some ways, that Locke's idea of equality in the state of nature specifically relies upon a certain kind of Christian theological context of argument. Locke's statement in paragraph four of the Second Treatise, his statement that "there being nothing more evident than that creatures of the same species and rank, promiscuously born to all the same advantages of Nature, should also be equal to one another." That Locke's statement that creatures of the same species and rank should be equal to one another, this is said to rely upon and depend upon a very specific religious argument. What it means to belong to a species and why belonging to the same species confers a special rank or dignity on each of its members only makes sense, according to this recent interpretation, if you believe or if it is believed that the species in question has a specifically moral relation to God. The question, I think, is whether Locke's idea of equality in the state of nature, or his idea of the moral law in the state of nature, relies upon this belief, or whether it can be inferred from such things as the basic principles of freedom, whether this can be inferred, as it were, from purely non theological, naturalistic premises or grounds. Locke, to be short, is silent in the Second Treatise about the theological foundations of his position. There are no discussions of important theological figures, such as Jesus or St. Paul or the New Testament, at least in the Second Treatise; he discusses these issues at length elsewhere. These may be, in some way, thought of as background considerations, but the question remains, I think, for us whether these are deeply embedded in Locke's arguments about divine workmanship or whether or not that language of divine workmanship simply serves as a kind of window dressing, again, for a purely secular naturalistic theory of human nature and political authority. Very important issue, I think, in coming to understand Locke and indirectly, very important for how we come to think of the American regime because--I'll just say, simply as a kind of footnote to what I've been saying--if Locke is thought of in some ways, as his doctrine as being, in some ways, at the founding principles of the American regime, the Declaration of Independence most notably, it becomes very important. It becomes part of a contemporary public argument whether those foundations owe their authority to some kind of theological doctrine, as Jefferson calls it in the opening of the Declaration, "the Laws of Nature and Nature's God," seems in some way to have Lockean overtones to it. Do our founding documents imply a theology of some kind, "the Laws of Nature and Nature's God," or are those principles, again, purely of a naturalistic secular kind that can do without theology altogether? That is an argument, a kind of scholarly and academic argument, to be sure, but it spills over into many of our public debates over the role or place of religion in public life, whenever we talk about issues of the appropriateness of issues like school prayer or should the Ten Commandments be publicly displayed in courthouses or in other public places? Or if you want to take another famous Jeffersonian position, is there a kind of absolute firewall, a wall of separation, between religion and the state? These issues that we very much work on today and think about are, you can see, deeply embedded in how we think about Locke and those opening sections of the Second Treatise dealing with natural law and the state of nature. So you can see, again, how these ideas penetrate deeply into the marrow of our public or political culture. Are we so different? Have we become so different? Will people living three-hundred years from now think of us as so different, and our public debates so different, from those that animated the public issues in the time of John Locke? Maybe not. Maybe we aren't that different. So enough for contemporary. Let me go back to Mr. Locke. The core of Locke's theory of natural law in the state of nature is arguably lodged in his account of property, chapter 5 of the Second Treatise. If you remember anything about Locke after this class, remember chapter 5. It is, by all accounts--maybe chapter 19 as well, "The Theory of Revolution," but chapter 5, account of property; certainly, in many ways, one of the most characteristic doctrines of Lockean political thought. Locke's view of human nature is that we are very much the property-acquiring animal. Aristotle had said we were political by nature; Locke says we are property-acquiring beings. Our claims to property derive from our work. The fact that we have expended our labor, our work, on something gives us a title to it. Labor confers value and is the source of all values. The state of nature is a condition, he tells us, of communal ownership, what Karl Marx would have called "primitive communism." The state of nature is given to all men in common, Locke says. Parts of it become private property only when we add our labor to something. Let me read a famous formula from sections 27 and 28. "Every man," Locke says, "has property in his person: this no body has any right to but himself." We all, in other words, come into the world with a certain private property, property in our person. No one else has a right to that. "The labour of his body," Locke continues, "and the work of his hands, we may say, are properly his for labour being the unquestionable property of the labourer, no man but he can have a right to what is once joined to, at least where there is enough, as good left in common for others." "That labour," Locke says, "puts a distinction between him and the common: that added something to them more than nature, the common mother of all, had done; and so they become his private right." So we have moved here, in this one paragraph, from the state of nature, which he says is common to all, to a condition of rudimentary private property, which we have in our body, our person, which he says also includes the labor of the body and the work of the hands, how we expend our activity. That labor, he says, which puts something between us and the common, becomes the source of ownership of things around us, and that ownership then, in turn, becomes a right. So they become, he concludes there, his private right, the source of a right to property. The natural law, as Locke seems to be saying, dictates a right to private property and it is to secure that right that governments are ultimately established. In a striking formulation, Locke tells us that the world was created in order to be cultivated and improved. Those who work to improve and develop nature, who add to nature through the labor of their body and the work of their hands, those who develop and improve nature are the true benefactors of humanity, of humankind. "God gave the world to men in common," he says, section 34, "God gave the world to men in common, but since He gave it to them for their benefit and the greatest conveniences of life that they were capable to draw from it," he writes; the world was given for our convenience, he says, to be drawn from, "it cannot be supposed He meant it should always remain common and uncultivated." And then he adds, "He gave it to the use of the industrious and the rational and not to the fancy or covetousness of the quarrelsome and contentious." God gave the world for our improvement of it and therefore, He gave it to the industrious and the rational. Locke seems to suggest in that very phrase that the state will be a commercial state, that the Lockean republic or the Lockean state will be a commercial republic. Think of that. Ancient political theory, Plato, Aristotle, regarded commerce, regarded property, as in many ways subordinate to the life of a citizen. Plato would have instituted a kind of communism of property among the guardians of his Kalipolis. Aristotle thought of the necessity of private property, but simply as a means to allow a few of those citizens, to engage in political life. Economy, you might say, was always subordinate to the polity. Locke turns this ancient and medieval doctrine, as well, on its head in many ways. The world belongs to the industrious and the rational, those who, through their own efforts, through their labor and work, increase and enhance the plenty of all. It is only a relatively short step from John Locke to Adam Smith, in that respect, the great author of The Wealth of Nations, again, just under a century after Locke's Second Treatise. For Locke--and let me just go on a little more about this--there are no natural limits to property acquisition. And this is, in a way, the essential point. The introduction of money or coinage into the state of nature, an issue I'm not going to talk much about here, but that becomes an important moment in his chapter 5 in his account of the state of nature, the introduction of money makes unlimited capital accumulation not only possible, but even a kind of moral duty. It becomes our duty to enhance and work upon the raw materials of the natural world around us. By enriching ourselves, we unintentionally work for the benefit of others. Consider the following remarkable sentence: "A king of a large and fruitful territory in America," he says, "feeds, lodges, and is clad worse than a day labourer in England." Because, of course, our work, Locke thinks, has enhanced the plenty of all in some way. The creation of a general plenty, the common wealth--and think of the way in which the revealing use of that term "common wealth," the wealth of all--is due, in many ways, to the emancipation of labor from the previous kinds of moral and political restrictions imposed upon it by the ancient philosophical, as well as religious, traditions. Labor becomes, for Locke, his source of all value and our title to common ownership and in a remarkable rhetorical series of shifts, he makes not nature, but rather human labor and acquisition the source of property and of unlimited material possessions. He begins this chapter, chapter 5, with the assertion, think about it, that "God hath given the world to men in common," once again suggesting that the original state is one of common ownership. He then suggests that every person is the owner of their own bodies and that one acquires a title to things through labor that we have mixed with that common world. But what starts as a very, very modest title to the objects that we have worked on, his example is something as simple as picking apples from a tree, the act of picking gives us a title to the apple, that very simple or rudimentary form of property soon turns into a full scale explanation of the rise of property and a kind of market economy in the state of nature. "Labor accounts," he tells us, "for ten times the amount of value that is provided by nature alone," he says at section 37. Our labor enhances the value of nature ten times. But he then goes on to add very quickly, "I have here rated the improved land very low in making its product but as one to ten when it is much nearer a hundred to one." Our labor advances things a hundred-fold. Shortly later, in section 43, he says that the value of anything is improved a thousand-fold due to labor. Again, what began as a fairly rudimentary discussion of the origins of private property at the beginning of chapter 5, limited by the extent of our use and spoilage, has, by the end of that same chapter, you might say, morphed into an account of large scale ownership with considerable inequalities of wealth and possession. By the end of chapter 5, there appears to be almost a direct link between Locke's dynamic theory of property in chapter 5 and James Madison's famous statement in Federalist No. 10. As Madison says, "the protection of different and unequal faculties of acquiring property is the first object of Government." Seems a very Lockean proposition in The Federalist. Locke gives, in other words, to commerce, to money-making, to acquisitiveness, a kind of pride of place and a sort of moral status, you might even call it, that it never enjoyed in the ancient and medieval worlds. The new politics of the Lockean state will no longer be concerned with glory, honor, thumos, virtue, but Lockean politics will be sober, will be pedestrian, it will be hedonistic, without sublimity or joy. Locke is the author of the doctrine that commerce softens manners, that it makes us less warlike, that it makes us civilized, something that reaches its, you might say, highest expression in the twentieth chapter or the twentieth book of Montesquieu's Spirit of the Laws. Commerce does not require us, for Locke, to spill blood or risk life. It is solid, reliable, thoroughly middle class in some ways. Locke is, again, the great author of the idea that the task of government is to protect not just the rights of property, but the right to acquire and build upon the property that we already own. So I want to end on this note and begin on Wednesday talking a little bit about what we might call John Locke and the spirit of capitalism.

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Professor Steven Smith: Last time, I believe I said I wanted to discuss three features that Tocqueville regarded as central to American democracy. That is not to say they were central to the democratic experience, but they are central features of the American democratic experience and to what degree these can be or could possibly be translated to other contexts in other emerging democracies remains very much an open question. But of these three features, the first I talked a little bit about on Monday is the importance of local government, the township as it's translated in this edition, what Tocqueville calls the "commune," the community, community spirit, local government. In some way, connected to what he calls later in the book "the spirit of the city," using "the city" here in the context of the ancient sense of polis, l'esprit de cité, a kind of polis-like character in these small New England townships, very important, Tocqueville believes, for the sustaining a democratic country and a democratic society. But the second, and probably the aspect of Tocqueville's account of democratic America that has received the most attention at least recently, is the aspect of what he calls throughout the book "civil association," civic association. It is what one might think of as intermediary groups, voluntary groups, civic organizations of all kinds that Tocqueville is immensely impressed with and which he turns into one of the central pillars of the democratic experience. He writes that, "in democratic countries," one of the most famous sentences from the book, "In democratic countries, the science of association," he says, "is the mother science. The progress of all the others depends on the progress of that one." And it is through uniting and joining together in common endeavors, he believes, that people develop a taste for liberty, a taste for freedom. "In America," if I can just quote him again, "In America, I encountered all sorts of associations, of which, I confess, I had no idea and I often admired the infinite art with which the inhabitants of the United States managed to fix a common goal to the efforts of many men to get them to advance it freely." Struck by the immense variety and multiplicity and sheer number of these various kinds of civic association. It is important to see, perhaps, this is one area in which Tocqueville seems to most clearly depart from Rousseau, at least the Rousseau of the Social Contract after having said last time that his account of local democracy, township democracy owes so much to Rousseau's account of the general will. But remember that Rousseau in the Social Contract, would inveigh against, warned against what he called "partial associations," partial associations like interest groups of various kinds that had the tendency to frustrate the general will, that stand, as it were, between the individual and the general will. But Tocqueville, on the other hand, regards these kinds of voluntary associations, associations of all sorts as precisely the place where we learn habits of initiative, cooperation and responsibility with others. By taking care of our own interests or the interests of our association, we learn to take care of the interests of others. "Sentiments and ideas renew themselves," Tocqueville writes. "The heart is enlarged and the human mind is developed." So you can see from a passage like that how much weight he puts on these civic associations. "The heart is enlarged. The mind is developed." It is through these associations, PTAs, churches, synagogues and other civil bodies and associations that institutions are formed that can both resist in its way the power of centralized authority, central government. But they are also, as it were, the locus, the seedbed where citizens learn to become democratic citizens. It is very much important for Tocqueville that these associations, the absence of which he felt very acutely in France, which had already become a highly centralized society. It was these intermediary, voluntary associations that stand between the individual and central authority, the authority of the national government, which is what makes them, of course, so important for him. This argument about the importance of civic association--I say it has become, in a way, the most talked about passage or part of the book in recent years--is due in large part to the influence of political scientist, Robert Putnam, a man who teaches at another university, a book called Bowling Alone. You've probably maybe heard of that. Here, Putnam speaks about what he calls "human capital," what Tocqueville, in less social scientific jargon, calls "habits of the heart," mores, habits of the mind and heart. But Putnam argues that it is this social capital that is developed through civic association and his chief example, as the title of the book and the article from which it draws suggest, is that the bowling league is a kind of model of civic association. Particularly, he is concerned with the decline of these associations in contemporary American life. Hence the title of the book, Bowling Alone. The fact that Tocqueville himself describes these civic associations as the product of art suggests that, that is to say, that they are not natural. They are not somehow the result of some kind of instinctual behavior on us. Joining with others in voluntary associations is a learned activity. It is something that requires a certain kind of culture and is a learned activity. It is something also, it is an art, it's a skill, it is a craft that can also be lost. His argument is that more and more people are, so to speak, choosing to "bowl alone," something that shows an alarming tendency towards isolation and the subsequent kind of depletion almost of our civic capacities. The question is, taking Tocqueville to the present, have our capacities for joining with others been eroded by the forces of modern politics and technology? Are, in fact, we becoming more and more a nation of solitaries and couch potatoes? These are some of the serious questions and there is a big literature that has grown up around it. Some of this literature finds Putnam's conclusions to be overdrawn, that he exaggerates the influence of these associations or the decline of these associations. In fact, our civic state is not as bad off as he suggests. But what I want to do, suggest today, and this is where we're going to show a film and Jude's going to help me, just a couple of clips, is that there is a serious question, I think, in my mind, whether bowling leagues are a proper model for a democratic association. Now, one can say, and using the title "Bowling Alone" that Putnam is just speaking metaphorically, that he doesn't mean bowling leagues. He's just using it as a metaphor. But let's take him at his word and let's find out if bowling leagues are, in fact, the ideal transmitter for democratic mores and values. I want to take an example from a movie of which I'm very fond by the Coen brothers called The Big Lebowski, which is a movie about a bowling league, or at least three gentlemen who take their bowling and their bowling league very seriously. The three of them are "The Dude," who is a stoned hippie, "Walter," who's kind of a whacked out Vietnam vet and "Donny," who's a lost waif. They are very, very concerned with getting into the finals, into the bowling tournament. In their way stands a man named Jesus Quintana who happens also to be a sex offender. I want to show a couple of clips from this movie and I should warn you that there is some very bad language being used here. So if you think that is going to be offensive to you, you should leave. It won't take more than about four minutes or so. We're going to show a couple of clips about the ethos of men bowling. Professor Steven Smith: One more. Professor Steven Smith: Obviously, it goes to show that civic association alone is not enough to create democratic citizens. Again, otherwise, "Smokey" and "The Dude" and "Walter" would be a perfect example of democratic citizens. Tocqueville focuses on a third, another leg of the stool of democratic life and that is what he calls the "spirit of religion." Central, again, as the third and maybe a very important prop of the American democratic experience. "On my arrival in the United States," he observes, "It was the religious aspect of the country that struck my eye first." Very impressed with that. Like other European visitors to the United States, both then as well as now, Tocqueville was deeply struck with how democracy and religion seem to walk hand-in-hand with each other, precisely the opposite of what has occurred in Europe where religion and democracy or religion and equality were long on a collision course. What made the American encounter with democratic life unique? That is one of Tocqueville's big questions. In the first instance, you could say, or as Tocqueville notes that America is primarily a puritan democracy. "I see the whole destiny of America contained in the first puritan who landed on its shores," he says, "like the whole human race in the first man." Our experience was determined in crucial ways by early Puritanism. America was created by people with strong religious beliefs and habits who brought to the New World a suspicion of government and a strong desire for independence. This has been the foundation of the separation of church and state that has done so much both for religious and political liberty. Tocqueville drew from this two very important consequences, I think, about religious life in America. The first is that the thesis propounded by the great philosophers of the Enlightenment of the eighteenth century and still advanced in many, you might say, enlightened quarters today, that religion will disappear with the advance of modernity. As modernity advances, religious life will disappear. I suppose in the twentieth century, Max Weber gave voice most prominently to that point of view that would be a process of secularization within modernity and a sort of gradual withering away of religious belief. Tocqueville shows that to be demonstrably false, that religion will not simply disappear as modernity moves forward and that the Enlightenment and its contemporary heirs, theorists of development and modernization and so on have been all together wrong about their confident predictions about the decline and withering away of religious faith. Secondly, Tocqueville takes it to be a terrible mistake to try to eliminate religion or to secularize society all together. This is, in fact, probably a more controversial, a very controversial claim. It was his belief, and again, perhaps here he's influenced by Rousseau in the chapter on civil religion at the end of the Social Contract that free societies rest on public morality and that morality cannot be effective without religion. It may be true that individuals can derive moral guidance from reason alone, but societies can't. The danger of attempting to eliminate religion from public life is that the need or desire to believe will therefore be transferred to other and far more dangerous outlooks. "Despotism," he says, "can do without faith, but freedom cannot." A very arresting sentence. "Despotism can do without faith, but freedom cannot." "Religion is more necessary in a republic and in a democratic country than any other," he says. But why is religion necessary to a republic? Why does democracy require religion? Here, Tocqueville gives a variety of answers. One persistent theme running throughout his book as a whole is that only religion can resist the tendency toward materialism and a kind of low self-interest that he believes is intrinsic to democratic ages and societies. "The principal business of religion," he frequently writes, "is to purify, is to regulate, is to restrain the kind of ardent desire for well-being and particularly, material well-being that becomes particularly prominent during ages of equality." That's one reason. But secondly or in addition, Tocqueville operates, I find, with a very interesting, I might even call it a metaphysic of faith that regards religious belief as a necessary component for human action. "When religion is destroyed in a people," this is Tocqueville. "When religion is destroyed in a people, doubt takes hold of the higher portion of the intellect and half paralyzes all the others." When religion is destroyed, doubt takes over. It has a kind of a paralyzing effect on the will and our capacity for action. This paralysis of the will, this inability to act is a condition that later writers would choose to call "nihilism." Faith is a necessary component for our belief that we are free agents and not simply the play-thing of blind forces and random causes, so to speak. Our beliefs about freedom and the dignity of the individual are inseparable for him from religious faith and it is unlikely that these beliefs about the dignity of the individual can survive without religion. Just to take a contemporary example of that, think about the debates we have had over such things as cloning and the sense that many people have that the dignity of the individual, which is often connected with a kind of religious belief, sanctity of life, the dignity of the individual is somehow deeply violated by these advances of sort of scientific technology. Religion remains a crucial prop for our beliefs about human dignity. No more powerful challenge to the Enlightenment's faith in science and scientific progress can be found than in Tocqueville. One final issues remains, I would say. Tocqueville often writes, and I would say this is the dominant tone of his writing on religion. He often writes as if religion is only valuable or valuable primarily for the social function it serves. This is certainly consistent with lots of things he says about religion. He's only concerned about religion for its social and political consequences rather than from the deeper truths of religious belief. "I view religion," he says, "only from a purely human point of view," he says. He's only looking for its affect on society. But I would ask, how accurate is that statement, or does it describe or characterize all of Tocqueville's views about religion? I think not. Let me just say why for a minute. I think that sort of sociological or functionalist reading of religion, that he's interested in it only for its social affect, is only part of Tocqueville's very complex attitude towards this subject. Maybe you'll have a chance to talk about this in your section. Maybe you'll have an opportunity to write about it at some other time. But remember that Tocqueville was not only a student of Rousseau. As he said in that letter to Louis de Kergolay that I mentioned last time, his other two great sources of inspiration were Montesquieu and a seventeenth-century French philosopher named Blaise Pascal. Pascal was a religious philosopher, who more than any other, emphasized the emptiness of knowledge without faith. Man may be the rational animal, but reason is somehow unable to plumb or reason is unable to grasp the unfathomable depths of the universe. In one of his most famous statements, Pascal said, "A vapor," a drop of water is enough to kill him, speaking of us, humans. "A drop of water is enough to kill us. Man is a reed, a reed, the weakest in nature, but he is a thinking reed." We are weak. We think, but it is our weakness. It is our dependence, sense of dependence that struck Pascal. Tocqueville, you can find this in several passages throughout the Democracy. Tocqueville, I think discovered in Pascal a sense of kind of existential emptiness, an incompleteness of life that cannot be explained in terms of reason alone. There is also, he felt, something deeply hubristic about the way in which conditions of equality foster this idea of rational self-sufficiency. Tocqueville's purpose, in many ways, was to limit reason to make room for faith, and this is one of my favorite passages. Let me just read a sentence or two. "The short space of 60 years," he writes, almost as an aside. "The short space of 60 years will never confine the whole imagination of man. The incomplete joys of this world will never suffice for his heart." Incomplete joys of this world will never suffice for his heart. In other words, there is something we desire beyond the here and now that only faith and can supply. The soul exhibits a kind of longing, a desire for eternity and a kind of disgust with the world and the limits of physical existence. "Religion," he goes on, "is only a particular form of hope and it as natural to the human heart as hope itself. Only by a kind of aberration of the intellect and with the aid of a sort of moral violence exercised on their own nature do men stray from religious belief. An invincible inclination leads them back to religion. Disbelief is an accident. Faith alone is the permanent state of humanity." If anyone's interested, that's on page 284. But no one can possibly read that section and come away from Tocqueville by thinking he had only a kind of functionalist, sociological view of religion, concerned with its effects on human behavior and society. Disbelief is an accident. Faith is the permanent condition of humanity and only through a kind of moral violence, through moral violence can religious faith be eliminated. I think these passages show a much deeper, almost metaphysical dimension to Tocqueville's thought. It shows him to be, like Plato in many ways, of enormous psychological depth and subtlety and insight. But these are the three features, or three of the features, I think the three central features that remain for him crucial to democracy: local government, civil association and what he calls the spirit of religion. Yet, obviously, all is not well. All is far from being well. Too often, way too often we read Democracy in America as it if were simply a celebration of the democratic experience in America. It is not. Tocqueville, among other things, is deeply worried about the potential, I mentioned briefly about this last time, the potential of a democratic tyranny. Why is there a belief or why would one believe that the democratic government alone will eliminate various forms of arbitrary rule in tyrannical government? In fact, it might create new forms of tyranny, democratic tyranny of which previous societies had been, perhaps, unaware. This is an issue that he treats twice in two important parts in his work; one in Volume 1, the other in Volume 2. I'm going to talk for a little bit today about his account of tyranny of the majority in Volume 1 and I'm going to save the rest of the discussion for next week when he talks about what he calls "democratic despotism" in the second part of Democracy in America. In Volume 1, he treats what he calls the "tyranny of the majority" largely in terms, you might say, that are derived or inherited from Aristotle and even the authors of The Federalist Papers. As you remember in Aristotle's Politics, Aristotle associated democracy with the rule of the many. "Rule of the many," for all kinds of purposes, generally means rule of the poor and rule of the poor for their own interest. The danger with democracy for Aristotle was that it still represented the tyranny of one class of society over the society of a whole, the largest class ruling in its interests over the minority. Democracy for the ancients was always a form of class struggle between the rich and the poor. That was, in many respects, the way in which democracy came to be viewed even by The Federalist's authors who came up with their own solution to the problem of democracy or what they called "republican government." The problem of republican government was this problem of, you might say, majority faction and their answer to the problem of majority faction was in Madison's term, "to enlarge the orbit of government," to make societies and polities much larger in order not to try to eliminate faction, but to increase them. By increasing the number of factions, you decrease the possibility that any one of them will be able to represent or exercise a kind of permanent majority control, a kind of permanent tyranny of the majority. The greater the number of factions, the less likelihood that any one of them will be able to exercise despotic power over national politics. This is a question that Tocqueville returns to or turns to in that very important chapter from book one called "The omnipotence of the majority in the United States and its Effects," which is, in many respects, a response or provides his reading and critique of the classical or traditional theory of democratic tyranny. The U.S. constitution, he talks about, has enshrined the majority in its own Preamble--"We, the People." It has enshrined the majority even as it has sought to limit the powers of the people. Although Tocqueville devotes a great deal of attention in Volume 1-- we're not really reading these sections, I don't think they're all that important for our purposes--he spends a great deal of attention simply sort of describing the makeup of the federal constitution, the structure of the Houses of government and so on. One has to say he is far less impressed than Madison or the Federalist authors were, that the problem of majority faction has been solved in America. Again, the Federalist authors, following Locke and Montesquieu, believed all that was necessary was separation of powers, a system of representation, a system of checks and balances, that this could serve as an effective check on majority rule. But Tocqueville was less certain of that. He was less certain that these, as it were, institutional devices alone could check what he calls the "empire of the majority." The empire of the majority, a term that he uses that clearly has kind of theological connotations, denoting a kind of divine omnipotence, that the people have come to be the ultimate or final authority. Rather than regarding, as it were, the people in Madisonian terms simply as a kind of ongoing shifting coalition of interests, Tocqueville regarded the majority in democratic societies, the power of the majority, as unlimited and unstoppable. Legal guarantees of minority rights, he thought, were unlikely to be ineffective in the face of mobilized public opinion. Why does Tocqueville believe that, or what led him to express such skepticism about even American democracy's inability to check the prospect of democratic tyranny? In part, I think, Tocqueville's answer was that majority tyranny was inseparable from the threats of revolutionary violence and particularly charismatic demagogues and military leaders like Napoleon in France and America's counterpart to Napoleon, Andrew Jackson. Napoleon was in France, the man capable of mobilizing the masses into fits of patriotic zeal and to carry on war. Jacksonianism, for him, simply looked like an American form of Bonapartism, a military commander riding to political power on the wings of popular support. More than anything else, Tocqueville feared militarism combined with a kind of unlimited patriotic fervor. It is in these respects you can begin to see some of the less ennobling features of the democratic experience and the more ominous possibilities of democratic rule. The power of the majority, he says, makes itself feared especially through the dominance of the legislature. He believed, we could talk about whether this belief is still valid or true, he believed that the most, again, that democracy tends towards a dominance of the legislatures where the people's voice makes its will most clearly known. By having short elections or short cycles every two years in the House of Representatives, it was a way of making sure that the legislatures, the House, the Houses, are very close to public opinion and public control. He sees this as a dangerous thing, this kind of legislative dominance that he sees is one form in the way in which the tyranny of the majority expresses itself. But the most important and the most memorable aspects of tyranny of the majority have less to do with these institutional forms, you might, say. It has to do with the way in which the empire--again, I'll use his term-- the empire of the majority makes itself felt in the realm of thought and opinion, the influence of the majority over thought. In an always startling passage from the book, Tocqueville remarks, "I know of no other country where, in general, less independence of mind and genuine freedom of discussion reign than in America." There's no country where there is less independence of mind and freedom of discussion. He is, I suspect, overstating the case, but his argument here is that the dangers to freedom of thought in a democracy do not come from the threat of an inquisition. They do not come from something like that, but they are exercised in more subtle forms of exclusion and ostracism. Tocqueville is, perhaps, in that passage, one of the first and most perceptive analysts of what today might be called the power of political correctness, to control and to eliminate certain kinds of ideas and opinions from being thought. It is the fear of ostracism, in some sense, the fear of being socially ostracized through which the majority exercises its control. Tocqueville's statement here is, of course, that persecution can take many forms under a democratic people, from the cruelest to the most mild. He gives various examples of the crueler forms of the way in which the majority have expressed itself. In a lengthy footnote to the book, for example, in some of these parts, he gives two examples; one in which during the War of 1812, he says there were some anti-war journalists in Baltimore--maybe you read that passage--who were taken out. Their newspaper press was burnt down and I think they were hung, he says. This is a way in which mob mentality took over. He also uses the example of the way in which black voters in the state of Pennsylvania, and he focuses on this particularly, have been disenfranchised. He mentions Pennsylvania in particular because Pennsylvania is a Quaker state, that is to say a state where one would have thought liberal opinion towards questions of racial justice would have been most advanced. Even there, he says, the majority constrained African American voters from, free blacks, from voting. So these are ways in which, again, some overt and cruel and persecutory, others milder and through the form of ostracism that he wants to say that democratic sovereignty can exercise itself. "Chains and executions are the coarse instruments," he writes, "that tyranny formerly employed. But in our day, civilization has perfected despotism itself, which seemed to have nothing more to learn." We have perfected despotism, he says. "Under the absolute government of one man, despotism struck crudely at the body so to reach the soul," no doubt thinking about the Inquisition and things like this in Spain and in parts of Catholic Europe. He writes, "and the soul escaping from those blows rose gloriously above it. But," he goes, "in democratic republics, tyranny does not proceed in this way. It leaves the body alone and goes directly for the soul." Well, there's a wealth of commentary you might think about when you read that passage that's implied there. Oh, God. The time's moving so quickly. There's so much more. So that, for Tocqueville, is one of the other sides of the democratic experience. Again, I want to return to a piece of that on Wednesday, next week rather, Monday, because I think you will see in Volume 2, Tocqueville has something of a change of heart. He doesn't become more optimistic. In fact, he becomes far more pessimistic about this. But there's certainly a change of tone in what Tocqueville has to say about the potentiality of majority tyranny. Well, we had so much fun watching the movie, I didn't get a chance--There's a little more I wanted to say, but this seems like a good note to break on. I'll try to finish whatever I can with Tocqueville on Monday and Wednesday I'm going to try to wrap things up and tell you what you should be thinking about. So anyway, enjoy yourselves and I'll see you next week.

Introduction_to_Political_Philosophy_with_Steven_B_Smith

20_Democracy_and_Participation_Rousseaus_Social_Contract_III.txt

Professor Steven Smith: There's so much to say and so little time. Today, I want to talk about the general will, Rousseau's most important contribution to political science and I will also want to talk about the legacies of Rousseau and what he's meant for the world that he did so much to shape. But I want to start first with the general will which is his answer to the problems of civilization or the political problem of the Second Discourse that we talked about last week, the problems of inequality, the problem of amour-propre, the problem of our general discontent. Social contract is his answer to the problem of natural freedom. This is so, in a way, because for Rousseau nature, he tells us, provides no standards or guidelines for determining who should rule. Unlike Aristotle, man is not here a political animal, and notice that when Rousseau speaks of the social contract in the general will as the foundation of all legitimate authority, he means, literally, that all standards of justice and right have their origins in the will in this unique human property of the will or free agency. It is this liberation of the will from all transcendent sources or standards, whether those be found in nature, in custom, in revelation, in any other source. The liberation of the will from all of these sources that is the true, as it were, center of gravity of Rousseau's philosophy. It is a world that begins to emphasize the primacy and the priority of the will, a moral point of view that I want to indicate a little later, finds its, in many ways, very powerful expression in the philosophy of Immanuel Kant. But given Rousseau's, let's call it libertarian conception of human nature, his description of the actual mechanism of the social contract may come as something of a surprise to us. The problem, to which the formula of the general will is the answer, is stated succinctly by Rousseau in Book I, chapter 6 of the Social Contract. "Find a form of association," he writes, "which defends and protects with all the common force, the person and goods of each associate and by means of which each one while uniting with all obeys only himself and remains as free as before." This, he calls, the fundamental problem for which the social contract is the solution. That statement, a famous statement, Book I, chapter 6, really contains two parts that merit close attention. The first part says--the first part of that clause says that the aim of the contract is to protect and defend with the common force the goods and person of each member. So far, think of that, this is entirely consistent with Locke's claim or even Hobbes' claim that the purpose of society is to protect the security or the life, liberty, and estate of each of its members. Yet, Rousseau adds to this Lockean or liberal clause you might say, a second and more distinctly Rousseauian claim, namely, that the contract must ensure not only the conditions for mutual protection and the preservation of self and property, but rather also that in uniting with one another, he says, each person obeys only himself and then he says, "remains as free as they were before." But how is this possible, we want to know. Isn't the essence of the social contract that we give up some part of our natural freedom to guarantee mutual peace and security? How can we remain as free as we were before, and as he says, obey only our--that the participant obey only himself. That is the paradox, in many ways, or the fundamental problem, as he calls it, to which his contract is a solution. Rousseau provides an answer as follows; he says, "Properly understood these clauses are all reducible to one. Namely," he says, "the total alienation of each associate together with all of his rights to the entire community." The total alienation of each associate with all of his rights to the entire community. And those two phrases, "total alienation" and "entire community" are obviously central here. In the first place, all persons must give themselves entirely over to the social contract to ensure that the terms of the agreement are equal for all. The total alienation clause as it were, is Rousseau's manner of ensuring that the terms of the contract are the same for everyone. But secondly, when we alienate ourselves, it is crucial, he says, that this be done or given to the entire community, for only then he wants to argue, is the individual beholden not to any private will or any private association, or to some other person but to the general will, the will of the entire community. The social contract is the foundation of the general will which is, for Rousseau, the only legitimate sovereign. Not kings, not parliaments, not representative assemblies, not presidents, but the general will of the entire community is the only general sovereign, the doctrine of the famous doctrine of what we call the sovereignty of the people or popular sovereignty. Since everyone combines to make up this will, when we give ourselves over to it entirely, he wants to argue, we do nothing more then obey ourselves. The sovereign, in other words, is not some distinct third party that is created by the contract, but rather the sovereign is simply the people as a whole acting in its collective capacity, the people in their collective capacity. Now, you might suggest that there is something deeply amiss here. That is to say, from a highly individualistic set of premises where each person is concerned only in the state of nature, or in the pre-contract tradition, only with the protection of their lives, persons and property, Rousseau seems to be leading us to a highly regimented and collectivized conclusion, where the individual has given over virtually his or her entire being to the will of the community. In what way does this render us as free as we were before? In what way do we remain free and obey only ourselves? That seems to be the problem. Is Rousseau's formula for the general will, a recipe or a formula for freedom, or is it a recipe for the tyranny of the majority of the type later analyzed by Tocqueville that we'll be seeing after the break? Rousseau wants to say, paradoxically, only through this total alienation do we remain free. Why is this? Why is this? Because he wants to argue no one is dependent upon the will of another. The people established through their act a new kind of sovereign, the general will, which he says is not strictly speaking the sum total, the additive total of the individual wills or the individual parts, but is more like the general interest or the rational will, if you want to use that kind of Kantian formulation, the rational will of a community. Since we all contribute to the shaping of this general will, when we obey its laws we do, he wants to say, no more than obey ourselves. Rousseau describes this new kind of freedom that we achieve under the general will. He wants to say that this brings about, in some ways, a radical transformation of human nature in itself. The freedom of the citizen under the general will is not the freedom of the state of nature, it's not the freedom to do anything we like, anything that our will and power allows us to do, but it is a new kind of freedom that he calls moral freedom, a freedom to do what the law commands. The passage from the state of nature, he writes, to the civil state, the passage from the state of nature to the civil state produces a remarkable change in man. For it substitutes justice for instinct in his behavior and gives his actions a moral quality that they previously lacked. And Rousseau continues that statement as follows. "What man loses through the social contract is his natural liberty and unmitigated right to everything that tempts him and he can acquire. What he gains is civil liberty and proprietary ownership of all he possesses, but--and here I think is the crucial argument or the crucial clause, but he writes--to the preceding acquisitions," that is to say civil liberty, "could be added the acquisition of moral liberty which alone," he says, "makes man truly the master of himself. For it to be driven by appetite alone is slavery and obedience to the law one has prescribed for oneself is freedom." That is a remarkable statement. "Obedience to the law that one prescribes for oneself is freedom." That is moral liberty, which is only created and possible through the social contract, and the implications of this, the moral and political implications of that statement are massive. It is here, in many ways, where Rousseau departs most powerfully, most dramatically from his early modern predecessors. Consider the following. For Hobbes and Locke, liberty meant that sphere of human conduct which is unregulated by the law. Remember chapter 21 of Leviathan, where Hobbes says, "where the law is silent" praetermitted in his term, "where the law is silent, the citizen is free to do what ever he or she chooses to do." Freedom begins, so to speak, where the law is silent. But for Rousseau, law is the very beginning of our freedom. Where the law is silent, we may have a kind of natural freedom, but our moral freedom, we are free to the extent that we are participants in the laws that we in turn obey. Freedom means acting in conformity with self-imposed law. A radically different understanding of what freedom consists and it seems underlying the difference between, one could say Hobbes and Locke on the one side, and Rousseau on the other; it's a difference between two very different conceptions of liberty. One might call them liberal and republican respectively, small "r" republican of course or democratic even, if you like. For liberals, following in the tradition of Hobbes and Locke, again, freedom has always meant a sphere of privacy where the law does not intrude or where other people do not intrude. This is why the separation of the public and the private sphere has always been so sacred to liberals, because only in the private sphere, only in that area of civil society where the state does not intrude is the individual really and truly free. But for the republican theory of liberty of which Rousseau is a most powerful modern exponent, this separation of public and private is only an exercise in what might be thought of as private selfishness. The task is rather to create a community where the individual and the public interest are not in conflict with one another, where the individual does not think of him or herself as a being apart from the social body. This is the freedom of the citizen, for Rousseau, who takes an active role in the determination of the laws of one's own community. Rousseau's purpose in saying this and in writing this seems to be to bring back to life a concept that he believes has been dormant, had laid dormant for centuries and that concept is the citizen. The last people who really knew what a citizen meant, he says, were the Romans. In a footnote, again to Book I, chapter 6, he indicates to what degree the true meaning of citizen has been lost on modern subjects. "Most modern men," he writes, "mistake a town for a city, and a bourgeois for a citizen." Think of that. Most mistake a bourgeois for a citizen. The modern world furnishes almost no examples of what a citizen is, and this is why it is necessary for Rousseau to return to the histories of antiquity, especially Rome and Sparta to find models of citizenship. Only in these societies can one find the spirit of self-sacrifice and devotion to the common good, a kind of patriotic devotion upon which citizenship is founded. If I could take perhaps Rousseau's most memorable example of the true citizen it comes from an example he lifts from the Roman writer, Plutarch that he uses in the opening pages of his book, The Émile, which I hope you will have a chance to read at some other time. Here, he tells an unforgettable story for anybody who ever reads Émile. "A Spartan woman," he writes, "had five sons in the army and was awaiting news of the battle," had five sons in the army and was awaiting news of the battle. "A helot, slave arrives trembling she asks him for news. ‘Your five sons were killed,' the helot replies. ‘Base slave, did I ask you this?' ‘We won the victory,' he says. The mother runs to the temple and gives thanks to the gods." Here, for Rousseau, was the ancient citizen. An example that is both terrible and sublime, which of course he wants it to be, he intends it to be. There is the example of what the true citizen is. The question, when you consider this possibility, is whether Rousseau's idea of the freedom of the citizen, freedom to live under self-imposed law, leads to a higher form of nobility, higher than the kind of low minded pursuit of one's self-interest as Rousseau wants. He wants to dignify politics again by leading to a higher form of nobility or does it result in a new kind of despotism, the despotism of law, the despotism of obedience to the general will and of course underlying that sinister reading of Rousseau is the famous or maybe infamous statement that not only that the general will is the source of freedom, but that anyone who obeys, who refuses to obey, the general will may be in his famous formulation, may be forced to be free. That anyone who disobeys it and being chastised or can be, as it were, forced to be free. Recall that this is a conception of freedom which, again, is almost the opposite of that of what we might again call the liberal tradition. A view, which, and again in a slightly paradoxical way, was given, a very powerful formulation by Hobbes. I want to read a passage that I read a couple of weeks ago from Hobbes which I think stands as a striking contrast to that of Rousseau's. Again, in chapter 21 of Leviathan Hobbes writes, "The Athenians and the Romans were free, that is, they were free commonwealths. Not that any particular men had liberty to resist their own representatives, but their representatives had the liberty to resist or invade other people." Hobbes clearly says that the ancient freedom was the freedom of the collective; it wasn't the freedom of the individual. "The freedom of the authorities," as he says, "to resist or invade other people." There is written, on the turrets of the city of Lucca, remember that, in the great characters at this day the word libertas and yet, he goes on to say, "no man can thence infer that a particular man has more liberty or immunity from service to the commonwealth there than in Constantinople." That is to say, freedom for Hobbes consists of, as he puts it, immunity from service, immunity from service and for this reason there is no reason to believe that anyone is freer in the republican city of Lucca, which has libertas on the wall than in Constantinople. That seems to, already a 100 or so years before Rousseau, suggest a powerful alternative to his view of freedom. Hobbes' point, like Rousseau's, is extreme and that in many ways is the power of these two views. Hobbes' view of freedom is immunity from service, Rousseau's view is that freedom consists, you might say, only in service. Our freedom starts where the law begins. Again, at the basis of this are two radically different views of the role of political participation in lawmaking. For Rousseau, again, laws are legitimate only if everyone has a direct share in making them. It doesn't mean we all agree with the outcome but only if we have some kind of share or voice in making them. For Hobbes, for Locke, for the authors of the federalist papers, on the other hand, the direct involvement of the citizen in lawmaking is clearly a subordinate or a secondary good. Legislation is better handled by persons chosen from the electorate who are, so to speak, the agents or representatives of the people. This was what the federalist authors argued was the great advance of modern political science, the doctrine of representation. What is far more important for the federalist authors, as well as for Locke, Hobbes and that tradition is that laws be generally known, that they be applied by impartial judges, rather than they be the direct expression of the general will. In many ways underlying the, again, liberal conception of law is a certain distrust of the collective wisdom or the collective sovereignty of the people. It is too cumbersome, in many ways, and also too dangerous a mechanism to call people together to decide on matters over public concern. This is better left according to this tradition to representatives. Rousseau obviously could not disagree more. One could say that Rousseau makes heroic and unreasonable assumptions about human nature. Why do we want to gather together constantly or often to decide, to deliberate, and to debate over questions of public concern? Most people, it's hard enough just to get most people, as we know, to go out to vote, why do we want to engage in endless debate of something like a college council meeting trying to discuss what to do, whether to buy or not a new set of dumbbells for the weight room. This is a debate that will go on for hours and hours and maybe even weeks. Don't people simply want to be left alone? Rousseau, again, he seems in some way, to make unreasonable assumptions about human nature and our capacity to engage in debate. But Rousseau will tell you he is not being idealistic at all. He is starting from the assumption of men as they are, he says. Unless everyone he wants to say is engaged in the process of legislation, there is no way for you know that the laws will be an expression of your will rather than simply the private will or corporate will of some individual or intermediary body. You will find yourself in a condition of dependence and slavery on the will of others. And what is really at issue for Rousseau is freedom from dependence on some faction, some interest, or some association that we have come to call today interest groups, in some way. Rousseau's appeal is not to our altruism, but rather, to our selfishness, in some way. Our desire, our private or selfish desire to preserve our freedom and resist the willful domination of others upon it. So far, this all, in many ways, is very abstract and Rousseau deliberately sets out his plan for the general will in a highly abstract and semi-technical language. But he turns to questions particularly in Book III about how is the general will actually applied. How is it applied? Here Rousseau is far more specific sociologically and so on about the conditions under which the general will, or a general will, can come about. In the first place, the general will can only operate in small states, much like the size of an ancient republic. In one place, one particularly notable place in the social contract, he says, only the island country of Corsica is today a place where the general will might be established. The modern nation-state, as we have come to think of it, is far too large and diffuse to determine the general will. Such a state, such large states will necessarily entail considerable degrees of social inequality of wealth, of status and with such inequalities, there can be no general will. Finally, or in addition, such a state where the general will is operative would be one that would have to, in some sense, eschew the temptations of commerce and luxury for these bring with them, again, large scale inequalities. His ideal city seems to be a kind of agrarian democracy, a small-scale agrarian society. Yet, at the same time, we might get the impression that only a direct democracy would satisfy Rousseau's requirements for the general will and yet we find out this is not quite the case. In Book III, which I hope you will look at with some care, he shows surprising flexibility about the forms of government that may be appropriate to different physical and different climates and different topographies and so on. In the chapter on democracy, he remarks even, "were there a people of God's that would govern itself democratically," and then he adds, "so perfect a government is not suited to men." So he is skeptical about the possibility of a direct democracy and by that he means a democracy not only where the people legislate, bring, create law but where they are in charge also with the administration of law as well, the execution or enforcement of the law. He is very skeptical about that kind of democracy. Again, democracies are only possible under very, very special unique circumstances; otherwise, aristocracy, monarchy, some kind of even mixed government is possible or even desirable. He insists on the separation of powers for much the same reasons that one finds in Locke. The people who make the law should not be charged with, responsible for executing and enforcing them, and throughout this part of the book Rousseau seems to be in dialogue with an unnamed rival whom he sometimes simply refers to as a famous author. That author is of course, Montesquieu, the author of The Spirit of the Laws. That came out in 1755 and Montesquieu was, of course, famous for arguing that different forms of government must be tailored to different climates, different geographies, different circumstances. In many ways, in Book III, Rousseau seems to indicate or to introduce a very, very almost un-Rousseauian emphasis on prudence, moderation, flexibility that seems at odds with the dogmatic claims of the first two books with its emphasis upon the absolute inviolability of sovereignty. But most important for Rousseau, it is important that legislative authority, in whatever kind of constitution and under whatever kind of government, that legislative authority is only, is always held by the people in their collective capacity. This is why, in a very powerful chapter, Book III, chapter 15, Rousseau rejects altogether the legitimacy of representative government. That passage or that chapter could be taken already as a kind of repudiation, not just of Locke, Locke's theory of the representative but also twenty years or so before the fact of the federalist argument for representation. "Sovereignty," he says, "can never be delegated." Sovereignty can never be represented, it can only be expressed. The general will can never be delegated to someone else. If you do that, if you delegate the authority for making law you are, he wants to say, on the first step down the road to tyranny because you give someone else, some partial body or association the power to make law over you. The lawmaking function can only legitimately be held by the people themselves. I'm going to skip over a bunch of stuff that has to do, very interestingly, I hope you can discuss it in your sections perhaps, with Rousseau's account of the legislator, the extraordinary individual who was responsible at the beginnings of regimes, for shaping the general, for molding a people and for, as it were, giving the general will a kind of shape and distinctive direction, and I'm also going to skip, for our purposes, the very interesting discussion of civil theology, which occupies the theme, the very last chapter of the book, Book IV, chapter 8, where Rousseau talks about the way in which a civil religion must be tailored to bring about love and obedience to the general will. It was that chapter, I should say, that more than anything else led to the books being burned and banned in Geneva and other places and for its powerful attack on Christianity in that chapter. I'm going to pass over that for the time being to look at the legacies of Rousseau and to talk about what--and I just deliberately use that word in the plural--the legacies of Rousseau, because there is virtually no part of modern life, political, cultural, intellectual, moral that does not in some sense bear the stamp or fingerprints of Jean-Jacque Rousseau. Rousseau's description of the legislator, the kind of political founder creating a people, was for many, in many ways, the closely connected with the French Revolution and particularly the claim of the revolutionaries to create a new nation, a new people, a new sovereign, a new popular sovereign in France. Consider the following words of the famous revolutionary Robespierre in his homage to Rousseau written in 1791. Divine man, this is Robespierre, "Divine man you taught me to know myself. While I was still young, you made me appreciate the dignity of my nature and reflect upon the great principles of the social order. The old edifice is crumbling, the portico of the new edifice is rising up on its ruins and thanks to you I have brought my stone to it. Receive my homage, as weak as it is, it must be please you. I wish to follow your venerable footsteps, happy if in the perilous career that an unprecedented revolution just opened up before us. I remain constantly faithful to the inspirations that I found in your text." People might tell you that the writings of Rousseau had no influence on the French Revolution, that the French Revolution was brought about by bread crises and economic problems and so on; absolute baloney. The writings of Rousseau had this powerful influence on the idea of creating a new people and a new nation. Yet, despite what appears to be a utopian and impractical in his politics, again, Rousseau had a profound influence on the politics of his era. He was approached during his lifetime to write constitutions for Poland and for Corsica, and of course that was the island where a generation later a man named Napoleon Bonaparte was born who attempted to, you might say in some way, extend Rousseau's teaching, not just to France but to all of Europe at the point of a gun to bring democracy to all of Europe at the point of a gun. Does that sound familiar at all in any way related to events going on now? Where we have a new kind of Bonapartism, perhaps. In many ways, although Rousseau's attack on representative government would seem to put him strongly at odds with the American Constitution, his glorification of the rural republic based on equality, moral simplicity, skepticism of commerce and luxury, this was to be re-echoed in the writings of Jefferson, with his ideal of a nation of small Yeoman farmers and certainly any reader of Tocqueville's depiction in celebration of the independent townships of New England. Tocqueville's account of this was directly dependent on his reading of Rousseau, the small-scale experiment in direct democracy that Tocqueville saw was a real world example of a kind of politics governed by the general will. And when you read those early chapters from Tocqueville's democracy in America about the New England township you will very much see Tocqueville looking at America through the lenses that were in some ways crafted or shaped by Rousseau. That influence was palpable on a whole host of later nineteenth-century writers. Like Tolstoy, for instance, whose celebration of Russian peasant life was inspired by Rousseau and through Tolstoy, Rousseau influenced the establishment of the Israeli kibbutz movement that was also founded by Russian Jews who had been influenced by Tolstoy, so you have a sort of self-reinforcing cycle of influence. These, you might say, small rural socialistic experiments in communal living exhibit the same kind of equality, self-government devotion to the common good that Rousseau helped people imagine might be possible. Yet, Rousseau's influence was not limited to politics. If he was a divine man, as Robespierre called him, he was no less so to Immanuel Kant, who claimed that it was his reading of Rousseau that led him to learn respect for the dignity and the rights of man; that's what Kant said. He called Rousseau "the Newton of the moral universal." Kant's entire philosophy and I hope you also have a chance to read Kant's critique of practical reason in some later philosophy course, Kant's entire moral philosophy is a kind of deepened and radicalized Rousseauianism where what Rousseau called the general will is transmuted into what Kant calls the rational will and the categorical imperative. It was not the least of Rousseau's legacies that after his death he became a hero both to the revolution and to the counter-revolution, both to a revival of Roman-style republicanism as well as to Romanticism, or if I can use the words of great Rousseauian, Jane Austin, he became both an advocate of sense and sensibility. Emerson, Thoreau, American transcendentalism with its worship of nature and its protests against the kind of deadening and corrupting influence of society, all of these people were the direct heirs of Rousseau. Rousseau's last work, a book called The Reveries of a Solitary Walker, set the stage for later American classics like Walden Pond and generations of nature writers that have come after it and imitated it. Only by turning away from the noise and business of society can one return to what precedes society, to the feeling of existence, to the feeling of or sweetness of mere existence, to the sentiment of existence, the le sentiment de soi, as Rousseau calls it, the sentiment of the self. There is a kind of union that he celebrates with nature that puts the solitary, the solitary walker either above humanity or below it. That type of man foreshadowed by Rousseau, the solitary is no longer a philosopher in any sense that we would understand. It might be better understood as an artist or a visionary. He can claim a privileged place in society because such a person regards him or herself as the conscience of that society. His claim to privilege is based on a heightened moral sensitivity rather than his wisdom or his rationality, and it is this kind of radical individualism, the radical detachment of the solitary from the interests of society that is perhaps Rousseau's deepest and most enduring legacy for us today. So, on that note I wish you a good break. I hope you have a lot of turkey to eat and you come back well rested and most, most, most importantly we come back with a win over that evil empire to our north. Thank you very much.

Introduction_to_Political_Philosophy_with_Steven_B_Smith

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Professor Steven Smith: I've always been told that any serious introduction to political philosophy has to start with a big piece of Plato. We've made some effort to do that. Now, we have to move on. So we move to Plato's son, his adopted son, in a manner of speaking, Aristotle. There's a story about the life of Aristotle. It goes something like this. Aristotle was born. He spent his life thinking and then he died. There is, obviously, more to his life than that. But, to some degree, this captures some of the way in which Aristotle has been perceived over the centuries. That is to say, the ultimate philosopher. Aristotle was born in the year 384,15 years after the trial of Socrates. He was born in the northern part of Greece, in a city called Stagira, which is part of what is now called Macedonia. It was called that then. When he was about your age, when he was 17 or thereabouts, maybe slightly younger than many of you, he was sent by his father to do what you are doing. He was sent by his father to go to college. He was sent to Athens to study at The Academy, the first university, spoke about and established by Plato. Unlike most of you, Aristotle did not spend four years at the Platonic Academy. He remained attached to it for the next 20, until the death of Plato. After the death of Plato, perhaps because of the choice of successors to The Academy, Aristotle left Athens, first for Asia Minor and then to return to his home in Macedonia where he had been summoned by King Phillip to establish a school for the children of the Macedonian ruling class. It was here that Aristotle met and taught Phillip's son. Who was Phillip of Macedonia's son? Student: Alexander. Professor Steven Smith: Alexander. You all remember the recent movie of a year or two ago about Troy with Colin Farrell about Alexander. Who played Aristotle in that film, do you remember? Student: Anthony Hopkins. Professor Steven Smith: Anthony Hopkins, excellent. Was it Anthony Hopkins? I have in my notes here it was Christopher Plummer. I'll have to check. I'll have to Google that when I go home. Maybe you're right. I have a feeling it was Anthony Hopkins. Whoever, he was an excellent Aristotle, didn't have a large enough part in the film. In any case, Aristotle returned to Athens later on and established a school of his own, a rival to the Platonic Academy that he called the Lyceum. There is a story that near the end of his life, Aristotle was himself brought up on capital charges, as was Socrates, due to another wave of hostility to philosophy. But rather unlike Socrates, rather in staying to drink the hemlock, Aristotle left Athens and was reported to have said he did not wish to see the Athenians sin against philosophy for a second time. I'll go back to that story in a minute, because I think it's very revealing about Aristotle. In any way, this story helps to underscore some important differences between Plato and Aristotle. At one level, you might say there is an important difference in style that you will see almost immediately. Unlike his intellectual godfather, Socrates, who wrote nothing but conversed endlessly, and unlike his own teacher, Plato, who wrote imitations of those endless Socratic conversations, Aristotle wrote disciplined and thematic treatises on virtually every topic, from biology to ethics to metaphysics to literary criticism and politics. One can assume safely that Aristotle would have received tenure in any number of departments at Yale, whereas Socrates could not have applied to have been a teaching assistant. These differences conceal others. For Plato, it would seem, the study of politics was always bound up with deeply philosophical and speculative questions, questions of metaphysics, questions of the structure of the cosmos. What is the soul? What is the soul about? Aristotle appears from the beginning to look more like what we would think of as a political scientist. He collected constitutions, 158 of them in all, from throughout the ancient world. He was the first to give some kind of conceptual rigor to the vocabulary of political life. Above all, Aristotle's works, like the Politics and the Nicomachean Ethics, were explicitly intended as works of political instruction, political education. They seem to be designed less to recruit philosophers and potential philosophers than to shape and educate citizens and future statesmen. His works seem less theoretical in the sense of constructing abstract models of political life than advice-giving, in the sense of serving as a sort of civic-minded arbiter of public disputes. Unlike Socrates, who famously in his image in Book VII of the Republic, compared political life to a cave, and unlike the Apology where Socrates tells his fellow citizens that their lives, because unexamined, are not worth living, Aristotle takes seriously the dignity of the city and showed the way that philosophy might be useful to citizens and statesmen. Yet, for all of this, one might say there is still a profound enigma surrounding Aristotle's political works. To put it simply, one could simply ask, what were the politics of Aristotle's Politics? What were Aristotle's own political beliefs? Aristotle lived at the virtual cusp of the world of the autonomous city-state of the Greek polis. Within his own lifetime, Aristotle would see Athens, Sparta, and the other great cities of Greece swallowed up by the great Macedonian Empire to the north. What we think of as the golden age of Greece was virtually at an end during the lifetime of Aristotle. Other Greek thinkers of his time, notably a man named Demosthenes, wrote a series of speeches called Philippics, anti-Phillip, to the north to warn his contemporaries about the dangers posed to Athens from the imperial ambitions of Macedon. But Phillip's warnings came too late. Again, the autonomous Greek polis that Plato and Glaucon, Adeimantus and others would have known came to an end. What did Aristotle think of these changes? What did he think was going on? He is silent. Aristotle's extreme reluctance, his hesitance to speak to the issues of his time, are perhaps the result of his foreignness to Athens. He was not an Athenian. Therefore, he lacked the protection of Athenian citizenship. At the same time, you might think his reticence, his reluctance to speak in his own voice may have also been a response to the fate of Socrates and the politically endangered situation of philosophy. Yet, for a man as notoriously secretive and reluctant as Aristotle, his works acquired over the centuries virtual canonical status. He became an authority, really one could say the authority on virtually everything. For Thomas Aquinas, who wrote in the thirteenth century, Aristotle was referred to, by Aquinas, simply as "the philosopher." There was no reason even to say his name. He was simply The Philosopher. For the great Jewish medieval philosopher, Moses Maimonides, Aristotle was called by him "the Master of those who know." Think of that, "the master of those who know." For centuries, Aristotle's authority seemed to go virtually unchallenged. Are you with me? Yet, the authority of Aristotle obviously no longer has quite the power that it once did. The attack began not all that long ago, really only as late as the seventeenth century. A man, who we will read later this semester, named Thomas Hobbes, was one who led the pack, led the charge. In the forty-sixth chapter of Leviathan, a chapter we will read later, Hobbes wrote, "I believe that scarce anything can be more repugnant to government than much of what Aristotle has said in his Politics, nor more ignorantly than a great part of his Ethics." Think of that – "nothing more repugnant to government than what Aristotle wrote in his Politics." Naturally, all thinkers, to some degree, have read Aristotle through their own lenses. Aquinas read Aristotle as a defender of monarchy. Dante, in his book, De Monarchia on monarchy, saw Aristotle as giving credence to the idea of a universal monarchy under the leadership of a Christian prince. But Hobbes saw Aristotle quite differently. For Hobbes, Aristotle taught the dangerous doctrine of republican government that was seen to be practiced particularly during the Cromwellian Period in England, during the civil war. Aristotle's doctrine that man is a political animal, Hobbes believed, could only result and did result, in fact, in regicide, the murder of kings. There are certainly echoes of this reading of Aristotle as a teacher of participatory republican government in the later writings of democratic thinkers from Tocqueville to Hannah Arendt. Anyway, this returns us to the enigma of Aristotle. Who was this strange and elusive man whose writings seem to have been enlisted both for the support of monarchy and for republics, even for a universal monarchy and a smaller participatory democratic kind of government? Who was this man and how to understand his writings? The best place to start is, of course, with his views stated in the opening pages of the Politics on the naturalness of the city. His claim that man is, by nature, the political animal. That's his famous claim. What does that mean--we are the political animal. Aristotle states his reasons succinctly, maybe too succinctly. On the third page of the Politics where he remarks that every city or every polis exists by nature, and he goes on to infer from this that man is what he calls the zoon politikon, the political animal, the polis animal. His reasoning here, brief as it is, is worth following. Let me just quote him. "That man" he says "is much more a political animal than any kind of bee or herd animal is clear." Why is it clear? "For we assert," he says, "nature does nothing in vain and man alone among the animals has speech. While other species," he notes, "may have voice, may have sounds and be able to distinguish pleasure and pain, speech"--logos is his word for it. Man has logos--reason or speech. The word can mean either.-- "is more than the ability simply to distinguish pleasure and pain." He goes on. "But logos," he writes, "serves to reveal the advantageous and the harmful. And hence," he writes, "the just and the unjust. For it is peculiar to man as compared to other animals that he alone has a perception of good and bad, just and unjust and other things." In other words, he seems to be saying that it is speech or reason, logos, that is able to both distinguish and create certain moral categories, certain important moral categories that we live by--the advantageous, the harmful, the just and unjust, and things of this sort that constitute, as he says, a family and a polis. But that's Aristotle. In what sense, we could ask ourselves and I think you probably will be asking in your sections, in what sense is the city by nature? In what sense are we political animals by nature? Aristotle appears to give two different accounts in the opening pages of the book that you might pay attention to. In the literal opening, he gives what looks like a kind of natural history of the polis. He seems there to be a kind of anthropologist writing a natural history. The polis is natural in the sense that it has grown out of smaller and lesser forms of human association. First comes the family, then an association of families in a tribe, then a further association in a village, and then you might say an association of villages that create a polis or a city. The polis is natural in the sense that it is an outgrowth, the most developed form of human association, in the way that one used to see in natural history museums, these kind of biological charts of human development from these lesser forms of life all the way up to civilization in some way. That is part of Aristotle's argument. But there is a second sense for him and, in some ways, a more important sense in which he says the polis is by nature. It is natural. The city is natural in that it allows human beings to achieve and perfect what he calls their telos. That is to say their end, their purpose. We are political animals, he says, because participation in the life of the city is necessary for the achievement of human excellence, for the achievement of our well-being. A person who is without a city, he says, who is apolis--without a city--must either be a beast or a god. That is to say, below humanity or above it. Our political nature is our essential characteristic. Because only by participating in political life do we achieve, can we acquire the excellences or the virtues, as he says, that make us what we are, that fulfill our telos or fulfill our perfection. When Aristotle says that man is a political animal by nature, he is doing more than simply asserting just a truism or just some platitude. In many ways he is advancing a philosophic postulate of great scope and power, although the full development of the thesis is only left deeply embedded. He doesn't fully develop it in this work or in saying. He isn't saying that man is political by nature. Note that he is not saying, although he is sometimes taken to be saying this, that he is not saying that there is some kind of biologically implanted desire or impulse that we have or share that leads us to engage in political life. That is to say we do not, he wants to say, engage in politics. To say it's natural for us to do so is not to say we engage in political life spontaneously and avidly, as you might say spiders spin webs or ants build anthills. He is not a kind of socio-biologist of politics, although he sometimes appears this way when he says that man is a political animal. In some ways, to the contrary. He says man is political not because we have some biological impulse or instinct that drives us to participate in politics, but, he says, because we are possessed of the power of speech. It is speech that makes us political. Speech or reason in many was far from determining our behavior in some kind of deterministic biological sense, speech or reason gives us a kind of freedom, latitude, an area of discretion in our behavior not available to other species. It is a reason or speech, not instinct, that makes us political. But then the question is, for Aristotle, the question he poses for us is: What is the connection between logos, the capacity for speech of rationality, and politics? How are these two combined? Why does one lead to or entail the other? In many ways, he's not making a causal claim so much. He's not saying that it is because we are rational creatures possessed of the power of speech that causes us to engage in politics. He has more of an argument of the kind that this attribute of logos actually entails political life. He makes his argument, I think, because logos entails two fundamentally human attributes. First, the power to know, you could say. The power to know is our ability to recognize, by sight, members of the same polis or city. It is, above all, speech that in a way, ties us to others of our kind. That we share not just the capacity for language in the way a linguist might assert, but that we share a certain common moral language. It is this sharing of certain common conceptions of the just and unjust that make a city. It is the capacity to know and to recognize others who share this language with us that is the first sense in which logos entails politics. But reason or logos entails more than this capacity. It also entails for Aristotle, interestingly, the power of love. We love those with whom we are most intimately related and who are most immediately present and visible to us. In many ways, Aristotle believes our social and political nature is not the result of calculation, as we will see in Hobbes, Locke, and other social contract theorists, but such things as love, affection, friendship, and sympathy are the grounds of political life and are rooted in our logos. It is speech that allows a sharing in these qualities that make us fully human. But to say, of course, that man is political by nature is not just to say that we become fully human by participating with others in a city. It means more than this. The form of association that leads to our perfection is necessarily something particularistic. The city is always a particular city. It is always this or that particular city. The polis, as Aristotle as well as Plato clearly understand, is a small society, what could be called today a closed society. A society that leads to our perfection that leads us to complete and perfect our telos must be held together by bonds of trust, of friendship, of camaraderie. A society based simply on the mutual calculation of interests could not be a real political society for Aristotle. We cannot trust all people, Aristotle seems to say. Trust can only be extended to a fairly small circle of friends and fellow citizens. Only a small city, small enough to be governed by relations of trust, can be political, in Aristotle's sense of the term. The alternative to the city, the empire, can only be ruled despotically. There can be no relations of trust in a large, imperial despotism. It follows, in one sense, that when Aristotle says that man is by nature a political animal and the city is by nature, the city can never be a universal state. It can never be something that incorporates all of humankind. It can never be a kind of cosmopolis, a world state or even a league of states or nations. The universal state will never allow for or does not allow for the kind of self-perfection that a small, self-governing polis will have. The city, as Aristotle understands, will always exist in a world with other cities or other states, based on different principles that might be hostile to one's own. That is to say not even the best city on Aristotle's account can afford to be without a foreign policy. A good citizen of a democracy will not be the good citizen of another kind of regime. Partisanship and loyalty to one's own way of life are required by a healthy city. To put the argument in terms that Polemarchus, from Plato's Republic would have known, friend and enemy are natural and ineradicable categories of political life. Just as we cannot be friends with all persons, so the city cannot be friends with all other cities or the state with all other states. War and the virtues necessary for war are as natural to the city as are the virtues of friendship, trust, and camaraderie that are also necessary. Note that in the opening pages of the book, Aristotle doesn't say anything yet about what kind of city or regime is best. All he tells us is that we are the polis animal by nature and that to achieve our ends, it will be necessary to live in a polis. But what kind of polis? How should it be governed? By the one, the few, the many, or some combination of these three categories? At this point we know only the most general features of what a polis is. It must be small enough to be governed by a common language of justice. It is not enough merely to speak the same words, but in a sense, citizens must have certain common experiences, certain common memory and experience that shape a city and the people. The large polyglot, multiethnic communities of today would not, on Aristotle's account, allow for sufficient mutual trust and friendship to count as a healthy political community. So Aristotle seems to be offering, in some respects, a kind of criticism of the kind of states with which we are most familiar. Think about that when you have your sections or when you talk about this text with your friends. What is Aristotle saying about us? The citizens of such a city can only reach their telos or perfection through participating in the offices, in the ruling offices of a city. Again, a large cosmopolitan state may allow each individual the freedom to live as he or she likes, but this is not freedom as Aristotle understands it. Freedom only comes through the exercise of political responsibility, which means responsibility for and oversight of one's fellow citizens and the common good. It follows, for him, that freedom does not mean living as we like, but freedom is informed by a certain sense of restraint and awareness that not all things are permitted, that the good society will be one that promotes a sense of moderation, restraint and self-control, self-governance, as Adeimantus says, that are inseparable from the experience of freedom. In many ways Aristotle there offers, as does Plato, a certain kind of critique of the modern or even the ancient democratic theory of freedom, which is living as one likes. You can see these opening pages of the book, dense argument being condensed in very deep ways, carry a great deal of freight. There's a lot in there that needs to be unpacked. I've only tried to do a little of that here with you today, to go over what Aristotle is suggesting in this idea of man, the polis animal. Whatever we may think about this view, whether we like it or don't like it or whatever your view might be, you must also confront another famous, more like infamous, doctrine that is also very much a part of Book I. I refer to his arguments for the naturalness of slavery. Aristotle tells us that slavery is natural. The naturalness of slavery is said to follow from the belief that inequality, inequality is the basic rule between human beings. Aristotle and Thomas Jefferson seem to disagree over the basic fact of human experience, whether it's equality or inequality. If this is true, Aristotle's Politics seems to stand condemned as the most antidemocratic book ever written. Is that true? Aristotle's claim about naturalness seems to require, as he told us, slavery, the categorical distinction of humanity into masters and slaves. How to understand that? Again, Aristotle's argument is deeply compact and will be easily misunderstood if you only read it once. It will just as likely be misunderstood if you read it three, four, five, or ten times, if you are not attentive to what he's saying. You must learn to read closely. What was Aristotle saying? In the first place, it's important that we avoid, I think, two equally unhelpful ways of responding to this. The first, which one finds among many modern-day commentators, many kind of neo-Aristotelians, we might call them, is to simply avert our eyes from the harsh, unappealing aspects of Aristotle's thought and proceed as if he never actually said or meant such things. We need to avoid the temptation, in many ways understandable as it might be, to airbrush or sanitize Aristotle, to make him seem more politically correct for modern readers. Yet, we should also avoid the second, equally powerful temptation, which is to reject Aristotle out of hand, because his views do not correspond with our own. The question is what did Aristotle mean by slavery? Who or what did he think was the slave by nature? Until we understand what he meant, we have no reason to either accept or reject his argument. The first point worth noting about this, is that Aristotle did not simply assume slavery was natural, because it was practiced virtually everywhere in the ancient world. You will notice that he frames his analysis in the form of a debate. He says at the outset of his argument, "There are some," he says, indicating this is an opinion held by many people. "There are some who believe that slavery is natural, because ruling and being ruled is a pervasive distinction that one sees all societies practice." But he says, "Others believe that the distinction between master and slave is not natural, but is based on force or coercion." Even in Aristotle's time, it appears slavery was a controversial institution and elicited very different kinds of opinions and responses. Here is one of those moments when Aristotle, as I indicated earlier, seems most maddeningly open-minded. He's willing to entertain arguments, both for and against the debate. Aristotle agrees with those who deny that slavery is justified by war or conquest. Wars, he remarks, are not always just. So, those who are captured in war, cannot be assumed to be justly or naturally enslaved. Similarly, he denies that slavery is always or only appropriate for non-Greeks. There are no, he is saying, racial or ethnic characteristics that distinguish the natural slave from the natural master. In a stunning admission, he says--listen to this--that "while nature may intend to distinguish the free man from the slave," he says, "the opposite often results. Nature often misses the mark," he says. Now we seem to be completely confused. If slavery is natural, and if nature intends to distinguish the slave from the free, the free from the unfree, how can nature miss the mark? How can the opposite often result? I mention this because such complications should alert the careful reader. We're trying to read carefully. What is Aristotle doing in making this seem so complicated? At the same time, Aristotle agrees with those who defend the thesis of natural slavery. His defense seems to run something like this. Slavery is natural because we cannot rule ourselves without the restraint of the passions. Self-rule means self-restraint. Restraint or self-control is necessary for freedom or self-government. What is true, he seems to suggest, about the restraint over one's passions and desires is true of restraint and control over others, just as he appears to be saying there is a kind of hierarchy within the soul, restraint of the passion. So does that psychological hierarchy translate itself into a kind of social hierarchy between different kinds of people? The natural hierarchy, then, seems to be a sort of hierarchy of intelligence or at least a hierarchy of the rational. "How did this come to be?" Aristotle asks. How is it that some people came to acquire this capacity for rational self-control that is necessary for freedom and others seem to lack it? How did that come to be? Is this hierarchy, again, a genetic quality? Is it something we're born with? Is it something that is implanted in us by nature in that sense, or is that distinction something that is created by nurture and education, what we would call today maybe socialization? If the latter, if this hierarchy of intelligence or this hierarchy of the rational is the result of upbringing, then how can slavery be defended as natural? Doesn't Aristotle call man the rational animal, the being with logos, suggesting that all human beings have a desire for knowledge and the desire to cultivate their minds and live as free persons. Isn't there a kind of egalitarianism, so to speak, built in to the conception of rational animal and political animal? He begins his Metaphysics, his great book the Metaphysics, with the famous opening statement, "All men have a desire to know." If we all have a desire to know, doesn't this connote something universal, that all should be free, that all should participate in ruling and being ruled as citizens of a city? Yet, at the same time, Aristotle seems to regard education as the preserve of the few. The kind of discipline and self-restraint necessary for an educated mind appears, for him, to be unequally divided among human beings. It follows, I think, that the regime according to nature, that is to say the best regime, would be what we might think of as an aristocracy of the educated, an aristocracy of education and training, an aristocratic republic of some sort where an educated elite governs for the good of all. Aristotle's republic, and I use that term to remind you of Plato as well, is devoted to cultivating a high level of citizen virtue where this means those qualities of mind and heart necessary for self-government. These qualities, he believes, are the preserve of the few, of a minority capable of sharing in the administration of justice and in the offices of a city. It seems to be a very elite teaching. Would you agree? Unappealing to us, perhaps, for that reason, very contrary to our intuitions and the way we have been brought up. Yes? You'll agree with me. But before we dismiss Aristotle's account as insufferably inegalitarian and elitist, we have to ask a difficult question, not just of Aristotle, but more importantly of ourselves. What else is Yale, but an elite institution intended to educate, morally and intellectually, potential members of a leadership class? Think about that. Can anyone get into Yale? Do we have an open admissions policy for all who want to come here? Hardly. Does it not require those qualities of self-control, discipline, and restraint necessary to achieve success here? I will leave aside, for the moment, what happens on Friday and Saturday nights. Is it any coincidence that graduates from this university and a handful of others not unlike it find themselves in high positions of government, of business, of law, and the academy? Is it unfair or unreasonable to describe this class, as Aristotle might, as a natural aristocracy? I leave you with this question to think about. Before we reject Aristotle as an antidemocratic elitist, take a look at yourselves. So are you, or you wouldn't be sitting here today. Think about that and I'll see you next week.

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Professor Steven Smith: Last time, I ended by talking about Machiavelli as both a revolutionary in many ways and a reformer of the moral vocabulary about virtue and vice, good and evil. Machiavelli seeks to replace, to transpose an older vocabulary associated both with Plato and certainly, perhaps more importantly, with biblical sources, wants to transform altogether the language of virtue, to give it a new kind of meaning, to change it from either Platonic or Christian otherworldliness to a greater sense of worldly power. Virtue is, for him, or to use his term again, virtù is related with manliness, with force, with power. He tells us, in chapter 25 of The Prince, the ethic of the prince must be one of audacity and even more audacity and that famous and very volatile image he uses, fortune is a woman and you must know how- the prince must know how to conquer the woman, must be used through policies of force, brutality, audacity. This is the language of Machiavelli. Virtue is associated with the quest for worldly glory, with ambition, with the desire to achieve success, and that's what I want to talk about at greater length today. I want to talk about what in the political and philosophical literature about this is called the problem of "dirty hands." And if you want to join the political game, you must be prepared to get your hands dirty, and what Machiavelli means by that, how he comes to this problem. In order, he argues, to effect a transformation of European morality, it is, in other words, to teach the prince, as he says in chapter 15, how not to be good, you have to go to the source of the morality. You have to go to the source of morality. To affect the maxims, to affect the standards that govern our lives, it is necessary to go to the source of those standards and those maxims and that can only be found in religion. Oddly, it seems in some ways, religion does not seem to be a major theme of The Prince. In a memorable passage from chapter 18, Machiavelli advises the prince always to cultivate the appearance of religion. The prince, he writes, should appear all mercy, all faith, all honesty, all humanity and all religion, he writes, adding nothing is more necessary to appear to have this last quality. The point is clear. The appearance of religion, by which he clearly means Christianity, is good while the actual practice of it is harmful. Think about the way in which that transforms what Plato says about justice in his answer to Glaucon in Book II of the Republic where…or Thrasymachus…where they both say it is more important, is it not more important to have the appearance of being just than the reality of it? And here, you see Machiavelli in a way adding his voice to that chorus. It is much better to have the appearance than the reality of religion. But in order to understand or to discover the core of Machiavelli's teachings about religion, I have to make a slight detour away from The Prince and to his Discourses on Livy and in maybe the most important chapter of that book, Book II, chapter 2, called "Concerning the Kinds of People the Romans had to Fight and how Obstinately they Defended their Freedom," a long title for a chapter to be sure, but here Machiavelli develops a powerful contrast between two opposed and mutually incompatible moral codes, the Christian and the pagan. "If one asks oneself," Machiavelli writes, "If one asks oneself how it came about that people of old," in olden--in the ancient world, "were more fond of liberty than we are today, I think the answer," he says, "is due to the same cause that makes men today less bold than they used to be," less bold, "and this is due I think to the difference between our education and that of bygone days." So what precisely is the difference that Machiavelli refers to here between our education and the education of bygone days that makes people or that made people in the ancient world more fond of liberty, as he says, than those of our contemporaries or Machiavelli's contemporaries? Machiavelli's emphasis here on education, particularly moral and religious education, is the key difference between the ancient times and his own. These two different ages, he believes, advanced two very different systems of moral and religious education, one based on pagan worldliness and the other based on Christian innocence. And it is that conflict, as it were, between what we might call worldliness and innocence that is the core of Machiavelli's moral code. Let me quote Machiavelli's passage from the Discourses at some length because I think it's very revealing: "Our religion," he writes, obviously thinking of the Catholic Christianity of his time. "Our religion," he writes, "has glorified humble and contemplative men, monks, priests, humble and contemplative men, rather than men of action. It is assigned as man's highest good humility, abnegation, and contempt for mundane things," whereas the other, that is to say the ancient moral code, "whereas the other identified it with magnanimity, bodily strength, and everything that conduces to make men very bold. And if our religion," he says, "demands that in you there be strength what it asks for is the strength to suffer rather than to do bold things." In other words, he says Christian strength, the strength of the Christian, is the strength to suffer, thinking of Jesus on the Cross rather than to, as he puts it, do bold things. And it is not for Machiavelli simply the existence of these two different moralities that is at stake. By softening morals, he believes, by making us gentler, Christianity has had some deeply perverse effects upon politics, so he claims. This pattern of life, Machiavelli continues, appears to have made the world weak and to have handed it over to the prey of the wicked. This pattern of life, this pattern of education, of moral education, introduced by the Bible and scripture and Christianity, has made the world weak. In other words, by teaching humility, self-abnegation, purity of heart, Christianity has made it difficult to develop qualities necessary for the defense of political liberty. Christianity has made the world weak or, if you want to use his again highly charged word for that, it has made the world effeminate. Machiavelli would no doubt be taken up against some board of offense today for using such a term but that's his language. What can I say? This is why he concludes there are fewer republics today than in the time of the ancients because we do not have the same love of freedom that they did. Now Machiavelli's explicit referencing of the ancient civil religions, the ancient civil theology, is a direct tribute to the role of Numa, N-u-m-a, in Livy's famous History of the Roman Republic. Justin, who is an authority on this text, can tell you more about it if you like, but in the opening books of Livy, he tells the story of how Rome was founded by Romulus, who had murdered his brother, Remus, but after this it required a second founding and the second founding was the work of a man named Numa, who, Livy writes, determined that Rome, which had originally been established through force of arms, should be reestablished through justice, laws and proper observances, in other words, religion. In order to complete the founding of the city, it was necessary to establish its gods and ensure proper respect for the law. Numa was the bringer of the Roman legal codes respecting religion, proper observances and the like. But Machiavelli uses Livy and in the story about Rome's second founding to bring home an important lesson about the utility of religion. "Religion," he tells the reader, "is not to be evaluated by its truth content but for its consequences for society." But the story of Numa or his use of that story tell us more than just a lesson about the social utility of religion. At the time of the founding of Rome, Machiavelli writes, religion was necessary to temper and control the warlike character of the Romans. Religion had to bring a softening effect upon against the violent and bestial character of the early Romans. But for us today, Machiavelli writes, religion has to serve the opposite purpose. It must instill something of a fighting spirit into people who have lost their instinct to resist encroachments on their liberty. In many ways, this is the deeper meaning of Machiavelli's slogan, "one's own arms." He uses in a variety of passages the formula that a good republic depends upon one's own arms and laws and in a deeper sense this idea of "one's own arms" means developing the capacities to resist encroachments on your freedom. The prince, in other words, has to use religion to encourage his subjects to rely upon their own arms rather than on divine promises and that again is the teaching of his retelling of the story of David and Goliath, the biblical story of David and Goliath, in chapter 13 of The Prince. You remember how Machiavelli retells and also rewrites that story. He writes the story saying that David went armed, went into battle with Goliath armed only, he says, with a sling and a knife, and those of you who know the story and checked against the biblical account of the story know that David only went into battle against Goliath armed with Saul's armor and his sling. Machiavelli gives him a knife. Where did this come from? Why does he add this? His subtle alteration of the biblical story is hugely revealing. Its moral seems to be "trust in God's promises, yes, but bring a knife just in case." It's like the old joke about the fighter who went in to the ring and before going in to the ring and he asked the priest to pray for him. He said, "I'll pray for him but if he can punch it'll help." In a small respect, that's Machiavelli. Machiavelli sensed that his own country was deeply deficient in these martial virtues, necessary to reassert greatness and this was a theme of a lengthy poem he wrote. Yes. You're surprised. Yes, Machiavelli wrote poetry and plays. His play, The Mandragola, is still performed, but he wrote an interesting poem, a lengthy poem called Ambizione, ambition, something like Platonic thumos, which lamented his countrymen's lack of civic spirit and their need to be reeducated in the art of war. I only want to read a small section to you from that poem: "If you perchance are tempted to accuse nature, if Italy, so wary and wounded, does not produce hard and bellicose people, this I say is not sufficient to erase our cowardice for education can supplement where nature is deficient. Stern education made Italy bloom in ancient days and made her rise and conquer the entire world and for herself make room. But now she lives, if tears can be called life, beneath the ruins and unhappy fate that she has reaped from her long lack of strife. But now she lives, if tears can be called life, beneath the ruins and unhappy fate that she has reaped from her long lack of strife." And just from this little section of the poem, you can see that the theme of a new kind of education and only that can remedy nature's defects, as Machiavelli calls them. It is this lack of strife, this long lack of strife, that makes people weak. People are weakened by prolonged peace and they are made strong, fierce and independent through war. Only by hardening themselves, he says, will it be possible for Italy, as he puts it, "to rise and conquer the entire world, in ancient days again and made her rise and conquer the entire world and for herself make room." His point seems to be this. If you want liberty, you have to know how not to be good, at least as Christianity has defined goodness. The Christian virtue of humility, turning the other cheek, forgiveness of sins, must be rejected if you want to do good as opposed to just being good. You have to learn, in other words, how to get your hands dirty. Between the innocence of the Christian and the worldliness of Machiavelli's new morality, there can be no reconciliation. These are just two incompatible moral positions that Machiavelli states but he goes further than this. The safety and security enjoyed by the innocents, our freedom to live blameless lives and to have untroubled sleep, depends upon the prince's clear-eyed and even ruthless use of power. The true statesman, the true prince for Machiavelli, must be prepared to mix a love of the common good, a love of his own people, with a streak of cruelty that is often regarded as essential for a great ruler in general, another part of knowing how not to be good, knowing when and how to use cruelty or what Machiavelli tellingly calls "cruelty well used." When it's well used, it's a virtue. This is simply another example of how moral goodness grows out of and even requires a context of moral evil. Machiavelli's advice to you is clear. If you cannot accept the responsibilities of political life, if you cannot afford to get your hands dirty, if you cannot accept the harsh necessities that may require cruelty, deceit and even murder, then get out of the way, then this is not for you. Don't seem to impose, don't seek to impose your own high-minded innocence, sometimes called justice, your own high-minded innocence on the requirements of statecraft because it will only lead to ruin. In the modern era, the presidency of Jimmy Carter, for example, is usually taken as exhibit A of the confusion between Christian humanitarianism and the necessities of reason of state. If you can't do the tough thing, if you can't do the harsh thing, Machiavelli says, then stay out of politics and don't attempt to impose your high-minded morality on the state. As I said at the beginning, in the philosophical literature, this has become known as the problem of dirty hands so named after a famous play written by the French philosopher Jean-Paul Sartre. The problem of dirty hands refers to the conflict of duties, again conflict of moralities between the harsh requirements of politics and the equally demanding desire for moral purity, to keep the world at a distance. Machiavelli doesn't deny that there is something deeply admirable about the desire to remain morally pure, morally decent, morally innocent, but he just wants to say this is a very different morality from the morality of politics. In Sartre's play, the action takes place in a fictional eastern European country during World War II, probably something like Yugoslavia, where a communist resistance fighter reproaches an idealistic young recruit to the resistance who is resisting or is balking at the order to carry out a political assassination. "Why did you join us?" the communist resistance fighter asks. "Purity is an idea for the yogi or the monk. Do you think anyone can govern innocently?" "Do you think anyone can govern innocently," the phrase taken of course from Saint-Just, one of the leaders of the Jacobin Reign of Terror during the French Revolution. What do you think politics is, a game of moral purity? The same kind of conflict is really very much at the core of the great political fiction of John le Carre, the great novelist of the Cold War and so on, and in his great, one of his early political thrillers, a book called The Spy who Came in from the Cold, he depicts there a British agent who was working undercover and who at the same time is carrying on a love affair with an idealistic young English librarian who has joined the communist party. In this case, she, the communist, is the idealistic one. She's joined the party because she believes it will aid the cause of nuclear disarmament and will bring international peace and when Lemas, the spy, reveals to her that he is a spy, he tells her his view of what politics is, the nature of politics. "There's only one law in the game," Lemas says, "the expediency of temporary alliances. Who do you think spies are, priests, saints, martyrs? They're squalid little men, fools, queers, sadists, drunkards, people who play cowboys and Indians to brighten their rotten lives. Do you think they sit like monks weighing up right and wrong?" So both of these cases, the Sartre case, the John le Carre case, in a way are interesting but they're also sort of cases of what I think of as faux Machiavellianism, kind of intellectuals engaging in tough talk to show that they have really lost their innocence, which is the sort of intellectual equivalent of losing your virginity, showing you're not really innocent about the world. Machiavelli of course likes to play that game and it suggests that the world is divided between the weak and the strong, between the realists who see things the way they are and the idealists who require the comfort of moral illusions. Yes, Machiavelli sometimes seems to corroborate this point of view. Does he not say that armed prophets always win, the unarmed prophets lose? Did he not say that he wrote to reveal the effectual truth of things and not just what people have imagined the case to be? Yet it seems inconceivable that Machiavelli wrote an entire book simply to prove the obvious, that is to say that the strong will always crush the weak and that politics is left to those who leave their scruples at the door. The question is, was Machiavelli really that kind of Machiavellian? Was Machiavelli a Machiavellian? Let's see. What kind of government did Machiavelli think best? As he indicates at the beginning of The Prince, there are two kinds of regimes: there are principalities and republics. But each of these regimes, he says, is based on certain contrasting dispositions or what he calls humors, umori, humors. "In every society," he writes, this is chapter 9 of The Prince, "two diverse humors are found from which this arise, that the people desire neither to be commanded nor oppressed by the great and the great desire to command and oppress the people." These are the two great political psychological dispositions, the popular desire not to be oppressed and the disposition of what he calls the great to command and oppress. Machiavelli uses these two psychological and even in some ways quasi-medical terms, humors, to designate two classes of people on which every society is based. His theory of the humors in chapter 9 seems in some ways to be reminiscent of Plato's account of the three classes of the soul or the three parts of the soul with one vivid exception. "Each class of the city," he says, "is bound or determined by a humor but neither humor is anchored in reason or rationality." Every state is divided into two classes expressing these two qualities, these two psychological qualities, the grandi, the rich and powerful who wish to dominate, and the popolo, the common people who wish merely to be left alone, who wish neither to rule nor be ruled. Now, one might expect that the author of a book entitled The Prince would favor the great, would favor the grandi, those who desire to rule. Are not these aristocratic goals of honor and glory precisely what Machiavelli seems to be advocating? Yet in many ways, Machiavelli proceeds to deprecate the virtues of the nobility, perhaps to our surprise. The ends of the people, the ends, the purposes of the people, is more decent than that of the great since the great want to oppress and the people want not to be oppressed, he says. His advice is that the prince should seek to build his power base on the people rather than on the nobles. Because of their ambition for power, the nobles will always be a threat to the prince and, in an interesting reversal of the Platonic and Aristotelian conception of politics, it is the nobles here who are said to be the more fickle and unpredictable and the people are more constant and reliable. Remember in the Platonic and Aristotelian view of politics the democracy, the rule of the people, the demos, was always criticized for it being fickle and unstable and subject to whim and passion and so on. Here, Machiavelli tells us it is the great who are subject to this kind of inconstancy and the people are more reliable. The worst, he writes, that a prince can expect from a hostile people is to be abandoned by them but from the great, when they are hostile, he must fear not only being abandoned but also that they may move against him. The grandi are more dangerous and fickle. So the main business of government consists in knowing how to control the elites because they are always a potential source of conflict and ambition. The prince must know how to chasten the ambition, to humble the pride, as it were, of the great and powerful, and this, we will see as early as Wednesday, becomes a major theme in the philosophy of Thomas Hobbes, humbling or chastening the pride of the few. The rule of the prince or sovereign requires the ability to control the ambition and to do so through selective policies of executions, of public accusations and political trials. Remember the example that we read at the end of class on Friday, I believe from chapter 7, the example of Cesare Borgia and Remirro d'Orco and how his execution, his bloody execution, left the people, Machiavelli says, stupefied and satisfied? Here is a perfect example of how to control the ambitions of the nobles and to win the people to your side. So Machiavelli's prince, while not exactly a democrat, recognizes the essential decency of the people and the need to keep their faith. And by decency he seems to mean their absence of ambition, the absence of the desire to dominate and control. But this kind of decency is not the same as goodness for there is also a tendency on the part of the people to descend into what Machiavelli calls idleness or license. The desire not to oppress others may be decent but at the same time the people have to be taught or educated how to defend their liberty. Fifteen hundred years of Christianity, he says, have left people weak, have left the people weak without their capacities to exercise political responsibility and the resources to defend themselves from attack. So just as princes must know how to control the ambitions of the multitude, how to control the ambitions of the nobles--excuse me--they, the princes, must know how to strengthen the desires of the common people. Some readers of The Prince, even some very astute readers of The Prince, have thought that Machiavelli's work is really, or Machiavelli's prince, is really a kind of democrat in disguise and that the prince is intended precisely to alert the people to the dangers of a usurpatory prince. This is for example what the great seventeenth-century political philosopher Spinoza believed about Machiavelli. In his book called, simply called, The Political Treatise, Spinoza wrote: "Machiavelli wished to show how careful a people should be before entrusting its welfare to a single prince. I am led," Spinoza continues, "to this opinion concerning that most far-seeing man because it is known that he was favorable to liberty." That's Spinoza on Machiavelli, because "he was favorable to liberty" and that the book, he says, is kind of a satire on princely rule. Or, if you don't believe Spinoza, if you don't believe his authority is sufficient, consider someone who you'll be reading in a couple of weeks, Jean-Jacques Rousseau, from the Social Contract. "Machiavelli was an honorable man and a good citizen," Rousseau says, "an honorable man and a good citizen who, being attached to the House of Medici, was forced, during the oppression of his homeland, to disguise his love of freedom." So, The Prince was written in a way that disguised the real teaching of the book, which is the love of freedom and presumably the freedom of the people, something of the type that Rousseau himself spoke about. Maybe these comments go too far. Maybe they are exaggerations and I think to some degree they are but it's revealing that both of these very serious readers of Machiavelli took him to be an apostle of freedom. Spinoza taking him, taking his book to be a warning to the people about the dangers of princely rule, Rousseau believing that he had deliberately disguised his love of freedom because he had to appeal to the tyrannical nature of the Medici family. In either case, they regard him as surreptitiously taking the side of the people against the nobles. In any case, whatever one makes of those examples, Machiavelli seems to be challenging important aspects of the classical conceptions that we've been talking about up to this point. In the classical republic, for the ancient republic of Plato and Aristotle, these republics were ruled by nobilities, gentlemen possessed of wealth and leisure, who were therefore capable of forming sound political judgment, who will dominate, while in Machiavelli's state it is the people who are going to be the dominant social and political power. Machiavelli wants to redirect power to some degree away from the nobles and toward the people. One wants to know why, why does he want to do that? In the first place, he judges the people to be more reliable, as he tells us, than the great. Once the people have been taught to value their liberty, have learned to oppose encroachments on their freedom, to be fierce and vigilant watchdogs rather than humble and subservient underlings, they will serve as a reliable basis for the greatness and power of a state. With the people on his side, the prince is more likely to achieve his goals of a robust civil life for his people and eternal glory for himself. And, as Machiavelli likes to say, the prince must know how to adapt to the times. What is true for princes is no less true for advisers to princes like Machiavelli himself. One must know the times and character of a people. In the ancient republic, it may have been necessary to find and impose restraints on the passions of the demos but in the modern world, he says, where republics have become a thing of the past, the people need to be taught how to value their liberty above all else. The most excellent princes of the past were those like Moses, he tells us, who brought tables of law and prepared people for self-government. It is fitting and proper that The Prince concludes, the last chapter, chapter 26, concludes with a patriotic call to his countrymen to emancipate themselves and liberate Italy from foreign invaders. So what did Machiavelli achieve? What were his actual accomplishments? Did he accomplish all he set out to do, to rewrite or to write a new moral code for political life, to found a new political continent, as he speaks about, to found new modes and orders along the lines of Columbus? Did he achieve this? First of all, one should not and cannot underestimate his unprecedented break with both classical and biblical antiquity. More than anyone else before him, and perhaps more than anyone else since, he sought to liberate politics from ecclesiastical control. The new prince, as we've seen, must know how to use religion but needs to learn how not to be used by religion, must not become a dupe of the religious. He must know how to use religious passions and sentiments but not be used by them. Politics must become a purely worldly affair. It should not be limited or constrained by any transcendent standards or moral laws that do not derive from politics itself, whether a law of God or some kind of transcendent moral order or code. Machiavelli's warning, we might say today, to the religious right, or his critique of the religious right, cannot make politics conform to transcendent moral law. But not only did Machiavelli bring a new worldliness to politics, he also introduced a new kind of populism, you might say. Plato and Aristotle imagined aristocratic republics that would invest power in an aristocracy of education and virtue. Machiavelli deliberately seeks to enlist the power of the people against aristocracies of education and virtue. He is a kind of proto-democrat almost who sought to re-create, not through accident and chance, but through planning and design a new kind of republic in the modern world. The republic that Machiavelli imagined, and it's interesting while he tells us he's only going to the effectual truth of things and not the imagination of it, nevertheless Machiavelli does himself imagine a new kind of regime, a new kind of republic in the modern world that would not be a city at peace but would be a city at war. It would be armed and expansive. Machiavelli's republic feeds on conflict, on war and conquest. It is aggressive and imperialistic. Does it sound familiar? Is it us? In fact, if you look at a brilliant article I think in this week's New Republic by Robert Kagan called "Cowboy Nation," Kagan demonstrates I think with a great deal of conviction that the American republic from its onset has been expansive, aggressive, imperialistic, from the conquest of the territories, the expropriation of the native Americans, the acquisition of Louisiana, wars of liberation against Mexico and Spain and so on, well into the twentieth and now the twenty first century, an aggressive, expansive, imperialistic republic. That, he says, has been our history and what it should say, what it doesn't quite say I think, is that it has been this history not because it is American but because it is a republic, because of its regime type, its regime character. That kind of behavior seems perhaps to be built in to the natures of republic. It was Machiavelli's admiration for the politics, what someone once called the lupine politics, the wolf-like politics, of republican Rome that led him to understand that all social and moral goods have been established by morally questionable means. Have we become or have we always been Machiavelli's republic, Machiavelli's desire? Think about that when you're in your sections or writing your papers and you will get those paper topics on Wednesday, by the way. And finally, Machiavelli is the author of a new amoral realism. "By whatever means necessary" I think is his motto or should be his motto, "by whatever means necessary," and oddly he claims to be merely stating out loud, merely stating aloud what all writers have known all along. It is necessary, he says, for the prince to know well how to use the beast and the man, he writes. "This role," he says, "was taught covertly by ancient writers. It was taught covertly by ancient writers," he says in chapter 18. The idea then that Machiavelli is doing no more than saying openly and overtly what ancient writers had wrapped in parable and enigma and myth says something about Machiavelli's new political science. What was previously taught only subtly and in private will now be taught openly and in public. What was once available only to a few, will now be available to all. Perhaps more than anything else, Machiavelli's new openness, his readiness to challenge received authority, and his willingness to consider authority as self-created, as self-made rather than bestowed by either nature or grace, is what fundamentally constitutes his modernity. So I'm going to leave it on that note and on Wednesday we will begin the study of one of Machiavelli's greatest and most profound disciples in the modern world, a man by the name of Thomas Hobbes.

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Professor Steven Smith: Well, today I'm going to finish Tocqueville or, to put it a different way, I'm going to say what I can about Tocqueville in 50 minutes, which is hardly finishing him. In fact, we've hardly begun but I want to talk about two things, two aspects of the book today, again, which will again only scratch the surface, and those two topics are the following. I want to talk about--a little bit about the moral and psychological components or features of the democratic state, which is largely the subject matter of Volume 2 of the Democracy and I also want to speak about the role of statesmanship. I mentioned earlier the issue of Tocqueville as educator, as a kind of political educator, and I want to talk today, end up today by talking a little bit about how he understands the role of the democratic statesman. But the first part--first subject is largely, again, the subject matter of Volume 2. Volume 1 of the Democracy, as you've probably noticed, focuses mainly, not exclusively to be sure but mainly on what I suppose we would call "the social and political institutions of democratic society," the institutional development of the democratic state. Volume 2 focuses much more on, so to call it, the moral and psychological components of the democratic individual. Tocqueville here shows himself more concerned with the internal develops, again, the moral and psychological determinants of democratic character, what is it to have a democratic soul, so to speak. That, I think, is Tocqueville's concern in the second volume, which in many ways, at least to my way of reading it, makes Volume 2 a sort of philosophically richer discussion than Volume 1, precisely because it focuses on what has the democratic social state done to us, how has it transformed us as individuals, how has it shaped us as individuals. These were, in many ways, Tocqueville's deepest problems and in this part of the book he shows himself to be a kind of moral psychologist of the democratic soul, very much along the same lines as we saw in Plato for example in Volume 8 of the Republic where Plato speaks about the different kinds of individuals, the different kinds of souls that are appropriate and have been shaped by different kinds of regimes. But I'd like to start with--I want to focus on three aspects, spend a little time on three of the components, aspects, psychological components of the democratic individual, and those in no particular order I want to discuss as compassion, what this translation has as restiveness, and self-interest. Taken together, I think, these three terms or these three concepts constitute, as it were, the sort of moral scope of the democratic state. In describing these character traits, Tocqueville is providing us with a kind of moral phenomenology, and excuse please a rather pretentious term, kind of moral phenomenology of democratic life, one in which we are invited to look and ask whether we see ourselves in this description and whether we like what it is we see. The first of these features that I want to focus on, the most important moral effect in some respects the democracy has had on its citizen, is for Tocqueville the constant tendency to make us gentler towards one another. This is an old eighteenth-century theme, to make us more compassionate, to make us gentler in our manners, habits, morals with one another. This is an old problem. Montesquieu, Tocqueville's great eighteenth-century precursor--Montesquieu had argued in the Spirit of the Laws, L'esprit des Lois, that it was commerce that instituted a kind of softening effects on manners and morals, moving us or taking us from a kind of warlike, aristocratic ethic to one of gentler manners and morals, and Montesquieu had attributed this largely to the influence of commerce. Rousseau, you will remember, in the Second Discourse, the Discourse on Inequality, made pitié or compassion, a repugnance to view the suffering of others, as a fundamental feature of natural man. Compassion, for Rousseau, remained a kind of remnant of our natural goodness, the fact that we can still cry or sympathize or empathize, as we might say, with the plight of others even with the growth of noisier and more powerful passions. This sort of capacity for sympathy or compassion remains even in civilized life a kind of remnant of our natural goodness. But for Tocqueville, this feature of compassion is not so much a feature of natural man as it was for Rousseau but it is for democratic life, a democratic social life. It is not nature but democracy that has rendered us gentler and led to the softening of morals and manners. What does Tocqueville mean by that, when he says, "life in democracy has become gentler"? In a very powerful chapter called "How Mores Become Milder as Conditions are Equalized," here he describes some of the moral and psychological consequences of the transition from the age of aristocracy to one of democracy. Under aristocratic times, he says, in aristocratic ages, individuals inhabited a world apart where members of one class or one tribe may have been similar to one another but they regarded themselves as being fundamentally different from the members of all other social classes or tribes. This did not so much render people cruel but it did render them indifferent to the pain and suffering of others outside their group. Under democracy, however, he says, where all are equal, all of us tend to think and feel in nearly the same manner. We no longer make or imagine these kinds of distinctions. The moral imagination, so called, of the democratic citizen, is able to transport itself into the positions of others more easily than individuals living in aristocratic times. All become alike or at least all are projected or perceived as being alike in our range of emotions, sensibilities, capacities for moral sympathies. As people become more like one another, Tocqueville says, they show themselves reciprocally compassionate regarding their miseries and the laws of nations become milder, the laws of nations become milder, they show themselves reciprocally compassionate to one another. That transformation of one of the key ethics of social life for Tocqueville has had profound effects on us. It has certainly made people gentler and more civil to one another. Such things, he tells us, as torture, deliberate cruelty, sort of spectacles of pain and humiliation that were once so much a part of everyday life have largely been eliminated from the world. I say largely, not entirely to be sure. We more readily identify ourselves with the pain or suffering of people possibly in very different parts of the world, world parts that we've never seen and may never visit. Consider, for example, our response to the victims of the tsunami in Indonesia or the genocide in Darfur. All of these events affecting people in places, again, where we may never go nevertheless seem to have a claim on our moral sympathies. President Bill Clinton profoundly captured this sense of enlarged moral sympathy when he told his audiences, "I feel your pain." Remember? I don't know. You probably won't remember that but you've probably heard the expression. It seems to show a kind of enlargement of the moral sympathies, being able to put oneself in the position of others who one doesn't know and may never meet. This is all a part of what Tocqueville understands, the softening of morals under a democratic way of life. And Tocqueville clearly regards this, in many ways, as a moral progress of sorts in our unwillingness to tolerate policies of deliberate cruelty in his statement, perhaps premature, that Americans of all the people in the world have succeeded or almost succeeded in abolishing the death penalty, not quite true but nevertheless maybe more truer than it is now. In democratic centuries, he says, men--but all of this compassion--here is--but here is the problem. All of this compassion comes still at a price. In democratic centuries, he writes, men rarely devote themselves to one another but they show a general compassion for all members of the human species. They rarely, he says, devote themselves to one another. This sort of generalized sympathy is genuine but soft. My ability to feel your pain does not really require me to do much about it. Compassion, you might say, turns out to be a rather easy virtue. It suggests sensitivity and openness. It implies caring without being judgmental. It is not entirely relativistic but it certainly refrains from imposing one's own moral judgments and way of life upon others. Does Tocqueville believe that democratic peoples are in dangers of becoming too soft, too morally sensitive, too incapable of exhibiting the kind of harsher, what we might call more aristocratic virtues of nobility, of self-sacrifice, of love of honor that formed the moral code of previous times? Well, the answer to that question is yes, he surely did believe that was becoming the case. Compassion is for Tocqueville in many ways an admirable sentiment and again it is one likely to expand our rage of moral sympathies but there is something called a kind of misplaced compassion that Tocqueville is very fearful about. Compassion is a virtue but it carries with us--with it, like every virtue, its own particular forms of misuse, for example, when compassion becomes a standard by which to express our forms of moral superiority to others. Consider the following. To be accused today, particularly in places like college campuses, to be accused of insensitivity to others, to some kind of moral insensitivity, is among many of us considered one of the worst moral crimes imaginable. We must all care or at least we must all pretend as if we care, yes, or must be seen to care about the plight of others much worse off than ourselves and the result of this, and I think this is Tocqueville's point, seems to be to create new moral hierarchies of compassion where one's superiority is demonstrated by our heightened sensitivity and feeling for others. And it is precisely this kind of misplaced compassion, asking the question who is the most sensitive among us, a very Rousseauian type question, this kind of misplaced compassion that is, I think, one of the psychological determinants of what we would call today "political correctness," obviously a term Tocqueville does not use, but you might think of the way in which the language of pity, compassion, sensitivity, has so much shaped our moral vocabulary, ways of thinking about ourselves and judging others. If you don't believe me, watch almost any daytime afternoon show like Oprah or any of these other shows and you'll see exactly what I'm talking about and of course you've all seen these shows, I think many more times than I have but nevertheless--compassion. This is the first or one feature of democratic social life but it is not the only one. It is connected or at least it exists alongside another. At the core also of the psychological life of modern democratic citizens, Tocqueville writes, is a profound sense of uneasiness, of anxiety, that Tocqueville calls by the French term inquietude, a word that maybe is difficult to translate into English, inquietude, anxiety. In earlier translation, this was called restlessness. In this particular translation, you have restiveness to indicate the sort of perpetually dissatisfied character of the democratic soul. In many ways, the democratic soul, like democracy itself, is never complete. It is always a work in progress. And this feeling of perpetual restlessness for Tocqueville is tied to the desire for well-being and by that he means particularly material well-being. It is the desire for happiness measured largely in terms of material happiness that is the dominant drive of the democratic soul. In many ways, Tocqueville brings to his analysis of democratic restiveness--you can see in it something of the aristocrat's disdain for the acquisition of you might say mere material goods for which most of us have to work so hard to acquire. Perhaps more than anything else this is what perplexes Tocqueville about democracy. Democracy meant for him predominantly a kind of middle class way of life, bourgeois life made up of people who are constantly in pursuit of some obscure object of their own desires. Consider the following passage, one of my favorite from the entire book, from a chapter entitled, "Why the Americans Show Themselves so Restive in the Midst of their Well-Being." Let me read it at some length. "In the United States," he says, "a man carefully builds a dwelling in which to pass his declining years and sells it while the roof is being laid. He plants a garden and he rents it out just as he was going to taste its fruits. He clears a field and he leaves it to others to care for the harvesting. He embraces a profession and quits it. He settles in a place from which he departs soon after so as to take his changing desires elsewhere. Should his private affairs give him some respite, he immediately plunges into the whirlwind of politics and when, toward the end of a year filled with some leisure still remains to him, he carries his restive curiosity here and there within the vast limits of the United States, carrying his restive curiosity wherever he may go. He will thus go 500 leagues in a day in order better to distract himself from his happiness." What a wonderful phrase, "to distract himself from his happiness. Death finally comes and it stops him before he has grown weary of this useless pursuit of complete felicity that always flees from him." Does that passage sound like anything we may have read here? Does it not sound as if it is modeled almost exactly after Plato's description of the democratic soul in Book VIII of the Republic, a person who is constantly moving, constantly restless, constantly unable to concentrate or to bear down on the one or very few things that give life a sense of wholeness and meaning and integrity? Here is the democratic man, restive in the midst of well-being, constantly moving ahead or moving to, as he says, distract himself from his own happiness. Tocqueville writes here, it seems, with a kind of disdain for a life understood as a constant and, in his view, self-defeating pursuit of happiness. The desire for well-being you might say becomes the right--almost the right of the democrat and the more one desires happiness the more it eludes our grasp. In the sentence just after the passage I just read, Tocqueville says, "One is at first astonished to contemplate the singular agitation displayed by so many happy men in the midst of their abundance." And you can sense Tocqueville's irony in his use of the term "so many happy," the distractions, the agitation, complete agitation displayed, he says, by so many happy individuals in the midst of their abundance. There's a world of social commentary condensed into those sentences. His combination of words like "agitation" and "abundance" in the same--again in the same context as the pursuit of happiness indicates for him that this way of life is more likely to bring frustration and anxiety than it is to bring us satisfaction and repose. And he traces this continual restlessness back to what seems to be for the democratic social--for the democratic individual the virtual obligation to be happy. I would ask you in this context if you have some time to read Darrin McMahon's wonderful new book on A History of Happiness to give you a little bit of an indication of the way this term has been used throughout its history and the way in which in many ways the obligation to pursue happiness, to restiveness, that kind of restiveness, is the source of so much, as he puts it, singular--the singular melancholy, he says, that the inhabitants of democratic lands often display amid their abundance. Life, liberty, the pursuit of happiness have become what one person once called a kind of joyless quest for joy and this is the second feature, this restless or restive character of democracy. And finally, the third feature of democratic psychology that I want to focus on is this idea of self-interest or self-interest well understood as Tocqueville calls it. This is a doctrine with which everybody is familiar from courses on moral psychology, on utilitarianism, to modern courses on--from--in economics and game theory and other things where the term "self-interest" is regarded almost as sort of a talismanic--has almost talismanic properties of explaining all kinds of human behavior. But Tocqueville means something very specific by self-interest or self-interest well understood. It is in one sense the kind of, you might say, everyday utilitarianism, not in any strict of the term, with which we are instinctively familiar when we hear or are told things like honesty is the best policy and things like this. It seems simple and obvious enough but it in fact is a very complex and difficult history. By the time that Tocqueville wrote these chapters in the Democracy, theories of self-interest had long been a kind of staple of European moral philosophy going back to the seventeenth century at least, going back to people like Hobbes and others. The question is what work does this idea, this concept, of self-interest rightly understood do for Tocqueville? In the first place, he understands it somewhat differently than, I think, we would. When we hear the term "self-interest," we are likely to think of it as opposed to or to think of its antonym as indicating some kind of altruism. While interest or self-interest is thought of as inherently self-regarding, altruism or something like that is an other regarded--an other regarding disposition, regarding the welfare, well-being of others. But when Tocqueville talks about self-interest, self-interested behavior was put forward by him as a kind of comprehensive antonym to behavior motivated by vanity, by honor and, above all, by the concept of glory, terms, remember, thinking--going back to Hobbes in some way and Hobbes' concern to replace ideas of vanity, vainglory and pride with a notion of fear of death, a kind of self-interested behavior. While glory was for Tocqueville and others associated with war and warlike pursuits, interest, self-interest, was invariably associated with commerce and peaceful competition. In contrast, in other words, to the aristocratic concern with fame and honor, interest was regarded--self-interest was regarded as a relatively peaceful or harmless disposition leading us to cooperate with one another for the sake of common ends. The pursuit of self-interest has a kind of unmistakably democratic and egalitarian impulse behind. The pursuit of self-interest is something literally everyone is able to follow even while such things as honor and glory seem to be by nature unequally available to different people. And into this debate between an ethic of honor and glory and an ethic of self-interest or self-interest rightly understood enters Tocqueville and his Democracy in America. He begins his chapter called, "How the Americans Combat Individualism by the Doctrine of Self-Interest Rightly Understood," with the following sentence, with the following observation. He writes, "When the world was led by a few wealthy and powerful individuals, these liked to form for themselves. They liked to form for themselves a sublime idea of the duties of man. They were pleased to profess that it is glorious to forget oneself and that it is fitting to do good without self-interest like God Himself. This was the official doctrine of the time in matters of morality, speaking of aristocratic ages. I doubt that men," he says, "were more virtuous in aristocratic centuries than in others but it is certain that the beauties of virtue were constantly spoken of. Only in secret," he concludes, "did men study its utility." You might think about that passage perhaps in section but note that Tocqueville adds to the concept of self-interest this idea or this modifier of well understood. What does this add? What is he intending that to say? Self-interest well understood is not the same thing as egoism or what Rousseau called amour-proper, for example. It is not the desire simply to be talked about, to be looked at, to be first in the race of life in that way. Rather, self-interest is connected, and self-interest well understood is connected to this passion for well-being and the desire to improve one's conditions that remained for Tocqueville a very important wellspring of human actions. But it is important to remember that these are not the only desires or these are not the only motives for action. Tocqueville probably is distinguished from many social scientists today by suggesting that self-interest well understood is not some kind of universal determinant of human behavior. It is not something universal. It is a product of a particular social state, a particular, we might say, the democratic social state. He is not in this sense a kind of moral or psychological reductionist who wants to see one cause of human behavior across all centuries and all climates. He is not saying that all behavior is self-interested. In fact, in that very chapter on self-interest rightly understood you will remember--you may remember, you probably don't remember, that he quotes in a footnote an essay by Montaigne, a name that I've mentioned before, an essay by Montaigne called Of Glory to remind the reader that the desire for fame and honor will always contend with the desire for well-being and happiness. And in many ways, these are two conflicting motives of human behavior. What did he believe that this ethic of self-interest well understood would bring about? Again, like compassion, the doctrine of self-interest has done much to sort of soften the harsher features of the aristocratic ethics, of warlike--of the warlike nobility. Self-interest well understood is a kind of antidote to an ethic of fame and glory and yet you can see throughout Volume 2 especially how Tocqueville laments the decline of this older aristocratic codes of honor and chivalry. By contrast, the doctrine of self-interest well understood is not lofty, he says, but it is clear and sure. It has characteristics of reliability and predictability. It is not itself a virtue, he says, but it can form people who are, and these are this terms, regulated, temperate, moderate, foresighted, masters of themselves, regulated, temperate, moderate, farsighted. What does that sound like? Think about that. What kind of person is this and what has it created? These are the virtues of the democratic republic. Again, these may not be heroic or extraordinary qualities but they do have the virtue of being within the range of everyone. But is such a code or is such a moral code desirable for itself? That's something that Tocqueville leaves a little bit up in the air. Of all philosophical theories, as he calls it, the doctrine of self-interest rightly understood is, he says, the most appropriate to the needs of men in our time. Think about that judgment: It is the most appropriate to the needs of men in our time. It doesn't seem to suggest that this is either universal or necessarily that it is the best. It is simply the best adapted to the needs of our time, to our level of humanity, to where we are now, and again there is a implicit- to be sure an implicit kind of critique suggested in those--in that phrase that you might think about as you read or as you go back to that important chapter on self-interest and its role in, again, the shaping of the modern democratic individual. So these three characteristics, compassion--What was the other one? What? Restlessness, yes. Well, good. I'm-- Yes. Yeah. I can't even remember what I'm talking about. Restlessness and self-interest. I was just--I was quizzing you. I was just checking. It doesn't have anything to do with short-term memory loss. These are what has shaped us and Tocqueville holds this up as a kind of portrait in a democratic individual and also of course primarily to--not so much to the democratic individual but to his readers back in France and saying this seems to be the future shape of humanity, of democratic humanity. We need both to adapt to it in some ways. We have to both recognize that this is what's coming and adapt to it but we also have to be to some degree wary of what is coming and what kind of people we may create out of ourselves, what may be created. And this brings me to the theme that I mentioned at the beginning about democratic statecraft, democratic education. What is the role of the statesman in a democratic age? How should one adapt as well as try to guide these features? Democracy in America is a work of political education, a supreme work of political education addressed to leaders or potential leaders not only for Tocqueville's time, but for the future. The possibilities of statecraft are, as they are always, dependent on what we understand politics and political science to be. What is it? In the introduction to the book, in one of those characteristically epigrammatic sentences and you should be attuned to these, Tocqueville often likes to give these one sentence paragraphs to highlight an idea, to really make it stand out. I don't recommend it for you but for him he takes one sentence and can make it--turns it into a paragraph. He talks about this book. He says, "Is a new political science for a world altogether new." That statement has to jump out at you off the page. What is this new political science? A new political science, again, in some ways following Machiavelli who departs from the ancients but perhaps also from his modern predecessors too like Machiavelli and Hobbes or Locke and Rousseau. What is the distinguishing feature of the political science for a new democratic age, for a world altogether new? Tocqueville's new political science, let me suggest to you, is based on a novel and profound understanding of the relationship between history or historical forces and human agency, between individual power- individual powers or agency and historical forces. Let me try to explain what I mean by that. As any reader of the Democracy quickly notes, even from the opening pages of the book, Tocqueville attributes a kind of providential power to history. The immense, centuries-long progress or transition from the aristocratic to the democratic era seems to be, as he describes it, almost an act of divine providence, almost of divine will. He warns his readers that it is a mistake, it is self-defeating to try to resist or to turn back this movement. This would be futile. It would not only be futile. He even suggests it would be impious, it would be in some ways to go against the will of God, as if the hand of God were behind this immense historical progress or process. Tocqueville no doubt deliberately overstates that argument but he does so, I think, in order to make a serious and profound point. Our politics are deeply embedded within long structures of human history that we can do little to alter and escape. We seem to be deeply embedded, we as individuals, deeply embedded within these structures. This is, to use a term that modern political scientists often use to describe this, it is an argument from what is often called path dependency, that we are again deeply embedded within historical processes, tendencies, paths of development that we can do little to resist or control. And in many ways Tocqueville often--you will find Tocqueville often writing as if he is some kind of historical or sociological determinist allowing little room for individual initiative or agency. Words like "fate," "destiny," "tendency" are frequently used throughout the book to underscore the limits of political action. It would even be an interesting experiment to go through the book, page by page, and find how many examples of those kinds of words, "tendency," "fate," "destiny," these kind of deterministic words that suggest irresistible movements, movement of history, how many times he uses these and in what context. And he frequently offers predictions throughout the book on the basis of what he regards to be underlying historical and social trends. You can hardly read a page of the book, sometimes not even a paragraph, without finding in it some kind of prediction based on these trends. Again, I would ask you if you have time go through the book. You don't have time this semester. Maybe it'll be a great senior essay later on, to go through the book and find examples, as many as you can of again predictions that Tocqueville makes on the basis of these historical--these--this claim about historical forces. And much of this seems to deny, taken literally, much of this would seem to deny the role of independent human initiative or statecraft in history. Consider the following passage from pages 154 and 155 of your translation. Here is what Tocqueville writes about the statesman. He says, "Sometimes after a thousand efforts the legislator succeeds in exerting an indirect influence on the destiny of nations and then one celebrates his genius whereas often the geographical position of the country about which he can do nothing, a social state that was created without his concurrence, mores and ideas of whose origin he is ignorant, a point of departure unknown to him, in part irresistible movements to society against which he struggles in vain and which carry him along in turn." There you go. He gives us a list of all of the different determinants of human surrounding conditions, geography, mores, social position. These, he says, impart an irresistible movements to society. There is that kind of deterministic language again against which he, the statesman, can do nothing and yet he begins this--he begins that little statement by saying that after a thousand efforts he succeeds in exerting an indirect influence on the destiny of nations and then he is celebrated as a great genius. You can see Tocqueville's irony and what appears to be downplaying the abilities or the role of the legislator, the statesman, to effect change of any significant kind. I don't like to be political but one might wonder what our President would have made of that had he read that passage or thought about it or those around him had thought about it a couple of years ago before our current miseries began. Anyway, this passage almost seems to be mocking the claims of Machiavelli or Rousseau who saw the ability of a new prince or a legislator to found peoples and institutions. Tocqueville seems to regard that the legislator can do relatively little on his own but is strongly hemmed in by a host of factors, geography, social customs, morality, again, over which one can do little. The legislator is more like a ship's captain dependent on the external circumstances that control the fate of the ship and he even goes on to say the legislator resembles a man who plots his course in the middle of the ocean. Thus, he can direct the vessel that carries him but he cannot change its structure, create winds or prevent the ocean from rising under his feet. All of this seems to be on the side of those historical features that limit what we can do. Yet, if Tocqueville often writes as if the statesman is hemmed in by these kinds of circumstances, he also, and you see this especially throughout Volume 2, strongly opposes all systems, all intellectual or philosophical systems of historical determinism, that deny to us the power of human agency. While he sometimes writes to shame or to humble the pretensions of human greatness, he is just as concerned about the tendency, in fact the very dangerous tendency toward self-abnegation that denies the role of the individual in politics and history. He often writes as if it is the peculiarity of democratic times when all peoples are considered equal and therefore all of us considered equally powerless to effect or change anything. And again, I would ask who has not felt this way at some time, maybe all the time, that with all of us being, again, more or less equal no one seems to have the power, a kind of singular power, to effect any great social change. There is one wonderful chapter among others but I'll just mention one. Look at the chapter, and I can't recall offhand the exact number of the chapter, but the one called "On Historians," on the role of historians in democratic and aristocratic times, and how he shows us that in aristocratic ages--he's thinking particularly of the ancient world--historians attributed to extraordinary individuals all the power to affect nations and change nations but in democratic times, in modern times, we tend to think of historians, one might also take his term "historian" to include social science as well. We tend to project systems which deny the power, the unique power of the individual. We are all products of vast, you might say, historical or causal circumstances over which the individual has no control. Think of the way in which Marxism, again, denies the power of the individual, Freudian analysis sees all of our desires and motives as determined by forces over which we have little control, all kinds of economic theories of development, again, which see us all acting under certain kind of uniform rules of human behavior. Where is the room for the individual? That chapter is a wonderful illustration of Tocqueville's general point. So what is then his teaching and, more specifically, what is his advice for the statecraft of the future? And it seems by the end of the book Tocqueville is walking on a very narrow tightrope. He wishes to convince his contemporaries that the democratic age is upon us, that the transition from aristocracy to democracy is irreversible, that it cannot be resisted, and that what he calls the democratic revolution is an accomplished fact, and yet at the same time he wants to instruct us that what form democracy will take in the future will very much depend on will, on intelligence, on what he sometimes calls enlightenment, and especially on individual human agency, what form democracy will take. Democracy may be inevitable. Equality, the age of equality, may be inevitable but democracy is not of all one piece. It depends not just on impersonal historical forces but on what you might call "the active virtue and intelligence of individuals" ranging from self-interest rightly understood to honor and ambition. Democracy can still take many forms and whether it will favor liberty or be favorable to liberty or to some kind of collectivism is for him very much an open question, what form democracy will take. And Tocqueville returns to this theme, his very, very important theme, in the last, very last paragraph of his book. "I am not unaware," he tells his readers, "that several of my contemporaries have thought that peoples are never masters of themselves here below. There is little we can do. And that they necessarily obey I do not know which insurmountable and unintelligent force born of previous events, the race, the soil or the climate." "Those," he says, "are false and cowardly doctrines that can never produce anything but weak men and pusillanimous nations." That is to say, these doctrines of historical determinism have an actual effect on people. It makes us weak. It makes us cowardly. It makes us--it makes entire societies and it enervates entire society and yet he continues, "Providence has not created the entire race entirely independent or perfectly enslaved. It traces it is true," he is speaking about providence. "It traces, it is true, a fatal circle around each man that he cannot leave but within this vast limits man is powerful and free, so too, he says, with people. Tocqueville leaves us, in other words, not with a solution but rather with a paradox or I would say a challenge for us to consider. We are determined but not altogether so. The statesman must know how to navigate the shoals between historical, social and cultural forces over which we have no say and those matters of institutional design and moral suasion that are still within our power to effect. Politics, as intelligent people have always known, which is not to say all people to be sure but as intelligent people have known, is a medium that takes place within language. It is a matter of providing people with the linguistic and the rhetorical abilities both to construct their pasts and to imagine their futures. It is language, going back to Aristotle, it is logos, it is language that gives us a latitude, an ability to adapt to changing circumstances and to create new ones. Tocqueville provides us living in a democratic age with the language to shape the future of democratic societies. What we do with that language, how we apply it to new circumstances and conditions that Tocqueville could never have imagined, will be of course entirely up to us. And on that note I have to end Tocqueville and Wednesday I'll see you for our last class and I'm going to talk about where we go from there.

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13_The_Sovereign_State_Hobbes_Leviathan.txt

Professor Steven Smith: Where else are we? Today we're going to continue the state of nature, Hobbes' most famous discovery, his most famous metaphor, his most famous concept. At the end of class last time, I tried to identify Hobbes' central problem, is the problem of authority, what makes authority possible, what makes authority legitimate, and in order to answer that question, I suggested, he created this idea, this metaphor again, of a state of nature, a state in which he says we are naturally in. Hobbes' state of nature is virtually the opposite of Aristotle's conception of the natural end or the natural telos of man. It does not consist of our perfection, a condition of our perfection as Aristotle believed, but for Hobbes the state of nature is something like the condition of human life in the absence of authority, in the absence of anyone to impose rules, order, law on us. What would human beings be like in such a condition, a condition of the type that he imagined maintains in periods of crisis, civil war of the kind that was true of England in the 1640s? And I suggested at the end of last time that in many ways Hobbes' idea of the state of nature can be understood in a sense as an extension of his scientific methodology set out in the opening chapters of the book. Let's imagine, as he says, human beings as if they were in a sort of laboratory test tube. Let's strip human beings of all their social ties and customs and traditions. Let's see what they would be like in abstraction from all of the social and political relationships which they enjoy and see how they would interact with one another almost as chemical properties. And you can see Hobbes working along that line but I would say this as it were scientific or proto-scientific conception of the state of nature is not the whole answer to this story because underlying Hobbes' conception of the state of nature is a powerful moral conception, a moral idea of the human being, and that's what I want to talk a little bit about today. Hobbes is a moralist, which seems odd in some ways. How could grim and dour old Thomas Hobbes be regarded as a moralist or someone with a moral conception of human nature and the human condition? But that's what I want to suggest to you today. The term, in a sense in which we might better characterize his conception of the state of nature, is one of individuality. Hobbes shows us what it is to exercise the qualities of moral agency; that is, to say to do for ourselves rather than to have things done for us or for you. Hobbes introduced into our moral language the idiom of individuality. And this concept, the concept of what it is to be an individual, a moral agent, isn't really--is really not older than or at least not much older than the seventeenth century. Until the Renaissance or not much later, people considered themselves primarily not as individuals but as members, members of a particular family, of a caste, of a guild, of a particular religious order, of a city or so on. The idea that one is first of all a self with an "I," an ego, would have been regarded as unintelligible and even as late as the nineteenth century Alexis de Tocqueville in Democracy in America says, "individualism is a recent expression arising from a new idea." That idea appeared new to Tocqueville as late as the nineteenth century and this idea of the individual, I want to suggest, is at least in part and maybe in large part traceable back to Hobbes. What is Hobbes' individual? Hobbes conceived us through a process of abstraction from the web of attachments in which we find ourselves. We are beings, he argues again in the opening chapters, whose fundamental characteristics as human beings are willing and choosing. We are beings for whom the exercise of the will is a preeminent feature and much of our happiness as human beings depends upon our capacity to exercise our will and our ability for choice. Life for Hobbes is an exercise in continual willing and continual choosing that may be temporarily interrupted but can never come to an end except with the end of life itself. Hobbes' individuality or individualism is closely connected to this conception of a human being or human well-being as success in the competition for the goods of life. "Continual success," he writes in chapter 6, "continual success in obtaining those things which a man from time to time desireth is what is called happiness or felicity. Our well being depends on our ability to achieve the objects of our desires, the objects of our choices, for there is no such thing," he continues, "as perpetual tranquility of mind, no such thing as perpetual tranquility, while we live here, because life itself is but motion and can never be without desire nor without fear no more than without sense." These are the characteristics of human life, sense, fear and desire, continual desire for one thing after another, and for Hobbes this fact is not simply a physical or factual description of human behavior but it is a moral condition because we are each of us bundles of activity and initiative, of likes and dislikes, of desires and aversions. Life for Hobbes is competition or struggle not just over scarce resources, although that might be part of the struggle, but for honors, for anything else that a person might value or esteem. Hobbes is fascinated and, is again like Montaigne and a number of others, he is fascinated with the diversity, the sheer diversity, multiplicity of human desires. What leads one person to laughter, leads another person to tears, what leads one person to piety and prayer, leads another person to ridicule and so on and so on. Even moral terms, Hobbes says, terms like "good" and "evil," he says are expressions of our individual likes and dislikes. We like something, he says, not because it is good but we call something good because we like it and the same with other moral qualities and attributes. They are expressions for him of our psychological states and aspirations and it is this individualism that is the ground of the general competition that we all experience for the objects of our desires that he says the--or from this he infers that the natural condition is one of competition, of struggle, of enmity and of war. In a famous passage from chapter 11 he posits, as he puts it, "a general inclination of all mankind, a perpetual and restless desire of power after power that ceaseth only in death." This is, as he puts it, "a general inclination of all mankind," this constant restlessness and motion and expression of our individuality and what I have been calling Hobbes' individualism is connected, in fact even is underwritten by another attribute that is central to Hobbes. It is his skepticism. Like many of the great early modern philosophers, Montaigne, Descartes, Spinoza, Hobbes was obsessed with the question about what can I know or, maybe put a different way, what am I entitled to believe, and there are many passages in Leviathan that testify to Hobbes' fundamentally skeptical view of knowledge. Right? He is a skeptic not because he believes that we can have no foundations for our beliefs whatever but he is a skeptic in the sense that there can be no, on his view, transcendent or nonhuman foundations for our beliefs. We cannot be certain, he thinks, of the ultimate foundations of our knowledge and this explains, you may have wondered about this, this explains the importance he attributes to such things as naming and attaching correct definitions to things. For reason, he writes in a famous passage, "for reason is nothing but reckoning, that is adding and subtracting the consequences of general names agreed upon." Knowledge, in other words, is for Hobbes a human construction and it is always subject to what human beings can be made to agree upon and that skeptical view of knowledge or at least skeptical view of the foundation of knowledge has far reaching consequences for him. If all knowledge, according to Hobbes, ultimately rests on agreement about shared terms, he infers from that that our reason, our rationality, has no share in what Plato or Aristotle would have called the divine Noos, the divine intelligence. Our reason has within it no spark of divinity. Our reason does not testify to some kind of inner voice of conscience or anything that would purport to give it some kind of indubitable foundation. Such certainty as we have about anything is for Hobbes always provisional, discovered on the basis of experience and subject to continual revision in the light of further experience, and that again experiential conception of knowledge. That kind of skepticism about the foundations of knowledge has further implications for Hobbes' views on such things as religion and religious toleration. "There are no signs or fruit of religion," he says, "but in man only," he says in chapter 12. That is to say, the causes of religion can be traced back and are rooted in the restlessness of the human mind in its search for causes. And it is because, he says, we are born ignorant of causes, we are ignorant of the causes of things, that we are led to search out beginnings and origins and this leads us ultimately, he says, to posit the existence of God who is, so to speak, the first cause of all things. Hobbes does not, despite this kind of rationalistic view of religion and his view that religion has its origin again in the restlessness of the human mind, Hobbes doesn't deny the possibility of revelation or some kind of direct communication of God to us. But what he does deny is that anyone who has claimed to receive such a revelation, he denies that any such person has the right to impose that view on anyone else because nobody else can correctly have the means to verify a person's claim to revelation. No one can impose their claim of revealed knowledge on another. Does this make Hobbes an atheist, as many would have maintained in his day? No. It makes him a skeptic about revealed religion. So it is because of this individualism and skepticism, a view of life as willing and choosing, that there are in the state of nature so to speak no standards to adjudicate conflicts, that the central issue of politics arises, namely what makes authority possible, how are people who are biologically individually constituted, so to speak, how can any of them ever--any of us ever be capable of obeying common rules or having moral obligations to one another? How is that possible, Hobbes continues to ask in a manner of speaking on almost every page of the book. But before answering that question, consider a little further Hobbes' account of the state of nature and what makes it seem like a plausible starting point to answer the question of what makes authority possible. To say that the state of nature consists primarily of individuals with again diverse likes, dislikes, beliefs, opinions and the like is not to say that the state of nature is a state of isolation, as it sometimes attributed to him. People in the state of nature may have regular and continual contact with one another. It is just that their relations are unregulated. They are unregulated by law; they are unregulated by authority. The state of nature is simply a kind of condition of maximum insecurity, an unregulated market with no common laws or rules to sustain it. The emphasis on the individual is just another way of saying, again unlike Aristotle, that no one has natural authority over anyone else. Relations of authority exist only by, so to speak, the consent or the will of the governed. And the fact that relations in the state of nature are unregulated for him makes it--it's synonymous with making it a condition of war, of "all against all," in his famous formulation. Now, you might look at that formulation, the state of war is one against--of all against all and you might say that such a condition of civil war, of maximum insecurity, of the total breakdown of condition of rules and laws is if anything the state of the exception. How often does that really occur in our experience in human life? But Hobbes, like Machiavelli, as we saw, likes to take the exceptional situation and turn it into the norm. It becomes the normal condition, state of security, insecurity, fear, conflict and the like. This is not to say, again, that the state of nature for Hobbes is one of permanent fighting. But it is one of permanent fear and distrust and he asks his readers…there are so many wonderful passages in this book, this just happens to be one of my particular favorites, he asks his readers if you don't believe me, again think of his skepticism, don't believe me, he says, check your own experience and see if I'm not right. And this is what he writes. "Let him, the reader, therefore ask himself," Hobbes writes, "when taking a journey he arms himself and seeks to go well accompanied. When going to sleep, he locks his doors even when in his house, and even when in his house he locks his chests and this, when he knows, he says, there be laws and public officers armed to avenge all injuries shall be done to him. What opinion, Hobbes asks, he has of his fellow subjects when he rides armed? What does that say about your thinking about your fellow citizens when you arm yourselves going for a trip, of his fellow citizens when he locks his doors at night or of his children and servants when he locks his chests? Does he not therefore as much accuse mankind by his action as I do by my words?" You can see the mischievousness of Hobbes in that delightful passage. What about you, he says, and this is not in some kind of state of nature. This is in a completely fully functioning society when you go armed, when you lock your doors, when you lock your chests at night, don't your actions and your experience simply confirm what I'm saying? And this tells us another thing about the state of nature which it is easy to forget. The state of nature, at least for Hobbes, is not some kind of primitive anthropological datum that we find by going back in time somehow. Rousseau will speak about it more this way. For Hobbes, the state of nature exists, he says, whenever authority is not enforced. The state of nature fully continues, in many ways, oddly even in civil society, he says, whenever we have reason to believe that our lives or our properties or ourselves are not secure. In fact, we can never be fully free of the fear and of the anxiety and uncertainty of the state of nature, even within to some degree of fully constituted civil society. The only exception to this of course in Hobbes' account of the state of nature when he says "don't you lock your doors at night" are of course Yale students living here on campus who are so trusting that they never lock their doors at night in the entryways and so on and then of course are always stunned to find when something is stolen from them, how could this be? And I tell them lock your doors but they still don't believe me. Maybe you'll now believe Hobbes if you don't believe me. So the state of nature, it's a state of insecurity, it's a state of conflict. How do we get out of it? This is of course the huge issue that Hobbes asks for the rest of--for much of the book. What do we do to get out of this state of nature to enter a condition of civil society and civilized life? How do I give up my right to do whatever is in my power to secure my person or my possessions, when I have no expectation, you might say, that others around me are prepared to do so as well? This is sort of a classic example of what economists and other people like them call the prisoner's dilemma. Why should I act in such a way if I have no expectation or reasonable expectation that those around me will reciprocate? Hobbes' members of a state of nature seem to be in a classic prisoner's dilemma problem. Maybe we can say, we could say or Hobbes could say, that laying down our right to do all things in seeking peace with others is the rational thing to do in the condition of nature. We are all rational actors and therefore it is rational for us to seek and to desire peace, but note that that is exactly what Hobbes does not say, he does not say this. Far from having a sort of rational actor model of politics, he operates with an irrational actor model. He assumes that it is not reason but our passions that are the dominant force of human psychology, our desires, our aversions, our passions. And although I have said that Hobbes has emphasized the diversity of our passions there are still two main passions that he feels universally dominate human nature and these two passions are pride and fear. Pride and fear, these are the Hobbesian equivalents of the two great--what Machiavelli called humors you remember, the two humors of the two great social classes, the desires of the rich and powerful as it were to rule over others and the desire of the weak not to be ruled. Machiavelli called those the two umori, the two humors. And Hobbes similarly works with a kind of model. He's a great political psychologist, the two great passions of pride and fear. Pride, he says, is the passion for preeminence, the desire to be first and also to be seen to be first in the great race of life. Prideful people, he tells us, are those overflowing with confidence about their own abilities to succeed and we all know people like this, don't we, like Yale students? They're all overflowing with confidence, kind of alpha types. Machiavelli might call them sort of manly men who are fully confident about their abilities. And yet Hobbes is a great debunker of human pride. Pride is equivalent to what he calls vanity or vainglory. It is a kind of exaggerated confidence in one's own power and ability. It is pride, the desire to lord it over others and to have one's superiority acknowledged by others, that is the great problem for Hobbes to be averted. But if pride for him is one of his great universal passions so is its opposite, fear. Hobbes makes the fear of death that may come to us at any time in the state of nature, perhaps he exaggerates this, by making it appear that the state of nature is a kind of existential condition in which death can come to you at almost any moment. But there is more to fear than this, simply fear of death, although Hobbes emphasizes and dramatically perhaps overemphasizes this. Fear is not just the desire to avoid death but to avoid losing, you might say again, in the great race of life, to avoid losing and to be seen as a loser. It is the desire to avoid the shame of being seen by others as losing out somehow. There is a social quality clearly to both of these passions, pride and fear, one again the desire to have one's preeminence esteemed by others, fear, the desire to avoid shame and dishonor. How we are seen by others is a crucial cardinal part of Hobbes' moral psychology and each of us, he says, contain. These do not simply represent two classes of individuals, two classes of persons. Each of us contains these two warring, you might say, elements within us, both self-assertion and fear of the consequence of self-assertion. The question is for Hobbes, how do we tame these passions? It is most of all pride that Hobbes wants to tame and of course the very title of his book, Leviathan, he tell us later on comes from what? Do you remember? Where does it come from? Who remembers? Passage from what? Job, Book of Job, where he refers to Leviathan as king of the children of pride. The book is based on a biblical metaphor about overcoming or subduing pride. As the great Marsellus Wallace says in the film Pulp Fiction, pride never helps, it only hurts, if you remember that magnificent speech. Fear, Hobbes says, is the passion to reckon on, is the passion to be reckoned on. It is fear, not reason, that leads us to abandon the state of nature and sue for peace. The passions that incline men to peace, Hobbes writes, are fear of death. This is not to say that Hobbes believes fear to be the naturally stronger of the two passions; in fact, far from it. There are many people certainly even around us who Hobbes believes do not fear death as they should, the proud aristocrat who prefers death before dishonor, the religious zealot prepared to sacrifice his life and of course those of others in order to achieve the rewards of heaven and of course just the risk taking individual who seeks to climb Mount Everest just for the honor and esteem involved. And it is part of the broader educational or pedagogic function of Leviathan to help us see, Hobbes thinks, the dangers of pride and the advantages of peace. Properly directed, fear leads to peace. Fear is the basis, even of what Hobbes calls the various laws of nature, that lead us to civil society. The laws of nature for Hobbes are described as a precept or a general rule of reason that every man ought to endeavor peace and it is out of fear that we begin to reason and see the advantages of society; reason is dependent upon the passions, upon fear. The first and most fundamental law of nature, he says, is to seek peace and follow it. Not only should one seek peace but we have an obligation, he says, to lay down our arms, to lay down our right to all things on the condition that others around us are prepared to do so as well. And Hobbes goes on to enumerate 19 laws of nature, I won't go into all of them, 19 laws of nature that constitute a kind of framework for establishing civil society. These laws he even compares as his equivalent of the Golden Rule which he states in the negative: Do not do unto others what you would not have them do unto you. Here is Hobbes' rewriting of the Golden Rule in terms of these laws of nature but these raise a question for us as readers of Hobbes. Right? Don't they? What is the status of the laws of nature? What is the moral status, if any, of these laws? Hobbes, as we see, sometimes writes as a sort of scientist or proto-scientist for whom nature and one supposes the laws of nature operate with the same kind of necessity as the laws of physical attraction. That's how he often writes about human behavior, that we obey the same laws of physical attraction as do any other bodies that we might choose to describe. They describe how bodies in motion always and necessarily behave, these laws of nature. And yet at the same time, Hobbes writes as a moralist for whom the laws of nature, he calls "precepts of reason" or general rules according to which we are forbidden to do anything destructive of life." In this sense, the laws of nature, as he describes them, appear to be moral laws with moral commands, commands you not to do anything that is destructive of life, your own or that of others, and these moral laws, in this sense, we have presumably the freedom to obey them or disobey them. If they acted with a kind of mechanical necessity or even geometric necessity, they could not possibly be moral laws in that way. They can only be moral if there is some semblance of human choice or will expressed in the relationship, our ability to do otherwise. So these laws of nature, seek peace and so on, do not simply seem to be descriptive of how people do behave. They seem to be prescriptive of how people ought to behave and this Hobbes even suggests at the end of chapter 15 when he writes about the laws of nature, "these dictates of reason men used to call by the name ‘laws' but improperly for they are conclusions or theorems according to what conduces to the conversation of mankind." These used to be called laws of nature, he says, but improperly. So if they are only improperly laws of nature why does Hobbes continue to use the term? Why does he use this terminology of "laws of nature"? In a sense, this might simply be Hobbes' way of paying homage to the ancient tradition of natural law going back to the medieval scholastics, to the stoics, and perhaps even beyond them. The natural laws for Hobbes are not divine commands or ordinances, he says, but they are rules of practical reason figured out by us as the optimal means of securing our well-being. These laws of nature, as he describes them, do not issue categorical commands so much as sort of hypothetical rules. If you want X, do Y; if you want peace, here are the means to it. And he calls these laws, these 19 laws of nature, the true and only moral philosophy. So you can see in that passage Hobbes takes himself to be a moralist writing within the great tradition of moral philosophy. These laws of nature are for him the true and only moral philosophy. Well, this brings me to some criticisms or at least some questions about Hobbes' conception of the laws of nature. What are we to make of these laws, as I've asked before? In one sense, there seems to be a genuine moral content to Hobbes' laws of nature which can be reduced to a single formula: Seek peace above all other goods. Hobbes, more than anyone else, wants us to value the virtues of civility. Those, you might say, summed up in a word are what the 19 laws of nature command. The civility entails the virtues of peace, equity, fairness, playing by the rules. Peace is for Hobbes a moral good and the virtues are those qualities of behavior that tend to peace and vices are those that lead to war. Consider the disadvantages of war and the benefits of peace. Here is what Hobbes writes. "In such a condition, that is the state of nature, there is no place for industry because the fruit thereof is uncertain and consequently no culture of the earth, no navigation nor building nor instruments of moving and removing things as require much force, no knowledge of the face of the earth, no account of time, no arts, no letters, no society and which is worst of all continual fear and danger of violent death." This is again the sort of existential condition in which Hobbes wants to put us in the state of nature and all the benefits he lists there, he enumerates, that are denied to us in such a condition, again no knowledge, no geography, no cultivation of the earth, no navigation or building. All of these things are the fruits of peace, he tells us. But at this point, a careful reader such as all of yourselves no doubt, would no doubt be suggesting, I've gone too far in suggesting or calling Hobbes a moral philosopher whose motto in a way could be summed up in the phrase "Give peace a chance." Is that what Hobbes believed? Why is the peace the highest good anyway? Why not justice? Why not honor? Why not piety? Why not the examined life? What makes peace so good for Hobbes? Well, I've given a number of… have quoted him on a number of reasons but one suggestion might be that it is not so much peace alone that Hobbes cherishes as life. Peace is a means to life. Every creature, he says, has a built-in desire to preserve itself, to persevere in its own existence, to continue in its own steady state you might say, and to resist invasion or encroachment by others. We are all endowed, he says, with a kind of natural right to life and the desire to preserve oneself is not just a biological fact, although it is also that, it is for him a moral right, it is a moral entitlement, every being has a fundamental right to its own life. We not only have a right to our lives but to do whatever we regard as needful to protect our lives. And again, this is not simply a brute fact of nature. It is a moral entitlement for Hobbes, the source of human worth and dignity. But now you will suggest, I've really gone too far, attributing to Hobbes a doctrine of human dignity that one might expect to find in a philosopher like Kant or someone else. Didn't Hobbes cynically write in chapter 10, "the value or worth of a man is of all things his price," what price we will fetch in the marketplace no doubt, the value or worth of a man is his price, a phrase incidentally quoted by Karl Marx to indicate the sheer heartlessness of the kind of the bourgeoisie society that Hobbes was hoping to bring about. And doesn't Hobbes' materialism and his sort of mechanistic theory of nature seem to detract from any inherent worth of the individual? There seems to be something to that and yet Hobbes certainly regards life as a precious good, perhaps the most precious good of all, and he writes with a lively sense of how fragile and endangered life is. The work as a whole can be seen as an effort to dispel what he believes to be false beliefs, false doctrines and beliefs, that disguise the truth from us, truth about the value of life; for example, beliefs about the afterlife and all beliefs that detract from an appreciation for the value of life as it is. This provides the moral basis of what I would call Hobbes' humanitarianism and yet that humanitarianism seems to raise further problems. Doesn't Hobbes or does Hobbes' attempt to instill in us, the readers of his book, his attempt to instill in us an appreciation for life and the value of life, does this simultaneously create an aversion to risk, an extreme fear of conflict and challenge or disorder? You could say is this constant fear that Hobbes harps on fear of death and the value of life, to put it rather rudely, is this not another word for cowardice? Does Hobbes' emphasis on the preservation of life as the supreme moral value, does this turn his mighty Leviathan into a kind of commonwealth of cowards? Where Aristotle made the courage of men in combat a central virtue of his ethics, Hobbes pointedly omits courage from his list of the moral virtues. At one point, he even suggests that courage is really just a species of rashness and his example of courage comes from duels and duel fighting which he says will be always honorable but are unlawful. "For duels," he says, "are many times effects of courage and the ground of courage is always strength or skill though for the most part," he says, "they be effects of rash speaking and the fear of dishonor in one or both of the combatants." In other words, courage for him again is a form of vanity or pride, the desire not to appear less than another. It is a form of rashness, he says. And that suspicion is further carried out in Hobbes' very interesting treatment of military conscription which he talks about in chapter 21. There he describes battle, as he says, "a mutual running away" to armies confronting one another he describes as a mutual running away, and furthermore he says when it comes to conscription there should be allowance made for those that he calls "men of natural timorousness," cowards in other words. A man that has commanded as a soldier, Hobbes writes, to fight against the enemy though his sovereign has the right enough to punish his refusal with death may nevertheless, Hobbes writes, in many cases refuse without injustice as when he substituteth a sufficient solider in his place. In other words, Hobbes' view of this is why do the fighting yourself, if you can get someone else to do it for you? There is no intrinsic virtue in courage or battle, if you can get somebody else to do the job for you, a sort of perfect description, I think, of our volunteer army, how we pay people to do this difficult and dangerous work for us. But the question is, can even a Hobbesian society, one which insists on rules and so on, can a Hobbesian society do entirely without-- Professor Steven Smith: Anyway, can a Hobbesian society do without what we might call them the manly virtues, the civic virtues, pride, love of honor that Hobbes seems to condemn? Consider the case of Ralph Esposito. Who is Ralph Esposito, you ask? His name is not in the index of Hobbes' book but Mr. Esposito is a New York City fireman who came to Branford College to be a Master's Tea guest not long after 9/11 and at length he discussed there people like himself who daily risk their lives running into building burning--burning buildings to rescue total strangers. Why do people do this? Is it because some people have a kind of built in sense of thumos, that wonderful Platonic term, pride, courage, love of risk that no society, not even a Hobbesian one, can do without? Even Hobbes' society presumably cannot do without a fire department or a police department; yet, if one were to follow Hobbes' risk averse psychology, if one were to follow the 19 laws of nature that advise us to seek peace and to avoid conflict, why would anyone ever become a fireman, a soldier, a risk taker, a policeman of any sort? Why would anyone ever risk one's life for one's country or a cause just to help other people, people that we don't know and probably will never know? Even in the passage that I cited earlier, where Hobbes describes the benefits of civil society, he speaks of activities like navigation, exploration and industry. Presumably, these are activities that are all engaged in risk taking behavior of one kind or another that seem not to be able to be explained by Hobbes' law of nature alone. So the question I want to leave you with today and that I want to pick up again on Wednesday is, in the end, what do societies require more of? Do they require more of Hobbes' men of natural timorousness or do they require more Ralph Espositos? And on that we'll finish up Hobbes on Wednesday.

Introduction_to_Political_Philosophy_with_Steven_B_Smith

24_In_Defense_of_Politics.txt

Professor Steven Smith: Anyway, today, the last class, I had on the syllabus, I think it was called globalization and political theory or something to that effect and I guess since writing that I've changed the theme of this final lecture a bit and I want to talk about defending politics or in defense of politics. And I'll try to explain what I mean by that as kind of a wrap up and exhortation for this last class. In 1962, an English political scientist and journalist by the name of Bernard Crick wrote a short and very polemical and influential little book called In Defense of Politics, and by politics Crick meant a distinctive type of human activity where conflicts of interests among groups are adjudicated by discussion, persuasion and debate rather than by force or by fraud. A political society, as Crick understood it, is one where individuals and groups played by certain agreed upon rules that will determine how conflicts of interests are to be decided. Crick called this little book--very lively and still definitely worth reading--he called his book In Defense of Politics because he regarded the proper understanding of politics as being distorted by certain currents of thought and practice in his own day among which were for example the highly ideological style of politics found for example in the Soviet Union and its client state, the kinds of nationalist politics emerging in the developing world, and even in some aspects of the conservative politics of contemporary Britain of his time where that meant a kind of unreflective deference to customs and tradition. I think today it's important to try to reprise Crick's plea for a defense of politics although in a slightly different way. Politics again, as Crick understood it, is something that takes place within a certain territorially defined unit called a "state." This may seem almost too obvious to bear repeating. For centuries what is called the res publica has been regarded as the proper locus of the citizens' loyalty. It was thought to be the task of political philosophy or political science in its original sense to teach or to give reasons for the love of one's own country. Classical political philosophy regarded patriotism as an ennobling sentiment. Consider just a few of the following passages that I asked Justin to put on the board from Cicero, from Burke, from Machiavelli, from Rousseau, and from Lincoln, writers from the ancient and the modern world from many different countries and times. All make important expressions, some more extreme than others like Machiavelli's--what else would one expect from an extremist like Machiavelli's--to simpler and more dignified statements like that of Burke or Lincoln but anyway, all expressing the view that politics has something to do with providing reasons for the love of country. Today, however, the idea of patriotism, at least among philosophers, seems to have fallen upon hard times. This isn't to say that patriotism, as a phenomenon of political life, is likely to disappear. To the contrary. Go drive 20 miles or so outside of any urban area and one is likely to see flags being waved, bumper stickers on cars proclaiming the driver's love of country, country music stations playing music that tells us to support our troops and keep driving our SUVs, all signs of American patriotism to be sure. But the issue seems quite different in universities and in educated circles, you might say, where patriotism has come to appear to be a morally questionable phenomenon. Tell someone at any Ivy League university that you are interested in patriotism and you will be treated as if you have just expressed a kind of interest in child pornography. Raise the issue and one is likely to hear very quickly repeated Samuel Johnson's famous barb that patriotism is the last refuge of a scoundrel or you might even hear, if the person's read a little bit more, E.M. Forster's famous statement that if he had to choose whether to betray his friend or his country that he, Forster, wished he had the courage to betray his country. Forster, the famous English novelist, author of Howards End and other important books, Forster presents the choice between friendship over country, of private over public goods, as a kind of tragic and even noble decision that one has to make. But Forster, in some respect, has given us, I would suggest, a false dilemma. Loyalty is a moral habit just as betrayal is a moral vice. People who practice one are less likely to indulge in the other. Consider the following example. A few years after Forster made his statement at Cambridge, I believe, three young Cambridge undergraduates in the 1930s by the names of Kim Philby, Donald Maclean, and Guy Burgess, I don't know if those are names that are familiar to people here any longer but they were very, very famous names at one point, they chose to betray their own country. That is to say they acted for many years as Soviet agents and for years passed on vital secrets, English secrets, to Moscow, as they all ascended up the ladder of British intelligence services until they were finally exposed in the 1950s. And it was not long after they were exposed and they had all fled to Moscow that they began to betray one another. Loyalty it seems, like betrayal, is not a bus that one can simply get off at will. Rather, people who betray others in one area of life are likely to do so as well in others. So Forster has given us a false choice between choosing friendship over country or country over friendship and as with most matters, I think it probably makes greater sense to examine the problem through the lenses of Aristotle who tells us everything we need to know about most questions. In the Nicomachean Ethic, Aristotle taught us that all virtues, that is to say, all excellences of mind and heart, are best understood as a mean along a continuum of excess and deficiency. It is a matter of finding a balance, the proper balance, between extremes. So it might be useful to regard patriotism in this light. If patriotism is a virtue, and I ask the question "if it is," it would be important to see it as a midpoint between two contending extremes, two contending vices. What are these vices, you might say, that obscure from us the meaning of--the proper meaning of the political today? On one side, you could say, the excess of patriotism is a kind of nationalistic zeal that holds absolute attachment to one's country and one's way of life as unconditionally good. This is the kind of loyalty expressed in sentiments like, "My country right or wrong," but was given powerful expression, perhaps the most powerful expression, in a short book- another short book in this case by a German legal philosopher of the early twentieth century named Carl Schmitt. Carl Schmitt wrote a short book called The Concept of the Political in 1921 and here Schmitt drew extensively on Hobbes but rather to defend a view of the political, but rather than tying the state of war, Hobbes' state of war, to a pre-political state of nature, Schmitt saw war and also which includes the preparation for war, as the inescapable condition of human life, of political life. Man, he believed, is the dangerous animal because we can kill one another and individuals, and more importantly groups of individuals, stand to one another in a virtually continual state of conflict and war. Schmitt believed Hobbes was right in many crucial respects but where he fell down was in believing that the social contract could create a sovereign state that would put an end to war. Quite the contrary, he thought. The inescapable political fact is therefore the distinction between what he called friend and enemy, those who are with us and those who are against us. To misunderstand that distinction, distinction that goes all the way back to Polemarchus' view in the Republic, where he talks about justice being doing good to friends and harm to enemies but might obviously go on much deeper or further than that. For Schmitt, that distinction was central to what he called the political. The political, he says, and he uses that word as a noun, we tend to think of political largely in its adjectival form, but in Germany you can often use it as a noun as well. The political, he wrote, is the most intense and extreme antagonism, becomes that much more political the closer it approaches to the extreme point, that of the friend, enemy grouping, he says. Friend and enemy are the inescapable categories through which we experience what he calls the political. Life consists of that fundamental distinction. Athens and Sparta, Red Sox and Yankees, Harvard and Yale--These are fundamental groupings, enemies, friends and enemies. All humanitarian appeals, he believed, appeals to the concept of human rights, to free trade or so on, all of these are, as it were, attempts to avoid the fundamental fact of conflict and the need for a politics of group solidarity. The politics of the future, he hoped, would be determined by those who have the courage to recognize this fundamental distinction and to act upon it. At the other end, however, of the continuum of excess and deficiency, the defect, you might say, of patriotism comes to light as a kind of today what we might call transpolitical cosmopolitanism. Present day cosmopolitanism is, to a very large degree, a product of another German philosopher named Immanuel Kant writing at the end of the eighteenth century. Kant stressed, on the other hand, that our moral duties and obligations respect no national or political or other kinds of parochial boundaries, whatever boundaries such as race, class, ethnicity, political loyalty, and the like. On this view, on Kant's view, that is, we owe no greater moral obligations to fellow citizens than to any other human beings on the face of the planet. Citizenship--if I can use language that is not exactly Kant's own, but is largely sort of identified with a kind of Kantian move in philosophy--citizenship is simply an arbitrary fact conferred on individuals through the accident of birth. But since birthright citizenship is an artifact of what you might call a pure sort of genetic lottery, there are no moral or special obligations attached to it. The Kantian emphasis on universality, that is to say that there is a moral law that can be universalized and held to be true for all human beings, stressed for Kant that we are all parts of what he called a kingdom of ends, a universal kingdom of ends where every individual is due equal moral value and respect because simply of their humanity alone. That idea of a cosmopolitan ethic of humanity, Kant believed, could only be realized in a republican form of government, today what we might call a democracy, or, to speak more precisely, what Kant believed it could only hold true in a confederation of republics overseen or ruled by international law. Kant was perhaps, I don't know if he was the first, but he gave the first, he gave the most powerful early expression to the idea of a league of nations, a league of nations that would put an end to war altogether between states for the sake of achieving what he called perpetual peace, the title of a famous essay of his. Hobbes and Locke, he believed, were wrong in attributing sovereignty, absolute sovereignty, to the individual nation state. For Kant, the state, the individual state, is merely a kind of developmental stage along the path to a world republic, a world republic of states organized around the idea of international law and peace. Only in, he believed, a league of republics would peace among the nations finally be realized and would individuals be able to treat one another as ends rather than means. If you want just some indication of how influential Kant's view has been, you can think that his idea of an international league of nations came to fruition over a century after his life in Woodrow Wilson's famous 14 Points issued after the first world war and elaborated more fully in the United Nations Declaration of Human Rights of 1948, all of which bear the unmistakable imprint of Immanuel Kant. Now, neither of these views, let me argue, either of these views, Schmitt's or Kant's, really captures the nature of the political. Let me start adequately so at least. Let me start with--return to Schmitt again. Schmitt's view is rooted, I believe, in a very important human truth, namely, the world is a dangerous, in fact, very dangerous place, like in many ways Hobbes or Machiavelli, Schmitt takes the extreme situation, that is to say, the situation of war and mobilization for war, and turns it into the norm, turns it into the normal situation. An extreme situation is one where the very survival, in fact, the very independence of a society, is at stake and for Schmitt every situation is potentially a life and death struggle against a kind of existential enemy where one must decide to choose up sides between friends and enemy. Politics, for him, is a kind of endless struggle for power guided by national self-interest alone. And yet, it would seem to me, a politics of unremitting war and preparation for war would be, have to be, self-defeating even in Schmitt's own terms. For example, why should the struggle between friend and enemy be exclusively what we might call an interstate rivalry? Wouldn't competition between individuals and groups just as easily become a feature of domestic politics as well? Why is war something that takes place exclusively between states rather than within them, as the logic of bitter rivalry and competition and friend and enemy cuts all the way down, so to speak? The logic of Schmitt's argument, at least as I understand it, points not only to war between states but ongoing civil war and civil conflicts within states, between rival groups expressing their own desire for power and their own loyalty to their individual groups. The result of this logic of conflict, it seems to me, would be the negation of politics, that is to say the destruction of the sovereign state as the locus of political power. Why should, again, the choice of friend and enemy be a choice between states rather than individuals. But let me then turn to Kant's view, cosmopolitanism, because if the effect of Schmitt's distinction between friend and enemy is to make politics identical with war, the effect of Kantian cosmopolitanism is to confuse politics with morality. Kant and his present day followers wish to transcend the sovereign state and replace it with known international rules of justice. If Schmitt believed that man is the dangerous animal, Kant believed man is simply the rule following animal. But Kant's desire, it seems to me, to transcend the state with a kind of international forum, is both naive and anti-political. If Hobbes was right when he said that covenants without the sword are but words, the question is who will enforce these international norms of justice? Kant's conception of a kind of global justice is to wish a world without states, a world without boundaries, a world, in short, without politics. International bodies like the United Nations have been notoriously ineffective in curbing or restraining the aggressive behavior of states and international courts of justice like that in the Hague have been highly selective in what they choose to condemn. It would seem that reliance on such bodies would have the further disadvantage of uprooting people from their traditions, from their local arrangements that most people find as a source of reverence or awe. There seems to be little room for reverence for the sacred, in the cosmopolitan ideal. The logic of this view, the logic of Kant's view for perpetual peace, necessarily leads to a world state, world government. Even Kant admitted that a world state would be what he called a soulless despotism. He was opposed to the idea of a world state, but the logic of his argument leads him inescapably in that view, in that vein. The idea underlying perpetual peace is that human life as such, human life independent that is of the kind of life one leads, is an absolute good. Such an idea, I think, can only lead in the long run to moral decay, that is to say, to a kind of inability or unwillingness to dedicate one's life to ideals, to the relatively few things that give life wholeness and meaning. The cosmopolitan state would be--the world state would be the home of what Nietzsche called the last man, a world where nothing really matters, where there is nothing really of importance left to do, a world of entertainments, a world of fun, a world void of moral seriousness. So these two extremes, nationalism and cosmopolitanism, are today the two doctrines or tendencies that tend to obscure the true nature of the political. Each of these extremes contains at best a part of the truth, a partial truth. The nationalist is surely correct in some respect, to see that politics is always a matter of the particular, particular states, particular nations, particular peoples and traditions. For the nationalist, the particular stands for something infinitely higher and more noble than the cosmopolitan or the universal. We enter the world as members of a particular family, in a particular neighborhood, in a particular state, in a particular part of the country and so on. We are a composite of particularities and these attachments, these particularities, are not something extraneous or accidental to our identities. They are what make us who we are. The demand that we give up our particular identities and assume a kind of cosmopolitan point of view would be the same thing to ask us, at least those who are native English speakers, to give up speaking English and adopt Esperanto, the artificial false language. I would ask, who was the Shakespeare or Milton of Esperanto? In other words, everything great derives from something rooted and particular. This is the morality of what you might call common ties. But there is also some truth on the cosmopolitan side, on the other hand. Are we simply determined or condemned by the accident of birth to live by the traditions of the particular nation in which we happen to have been born? Doesn't this deny what seems to be highest in us, that is to say our capacity for choice, to detach ourselves from our surroundings, to determine for ourselves how we will live and who we will be? This idea of choice, of being able to choose for oneself, is, I think, at the bottom of our experience of human dignity. We experience our moral worth as human beings through our ability to choose how we will live, with whom to live, and under what conditions. This kind of ideal, this cosmopolitan ethic, has the virtue of allowing us to stand outside of our particular situation and view ourselves from, what you might call, the standpoint of the disinterested spectator, from a higher or more general point of view. And clearly, such a morality gives us a kind of critical distance or vantage point on how we can judge ourselves and our society. From this point of view, our local and particular attachment to family, friends, fellow citizens, again carries no overwhelming moral weight. We must view them as we would view anyone or anything else, disinterestedly, objectively, and this one might call the morality of cosmopolitanism. Each of these ethics, the ethic of communal ties, the ethic of cosmopolitan individualism, express, again, an important piece of the truth of politics although neither is alone complete in itself. How to combine them or what should we do? In many respects, I think these two ethics, these two forms of ethos, are very much combined already in the American regime and how the American way of life should be properly understood. Consider the following. American regime is the first truly modern nation, that is to say, a nation founded upon the principles of modern philosophy. Our founding document, the Charter of American Liberties, the Declaration of Independence, is dedicated to the proposition that all men are created equal. It is fair to say that the American regime requires more than loyalty, that is to say it requires understanding, it requires understanding of that founding principle or that proposition, and the various texts and debates in which that proposition was later articulated as well as the range of responses and alternatives to it. To believe for example, as you all now know, to believe that "all men are created equal and endowed with unalienable rights" requires us to consider the opposite proposition contained in books like Plato's Republic or Aristotle's Politics that believe that human beings are not equal and that the best regime is one governed by a philosophical aristocracy. So to consider our regime means in some ways to consider it in the light of these universal alternatives. But ours is also a regime that contains elements of both the universal and the particular. Again, the American regime is one founded on what Jefferson called "a self-evident truth," the truth that there are certain unalienable rights, that these principles are not simply true for Americans but believed to be good for all human beings, always and everywhere. Consider Tom Paine in The Rights of Man where Paine writes, "The independence of America was accompanied by a revolution in the principles and practice of government, government founded on a moral theory," he says, "on the indefeasible hereditary rights of man that is now revolving from west to east." In other words, far from suggesting a traditional form of communal morality, American politics, as Paine suggests there, requires a commitment to the highest, most universal moral principles. That seems to be the cosmopolitan dimension upon which the very nature of the American regime rests. But the question does not end there. The principles of Jefferson and Paine once again did not arise sui generis. Anyone knows Jefferson's principles about equality and rights have their profound source in the philosophy of John Locke and particularly in his Second Treatise of Government. Recall that Locke occupies a central moment in the development of the modern state and his new idea of a kind of industrious and rational citizen. Locke's philosophy emerged not only in conversation with the other great founders of modernity like Machiavelli and Hobbes but, in some important sense, it emerged in opposition to the tradition of the classical republic whose greatest representatives were Plato, Aristotle, Cicero, and Polybius. It would seem then, in other words, to be an American citizen in the fullest sense of the term requires an immersion in the philosophical tradition because only in America, of all the countries in the world I believe, does the philosophical tradition remain most deeply and truly alive. And yet at the same time, the American regime requires an understanding and appreciation not only for a set of abstract philosophical ideas and debates but for a constitution, its history and a distinctive way of life. A regime is obviously more than a catalog of philosophical doctrines and abstract propositions but is embedded within a particular set of moral, legal, political, constitutional practices that give it color and distinguish it from all others. A proper understanding of the particular regime requires today, or requires at any time, an immersion in history, not only philosophy but in history, and I mean by history not social history, economic history or even cultural history, but history in the proper sense of the term, that is political history. Political history presupposes the centrality of politics, of how the constitution of any society and its most fundamental laws shape the character and choices of its citizen body. Political history concerns the struggle of individuals and groups for power. It concerns the political uses of power or, maybe to speak a little more clearly, of the two great ends to which power can be put, namely freedom and empire. Political philosophy is related to political history. In fact, political history or political philosophy presuppose one another in the same way or in the same relation of the universal to the particular. While the political philosopher studies the principles, the underlying principles of the regime, the political historian examines the way those principles have been applied in practice. While the philosopher is concerned with the best regime, the regime that is best according to unchanging principles, the historian is concerned with what is best for a particular people at a particular time and place, Athenians, Frenchmen, Americans and so on. And this is what the greatest political historians, Thucydides, Theodor Mommsen, Lord Macaulay, Henry Adams, this is what they have done. They have examined how different regimes, both express but also depart from fundamental principles. When Adams, for example, examines in painstaking detail the acquisition of the Louisiana Territory under the Jefferson administration, he does so always against the backdrop of Jeffersonian ideals about democracy and limited government. But that leads us to the final question that I want to end with, is the proper understanding and appreciation of the political is not something we inherit but obviously something we must be taught. Like anything that must be taught, it requires teachers. But where are such teachers to be found at least today? It would seem only very rarely in universities and rarer still in departments of history, political science or economics. Excuse my polemic. Modern professors of history, for example, often appear to teach everything but a proper respect for tradition. One would get the impression from many classes that America alone among the nations of the world is responsible for racism, homophobia, the despoliation of the planet and every other moral evil that one can imagine. In my own field, political science, that once designated the skill or art possessed by the most excellent statesmen or politician, civic education has been replaced by something called "game theory" that regards politics as a marketplace where individual preferences are formed and utilities are maximized. Rather than teaching students to think of themselves as citizens as these members--individuals did, the new political science treats us as something called rational actors who exercise our preferences, but the question is, what should we have a preference for, how should rational choice be exercised? On these questions, that is to say the most fundamental questions, our political science is sadly silent. It has nothing to offer and nothing to say. By reducing all politics to choice and all choice to preference, the new political science is forced to accord legitimacy to every preference however vile, base or indecent it may be. That kind of value neutrality towards preferences is akin to the philosophic disposition that we know as nihilism, that is to say the belief that our deepest principles and convictions are nothing more than blind preferences. So the purpose of political science is not to stand above or outside the political community as an entomologist observing the ant behavior but rather to serve as a civic-minded arbiter and guardian of disputes in order to restore peace and stability to conflict ridden situations. We are in danger today of losing touch with those questions and those insights that are the original motivation for understanding politics. In place of these questions has arisen a kind of narrow-minded focus on methodology often at the expense of the life and death issues that make up the substance of the political. So I end with this question. Where should the study of political science be now? You have sat through 13 weeks of an introductory course. Where do you go from here? To ask a question posed brilliantly by Karl Marx, he asked, "Who will educate the educators?" the best question he ever asked. How can we begin a comprehensive reeducation of today's political science? The only answer and the best answer I can give you today is simply to read old books. These are our best teachers in a world where real teachers are in short supply. In addition to what you have read here, I would include front and center in your future reading books like Plato's Laws, Machiavelli's Discourses on Livy, and Montesquieu's incomparable Spirit of the Laws, and of course, The Federalist Papers. To read these books in the spirit in which they were written is to acquire an education in political responsibility. This, of course, or these should be supplemented by a study of the deeds and writings of the most important American statesmen from Jefferson, Madison, Lincoln through Wilson and Roosevelt. And these, in turn, should be supported by the study of our leading jurisprudential thinkers from Marshall, Holmes, Brandeis, and Frankfurter. And finally, this should be completed by an examination of the most important statesmen and leaders from world history from around the world, from Pericles to Churchill. Once you have completed those readings, once you have done that, and I would say only when you have done that, can you say that you are living up to the highest offices of a Yale student aptly summarized on the memorial gate outside of Branford College which says, "For God, For Country, and For Yale." Thank you for your time and patience over this semester and good luck to you in the future.

Introduction_to_Political_Philosophy_with_Steven_B_Smith

6_Philosophers_and_Kings_Platos_Republic_V.txt

Professor Steven Smith: Today I have the impossible task of finishing the parts of the Republic that I have assigned for the class. And in the past sometimes, I've assigned a full two weeks to the Republic, which would be four lectures, but because I wanted to do some other things with the course as well, I had to cut the Republic by one lecture, and now I'm paying for that today. So I'm going to try to rush through, unfortunately, a number of the major themes regarding the creation of the just city, the creation of Kallipolis and then try to end the class by talking about, as I like to do for every thinker, what does in this case, what does Plato, what are his views on modern America. What does Plato say to us today? But I want to start with what is one of the grand themes of the Republic, it is indicated in Book II by Adeimantus' speech about self-control. It is introduced further by the claims of Socrates to control, to censor, to control the poetry and the arts of the city. And this is the big theme of what one might call "the control of the passions." This is the theme of every great moralist from Spinoza to Kant to Freud. How do we control the passions? And it is certainly a large theme of Plato's theory of justice in the Republic. Every great moral philosopher has a strategy for helping us submit our passions to some kind of control, to some kind of supervening moral power. And again, recall this is the theme raised at the beginning of Book II by Adeimantus, who puts forward an idea of self-control, or what he calls self-guardianship as his goal. How can we protect ourselves from the passion for injustice? And one of the things Socrates emphasizes is that the most powerful of those passions, the most powerful passion is that Socratic passion that he calls thumos, or what our translator has as spiritedness, anger, maybe what biblical translators call heart, having a big heart, having thumos and all of that implies. This is for Plato, the political passion par excellence. It is a kind of fiery love of fame, love of distinction that leads men and women of a certain type to pursue their ambitions in public life, in the public space. It is clearly connected this notion of spiritedness or this thumotic quality to our capacities for heroism and for self-sacrifice. But it is also connected to our desires for domination and the desire to exercise tyranny over others. Thumos has a kind of dual component to it. It can lead us to a sense of kind of righteous indignation and anger at the sight of injustice, but it can also lead us in a rather contradictory way to desire to dominate and tyrannize over others. This is the quality that Socrates regards as being possessed by every great political leader and statesman, but it is also clearly a quality possessed by every tyrant. And the question posed by the Republic, in many ways, the question around which the book as a whole gravitates, is whether this thumotic quality can be controlled. Can it be re-directed, can it be re-channeled in the service of the public good? Socrates introduces the problem of thumos by a story, a particularly vivid story that I hope you all remember, where in Book IV he tells the story about Leontius at the walls. "Leontius," he writes, "was proceeding from the Piraeus outside the north wall when he perceived corpses lying near the public executioner. At the same time, he desired to see them. He wanted to see this grotesque sight, these dead bodies lying there. And to the contrary, he felt disgust and turned himself away and for a while he battled with himself and hid his face. But eventually overpowered by desire, he forced his eyes open and rushing towards the corpses said 'see you damn wretches, take your fill of this beautiful sight'" 439c. That story that Socrates tells here is not one of reason controlling the passions, but rather one of intense internal conflict that Leontius felt. We see his conflicting emotions both to see and not to see, a sense that he wished to observe and yet he is at, in some ways, at war with himself, knowing to gawk, to stare at this sight. There's something shameful about it and he felt shame. One example I particularly like of this was suggested last year, I think, by Justin Zaremby who said it's the emotion we all feel when we're driving down the highway, right, and we see a car crash or we go by a wreck and everybody slows down, right, they all want to see. What are they hoping to see? Well, they want to see blood, they want to see if there's a body, they want to see how much damage has been caused. And we've all been in this, where we know that it's shameful to look at this, just drive on, as Socrates would say "mind your own business," and yet at the same time we feel, even against our will, compelled to look and think about that. And think about that and this case of Leontius the next time you, for those of you who have driver's licenses, are next driving on the highway and see something like that. It is the thumos that is the cause of--that should be the cause of your shame at slowing down to look. Sometimes we can't help but slow down because everybody is slowed down in front of us, we have no choice. But anyway, that incident, that story that Socrates relates is connected to the fact that Leontius is a certain kind of man. He regards himself as proud, independent, someone who wants to be in control of his emotions but isn't. He is a soul at war with himself, and potentially therefore, at war with others. And what the Republic tries to do is to offer us strategies, maybe we might even call it a therapy, for dealing with thumos, for submitting it to the control of reason and helping us to achieve some level of balance, of self-control and moderation. And these are the qualities taken together that Socrates calls justice, that can only be achieved when reason is in control of the appetites and desires. Again, a question the book asks is whether that ideal of justice can be used as a model for politics. Can it serve as a model for justice in the city? This connection he has established between justice in the city and justice in the soul, what are the therapies or strategies for solving injustice in the soul or imbalance of some kind in the soul? Can those be transferred or translated in some way to public justice, to political justice, justice in the polis? Right? You with me on that so far? So, on the basis of this, Socrates proposes how to proceed with the construction of Kallipolis, and he does so through what he calls three waves. There are three waves, three waves of reform, so to speak, that will contribute to the creation of the city. The first of these waves is, you remember, the restrictions on private property, even the abolition of private property. The second, the abolition of the family, and the third wave being the establishment of the philosopher kings. Each of these waves is regarded as in some way necessary for the proper construction of a just city. And I'm not going to speak about all of them, but I do want to speak a little bit about, because it has particular relevance for us, his proposals for the co-education of men and women that is a great part of his plan, especially related to the abolition of the family, that men and women be educated in the same way, right. The core of Socrates' proposal for equal education is presented in a context that he knows to be or suggests will be laughable. It will certainly be seen that way, he suggests, by Glaucon and Adeimantus. There is no job, he states, that cannot be performed equally well by both men and women. Is Socrates a feminist? Gender differences, he says, are no more relevant when it comes to positions of political rule than is the distinction between being bald and being hairy. Socrates is not saying that men and women are the same in every respect, he says, but equal with respect to competing for any job at all. There will be no glass ceilings in Kallipolis. The first, in many ways, great defender, the first great champion of the emancipation of women from the household. But this proposal comes at certain costs, he tells us. The proposal for a level playing field demands, of course, equal education. And here he says that men and women, being submitted to the same regime, will mean, among other things, that they will compete with one another in co-educational gymnasia. They will compete with each other in the nude because that is the way Greeks exercised. They will compete naked in co-educational gymnasia, think of that. Furthermore, their marriages and their procreations will be, he tells us, for the sake of the city. There is nothing like romantic love among the members of the guardian class. Sexual relations will be intended purely for the sake of reproduction and unwanted fetuses will be aborted. The only exception to this prohibition is for members of the guardian class who are beyond the age of reproduction, he tells us, and they, he says, can have sex if they're still able, with anyone they like. A kind of version of recreational sex as a reward for a lifetime of self-control. Child-bearing may be inevitable for women but the rearing of the child will be the responsibility of the community or at least a class of guardians and common daycare centers. A sort of variation of Hillary Clinton's book that "it takes a village to raise a child," comes right out of Plato apparently. No child should know their biological parents and no parent should know their child. The purpose of this scheme being to eliminate senses of mine and me, to promote a kind of common sense of esprit de corps among the members of the guardian class, "a community of pleasure and pain," Socrates calls it at 464a. What we are creating is a community of pleasure and pain. I will feel your pains, and of course you will feel mine. The objections to Socrates, are of course, you know, raised as early as by Aristotle himself, in the very next generation. How can we care for things, how can we truly care for things that are common? We learn to care for things that are closest to us, that are in some way our own. We can only show proper love and concern for things that are ours, not things that are common. Common ownership, Aristotle argues, will mean a sort of common neglect. Children will not be raised better by putting them under the common care of guardians or in daycares but they will be equally neglected. But it is in this, and you can think about that, about whether that's true or not, but it is in the same context of his treatment of men and women that something else often goes unnoticed and that is Socrates' efforts to rewrite the laws of war, because of course the guardians are being trained and educated to be guards, to be warriors, to be members of a military class. In the first place, he tells us, children must be taught the art of war. This must be the beginning of their education, Socrates says, making the children spectators of war. Children will be taken, he seems to suggest, to battles and to sites of where fighting is going on, to be spectators for them to become used to and habituated to seeing war and what everything that goes on. Not only is expulsion from the ranks of the guardians penalty for cowardice, but Socrates suggests there should be, listen to this, "erotic rewards for those who excel in bravery." Erotic rewards for excellence in bravery. Consider the following remarkable proposal at 468c, "and I add to the laws of war," Socrates writes, "that as long as they, the guardians, are on campaign, no one whom he wants to kiss should be permitted to refuse. So that if a man happens to love someone, either male of female, he would be more eager to win the rewards of valor." That is to say as a reward for bravery, exhibited bravery, the hero should be allowed to kiss anyone they like while they are on patrol, male or female. A particularly puritanical editor of Plato from the twentieth century writes in a footnote to that passage, "this is almost the only passage in Plato that one would wish to blot out," his sensibilities were offended by this notion. But I wonder what kind of, if this might even make a powerful incentive for military recruitment today. What do you think? Well, think about it. I don't know. So, at long last, we move from the education of the guards to justice. What is justice, we've been questioning asking ourselves throughout this book in which Plato has been, Socrates has been teasing us with. At long last we come to this thing. The platonic idea of justice concerns harmony, he tells us, both harmony in the city and harmony in the soul. We learn that the two are actually homologous in some way. Justice is defined as what binds the city together and makes it one. Or he puts it another way, consists of everyone and everything performing those functions for which they are best equipped. Each of the other citizens, Socrates says, must be brought to that which naturally suits him, which naturally suits him, one man, one job, he says. So that each man practicing his own which is one, will not become many but one. Thus you see, he says, the whole city will naturally grow up together. Justice seems to mean adhering to the principal, justice in the city, adhering to the principal of division of labor. One man, one job, everyone doing or performing the task that naturally fits or suits them. One can, of course, as you've already imagined, raise several objections to this view and again Aristotle seems to take the lead. Plato's excessive emphasis on unity would seem to destroy the natural diversity of human beings that make up a city. Is there one and only one thing that each person does best? And if so, who could decide this? Would such a plan of justice not be overly coercive in forcing people into predefined social roles? Shouldn't individuals be free to choose for themselves their own plans of life wherever it may take them? But however that may be, Plato believes he has found in the formula of one man, one job, a certain foundation for political justice. That is to say, the three parts of the cities, workers, auxiliaries, guardians, each of them all work together and each by minding their own business, that is doing their own job, out of this a certain kind of peace and harmony will prevail. And since the city, you remember, is simply the soul at large, the three classes of the city merely express the three parts of the soul. The soul is just, he tells us, when the appetites, spiritedness, and reason cooperate with reason, ruling, spirit and appetite, just as in the polis, the philosopher-king rules the warriors and the workers. The result, he tells us, is a kind of balance of the parts of the whole, right. Justice is a kind of harmony in which the three parts of the city and the three parts of the soul are direct expressions of one another. But that formula forces us to return to the original Socratic question about the harmony of the soul and the city. Is the structure of a city identical to the structure of a soul? Are they really identical? Well, maybe, maybe not. For example, every individual consists of three parts, of appetite, spirit, and reason. Yet each of us will be confined it seems to only one task in the social hierarchy. I assume what Socrates means by that is though each individual will, each of us, embody all three features of soul, appetite, spirit, and reason, only one of these will be the dominate trait in each of us. Some of us will be dominantly appetitive personalities, others will be dominantly spirited and so on. But even still when we think of it, if I am a member of the money making class, I am still more than simply a bundle of desires and appetites, just as a member of the warrior class would be clearly more than mere thumos or mere spiritedness. So, to confine the individual, it seems, to one and only one sphere of life would seem to do an injustice to the internal psychological complexity that makes each of us who we are. Let's examine that problem from a slightly different point of view. Socrates tells us repeatedly that justice in the city consists of each member, each citizen fulfilling his task in the social division of labor, in the social hierarchy. But this seems to be a very far cry, does it not, from the kind of justice he talks about in the soul that consists in what we might think of as sort of rational autonomy or self-control where reason controls the passions and the appetites. In fact, the vast majority of citizens in even the platonically just city will not necessarily have platonically just souls. The harmony and self-discipline of the city will not be due, it seems, to each and every member of the city but rather will rely on the guardian class, that special class of philosopher kings who will rule, let it be recalled, through selective lies, myths, and other various kinds of deception. So how can it be the case if at all, that you could have a just city, that is to say a city where everyone is performing their own task, they're following the division of labor, and yet very few of those members will have, so to speak, platonically just souls, that is to say, souls dominated by a kind of self-control or self-guardianship? That would certainly not be true of the members of the artisan class or the military class for that reason. So the question, that question is posed, that objection is posed by Adeimantus, you remember, at the beginning of Book IV. "What would your apology be Socrates," Adeimantus says, "if it were objected that you're hardly making these men happy, these people just," he says at 419a. Adeimantus is concerned that Socrates is being unfair to the auxiliaries and the guardians, giving them all the responsibilities but none of the rewards, none of the pleasures that would seem to be the reward of responsibility. How can a citizen of Kallipolis live a just or happy life if he or she is deprived of most of the goods or pleasures that we seek? Socrates gives a rather lame response. In founding the city, he says, we are not looking to the exceptional happiness of any one individual or any group but rather to the city as a whole. And Adeimantus appears to accept that response, oh yes, I forgot we are concerned with the happiness, the justice of the whole. But his question is still one that lingers and one that Plato includes for a purpose. What about, how can you have a platonically just city if most people in it, certainly most people of the auxiliary class are deprived of the pleasures and the goods that we desire? It's a question that lingers and one might wonder whether Socrates ever successfully answers that question. He silences Adeimantus in some way as he silences Thrasymachus earlier; that is not always to say that their objections have been answered. And that leads, as it were, to the third and final wave of paradox of the Kallipolis which is the famous proposal for the philosopher-king. What is Plato without the philosopher-king? What is the Republic without the philosopher-king? Unless the philosophers rule as kings or those now called kings, genuinely philosophize, there will be no rest from the ills for the cities, he says, right? Socrates presents this proposal, again, as outlandish. He says he expects to be drowned in laughter. And this has led some readers to suggest that the proposal for philosophers' kings is ironical. That it is intended as a kind of joke to, in many ways, discredit the idea of the just city or at least to indicate its extreme implausibility. The question is why does Socrates regard philosophic kingship as required for Kallipolis, for the just city? Let me say, I am by no means convinced that the idea for the philosopher-king is an impossibility or is intended as a kind of absurdity. Plato himself, remember, made a number of trips to Sicily to serve as the advisor to a king there, Dionysius, and all of these missions failed and left him deeply dispirited. The ambition in some ways to unite philosophy and politics has been a recurring dream of political philosophy ever since Plato. Socrates says he will be drowned in laughter but many other people have taken this dream or this aspiration very seriously. Consider one thinker, and I will, I'm going to read you a short passage and I'm going to come back to this again later in the semester, from Thomas Hobbes' Leviathan, chapter 31 of Leviathan, where Hobbes gives us a very personal statement about his intention in writing this book. Hobbes wrote, "I am at the point of believing that my labors will be as useless as the commonwealth of Plato." He seems to be rather despairing about whether this book is actually going to have any affect. "I'm in the point of believing it will be as useless as the commonwealth of Plato," for he also is of the opinion that it is impossible for the disorders of state and change of government by civil war ever to be taken away until sovereigns be philosophers. But after admitting his despair about the possibility of realizing his ideas and practice, Hobbes continues as follows, "I recover some hope," he says, "that one time or other, this writing of mine may still fall into the hands of a sovereign who will consider himself without the help of any interested or envious interpreter. And by the exercise of entire sovereignty in protecting the public teaching of it, convert this truth of speculation into the utility of practice." So there you have Hobbes talking about his own book, expecting or at least hoping it will fall into the hands of a sovereign who one day, again, without envious or self-interested interpreters may, may one day become a practical source of guidance for statecraft. Here it is Hobbes taking Plato's suggestion very seriously, and we see this again very much in the history of political philosophy in thinkers like Rousseau, or Marx, or Nietzsche, or Machiavelli all of whom sought to gain the ear of political leaders and convert their ideas into some kind of practice. But most of the objections to Plato's particular form of the philosophic kingship really are centered on the practicality of his idea. And beyond this, there is the problem with the very cogency of the idea itself. Consider the following, can philosophy and politics actually be united? It would seem that the needs of philosophy are quite different from the demands or requirements of political rule. Can you imagine Socrates willingly giving up one of his conversations for the tedious business of legislation and public administration? Can one imagine that? The philosopher is described by Plato as someone with knowledge of the eternal forms, lying behind or beyond the many particulars. But just how does that kind of knowledge help us deal with the constant ebb and flow of political life? It seems not enough that the philosopher have knowledge of the forms but this knowledge has to be supplemented by experience, by judgment and by a kind of practical rationality. Was Plato simply unaware of this, I can't believe that. I don't believe that. So the question is, what kind of unity was he expecting of philosophy and politics? Anyway, philosophers are not purely thinking machines but they are also human beings composed of reason, spiritedness, and appetite. Will not even philosophers, one might ask, given the possibility of absolute power be tempted to abuse their positions? Maybe, maybe not, who knows. So these are the questions, these are at least among the questions that Socrates or Plato, the author of the book, deliberately poses for us to consider. So what is the doctrine of the philosopher-king intended to prove? Must the massive effort to construct the city in speech in order to understand justice in the soul? Is it a philosophical possibility? Does he hold it out as a real possibility or must it be considered a failure in some way or that if the dialogue does end in failure what can we learn from that? Those are questions I want you to consider. But for now, what I want to do is talk about Plato's democracy and ours. What does Plato teach us about our own regime? Could Plato have imagined such a regime? I think in many ways he can and he did. In one sense, the Republic, and I've given some indications of this today, seems to be the most anti-democratic book ever written. Its defense of philosophic kingship is itself a direct repudiation of Athenian democracy. Its conception of justice, minding one's own business, is a rejection of the democratic belief that citizens have sufficient knowledge to participate in the offices of government. To be sure, Athenian democracy is not American democracy. Plato thought of democracy as a kind of rule by the many that he associated with the unrestricted freedom to do everything that one likes. This seems in many ways to be quite far from the American democracy based on constitutional government, systems of checks and balances, protection of individual rights, and so on. The differences between Athens and Washington seem to be very far. And yet, in many ways, Socrates diagnoses very powerfully an important condition of modern democratic life with which we are all familiar. Consider this passage in Book VIII of the Republic that I encourage you to read but is not on your assigned list. Socrates writes in Book VIII, 561c, "speaking of the democratic soul, the democratic man, he also lives along day by day gratifying the desire that occurs to him, at one time, drinking and listening to the flute." Today we have different kinds of music to substitute for the flute but you get the point. Drinking and listening to the flute, at another time downing water and dieting, now practicing gymnastics and again idling and neglecting everything, and sometimes spending his time as though he were occupied with philosophizing. Often, he engages in politics and jumping up, says and does whatever chances to come to his mind. And if he ever admires any soldiers, he turns in that direction. And if money-makers in that one and there is neither order nor necessity in this life but calling this life sweet, free, and blessed, he follows it throughout. Is that image of life at all familiar to us? Doing anything you like, it seems to be the opposite of the platonic understanding of justice as each one doing a special function or fulfilling or doing a special craft. Just doing whatever you like and calling that sweet, free, and blessed throughout. This account should be instantly recognizable as the state of modern democracy in some ways. There exists, as Plato and Socrates clearly understand, a very real tendency within democracy to identify the good human being, the good man with, you might say, the good sport, the regular guy, the cooperative fellow, you know, someone who goes along and gets along with others. By educating citizens to cooperate with each other in a friendly manner, democracy seems, so Plato is suggesting, they stand in danger of devaluing people who are prepared to stand alone, of rugged individualists who will go down with the ship if need be. It is precisely this kind of creeping conformism, this kind of easy going toleration, this sort of soft nihilism that democracies tend to foster in which not only Plato, but modern thinkers like Emmerson, and Tocqueville, and Mill, John Stewart Mill, very much warned about. What bothers Socrates most about our democracy is a certain kind of instability, its tendency to be pulled between extremes of anarchy, between lawlessness and tyranny. It is in this section of the Republic, Adeimantus asks, won't we with Aeschylus say whatever comes to our lips? Won't we say with Aeschylus whatever comes to our lips? The idea of having the liberty to say whatever comes to our lips sounds to Plato like a kind of blasphemy. The view that nothing is shameful, that everything should be permitted, to say whatever comes to our lips… There is a kind of license that comes from the denial of any restraints on our desires or a kind of relativistic belief that all desires are equal and all should be permitted. Plato's views on democracy were not all negative, to be sure. He wasn't only a critic of democracy. It was, after all, a democracy that produced Socrates and allowed him to philosophize freely until his seventieth year. Would this have been permitted in any other city of the ancient world? And he surely would not be allowed to philosophize in many cities and countries today. Remember the letter that Plato wrote near the end of his life, when he compares the democracy to a golden age, at least in comparison to what went after. Plato here seems to agree with Winston Churchill that democracy is the worst regime except for all the others. It's the worst that's been tried except for everything else. So what is the function of Kallipolis, this perfect, this beautiful city? What purpose does it serve? The philosopher-king, he tells us, may be an object of hope or wish but Plato realizes that this possibility is not really to be expected. The philosophic city is introduced as a metaphor to help us understand the education of the soul. The reform of politics may not be within our power but the exercise of self-control always is. The first responsibility of the individual who wishes to engage in political reform is to reform themselves. All reform seems to begin at home. And we see this very vividly when we look at so many politicians today in public scolds who teach us and who are hectoring us about living a certain way of life, living a certain, living according to their likes and then we will find out of course something very shameful about them. I'm thinking of a couple of people in particular, I won't mention any names in the public sphere. Plato's judgment seems to be "you need to reform yourself before you can think about reforming others." This is a point that is often lost in the Republic, that it is first of all a work on the reform of the soul. That is not to say at all that it teaches withdrawal from political responsibilities, it does not. Philosophy and certainly Socratic philosophy requires friends, comrades, conversations. It is not something that can simply be pursued in isolation. Socrates understands that those who want to reform others must reform themselves, but many who've tried to imitate him have been less careful. It is easy to confuse, as many people have done, the Republic, with a recipe for tyranny. The twentieth century, and even the beginnings of our own, is littered with the corpses of those who have set themselves up as philosopher-kings, Lenin, Stalin, Hitler, Mao, Khamenei, to name just some of the most obvious. But these men are not philosophers. Their professions to justice are just that, they are professions or pretensions expressing their vanity and their ambition. For Plato, philosophy was in the first instance, a therapy for our passions in a way of setting limits to our desires. And this is precisely the opposite of the tyrant who Plato describes as a person of limitless desires who lacks the most rudimentary kind of governance, namely self-control. The difference between the philosopher and the tyrant illustrate two very different conceptions of philosophy. For some, philosophy represents a form of liberation from confusion, from unruly passions and prejudices, from incoherence. Again, a therapy of the soul that brings peace and contentment and a kind of justice. And yet for others, philosophy is the source of the desire to dominate. It is the basis of tyranny in the great age of ideologies through which we are still passing. The question is that both tendencies are at work within philosophy and how do we encourage one side but not the other. As that great philosopher Karl Marx once asked, "Who will educate the educators?" It's the wisest thing he ever said. Who will educate the educators, who do we turn to for help? There is obviously no magic solution to this question but the best answer I know of is Socrates. He showed people how to live, and just as importantly, he showed them how to die. He lived and died not like most people but better, and even his most vehement critics will admit to that. Thank you very much. I'll see you next Wednesday, and we'll start Aristotle.