How does GNU software development sustain economically? I apologise if this question is out of topic, but it is simultaneously an economy and a programming question. If it should go to another SE communality, please indicate me.

In theory, GNU software is entirely developed by volunteers during their free time, or by companies that voluntary fund programmers to develop GNU software (by using income from another sector of their activity).

I understand how it can work perfectly fine for small-scale project that can be done in a couple of rainy weekends by a single individual (let's say for example a sudoku game), because after all computer programming is an extremely fun and rewarding hobby, and I have no problem seeing people developing small or medium programs during their free time and sharing them to the world.

The problem is that this scales extremely poorly for larger programs for the following reasons:

As fun as programming is, as the project that have to be implemented becomes larger, the time it takes to implement the desired functionality grows extremely quickly. A larger scale program takes an incredible amount of time to develop, for instance it could easily take 15 years of free time and vacation time for an individual to program an operating system, and by the time his software is released it'll be completely obsolete.
As other people write programs in another way that the way you would have done it, reading and understanding someone else's code takes a lot of time, in most cases as much as would writing your own code from scratch. Modifying another people's code and try to improve it, as it is encouraged by the GNU philosophy, is almost just as time consuming as developing your own clone of the said program with the functionality you'd like to add.
As soon as 2 or more people will have to collaborate to develop a larger program, this creates lots of decision-taking issues that would never arose on a single-developer project. The result is that, for example if a group of 2 programmers collaborate for a project that would take 10 years for a single man to make, they won't make it in 5 years but probably in 8.
If people that collaborate for the same project meets on the internet solely, it is easy for one member of the project to vanish suddenly (either because he lost interest or because he physically can't be on the internet anymore), thus making collaboration even harder

So, while I understand perfectly how simple programs can be developed with the GNU mindset, I absolutely don't see how such huge programs such as GNU/Linux or gcc are possible on this model. gcc is around 7 million lines of code. I know lines of code does not mean much, as in a later stage of a project the more productive programmer is the one that will actually remove lines of code (simplifying and/or optimizing the project), but this gives an overwiew how massive of a project gcc is.

So in theory, anyone can freely modify gcc during their free time, but in practice? It was developed by very professional people as a job, not as a hobby. Anyone making a compiler as a hobby will eventually give up as the cost/benefit is not worth it:

Developing a large program is such a long term huge project, they'd rather use their free time to have other activities that are more rewarding or more enjoyable in the short term
If they were to develop a large program anyway, they'd rather do it for a company that will pay them than doing it for free

In order to get people interested at developing a program such as GNU/Linux, gcc or Open Office in the long run, it should be rewarding. So my question is: Why is there people contributing to large GNU project, if they aren't getting a salary for it?

I would like to start by saying that I'm not a programmer and I have never contributed to any open source project. However, I have been interested in open source for a long time and I believe that I understand the general concepts of open source and how it works.

To start of, I would like to say that open source does not mean that you cannot make money on the software. It just means that the code has to be publicly available. Companies like Red Hat and Canonical make money not by selling the software, but by selling their expertise. If i wan't my company to run a Linux server, I can get the software for free. But I need somebody to install it, set it up and give support. This is where specialist from e.g. Red Hat comes in and makes money. For the company it makes sense, because hiring their own specialist would probably be much more expensive. This also gives these companies an incentive to contribute the the code. They want their product to be good so people will use it and by their services.

But lets talk about your points about scalability.

The cool thing about open source is that you do not have to develop everything from scratch. An operating system like Ubuntu has not been build by a single person. Instead a lot of people has contributed to different parts of the system (actually I think it would be hard to find one person with all the skills to make and effective operating system). For example, the Ubuntu people does not develop the Linux kernel. They just use one developed by others. So what was without open source probably impossible, is now possible,because you can build on other peoples work.
Reading and understanding others code it not more time consuming than writing your own. At least not in many cases. Beyond that, you do not have to understand all the code you use. If I want to write a program for Linux, I do not have to understand how all the parts in that program works in detail. I just have to know what they do. I can then take these parts and put them together with other parts to create my program. Or I can take an existing program and modify it for my needs.
tools like git and github makes it incredibly easy to collaborate. You just get the code and make modifications. You then submit them to the person in charge of the project. If it is good, it will be accepted.
people go in and out of projects all the time. But if the project is popular, enough will be working on it.

Here are some reasons why open source works.

I think the main reason that open source software has become so good is that the  large number of people working on the a project, insures a level of expertise that i hard to archive in a small team of developers. While it might seem weird, this single fact, seems to outweigh all the negative problems that can arise in open source.
In commercial programming, the project dies with the firm. Lets say you  by some software from a company that then closes. Then your screwed, as you will not receive updates and bug fixes, and you will need to by new software to keep up. With open source you can just find another company to support you software or develop it yourself.

If you are still interested, I suggest you read
The Cathedral and the Bazaar


Did previous researchers fail to detect the hot hand simply because of a statistical fallacy? Many basketball fans/players believe that having made several shots in a row,  the next shot is more likely to go in. This is sometimes called the hot hand.

Starting (I think) with Gilovich, Mallone, and Tversky (1985), it was "shown"  that this was in fact a fallacy. Even if several shots in a row have gone in, the next shot is no more likely to go in than your average shooting percentage would dictate.

Miller and Sanjurjo (2015) argue that the hot hand does in fact exist and previous researchers had simply fallen prey to a fairly basic statistical fallacy. Their argument is something like this:

Flip a coin four times. Compute the probability that H follows H. To give a few examples: HHTT would have probability 1/2, HTHT would have probability 0/2, TTHH would have probability 0/1 1/1, and both TTTT and TTTH would be N.A.

Miller and Sanjurjo's punchline is that the expected value of this probability is not 0.5, but ≈0.4. And the error made by previous researchers was to incorrectly assume that the expected value of this probability is 0.5. So if for example these previous researchers conducted the above coin-flipping experiment and found the average probability to be say 0.497, they incorrectly concluded that there was no evidence of a hot hand (not significantly different from 0.5), when in fact there was very strong evidence of a hot hand (significantly different from 0.4).

My question is this: Are Miller and Sanjurjo correct that previous researchers failed to detect the hot hand simply because of this mistake? I have only skimmed one or two papers on this so I wanted to get some confirmation from someone here who might know this literature better. This seems like a surprisingly silly error to have persisted for three decades or more.

(This answer was completely rewritten for greater clarity and readability in July 2017.)
Flip a coin 100 times in a row.
Examine the flip immediately after a streak of three tails. Let $\hat{p}(H|3T)$ be the proportion of coin flips after each streak of three tails in a row that are heads. Similarly, let $\hat{p}(H|3H)$ be the proportion of coin flips after each streak of three heads in a row that are heads. (Example at bottom of this answer.)
Let $x:=\hat{p}(H|3H)-\hat{p}(H|3T)$.
If the coin-flips are i.i.d., then "obviously", across many sequences of 100 coin-flips,
(1) $x>0$ is expected to happen as often as $x<0$.
(2) $E(X)=0$.
We generate a million sequences of 100 coin-flips and get the following two results:
(I) $x>0$ happens roughly as often as as $x<0$.
(II) $\bar{x} \approx 0$ ($\bar{x}$ is the average of $x$ across the million sequences).
And so we conclude that the coin-flips are indeed i.i.d. and there is no evidence of a hot hand. This is what GVT (1985) did (but with basketball shots in place of coin-flips). And this is how they concluded that the hot hand does not exist.

Punchline: Shockingly, (1) and (2) are incorrect. If the coin-flips are i.i.d., then it should instead be that
(1-corrected) $x>0$ occurs only about 37% of the time, while $x<0$ occurs about 60% of the time. (In the remaining 3% of the time, either $x=0$ or $x$ is undefined — either because there was no streak of 3H or no streak of 3T in the 100 flips.)
(2-corrected) $E(X) \approx -0.08$.
The intuition (or counter-intuition) involved is similar to that in several other famous probability puzzles: the Monty Hall problem, the two-boys problem, and the principle of restricted choice (in the card game bridge). This answer is already long enough and so I'll skip the explanation of this intuition.
And so the very results (I) and (II) obtained by GVT (1985) are actually strong evidence in favor of the hot hand. This is what Miller and Sanjurjo (2015) showed.

Further analysis of GVT's Table 4.
Many (e.g. @scerwin below) have — without bothering to read GVT (1985) — expressed disbelief that any "trained statistician would ever" take an  average of averages in this context.
But that is exactly what GVT (1985) did in their Table 4.
See their Table 4, columns 2-4 and 5-6, bottom row. They find that averaged across the 26 players,

$\hat{p}(H|1M) \approx 0.47$ and $\hat{p}(H|1H) \approx 0.48$,
$\hat{p}(H|2M) \approx 0.47$ and $\hat{p}(H|2H) \approx 0.49$,
$\hat{p}(H|3M) \approx 0.45$ and $\hat{p}(H|3H) \approx 0.49$.

Actually it is the case that for each $k=1,2,3$, the averaged $\hat{p}(H|kH)>\hat{p}(H|kM)$. But GVT's argument seems to be that these are not statistically significant and so these are not evidence in favor of the hot hand. OK fair enough.
But if instead of taking the average of averages (a move considered unbelievably stupid by some), we redo their analysis and aggregate across the 26 players (100 shots for each, with some exceptions), we get the following table of weighted averages.
Any                     1175/2515 = 0.4672

3 misses in a row       161/400 = 0.4025
3 hits in a row         179/313 = 0.5719

2 misses in a row       315/719 = 0.4381
2 hits in a row         316/581 = 0.5439

1 miss in a row         592/1317 = 0.4495
1 hit in a row          581/1150 = 0.5052

The table says, for example, that a total of 2,515 shots were taken by the 26 players, of which 1,175 or 46.72% were made.
And of the 400 instances where a player missed 3 in a row, 161 or 40.25% were immediately followed by a hit. And of the 313 instances where a player hit 3 in a row, 179 or 57.19% were immediately followed by a hit.
The above weighted averages seem to be strong evidence in favor of the hot hand.
Bear in mind that the shooting experiment was set up so that each player was shooting from where it had been determined he/she could make roughly 50% of his/her shots.
(Note: "Strangely" enough, in Table 1 for a very similar analysis with the Sixers' in-game shooting, GVT instead present the weighted averages. So why didn't they do the same for Table 4? My guess is that they certainly did calculate the weighted averages for Table 4 — the numbers I present above, didn't like what they saw, and chose to suppress them. This sort of behavior is unfortunately par for the course in academia.)

Example: Say we have the sequence $HHHTTTHHHHH…H$ (only flips #4-#6 are tails, the remaining 97 flips are all heads). Then $\hat{p}(H|3T)=1/1=1$ because there is only 1 streak of three tails and the flip immediately after that streak is heads.
And $\hat{p}(H|3H)=91/92 \approx 0.989$ because there are 92 streaks of three heads and for 91 of those 92 streaks, the flip immediately after is heads.

P.S. GVT's (1985) Table 4 contains several errors. I spotted at least two rounding errors. And also for player 10, the parenthetical values in columns 4 and 6 do not add up to one less than that in column 5 (contrary to the note at the bottom). I contacted Gilovich (Tversky is dead and Vallone I am not sure), but unfortunately he no longer has the original sequences of hits and misses. Table 4 is all we have.


How is currency depreciation related to import tariffs? In the news today they were talking about how the US is "threatening to increase proposed tariffs on Chinese imports from 10 percent to 25 percent."
One of the reasons they gave for this was that Chinese currency has depreciation 8% which rendered the previous 10% tariffs ineffective. The news even described China as "fiddling" with their currency as if it was intentional to get around tariffs.
I'm wondering how these two are related. How does a currency being worth less effect the percent based tariffs?

Assume that a chinese company is selling a product for 100 RMB or 16 USD.

With the 10% tariff the cost for an american importer rises to 17.6 USD.

If the the RMB depreciates 8% and the price in RMB is unchanged the importer pays 8% less in USD or 16.192 USD.

Depreciating the RMB by 8% offsets (almost all of) the effect of the 10% tariff.

This calculation does not account for the fact that chinese imports get more expensive. If the good is manufactured using imported oil the cost of that oil will not be offset by the depreciated RMB.


Abraham (1987) Simple Job Market Matching Model I have a question about a derivation in Abraham (1987)'s simple job market matching model (equations 3 through 7):

She begins by writing down tautologies:

J - V = L - U = E

where J is the number of jobs, V is the number of vacancies, L is the total labor force, U is the total unemployed, and E is the total employed. Consider a probability function that a particular vacancy will be filled:

p(V/U) where $p_u > 0, p_v <0, p_{uu} < 0, p_{vv} > 0$.

Now, letting the total quits and firings from the current stock of employees be sE we must have:
\begin{equation}
 p(V/U)V - sE =0
\end{equation}
Letting U/E = u, V/E = v, and dividing through by E we get:

\begin{equation}
p(v/u)v = s
\end{equation}

Now things get tricky. She writes "along the set of points satisfying (the above)" we have:

\begin{equation}
\frac{dv}{du} = - \frac{p_u}{p+p_v v}
\end{equation}

My question is: how? Suppose I treat this as an implicit differentiation problem. Write $F(v,u) = p(v/u)v-s = 0$ then we know we know:

\begin{equation}
\frac{d v}{d u} = - \frac{\partial F}{du} / \frac{\partial F}{dv}
\end{equation}

I obtain $F_v = p + (v/u)p_v $ and $F_u =  - (\frac{v}{u})^2 p_{u}$ which yields:

\begin{equation}
\frac{dv}{du} =  \frac{(\frac{v}{u})^2 p_{u}}{p + (v/u)p_v}
\end{equation}

This doesn't look very close. Am I missing something? What's going on here?

EDIT:

Inspired by the answer below, I agree it is as simple as totally differentiating P(u,v) = p(v/u).

\begin{equation}
p(v/u) = P(v,u) \rightarrow P(v,u)v - s = 0
\end{equation}

Now, let's totally differentiate this expression with respect to v,u, and s.

\begin{equation}
\left[P_v(v,u)v + P\right] dv + v P_u du - ds = 0
\end{equation}

Consider $ds = 0 $ e.g. no change in the separation rate. This yields:

\begin{equation}
\frac{dv}{du} = - \frac{vP_u}{vP_v + P}
\end{equation}

Substituting $p(v/u)$ for $P(v,u)$ gives the expression off by a factor v.

Note that function $p$ actually depends upon a single variable $u/v$ and I prefer avoiding the abuse of notation
(source of confusion) and write $p(u/v) = P(u,v)$. This implies:

\begin{equation}
p'(u/v) = P_u(u,v)v = P_v(u,v) \cdot (-v^2/u).
\end{equation}

So if $P(u,v)v = s$ total differentiation along $ds=0$ yields

\begin{equation}
P_u(u,v)vdu + (P+P_v(u,v)v)dv = 0,
\end{equation}

or equivalently

\begin{equation}
\frac{dv}{du} = - \frac{P_uv}{P+P_v v}.
\end{equation}

This result is still slighly different from those given in your question... Who can help further?


Condorcet's paradox: Is the majority rule transitive? From this wikipedia link I would say that the majority rule is not transitive. Also I'm not sure I understand exactly what is transitivity in this situation... With a usual preference relation $x\succsim y$ and $y\succsim z$ then $x\succsim z$.

However, from the wikipedia, we could get a paradox ?

As you stated, transitivity is that overall $x \succeq y$ and $y \succeq z$ implies $x \succeq z$. I will show an example where majority rule isn't transitive and hopefully it will answer your question.

Imagine that we live in a world with three people: Person 1, Person 2, and Person 3. Each of these people have preferences over three outcomes $x$, $y$, and $z$. Every decision in this world is made according to majority rules, i.e. The outcome chosen is the one preferred by at least two of the individuals.

Now imagine that the three people have preferences according to:

Person 1: $x \succeq y \succeq z$
Person 2: $y \succeq z \succeq x$
Person 3: $z \succeq x \succeq y$

The majority rule decision making implies that we rank the pairs according to the following:

$ x \succeq y$ because Person 1 and Person 3 prefer $x$ to $y$
$ y \succeq z$ because Person 2 and Person 3 prefer $y$ to $z$
$ z \succeq x$ because Person 2 and Person 3 prefer $z$ to $x$

Notice that this is exactly a violation of transitivity.


Find a subgame perfect equilibrium and a Nash equilibrium I want to know if my thinking is correct.
Look at the following game.

As the game has only one subgame (i.e., the game itself) then the Nash Equilibria will coincide with the subgame perfect equilibria.

In this case, we have two Nash equilibria: {U, u} and {D, d}. By my statement before, the subgame perfect equilibria will be {U, u} and {D, d} too.

Am I right?

As afreelunch only answered in the comment section, I'll answer it here.

The SPNE (Subgame Perfect Nash Equilibrium) is a refinement of the NE (Nash Equilibrium). So, let's call $S$ the set of all SPNE and $N$ the set of all NE in a game. Then:

$$S \subseteq N.$$

If the sequential game has only one subgame (see the subgame definition here), the game itself, then we will have.

$$S = N.$$

And my reasoning was right.


Utility representation of single peaked preferences Is it true that a single-peaked preference (with the peak at some finite point) over the set of real numbers, always has a utility representation ??
If yes, can you please hint towards the proof or references.
Is this result generalizable to multi-dimensional single-peakedness?

No. Basically, you can encode a form of lexicographic preferences, probably the most familiar example of non-representable preferences, as single-peaked preferences on $\mathbb{R}$.
Define $\succeq$ so that $x\succeq y$ exactly if either $|x|<|y|$ or $|x|=|y|$ and $x\leq y$. Basically, the closer to the peak of $0$ a number is, the better, and in case of a tie, the number to the left of $0$ is better.
Suppose for the sake of contradiction that there is a utility representation $v:\mathbb{R}\to \mathbb{R}$ of $\succeq$. For each $r\in\mathbb{R}_{++}$ (the strictly positive numbers), let $q_r$ be a rational number in the interval $\big(v(r),v(-r)\big)$. Since for $r\neq r'$, $\big(v(r),v(-r)\big)\cap\big(v(r'),v(-r')\big)=\emptyset$, we have an injection $r\mapsto q_r$ from $\mathbb{R}_{++}$ to $\mathbb{Q}$, which is impossible since $\mathbb{R}_{++}$ is uncountable and $\mathbb{Q}$ is countable.


Why might it make sense to buy bonds in a company facing restructuring? In this video, Sal Khan states, "If you really thought that Lehman Brothers in the long term was going to come back, what you might want to do is somehow try to become one of its bondholders, and then when it goes through bankruptcy, on the other side of the bankruptcy, you might end up in shares of the new bank, whatever it's called, you know, Goldman Brothers, or whatever."
My understanding is that, minor secondary market factors aside, you could only break even or lose via this strategy.  If the assets of the company being restructured were truly worth less than its liabilities, then the owners of each tranche of those liabilities would receive shares worth at most the value they were owed, and at least the equity tranche would receive less.  How could this be a profitable strategy?

There are experts in distressed debt. They buy the debt at a deep discount to face value, with the belief that the salvage value is greater than what they paid.
To do this successfully, knowledge of how bankruptcy works is required. Different types of debt have different priorities in bankruptcy. In most cases, bonds are deeply subordinated, and so often have extremely low recovery rates versus senior debtors.


Comparative statics question with an application In the state of Mexas, two politicians (Mr. BO, or "Politician 1" and Mr. TC, or "Politician 2") are competing intensely for a senate seat. The two politicians spend on advertising to increase the number of supporters. A political consultant finds that the optimal advertising expenditure of Mr. BO, $S_{1}$, depends on the spending $S_{2}$ by Mr. TC and a "likability" parameter $\alpha$ that influences the popularity of Mr. BO in Mexas: $$\text{Equation 1:  }S_{1} = f\left ( S_{2}, \alpha \right ),$$
where $f:\mathbb{R}_{+}^{2}\rightarrow \mathbb{R}$ is twice continuously differentiable, $0<\frac{\partial f\left ( S_{2}, \alpha \right )}{\partial S}<1$ and $0<\frac{\partial f\left ( S_{2}, \alpha \right )}{\partial \alpha}<1$ for all $S_{2} \geq 0$ and all $\alpha \geq 0$.

The optimal expenditure of politician 2 depends on the spending $S_{1}$ of politician 1 and a "redness" parameter $\beta$, which influences how many people would stick to Mr. TC: $$\text{Equation 2:  }S_{2} = g\left ( S_{1}, \beta \right ),$$
where $g:\mathbb{R}_{+}^{2}\rightarrow \mathbb{R}$ is twice continuously differentiable, $0<\frac{\partial g\left ( S_{1}, \alpha \right )}{\partial S}<1$ and $0<\frac{\partial g\left ( S_{1}, \alpha \right )}{\partial \alpha}<1$ for all $S_{1} \geq 0$ and all $\beta \geq 0$.

The equilibrium values of $S_{1}$ and $S_{2}$ are given by the solution of the simultaneous equations (1) and (2). Suppose that there is a unique solution $S_{1}^{*}>0$ and $S_{2}^{*}>0$.

Does an increase in $\alpha$(holding $\beta$ constant) necessarily increase or necessarily decrease $S_{1}^{*}$? Explain.

My attempt: Here I have used Implicit Function Theorem to answer the question, as we are essentially looking for the comparative statics of $\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha }$.

Since from equation 2, $S_{2}^{*} = g\left ( S_{1}^{*}, \beta \right )$, I substituted $S_{2}^{*}$ in equation to obtain $$S_{1}^{*} = f\left ( g\left ( S^{*}_{1}, \beta \right ), \alpha \right ).$$ Then since $$S_{1}^{*} - f\left ( g\left ( S^{*}_{1}, \beta \right ), \alpha \right ) = 0 \equiv F,$$
I can apply the Implicit Function Theorem (IFT) to derive $\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha }$.

So $$\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha } = - \frac{\frac{\partial F}{\partial \alpha}}{\frac{\partial F}{\partial S_{1}}} = - \frac{\frac{\partial f\left (  \right )}{\partial \alpha}}{1 - \frac{\partial f \left (  \right )}{\partial g} \frac{\partial g \left (  \right )}{\partial S_{1}^{*}}} \frac{>}{<} 0.$$

Since we do not know the sign of the derivative $\frac{\partial f \left (  \right )}{\partial g}$, the effect of $\alpha$ on $S^{*}_{1}$ is ambiguous.

Does my answer make sense?

You can obtain a definite answer for $$\text{sign}\left\{\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha }\right\}$$ given the assumptions.

From

$$S_{1}^{*} - f\left ( g\left ( S^{*}_{1}, \beta \right ), \alpha \right ) = 0 \equiv F$$

and the Implicit Function Theorem

$$\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha } = - \frac{\partial F/\partial \alpha}{\partial F/\partial S^*_{1}} $$

we have

$$\frac{\partial F}{\partial \alpha} = -\frac{\partial f}{\partial \alpha}$$

and

$$\frac{\partial F}{\partial S^*_{1}} = 1-\frac{\partial f}{\partial S_2}\cdot \frac{\partial g}{\partial S^*_{1}}$$

For these expressions we know not only the signs but also the magnitudes. The result follows.


Question about Federal reserve open market operations - reducing money supply I'm trying to understand how open market operations work to reduce money supply.

So the Federal Reserve sells bonds... when this happens, the bank balances of the bond-purchasers goes down... the bank balance of the Federal reserve goes up.

So now what does the Federal reserve do with this money... is the money just eliminated from the system completely? Or does the Federal Reserve hold that money? Does it even make a difference... Thanks.

Have you ever gifted your parents a "I'll wash the house" ticket? Or maybe your parents just forced you to do it. In any case, money, or less colloquially a Federal Reserve note, is like a "I'll wash the house" ticket issued by the Federal Reserve. But instead of being able to redeem it for a house wash, you can redeem it to receive a certain amount of gold. That is if you lived pre-1971 (edit: pre-1933). Now you can't redeem it for anything, but it doesn't change the fact that Federal Reserve notes are a liability to the Federal Reserve.

So if your parents cash-in your "I'll wash the house" ticket, what will you do with the physical ticket? Store it as a souvenir? Rip it up? Re-gift it? Sell it to them in exchange for a new computer? I think "[d]oes it even make a difference" sums it up pretty well, especially since most of the "tickets" are digital not physical.

Footnote: The government guarantees you can pay taxes with Federal Reserve notes. So in that circular sense, Federal Reserve notes are redeemable for something.


What is the logic the Fed used for paying interest on reserves? During the last few years, the Fed has been paying banks interest on reserves. Its a strange policy if you want banks to invest their money in the real economy. Why did they do this? Is it a way to recapitalize the banks? Is it a way to guarantee their solvency irrespective of the availability of investment opportunities? Is it a way to limit the potential inflation of QE?

According to Bernanke himself, interest paid on reserves in excess of the penalty free upper bound are used as an additional tool with which the Fed can place a floor on the federal funds rate.The logic here is that banks have no incentive to lend in the overnight market at rates lower than this interest rate.

This link has some links to info on the topic. It's an old answer that got little attention and so here is my chance to get some use out of it :)

Federal Reserve Interest


Balance of Payment (BPM6) - Direct Investment: how to record I have a question related to recording in BOP
Let say we have two countries:

DOMES (domestic country)
EXT (external country)

a company in EXT invest a new factory in DOMES. I don't know how this would be record in BOP
I know that one side of transaction is recorded under:

FINANCIAL ACCOUNTS // DIRECT INVESTMENT // Net Incurrence of Liability // Equity and Investment fund shares // quity other than reinvestment of earnings (3AA100 L AA)

but I don't know where is the other side will be recorded.
Thank you
Phan

Assuming country EXT builds a factory in DOMES:

For country DOMES: this is direct investment liability
For country EXT: this is direct investment asset


Can you prove, and with what assumptions, that a market will allocate some minimum amount of basic resources to every worker? I'm looking for a summary/less technical answer, as I'm not very well versed in the details of economic theory. I'm curious if there is some way to show that under some circumstances in a market economy that every worker (assuming they work some minimum hours, are not disabled, maybe have a basic high school education, etc) who participates in the economy will be able to procure some amount of basic resources (i.e. food, shelter, clothing).
More directly: In a market economy, will everyone who participates be able to sustain themselves, or can even a fully participatory, albeit poorly educated and unskilled, worker become unable to sustain themselves, and this is just an accepted fact of economic theory?

There is no guarantee that anyone in market economy earns some subsistence minimum. Generally, wage depends on marginal product of worker's labor and if there are some quasi-rents to be had also on the relative negotiating power of worker and employee. Of course, there is a bit more nuance to it, for that you can see Borjas: Labor Economics 5th ed, but assuming that people roughly get wage equal to their marginal product is reasonable first approximation.
There is no reason to a priori assume that marginal product of some worker will be enough to procure some basic goods and services (food, shelter, clothing are goods not resources - labor and other factors are actually resources in economics). So it might very well be that there are people for whom wages are not enough to be able to sustain themselves just on their own without some sort of subsidy/charity.


How have central banks been injecting currency after crises since 2007? In an article by Jean-Michel Naulot about the 2007–2008 financial crisis, I read (my translation):

Although central banks had an exemplary reaction at the time of the [2007–2008] crisis, they subsequently carried out an unprecedented policy by massively and durably injecting currency into the financial system, as never before. A major part of these liquid assets was invested into risky assets and speculation. Private debt rose considerably again.
In the past 20 years, a fiendish debt feedback system has risen: as crises break, public spending is massively revived and tax revenue is reduced, hence public debt soars. Central banks keep interest rates low for extended periods of time and private debt rises considerably, especially in countries with high levels of inequality. Excessive private debt leads to a new crises, which drives public debt up, ...

The second paragraph suggests that public debt rises because of public spending programs and because lower tax revenue during hard times is not compensated by higher tax revenue during boom times. However the first paragraph suggests that there's more to public debt than public works (which benefits the public) and low taxes (which benefits whoever gets low tax rates). Where does the money that's “massively and durably injected” go?
Does this refer to stimulus packages? To governments compensating for defaulting financial institutions? Who's getting the money? Are these loans (that don't seem to be repaid) or gifts?

The question touches several points, but I can try to explain the "where did the money go in the crisis" with an example from Portugal:

Through automatic stabilizers, and
due to acquisition of bankrupt banks and companies, absorbing the losses in the public budget.

With the crisis several companies and financial institutions went bankrupt, meaning that the government increased spending with the automatic stabilizers such as unemployment subsidies => more public spending due to more subsidy and loss of tax collected from company and salaries

Allegedly due to international toxic assets, as well as some media speculation, some banks couldn't face their responsibilities anymore, declaring bankruptcy - see BES bank for example. People were in panic because the money they had in the bank could disappear suddenly, so the government nationalized the bank, guaranteeing everyone's deposits, assuming the losses of the bank. This means that the government increased its costs in millions of euros to save this and other bank, in the interest of national financial stability.

If you are more looking at how central banks inject money in the economy, take a look at money supply tag.


Optimal Top Marginal Tax Rates Some of you might know of US politician Alexandria Ocasio-Cortez, a member of the Democratic Party as well as the Democratic Socialists of America. In a recent interview she had on 60 Minutes, she suggested the idea of raising the top marginal tax to 70% for income above 10 million dollars. The recent conversation surrounding this proposal is unsurprisingly heated (though mostly unproductive, as people struggle to understand what a marginal tax rate entails).

  My question is what recent literature is there for determining the optimal top marginal tax rate of a country?

For example, in Piketty, Saez, and Stantcheva (2011), the optimal marginal tax rate they estimate is 83% (on a cursory search through the paper, I cannot find for what level of income though) looking at labor supply and tax avoidance elasticities, but also at a "compensating bargaining" elasticity that tends to make the optimal rate higher than other models they cite.

A good answer will briefly explain what the literature you cite argues for, with some criticisms and/or defenses of the literature and its possible shortcomings. Recent literature of course is preferred, but if an older model is needed to explain newer ones, it is fine to cite those too.

The literature on optimal income taxation kicked off with Diamond, Mirrlees (1971). In their model, people differ only by ability, i.e. the highest earning person is the most productive one. As a consequence, any marginal tax rate on this person will be detrimental to production and hence to social welfare (in this model). Therefore, the optimal top marginal tax rate is zero!

The main innovation was initiated by Emmanuel Saez (Saez (2001) and Piketty, Saez (2012)). The basic idea is that the social planner maximizes a social welfare function (SWF) that captures the fact that individuals are to some degree inequality averse. Plus, they allow the introduction of welfare weights, i.e. a normative decision on whose well-being matters for the society as a whole. The innovation of this literature is the formulation, that allows to calculate an optimal tax rate based on observable (or, rather, estimatable) figures. This is referred to as the sufficient statistic approach. See e.g. Eq.7 in Piketty, Saez (2012) for the optimal top tax rate $\tau$:

$$\tau = \frac{1-g}{1-g+a *e} $$

This depends only on three parameters: $e$ is the extent to which top income earners are able to dodge the tax. In the basic model, this is only labor supply, but later contributions (incuding the one you cite, but also Saez et al. (2012), [yes, Saez again]) acknowledge that there are many other avoidance strategies. $g$ is the social welfare weight of the top-income earners and $a$ is an indicator of the income distribution.

The optimal top marginal tax rates above 70% stems basically from the assumption that $g$ is zero, i.e. society as a whole does not really care about the richest. Another key factor is the magnitude of $e$. If I argue that with a sufficiently high top tax rate, all millionaires will leave the country, I assume a high $e$, which drives down $\tau$. There are also researchers that came up with different values for $a$. One quite well-known criticism is from Mankiw et al. (2009). They back up their arguments however, e.g. on less reliable data, which is why I don't find it very substantiated at least on the top tax rate issue.

For a general discussion on optimal taxation beyond the top tax rate, see also Diamond and Saez (2011).

Summing up, the Saez literature is a great tool to link data and economic theory and has been heavily used. Empirically, estimating the various margins of $e$ is very difficult and has been tried numerous times. The top tax rate of 70% or higher is debatable as it has some normative settings. But overall, this literature has made compelling arguments in favor of progressive taxation and higher taxation of inheritances. It might be my personal bias, but I'm not aware of any fundamental, substantiated criticism to the Saez approach.


Repeatedly playing an equilibrium in a repeated game What is a formal proof to this known fact about repeated games?

  The situation in which, in every time step, the players play a Nash equilibrium in the basic game unconditioned on history, is a Nash equilibrium in the repeated game.

Suppose all players play unconditioned Nash equilibria, except player $i$ who diverts and plays another strategy. The other strategy of player $i$ may depend on history. We have to show that player $i$'s payoff in the repeated game does not increase.

For every time-period t, define the following random variables:

$X^{t}$ - the utility of player $i$ in time t when all players, including player $i$, play the basic Nash equilibrium.
$Y_{h}^{t}$ (where $h$ is a history vector of length $t-1$) - the utility of player $i$ in time $t$ when player $i$ diverts and plays the alternative strategy while all other players continue to play the basic Nash equilibrium.
$Y^{t}$ - the utility of player $i$  in time $t$ when player $i$  diverts while all other players continue to play the basic Nash equilibrium:
$$Y^{t}=\sum_{h}Prob[h]\cdot Y_{h}^{t}$$

By definition of a Nash equilibrium, for every time $t$
and for every history $h$:

$$E[Y_{h}^{t}]\leq E[X^{t}]$$

Hence also:

$$E[Y^{t}]\,=\,\sum_{h}Pr[h]\cdot E[Y_{h}^{t}]\,\leq\,\sum_{h}Pr[h]\cdot E[X^{t}]\,\leq\,E[X^{t}]\cdot\sum_{h}Pr[h]\,\leq\,E[X^{t}]$$

Summing over the entire series, and using the fact that the expectation of a sum is the sum of expectations:

$$E[\sum_{t=1}^{T}Y^{t}]\leq E[\sum_{t=1}^{T}X^{t}]$$

So the player cannot gain by deviating.

Thanks to @denesp for verifying this proof!


What is the need for a special purpose entity/vehicle in mortgage backed securities What is the need to create a special purpose entity/vehicle (SPV) for mortgage backed securities? What potential risks are avoided due to the creation of SPV?

It is my naive idea that the holder of the security always has the first priority right for receiving repayment and interest on the loans. How is this affected in the scenario where the bank transfers the assets to an investment bank vs the SPV transfers the assets to an investment bank, which in turn is securitized?

Here are a few reasons that build on @Dismalscience's answer.

Capital requirements: Banks don't typically need to hold capital
against loans they originated but subsequently moved into an SPV. This might be regulatory arbitrage but it might be a socially efficient outcome meant to move assets and liabilities out of the banking system.
Market segmentation: Because SPVs can tranche (slice) their loans they can offer products with different assumed default correlations, prepayment risk, and aggregate credit risk. These products can appeal to customers with different beliefs and preferences over risk.
Bank business model: Banks may be better at finding customers, screening them, and handling the paperwork of making loans (broadly originating) then they are at proving inexpensive and patient capital to fund those loans. Securitization allows the bank to specialize in origination without also being the ultimate capital provider.
Information destruction: Because SPVs contain loans that are diversified over geography and borrower characteristics they can reduce market adverse selection, improving pricing and liquidity.


Finding equilibrium of perfect competition in the short run with a cost function I am having a lot of problems trying to find the equilibrium price when we are given a cost function and demand function, but no supply function.

  All firms have the following production function: $Q = \sqrt{K \cdot L}$

  Wages are w = 9, and the rental rate of capital is r = 36.

  In the short run, capital is fixed at ̅ = 3 units. Market Demand is given by P = 360 – 2Q

  There are 9 firms in the market, find P*, Q*, and Total Surplus

In solving the production function with a fixed K:

$MPL$ = K$^{1/2}$ \cdot L$^{1/2}$

$MPL$ = ${1/2} \cdot $K$^{1/2}$ $\cdot$ $L^{-1/2}$

= 1/2 \cdot (K$^{1/2}$ $\cdot$ L$^{1/2}$)

Substituting 3 for K:

= 1/2 \cdot ($\sqrt{3}$ \cdot $\sqrt{L}$)

$\sqrt{3}$/2 =  $\sqrt{L}$

$\sqrt{L}$ = .8660254

$L^*$ = .75

Where C = wL + rK

C(Q) = w($L^*$) + r($K^*$)

C(Q) = (9*.75) + (36*3)

C(Q) = 114.75

C(Q) @ 9 Firms =  (114.75 * 9) = 1032.75 (I think???)

But this doesn't have a slope, so how does this make sense? Obviously you would normally find the equilibrium through Qd = Qs, but what do I equal Qd to now?

In these problems, you are generally dealing with identical firms - all of whom will supply the market according to their marginal cost curve.  If we were solving for a long run equilibrium the first thing we would do is get the supply curve - which can be done by finding the optimal mix of capital and labor via:

MPL/W = MPK/r

This will get you L = 4K.

In this situation you can skip to:
$$
Q = \sqrt{3}\sqrt{L}\\
L = Q^{2}/3
$$

Plug those into $C(Q) = Kr + Lw$

$$
C(Q) = 108 + 3Q^{2}\,
$$

The fixed cost is \$108 (but that is sunk in the short run, so the firm will ignore it), the variable cost is $3Q^{2}$, making the marginal cost $6Q$.  For the market, this will be $\frac{Q}{1.5}$ (the horizontal sum of all firms' production)

Marginal revenue in this market is just the price since we assume it is perfect competition, so setting market supply = demand gives you
$$
360 - 2Q = \frac{Q}{1.5}\\
Q = 135\,
$$

$Q/n = q$  tells us that each of the 9 firms must produce 15 units for the market to supply 135 total units.  According to the demand curve, the market price is \$90 when 135 units are sold.

Total surplus can be found by calculating the area of two triangles:

$$
CS = (1/2)135*(360-90) = \$18,225\\
PS = (1/2)135*(90) = \$6,075\\
TS = \$24,300\,
$$


Book recommendations: National income accounting and IS-LM model Basically I'm interested in solving tricky thought provoking exercises based on National income and on IS LM model. I've read mankiw but it didn't satisfy my needs.

I would suggest:

1) Macroeconomics by Dornbusch and Fisher.

2) Macroeconomics by Richard Froyen

3) The Economics of money and financial markets by Mishkin.

Between the first two books that is Dornbusch and Fischer and Richard Froyen, I am indifferent.  Both are equally good and the only difference is that Dornbusch and Fischer has a slightly more mathematical rigor whereas Froyen is a more story like engaging macroeconomics narrative with fair amount of mathematics.

If you need a truly engaging book,  then go for Mishkin. But I will suggest don't read Mishkin as the first book to understand macroeconomics.


What kind of policy is minimum wage? Does it fall under fiscal, monetary, or supply-side, or perhaps some other group? Also (seeing as minimum wage is a form of price control policy), where does the broader "price control policy" fall?

Looking up each of those terms you ask about would have given you your answer.

Fiscal policy is economic policy enacted through government. The minimum wage is a type of fiscal policy.

Monetary policy is economic policy enacted through central banking. The minimum wage is NOT a type of monetary policy.

Supply-side is a macroeconomic theory that growth is best achieved through capital investment and lowering barriers to production. Usually the policy suggestions to come out of this are to remove government regulation and lower marginal tax rates. The minimum wage is a type of regulation, so minimum wage would not fall under the doctrine of classical supply side economics.

I don't understand what you mean when you ask where "price control policy" falls under. There's no real categorization for it besides also being a type of fiscal policy.


Should the "value function" be "utility function" in prospect theory? I have a background in mathematics rather than economics, and currently reading Choices, Values, and Frames[1]. The paper defines a "hypothetical value function" (the s-shape that is concave for gains, convex for losses).
Given the definition of value to be descriptive, and utility to be normative, should it not be a "hypothetical utility function" instead?
The expected value of a risky decision is defined by the sum of all of weighted possible outcomes, from Wikipedia e.g.:
$${\displaystyle \operatorname {E} [X]=\sum _{i=1}^{k}x_{i}\,p_{i}=x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{k}p_{k}}$$
The corresponding expected utility is then:
$${\displaystyle \operatorname {E} [u(x)]=\sum _{i=1}^{k}u(x_{i})\,p_{i}=u(x_{1})p_{1}+u(x_{2})p_{2}+\cdots +u(x_{k})p_{k}}$$
I understand (incorrectly?) one key difference between expected utility theory and prospect theory is in the way that $u$ is constructed - the former dependent on the total wealth, and the latter dependent on the gain/loss of the change itself. Nevertheless, they are both dealing with the "satisfaction" of a gamble, and not the expected value, which does not change - hence my question.
I feel that I'm missing a trick here. Any illumination would be greatly appreciated!
[1] Kahneman, D., Tversky, A., 1984. Choices, Values, and Frames. American Psychologist, Choices, Values, and Frames 39, 341–350.

The terminology in microeconomics is not completely unified but typically differs slightly from the mathematical one. For a real-valued random (outcome) variable $X$, the mathematical expected value $\mathbb{E}(X)$ would rather be called the expectation of $X$. The utility $u(x)$ of an outcome $x$ is understood to be given by a (Bernoulli) utility function in the sense of EUT, i.e. a function $u(.)$ such that the expected utility $\mathbb{E}(u(.))$, sometimes written as a (von-Neumann-Morgenstern) utility function $U(.)$ on lotteries on outcomes, represents the preferences over lotteries on outcomes, while in behavioral economics the term value or valuation $v(x)$ typically represents some kind of subjective valuation of $x$, as e.g. in prospect theory. (So here, value is not to be understood as the mere numerical value of $x$, as in the usual mathematical definition of a "function value" or an "expected value".)
Confusingly, however, $v(.)$ is sometimes also used to denote an alternative (Bernoulli) utility function. For example when explaining that a positive linear transformation of a utility function is again a utility function representing the same preferences, one of them is usually denoted by $u$ and the other by $v$. Moreover, in a utilitarian framework, $v(.)$ is often used to denote the willingness to pay for $x$, i.e. a monetary valuation of $x$.


In modern economics, what are the least restrictive conditions shown to theoretically achieve Pareto Efficiency? In researching this question, I've come across plenty of sources that regurgitate intro econ textbooks that say markets equilibrate to Pareto efficiency in conditions of perfect competition (where specific and minimal definitions are also hard to find). I'm sure they're correct that this happens in perfect competition, but I've also read that various economists have simplified and minimized the conditions necessary for a market to theoretically reach pareto efficiency, but I can't find any details on those.

So what are the least restrictive conditions necessary for all markets that fit those conditions to theoretically progress toward a Pareto efficient equilibrium?

Well, when considering the minimal conditions necessary for an allocation to be Pareto optimal we must go to the primitives. First, we need the fact that all agents have rational preferences. What this means is that preferences are complete and transitive. Another thing we need is for preferences to be strictly convex.

Let $\succcurlyeq$ be the preference relation "weakly preferred to":

$\textbf{(A) Transitivity:}$ Given allocations $x,y,z \in X$, if $x\succcurlyeq y$ and $y\succcurlyeq z$ then $x\succcurlyeq z$

$\textbf{(B) Completeness:}$ Given allocations $x,y\in X$, either $x\succcurlyeq y$ or $y\succcurlyeq x$ or both $(x\sim y)$

$\textbf{(C) Strict Convexity}:\forall\, x,y \in X$ and $\forall \lambda\in[0,1]$ we have:
$$\lambda f(x)+(1-\lambda) f(y)>f(\lambda x+(1-\lambda)y)$$

Also, let $X$ be the set of goods. We need $X$ to be finite.

These four axioms are sufficient for the existence of a competitive equilibrium.

Now, we must build a bare-bones theoretical market. What we need is the following:

$\textbf{(1) Complete Market:}$ Everyone has complete information and there are no transaction costs.

$\textbf{(2) Price Taking:}$ There is one price which everyone pays (and by the above statement, everyone knows this price).

$\textbf{(3) Locally Non-Satiated Preferences:}$ Basically, giving an agent more of any good will always increase their utility.

If we have $(A)-(C)$ and we have a finite number of goods, we will have a competitive equilibrium, and if we have $(1)-(2)$, we have a competitive market. Using these two and $(3)$, we can use the First Fundamental Theorem of Welfare Economics to say that our equilibrium is Pareto optimal.


Price difference of substitute goods when the supply increases? In a competitive market, there are 3 types of goods: X, Y and money (m). X and Y are substitute goods, and the utility functions are quasi-linear with respect to money, i.e:

$$U(x,y,m) = u(x,y) + m$$

The supply of X and Y is  determined exogeneously, and the prices $p_x$ and $p_y$ are then determined such that the system is in equilibrium (i.e. demands for X and Y equal their supply).

Now the supply of X is increased, and a new equilibrium is achieved with new prices $p_x'$ and $p_y'$. What can we say about the new prices?

Certainly $p_x'<p_x$, since we must have new consumers buying the newly available units of X.
Certainly $p_y'\leq p_y$, since X and Y are substitutes (when the price of X decreases, the demand for Y weakly decreases so the price of Y also weakly decreases).
My main question is: what happens to the price difference $p_x-p_y$? I did some anecdotal simulations, and it seems that $p_x'-p_y'\leq p_x-p_y$. I.e, even if there is a decrease in the price of Y, the decrease in the price of X is weakly larger.

Is this observation correct?

What really happens to the price difference when the supply increases?

To use a simple example, assume that consumer $i$ maximizes

$$U(x_i,y_i,I_i) = \alpha\ln x_i +(1-\alpha)\ln y_i + (\bar I_i-p_xx_i -p_yy_i)\\
s.t. \bar I_i \geq p_xx_i +p_yy_i$$

In other words, it may spend all his income on the two goods (but not more), or he may keep some for other goods, not modeled here. Due to the inequality in the constraint we need to use a Karush-Kuhn-Tucker non-negative multiplier, rather than the usual Lagrange multiplier.
Assuming a fixed income, the lagrangean function is

$$\Lambda = \alpha\ln x_i +(1-\alpha)\ln y_i + (\bar I_i-p_xx_i -p_yy_i) +\lambda_i(\bar I_i-p_xx_i -p_yy_i)$$

Assuming income is fixed, the first order conditions are

$$\frac {\alpha}{x_i} - (1+\lambda_i)p_x \leq 0,\;\; \frac {1-\alpha}{y_i} -  (1+\lambda_i)p_y  \leq  0$$

$$\lambda(\bar I_i-p_xx_i -p_yy_i) = 0 ,\; x_i\cdot \left(\frac {\alpha}{x_i} -  (1+\lambda_i)p_x\right)=0, \; y_i\cdot \left(\frac {1-\alpha}{y_i} -  (1+\lambda_i)p_y\right) =0$$

it follows that at the optimum both goods will be demanded in positive quantities, which in turn requires that the first-derivatives be set equal to zero. This gives us

$$\frac {\alpha}{p_xx_i} = \frac {1-\alpha}{p_yy_i} \implies x_i^D=\frac {\alpha}{1-\alpha}\frac {p_y}{p_x}y_i^D \tag{1}$$

from which we can obtain the optimal relation for the Expenditure $E_i$ on the two goods

$$E_i^* = \frac {1}{1+\lambda_i^*} , \;\; E_i^* \leq \bar I_i$$

Since $\lambda_i \geq 0 \implies  \max E_i^* \leq 1$ so if it so happens that $\bar I_i >1$ then not all income will be spent on the two goods, the constraint won't be binding and so $\lambda_i^* =0 $. (Is $\max E_i^* \leq 1$ a strange result?) On the other hand if $\bar I_i \leq 1$ then we will necessarily have $E_i^* = \bar I_i$, and $\lambda_i^* = (1-\bar I_i)/\bar I_i$.

In either case, optimal relation $(1)$ remains valid. So, assuming identical consumers with respect to preferences (not necessarily with respect to Income), aggregating $(1)$ we obtain the market-level relation

$$ X^D=\frac {\alpha}{1-\alpha}\frac {p_y}{p_x}Y^D \tag {2}$$

At equilibrium, we have $X^D = X^S,\; Y^D = Y^S \tag{3}$.

Equations $(2)$ and $(3)$ hold for any supply.
We now change the supply of $X$ but leave the supply of $Y$ unchanged. Indexing  the initial situation by $0$ and the second situation by $1$, they are described by

$$ X^D_0=\frac {\alpha}{1-\alpha}\frac {p_{y0}}{p_{x0}}Y^D_0,\;\;X^D_0 = X^S_0,\;\; Y^D_0 = Y^S_0 \tag {4}$$

$$ X^D_1=\frac {\alpha}{1-\alpha}\frac {p_{y1}}{p_{x1}}Y^D_1,\;\;X^D_1 = X^S_1>X^S_0,\;\; Y^D_1 = Y^S_1= Y^S_0\tag {5}$$

Then we have

$$X^D_1 - X^D_0 = \frac {\alpha}{1-\alpha}\frac {p_{y1}}{p_{x1}}Y^D_1 - \frac {\alpha}{1-\alpha}\frac {p_{y0}}{p_{x0}}Y^D_0   $$

and using the various relations

$$\implies X^S_1 - X^S_0 = \frac {\alpha}{1-\alpha}\frac {p_{y1}}{p_{x1}}Y^S_1 - \frac {\alpha}{1-\alpha}\frac {p_{y0}}{p_{x0}}Y^S_0 >0   $$

$$\implies \frac {\alpha}{1-\alpha}Y^S_0\cdot \left[\frac {p_{y1}}{p_{x1}}-\frac {p_{y0}}{p_{x0}}\right] >0$$

$$\implies \frac {p_{y1}}{p_{x1}}-\frac {p_{y0}}{p_{x0}} >0 \implies \frac {p_{y1}}{p_{y0}}>\frac {p_{x1}}{p_{x0}} \tag{6}$$

$(6)$ tells us that the price of $Y$, if it falls, it will fall certainly less than the price of $X$, but in proportional terms.

It appears that in this benchmark example, we cannot say anything about the price changes in terms of levels, as the OP asks about.

If the levels of the two prices are very different we can easily have a smaller proportional fall for $p_y$, while at the same time a higher fall in monetary units. Set for example $p_{y0} = 100, p_{x0} = 1$, and assume that the price of $Y$ falls only by $10$% while the price of $X$ falls by $20$%.

In turn the levels of prices depends on the magnitude of the supply of the two goods also, which are treated as exogenous here.


Why do commercial banks buy government bonds at negative interest rates instead of keeping cash? I know commercial banks need to meet liquidity requirements, but why would they choose to invest their money in an instrument that is guaranteed to give them a negative return, instead of just keeping their clients' money in "cash"?

Is there some legislation forcing them to use government bonds?

For a bank , the basic choice of assets is between (a) loans to customers (b) invest in financial asset such as government bond (c) deposits with the central bank.  Note that (c) may return a negative interest rate, so (b) may be better than (c).  Holding a large amount of physical cash in the vault is not practical for a bank of any decent size, due to the cost of storage and insurance.  It would also be discouraged by the bank regulators , because it would frustrate the negative interest rate policy the central bank is trying to apply.


Adverse selection in competitive markets- labor market I'm referring to the Figure 13.B.1 titled 'A competitive equilibrium with adverse selection' from 'Microeconomics Theory' by Mas Colel et al. It basically graphs the expected value of the workers' productivity ($E[\theta|r(\theta)\leq w])$ against the wage offered $w$ in an attempt to explain the wage equilibrium when the opportunity cost $r(\theta)$ is lower than $\theta$ for all $\theta$.. For the life of it, I cannot understand why this function is a curve (as opposed to straight line) in the graph. What is the distributional assumption for worker's type $\theta$ that is made in order to obtain the curve?

Thanks.

Suppose that $r'(\theta)>0$. The following figure should make clear that saying $r(\theta)\leq w$ is equivalent to saying that $\theta\leq r^{-1}(w)$ (where $r^{-1}(\cdot)$ is the inverse of $r$):

So we can rephrase your question as 'why should $E(\theta|\theta<r^{-1}(w))$ be non-linear?'

Let's calculate this expectation (assuming that $\theta$ is distributed on support $[\underline{\theta},\overline{\theta}]$ according to CDF $F$):

$$E(\theta|\theta<r^{-1}(w))=\frac{1}{F[r^{-1}(w)]}\int_\underline{\theta}^{r^{-1}(w)}\!\theta F'(\theta)d\theta.$$

For $E(\theta|\theta<r^{-1}(w))$ to be linear in $w$ we would need $\partial E(\theta|\theta<r^{-1}(w))/\partial w$ to be constant.

Differentiating with respect to $w$ we have
$$\frac{\partial E(\theta|\theta<r^{-1}(w))}{\partial w}=\frac{F'[r^{-1}(w)]\frac{\partial r^{-1}(w)}{\partial w}\left[F[r^{-1}(w)]r^{-1}(w)-\int_\underline{\theta}^{r^{-1}(w)}\!\theta F'(\theta)d\theta\right]}{F[r^{-1}(w)]^2}$$

The above expression isn't the prettiest thing to look at, but a quick glance should convince you that it will only be constant under very special circumstances. For example, even if we take a highly linear environment like $r(\theta)=\theta$, $F(\theta)=\theta$, we have
$$\frac{\partial E(\theta|\theta<r^{-1}(w))}{\partial w}=\frac{w^2+\underline{\theta}^2}{2w^2},$$
so $E(\theta|\theta<r^{-1}(w))$ is non-linear.


Is Cobb-Douglas the only output function corresponding to a competitive economy? Apologies if this is a rather simple question, I appreciate any guidance.

  $$ Q(K,L) = AK^\alpha L^{\beta} $$

  where A is a constant. Identify the conditions on $\alpha$ and $\beta$
  for output function Q to correspond to a competitive economy.

So a competitive economy is one where the factors of production are paid their marginal product. I am aware that the Cobb-Douglas production function alignes with this, so I would have: $ \alpha + \beta = 1$ and $\alpha, \beta > 0$.

But is this really the only and most general answer? I cannot see why we couldn't have different exponents and still pay the factors of production their marginal product.

Let $a+b<1,\;\; a,b>0$.  Pay the factors of production their marginal product:

$$rK = \frac {aQ}{K} K= aQ,\;\;\ wL=\frac {bQ}{L} L=bQ$$

So total payments to factors of production will be

$$rK +wL = aQ + bQ = (a+b)Q < Q$$

And the question is: who gets the rest of the output that has been produced?


How do we find pareto optimal points in a 2 goods simple exhange economy? Suppose there are 2 individuals in a simple exchange economy with utilities $U_{1}= ax_{1} + x_{2}$ and $U_{2}= y_{1}y_{2}$.

Endowments are $(x_{1},x_{2})=(4,0)$ and $(y_{1},y_{2})=(1,5)$.

We are asked the values of $a$ such that the above allocation is pareto optimal.

The answer as given is $MRS_{A} \ge MRS_{B}$ implies pareto optimal.

How did we arrive here?

URL for reference: http://www.econschool.in/stuff-of-interest/anotherpost/dse-2013-q34

URL for actual problem: http://www.econschool.in/stuff-of-interest/anotherpost/dse-2013-q34

The following plot has the answer to your question, observe it carefully:


Why have so many legitimate smaller biotechs sprung up and thrived? I stumbled this question on r/AskScienceDiscussion, but I'd like to ask it from a microeconomics or competition standpoint.
Why can't Big Pharma accomplish what legitimate small biotechs have? Unquestionably Big Pharma has more money and can corner the market. Thus how could smaller biotechs have dawned at all? E.g. Regeneron collaborates with with Bayer and Sanofi. I'm not referring to potential pump-and-dumps like Moderna.

In Q1, Regeneron made over \$281 million from its Bayer partnership, primarily stemming from Bayer's sales of Eylea outside the U.S. It made nearly $247 million from its Sanofi collaboration, with immunology drug Dupixent the primary contributor.

FDA approved Neurocrine Biosciences`s valbenanzine in 2017. Neuocrine has collaborated with Eli Lilly and Company in 1996, GlaxoSmithkline in July 2001, Pfizer in Dec 2002, AbbVie Inc in 2010. But why didn't these Big Pharmas acquire Neurocrine?
On Jul 13 2020, Amgen doubles down on BeiGene investment. On Jul 15 2020, Nabriva inks Sivextro distribution with Merck.
On Aug 6 2020, FDA lets NeuroRx, Relief Therapeutics test RLF-100 in COVID-19 patients, but why haven't Big Pharma bought out or hostilely taken over these biotechs?
In R&D, small biotechs hold their own against big pharma | BioPharma Dive

Despite spending billions on R&D, big pharma doesn't punch its weight in discovering the new molecules that eventually make it to market.
Instead, it's often the smaller biotechs which are responsible for uncovering promising new therapeutic approaches, an industry maxim backed up by an April report from research group Iqvia.

I would prefer to take a broader tact with this: Why aren't all firms bought at a smaller size? Why are there any small R&D firms?
One answer is related to uncertainty.

Some firms look like they will be profitable, but they are not. Perhaps their research fails: they cannot get approval, the drug doesn't work, or it has severe undesirable side effects. This uncertainty is part of the business.
Firms produce signals of their quality. The signals reduce uncertainty and it becomes clearer which startups will be profitable, increasing their prices. Owners will be less willing to sell, and other buyers will make competing offers.
The increased price of the firm leads to a larger asking price, so any net profit of purchasing the new firm is reduced.
Since the profit margins are reduced, the remaining uncertainty leads to continued hesitation in purchasing.
This process continues until the small startup is no longer small enough to buy.

Sometimes, a profitable firm can be identified with sufficient certainty before the final step 5, and that is where you see purchases.


A world with total automation I was wondering. Suppose that in the whole realm of economic activities, all those activities were fully automized. From the finding of raw resources to the making of products that leave the factories, so the owners of the companies that perform the many tasks from resources to end-product don´t need employees anymore; the maintenance of infrastructures; social- and health-care; in short, every kind of work were we need in this time people for (I´m aware it´s rather far fetched, but often when you push things to the limit some things can become clear).

Would the world be a place of milk and honey, or will conflicts arise?

I would strongly recommend you to read Erik Brynjolfsson, Andrew Mcafee et al for some serious and real insights about this topic. The world economic forum this year had the same theme.
In my view (which are heavily derived from the works of the above mentioned dignitaries) the problems arising from complete automation will be the lowering or stagnation of income levels. As Erik mentions in his Ted talk, productivity levels are at all time high but income levels are not. This was not the case in previous technological revolutions. People and systems adapted to new technologies and income rose along with productivity. Now they don't share the same relationship. One of the reasons is that we are not changing our systems well to absorb the technology. For example, if people are saving a lot of time because of automation, they should be able to spend it in something constructive. May be entrepreneurship, maybe knowledge work. To make all those automation work, people are required to program, design, develop and create. There needs to be a shift from typical work to knowledge work. If such a shift is not there and that too at a speed that matches with the speed of technological changes then income levels will continue to fall, unemployment will rise.
In Andrew Mcafee's words, "we need to RACE WITH THE MACHINE".


The relationship between profit and wages It seems obvious that an individual company increases their profits when they reduce wages or increase automation or some such.

But if wages fell universally because of this, wouldn't that decrease profits, since ultimately the earnings of all those companies rely on wage-earning consumers?

It would seem to me, then, that an industrialist, in the sense of an owner of a particular company, should want to reduce wages, but a true capitalist, in the sense of someone with their finger in many pies, would actually want to increase wages - and that their interest would be to prevent other classes from accumulating savings by making sure that those increased wages are channeled into as much additional consumption as possible.

Is this correct? How can I understand this better?

In my innumerate way I have tried to reduce this to a simple thought experiment and here is where things get messy and confusing for me:

Suppose there is just 1 capitalist and 1 worker & consumer.

Suppose that the worker produces $100 worth of goods and services.

In order for them to be purchased from the market, that worker will need to also earn $100.

So there doesn't appear to be any profit for the capitalist at all in this picture. And yet isn't profit surplus value? Where is that coming from?

Clearly I am getting this wrong, both because the above situation could not exist, and because if it did, it could not grow/change. But what is my error?

My gut sense is that factors involved here include:

That a proper consideration of the role of currency is needed to pair with the goods, since we are not just dealing with the value of the goods, but the denotation of that value in money, the circulation of which behaves differently than that of the goods.
That time intervenes between production and consumption causing a circulating effect that somehow (through market & financial intermediaries?) magnifies apparent value involved.
Consumer credit - like credit cards - would seem to have a distorting effect on this process, but that's still in expectation of future wages, right?
The existence of different tiers of wage earners and consumer, whether by class in the same economy, or maybe more importantly, in different economies - high earning "consumers" in Western nations having a codependent relationship with low earning "laborers" in developing nations. This one in particular seems relevant to me.
The spending of the capitalists, even if that's minuscule in comparison. But if the "surplus value" from the workers going to the capitalist is what's needed to be spent in the market to buy all the produced goods and services then they would only be recirculating what they have, not earning, saving and reinvesting.

What also seems relevant:

"Semi-fungibility" of different goods. What's an arch-capitalist going to do with a few million more bushels of grain? It would seem to me that money approximates fungibility to large extent but at the end of the day you can't eat a yacht.

But this is all inchoate musing and I feel helpless to understand much less really explain any of it.

Any direction you could give me would be much appreciated!

Thank you,

Jenna

At the end of the day, currency (such as $) is just a means of purchasing real goods and services in an economy. Roughly, utility corresponds to consumption. Since you assume the capitalist isn't allowed to consume, there is nothing for the capitalist to gain even if they capture all the currency.

Here is a slightly modified thought experiment.

Suppose there is one employer and one employee in an economy. Both are consumers. There is only one good: tomatoes. In other words, every credit is spent on purchasing tomatoes, and the real GDP of this economy is the number of tomatoes produced. If the employee earns less wages, then the employer has a larger slice of the revenue, and the employer now can purchase more tomatoes.

Now what if the employer tries to pay the employee almost nothing? The employee might just quit: no tomatoes are produced, and everyone is worse off as no one gets tomatoes. Even if the employee doesn't quit, at some point the employee will be less productive because they are starving from not being able to consume tomatoes. In other words, for the employer that wants to consume as many tomatoes as possible, there is a balance between taking as large a share from the existing pie of tomatoes and increasing the pie by allowing the employee to have enough to be productive.


Is there a super term for goods/products and services? A business can sell "goods" (/"products") and "services".

Is there a term that encompasses both of these?

Various words may fit (e.g. "offering"), but I'm looking for the technical term, if there is one.

I'm not aware of any such "technical" term in economics. Goods and services are referred to as goods and services in economics, period. From the example you give in the comments, it looks like you're trying to write an introduction/review of a business. In that case, not using a technical jargon, be it existing or not, would probably be better anyways. By the way, X = "offering" sounds pretty good to me.


Government's intervention question Given that demand for a good X is equal to $q_D=393-2p$ and market supply is $q_S=p/4-12$. Find equilibrium price and quantity, consumer and producer surplus and draw a diagram illustrating the situation. Given that:

a) $T=2q$, every single item sold is taxed.

b) $T=20\% TR$ total revenue is taxed

Obviously i have calculated the equilibrium price and quantity before taxation that is $p=180,q=33$.But i have no idea how to caculate those two values after taxation.

I apologise for the poor formatting and such, this is one of my first posts so please let me know if anything needs improvement (including, possibly, the correctness of my answer!).

For part a.), I'm assuming that a per unit tax of $2q$ is levied on the consumer in this market. The inverse demand function and inverse supply function are given by $p = \frac{393}{2} - \frac{q}{2}$ and $p = 4q + 48$ respectively. Thus, a per unit tax of $T = 2q$ will result in the inverse demand function being $p + 2q = \frac{393}{2} - \frac{q}{2} \iff p = \frac{393}{2} - \frac{5q}{2}$. Equating supply and demand, we find that the new equilibrium quantity is $q = 22.846$, the price that consumers pay is $185.077$ and the price producers receive is $139.385$.

For part b.), I'm assuming that a tax of $20$% of revenue is levied on the producer in this market. The inverse demand function and inverse supply function are given by $p = \frac{393}{2} - \frac{q}{2}$ and $p = 4q + 48$ respectively. Thus, a tax of $20$% of revenue will result in the inverse supply function being $p - 0.2p = 4q + 48 \iff p = 5q + 60$. Equating supply and demand, we find that the new equilibrium quantity is $q = 18.2$, the price that consumers pay is $187.4$ and the price producers receive is $120.8$.

I hope this makes sense (and is correct)! Feedback would be great!


Political theories of inflation A recent op-ed (paywall) in the Financial Times about inflation stated that (emphasis mine):

Across the developed economies, the Phillips curve has gone ignominiously flat. The world's leading central bankers are scratching their heads.
[...]
Rather than a response to tightness in the markets for products and labour, [in earlier theories] inflation was understood as a barometer of the political system’s ability (or lack of it) to resolve struggles over the distribution of income and wealth. Workers and bosses, creditors and debtors, fought one another for a larger slice of the pie. When negotiations failed, the path of least resistance was the excessive expansion of credit.
[...]
Stop dreaming of mythical beasts like the global output gap and the natural rate of interest, they would say. Wake up to the political realities.
Globalisation and the march of technology have strengthened the hand of employers and weakened that of workers. So it is no surprise that though unemployment has relentlessly fallen, wage growth remains in the doldrums.
[...]
What these unfashionable theories of inflation see — that the central bankers’ current models do not — is simply put. It is the dirty secret of economics: the central importance of power.

The author is referring to "old" political theories of inflation. I have struggled to find references about them. I wonder if someone has come across some of these political theories of inflation?

One book that may help you out is Macroeconomics and the Phillips Curve Myth by James Forder.  His narrative is similar to the one you refer to, but the terminology is a little different.  He argues that pre-1968, the economic literature had a more nuanced understanding of inflation that incorporated factors like profit and union power to a greater extent.  I'm not sure the ideas would be completely new to you, and their authors probably wouldn't have described them as "political theories", but Forder argues that there is a great deal of academic work, including empirical work, that quite suddenly fell out of historical accounts of the literature.  If nothing else, I think his reference list could get you started.


Regression - Testing for autocorrelation in the presence of heteroscedasticity I have constructed a linear time series regression model and estimated the parameters by applying OLS. I now want to test wether the assumptions for proper large sample inference (asymptotic Gauß Markov assumptions) are fulfilled.

Now, I am not sure how to test wether the residuals are autocorrelated or not. Since my model contains lagged dependent variables I can not use the Durbin-Watson test (since my independent variables are not strictly exogenous). Following Wooldridge I decided to apply the Breusch-Godfrey test. But the residuals are heteroscedastic, which I tested for via applying the Bresuch-Pagan test.

Wooldridge says that in the case of heteroscedasticity, one can not apply the usual Breusch-Godfrey test. How can I test for autocorrelation in the presence of heteroscedasticity? Is there any robust method? If that is of any interest - I am using R, so it would be helpful if there would be an implementation of the method (if there is one) in R.

EDIT: I have found a quite interesting paper that proposes a method of dealing with the topic: The modified Breusch-Godfrey test. Link: http://www.naun.org/main/NAUN/mcs/17-542.pdf.

Yet, I did not find any practical implementations of this test. As I am (just) an undergrad student, my possibilities regarding implementing such methods on my own are rather limited. So I am still looking for a general approach/test or method. (And I assume there has to to be a method, because the problem I am having strikes me as a rather common one.) Thank you!

General remarks: The BG test under homoskedasticity can be done using the bgtest command in the lmtest package of R. The $(n-p)R_{aux}^2$ version mentioned in link works only under homoskedasticity. In the presence of heteroskedasticity, Wooldridge (1991, JoE) gives a discussion (as noted in the Wooldridge textbook you mentioned).

What I think: I guess that what Wooldridge does is to use a heteroskedasticity-robust variance estimator. For this, (i) get the OLS residuals, (ii) regress e(t) on e(t-1), ..., e(t-p) and X, and test the joint significance of e(t-1), ..., e(t-p) using a heteroskedasticity-robust covariance estimate. If you want to use R, do the following for AR(2):

DF <- data.frame(y=rnorm(100), x1=rnorm(100), x2=rnorm(100))
ols <- lm(y~x1+x2, data=DF)
DF$e <- ols$resid
DF$e1 <- c(NA,DF$e[-100])    # are there better ways to lag a variable?
DF$e2 <- c(NA,DF$e1[-100])
aux <- lm(e~e1+e2+x1+x2, data=DF)
library(car)
lht(aux, c('e1','e2'), white.adjust='hc3')

Discussions: That said, there is the generated regressor problem in the aux regression above, that is, the some of RHS variables (e1 and e2) are generated using the OLS regression results. This usually causes trouble. However some tests can be done even if the regressors are generated. I guess this is one, but I haven't checked it formally.

Further discussions: The BG test is an LM test, while lht does a Wald test. The difference should be minor.

Simulation results: I did simulations. The ordinary BG test seems to fail. The robustified version seems working.

library(car)
iterate <- 1000
n <- 400

ans <- data.frame(ord=rep(NA,iterate), rob=rep(NA,iterate))
set.seed(1)
for (iter in seq_len(iterate)) {
  x1 <- rnorm(n+1)
  x2 <- rnorm(n+1)
  u0 <- rnorm(n+1)
  u0[1:floor(n/2)] <- 2*u0[1:floor(n/2)]
  u <- sqrt(1+abs(x1+x2))*u0

  ## y(t) = 1+x1(t)+x2(t)+0.5*y(t-1)+u(t)
  y <- filter(1+x1+x2+u, 0.5, method='recursive')

  y1 <- y[-(n+1)]      # y(1), ..., y(n)
  y <- y[-1]           # y(2), ..., y(n+1)
  x1 <- x1[-1]         # x1(2), ..., x1(n+1)
  x2 <- x2[-1]         # x2(2), ..., x2(n+1)

  ols <- lm(y~x1+x2+y1)

  e <- ols$resid
  e1 <- c(NA,e[-n])    # NA, e(1), ..., e(n-1)
  e2 <- c(NA,e1[-n])   # NA, NA, e(1), ..., e(n-2)
  aux <- lm(e~e1+e2+x1+x2+y1)
  tst0 <- lht(aux, c('e1','e2'), white.adjust=FALSE)
  ans$ord[iter] <- as.numeric(tst0$`Pr(>F)`[2] < 0.05)
  tst1 <- lht(aux, c('e1','e2'), white.adjust=TRUE)
  ans$rob[iter] <- as.numeric(tst1$`Pr(>F)`[2] < 0.05)
}

print(colMeans(ans))
##   ord   rob
## 0.090 0.046

Note: I edited the simulation a lot. I hope this one is OK.


intuitive interpretation of the marginal return/productivity of capital being less than one Let's assume you have a production function, f,  and you want to know how the output changes with respect to capital, everything else constant (ceteris paribus), so you want to know the marginal productivity or return on capital.

This is done by taking the partial derivative of capital to output.

$$  \frac{\partial y_t}{\partial k_t}  $$

Usually, in most models, the result will be a value between 0 and 1. However, this means that, all other variables being equal, for one extra euro (which comes out of nowhere, I guess, given that all other variables should be the same: ceteris paribus condition), output increases with less than this one euro..., right? I don't get the intuition that this value is less than one, I would it expect to be higher than one, at least. I mean, did 80% of it just vanish?

On wikipedia it is defined as: The marginal product of capital (MPK) is the additional output resulting, ceteris paribus ("all things being equal"), from the use of an additional unit of physical capital. Mathematically, it is the partial derivative of the production function with respect to capital.

I'm having some difficulty with the meaning of "additional" in this definition. It's not additional to this one extra euro of capital out of nowhere, right? This one euro is really transformed in only like e.g. half a euro in output: e.g. capital +1 ==> output + 0.5.

In contrast to this marginal productivity of capital, you can also calculate the direct partial effect of capital on output, which e.g. for a Cobb-Douglas function is given by the elasticity (relative change of output given 1% relative change of capital) parameter for capital (it's power).
Yet another way to express the influence of a change in capital is the full effect: in some models, also other variables change when capital changes, e.g. labour, technology and so on, this is called the full effect.

I just mention these effects to contrast them with the calculation and interpreatation of the return on capital.

Several thoughts in this area:

1) Capital stock lasts over time. I may trade 1 euro for 1 unit of capital. This capital returns 0.01 euros every period forever.  It pays itself off eventually.

2) If one has an incredible high rate of return and a sufficiently low discount rate, one might have divergent consumption paths, ex: I spend all of my money on capital, each unit of capital returns \$2 for the next period. I repeat forever, but never consume anything, because any consumption represents a loss of \$2 tomorrow.  After 1000 periods, I have a huge amount of capital but no consumption.

3) Cobb-Douglass is normalized so the exponents sum to one. This led to some off-the-cuff work. It has derivatives:

$MPK =\alpha_k * (\frac{L}{K})^{1-\alpha_k}=\alpha_k k^{-(1-\alpha_k)}$

$MPL= (1-\alpha_k) (\frac{K}{L})^{\alpha_k} = (1-\alpha_k) k^{\alpha_k}$

In optimal long-run state, these two will be equal.

$\alpha_k k^{-(1-\alpha_k)} = (1-\alpha_k) k^{\alpha_k}$

$\alpha_k = (1-\alpha_k) k^{\alpha_k --(1-\alpha_k)}$

$\alpha_k = (1-\alpha_k) k^{1}$

$k^{1} =  \frac{\alpha_k}{1-\alpha_k}$

Plug this back in and you get:

$MPL^{*} = \alpha_k \frac{\alpha_k}{1-\alpha_k}^{-(1-\alpha_k)}$

$MPL^{*} = \alpha_k \frac{\alpha_k}{1-\alpha_k}^{\alpha_k-1}$

$MPL^{*} = \alpha_k \frac{1-\alpha_k}{\alpha_k} * \frac{\alpha_k}{1-\alpha_k}^{\alpha_k}$

$MPL^{*} = (1-\alpha_k) * \frac{\alpha_k}{1-\alpha_k}^{\alpha_k}$

$MPL^{*} = (1-\alpha_k) * \frac{\alpha_k^{\alpha_k}}{(1-\alpha_k)^{\alpha_k}}$

$MPL^{*} = \frac{\alpha_k^{\alpha_k}}{(1-\alpha_k)^{\alpha_k - 1}}$

$MPL^{*} = \alpha_k^{\alpha_k}*(1-\alpha_k)^{1-\alpha_k}$

This looks like it's some sort of fraction. A quick screenshot on wolfram alpha suggests indeed that is the case.


Why does US equities valuation have the downward pressure on USD? While making readings on economics I came across a sentence:
..large valuation gap that has opened up between US equities and the rest of the world. This could help bid up non-US equity valuations and place further downward pressure on the USD.
To my understanding, holding everything else equal, high stock prices of US corporate encourage investors to buy them, thus it should place upward pressure on the dollar. Why would the authors argue that the direction of pressure is down?

I think what your quote is trying to say is: Investors have bid up US stocks to the point where their valuation ratios (e.g. pe) are high relative to other countries. This could lead investors to rebalance in favor of other stocks. Hence the downward pressure on the dollar.
Whether or not that argument makes sense is another questions (about which I have my doubts).


Is there a per country estimation of the GDP drop due to COVID? I've been looking around for some per-country estimation of the expected GDP drop due to the COVID pandemic.
I've found this answer in this site but it only mentions a few countries. I understand not all countries will have such an estimation, but maybe for the G20 or similar groups?

There are some cross-country studies but not that many of them yet. For example, Maliszewska, M., Mattoo, A., & Van Der Mensbrugghe, D. (2020) from world bank provide some cross-country estimates of the cumulative effect of covid-19 for various countries and regions across the world. Their estimated cumulative effect of covid on GDP (measured as deviation from their benchmark) is shown below.

However, I would advise to take these results with a pinch of salt. These are preliminary estimates that were published in April and better ones will be fore sure forthcoming in near future.


No Ponzi game condition and transversality condition are the same? Given the following non-stochastic planning problem with finite horizon,
\begin{align}
&\max_{\{k_{t+1}\}}\sum^T_{t=0}\beta^tU[f(k_t-k_{t+1})] \\
\text{s.t. } & 0\leq k_{t+1}\leq f(k_t)\\
& k_0 >0 \text{ (given)}.
\end{align}
I found that in order to make the first order conditions necessary and sufficient I have to add the so called no Ponzi game condition, i.e.
\begin{gather}
\lim_{T \rightarrow \infty} \frac{k_{T+1}}{R_{T+1}} \geq 0
\end{gather}

When written with the equal sign, this condition can be interpreted as the willingness of not keeping any capital at the end of life. And this is the same interpretation of the so called transversality condition.

Thus, is it right to interpret the no Ponzi game condition as a finite horizon version of the transversality condition? If not, which is the difference between them?

Is it right to interpret the no Ponzi game condition as a finite
  horizon version of the transversality condition?

No. The "No-Ponzi-Game" or "solvency" condition is an external constraint imposed on the individual by the market/other participants. The individual would very much like to violate it.

The Transversality condition must be satisfied in order for the individual to maximize indeed its intertemporal utility. It is an optimization condition.

So they are conceptually very different aspects of the problem.

Finally the No-ponzi-game/solvency condition is not inherently of finite horizon -it extends to the infinite horizon also.


gibbard-satterthwaite theorem and median voting The gibbard-saterthwatie theorem states that a social choice function where truthfullness is a dominant strategy must be dictatorial. However, in a median voting system, with single peaked preferences of voters, truthfullness is a dominant strategy yet it is not dictatorial.
what is the solution to this contradiction?

Gibbard and Satterthwaite insist that the social choice function must be defined over all rational preferences over outcomes. That is, if voters' preferences could be anything (subject to the constraint of completeness and transitivity), then we have the Gibbard–Satterthwaite theorem.

On the other hand, if preferences were single-peaked, then the assumptions they make do not hold, and the theorem does not apply.


Cost function from CES production function How can I find the cost function $c(w,p)$ given that the production is
$$ f(x)=(x_1^p + x_2^p)^{1/p} \ \ for\ \  0<p <1 $$
I tried to solve it and found that
$$TC(y) = \left\{
        \begin{array}{ll}
            w_1y & \quad w_1 < w_2 \\
            wy & \quad w_1=w_2 \\
            w_2y & \quad w_2 <w_1
        \end{array}
    \right.
$$
Can you say me if I'm on the right way?

If you are interested in the case where $\rho \geq 1$ then look at the post CES
$\ \ \rho \geq 1$. For the standard case where $0 < \rho < 1$ you should get a result like this
$$C(w_1,w_2,y)  = \left(w_1^{\frac{\rho}{\rho -1}} +w_2^{\frac{\rho}{\rho -1}}\right)^{\frac{\rho - 1}{\rho}} y.$$
To see this you should start by setting up the cost minimization problem
$$\min_{x_1,x_2} \ \ w_1x_1 + w_2x_2 \\[8pt]
s.t. \ \ (x_1^\rho + x_2^\rho)^{1/\rho} \geq y$$
for this problem the Lagrangian function is
$$\mathcal L(x_1,x_2,\lambda) = w_1x_1 + w_2x_2 - \lambda((x_1^\rho + x_2^\rho)^{1/\rho} -y).$$
From the first order conditions of the Lagrangian you can show the constraint is binding in optimum $(x_1^\rho + x_2^\rho)^{1/\rho} = y$ and get MRS equal to relative prices
$$(1) \ \ \frac{w_1}{w_2} = \frac{x_1^{\rho - 1}}{x_2^{\rho - 1}},$$
given this information you should be able to solve for $x_1$ and $x_2$ as a function of the parameters of the problem which in this case is $\rho,y,w_1,w_2$.
Try to get $(x_1^\rho + x_2^\rho)^{1/\rho}$ to appear in the MRS equal to relative prices. So manipulate (1) to get
$$w_1^{\frac{\rho}{\rho -1}}x_2^\rho =  w_2^{\frac{\rho}{\rho -1}}x_1^\rho,$$
then add $w_2^{\frac{\rho}{\rho-1}}x_2^\rho$ to both sides of equation
$$w_1^{\frac{\rho}{\rho -1}}x_2^\rho +w_2^{\frac{\rho}{\rho -1}}x_2^\rho =  w_2^{\frac{\rho}{\rho -1}}x_1^\rho + w_2^{\frac{\rho}{\rho -1}}x_2^\rho,$$
isolate factors on both sides and exponentiate with exponent $1/\rho$ to get
$$\left(w_1^{\frac{\rho}{\rho -1}} +w_2^{\frac{\rho}{\rho -1}}\right)^{1/\rho}x_2 =   w_2^{\frac{1}{\rho -1}}(x_2^\rho + x_1^\rho)^{1/\rho} = w_2^{\frac{1}{\rho -1}} y ,$$
from here you can solve for conditional demand $x_2^\star(w_1,w_2,y)$. However, it is easier to oberserve that the factor $\left(w_1^{\frac{\rho}{\rho -1}} +w_2^{\frac{\rho}{\rho -1}}\right)^{1/\rho}$ do not change when interchanging indexes - it is symmetric. So define $a := \left(w_1^{\frac{\rho}{\rho -1}} +w_2^{\frac{\rho}{\rho -1}}\right)^{1/\rho}$ and conclude that
$$ax_1 = w_1^{\frac{1}{\rho -1}} y \\[8pt]
ax_2 = w_2^{\frac{1}{\rho -1}} y,$$
multiply first equation with $w_1$ and second with $w_2$ and add them to get
$$a(w_1x_1 + w_2x_2) = (w_1^{\frac{\rho}{\rho -1}}+w_2^{\frac{\rho}{\rho -1}})y = a^\rho y$$
solve for $(w_1x_1 + w_2x_2)$ which are the costs to get the result that
$$C(w_1,w_2,y) = a^{\rho -1} y = \left(w_1^{\frac{\rho}{\rho -1}} +w_2^{\frac{\rho}{\rho -1}}\right)^{\frac{\rho - 1}{\rho}} y$$


Mathematic notation of scale returns I'm having trouble with returns to scale mathematical notation which i found counterintuitive even I'm fine with its definition. Any help will be very appreciated.

There are three possible types of returns to scale: increasing returns to scale, constant returns to scale, and diminishing (or decreasing) returns to scale.

If output increases by the same proportional change as all inputs change then there are constant returns to scale : $F(aK,aL) = aF(K,L)$.
If output increases by less than that proportional change in all inputs, there are decreasing returns to scale : $F(aK,aL) < aF(K,L)$.
If output increases by more than the proportional change in all inputs, there are increasing returns to scale $F(aK,aL) > aF(K,L)$.

If we define (with $a > 0$) :

$F(aK,aL)$ : simultaneous and proportional inputs increase
$aF(K,L)$ : output increase

Then, to me increasing returns to scale should be written in this way : $F(aK,aL) < aF(K,L)$ since the change in the output is greater in proportion than the change in the inputs. And the other way for decreasing returns to scale. This is not clear to me.

I hope you will help me to fix this misunderstanding.

Best,
Marc

Quoting from the question:

  $F(aK,aL)$ : simultaneous and proportional inputs increase
  $aF(K,L)$ : output increase

But $F(aK,aL)$ is an output increase, that is why it starts with $F$. It is the output increase due to the simultaneous and proportional increase of inputs. At the same time, $aF(K,L)$ is a hypothetical output, the output were this if it was increased by the same proportion as the inputs. So
\begin{align*}
F(aK,aL) & > aF(K,L)
\end{align*}
shows that if you increase the inputs simultaneously and proportionally, the increase in output is larger than proportional.


How does the limit of $U(x, y) = (ax^{-c} + by^{-c})^{-\frac{1}{c}}$ as c approaches 0 yield the Cobb-Douglas utlity function? \begin{equation*}
U(x, y) = (ax^{-c} + by^{-c})^{-\frac{1}{c}}
\end{equation*}

I ask this mainly because after logging both sides of the Utility equation (the first step to proving the assertion, I assume), I am left with:

\begin{equation*}
\lim_{c \rightarrow 0} \dfrac{-\ln(ax^{-c} + by^{-c})}{c}
\end{equation*}
I know that the bottom will go to 0, and I have a feeling that the top will go to 0 to.  However, all I am left with on the top is essentially $a + b$, and for it to go to 0, $a + b = 1$.

How can $a + b = 1$?  Is this the right direction?  What does $a + b = 1$ mean?  Why does $a + b = 1$?

Edit:  And once proven, what does this whole "limit" thing say about the original function?  What is so special about this particular equation such that its limit as $c \rightarrow 0$ is the Cobb Douglas function?

Edit 2:  Upon further research, I have discovered a suspiciously similar function known as the CES.  $a$ and $b$, however, are instead $a$ and $(1-a)$ !!  Now I'm even more confused.  How am I supposed to derive that complementary relationship from this equation?  This is supposed to be consumer theory!

It is not true that this function is equivalent to the Cobb-Douglas utility function when $c \sim 0$ for any values of $(a,b)$; you have to assume $a+b=1$ for that, i.e. $b=1-a$.

To see why it is true, fix $(x,y)$ and consider the following Taylor expansion of $U(x,y)$ when $c$ gets close to $0$. We have

\begin{align*}
(ax^{-c}+by^{-c})^{-\frac{1}{c}} & = e^{-\frac{1}{c} \ln{[ax^{-c}+by^{-c}]}} \\
& = e^{-\frac{1}{c} \ln{[ae^{-c\ln(x)}+be^{-c \ln(y)}]}} \\
& = e^{-\frac{1}{c} \ln{[a(1-c \ln(x) + o(c)) + b(1-c \ln(y) + o(c))}]} \\
& = e^{-\frac{1}{c} \ln{[a+b - c (a \ln(x)+b \ln(y)) + o(c))]}} \\
\end{align*}
If $a+b>1$, the term in the exponential converges to $-\infty$ when $c \rightarrow 0$ and therefore $U(x,y) \rightarrow 0$. If $a+b<1$, the term converges to $+\infty$ and therefore $U(x,y) \rightarrow +\infty$.

To obtain the convergence towards the Cobb-Douglas function, we must therefore assume $a+b=1$. In that case we have
\begin{align*}
U(x,y) & = e^{-\frac{1}{c} \ln{[1-c(a \ln(x) + (1-a) \ln(y))+o(c)]}} \\
& = e^{-\frac{1}{c} [-c(a \ln(x) + (1-a) \ln(y))+o(c)]} \\
& = e^{a \ln(x) + (1-a) \ln(y) + \frac{o(c)}{c}} \\
& \rightarrow_{c \rightarrow 0} e^{a \ln(x) + (1-a) \ln(y)} \\
& = x^{a} y^{1-a}
\end{align*}
which is the Cobb-Douglas utility function with parameters $(a,1-a)$.


Regressing (Very) Smooth Time Series What are the possible problems/issues of regressing smooth time series with almost no fluctuation?
Here is a specific example.

Is there anything I have to pay attention to when interpreting coefficients? Or is there no proper way to regress time series like these? If so, what are the reasons?

One thing to consider, is that it looks like you may have a unit root, though not necessarily.
An example of a unit root would be the stochastic process $y_k=y_{k−1}+\epsilon_{k−1}$, where the error term is mean-zero.
Unit roots can cause problems. For one, they are not stationary processes. Using OLS relies on stationarity. A violation can lead to a 'spurious regression': invalid estimates but with a high R-squared.
However, these problems are solvable. The first step is to run a unit-root test, of which there are many.
I suggest looking elsewhere for further material on this, e.g. here. There is lots out there and a full treatment would be outside the scope of one answer.


Are there any examples of tariffs working? Nations have imposed tariffs on goods for many reasons throughout the ages.  Are there any examples of a tariff actually succeeding at its intended goals?

When I research tariffs, I find countless arguments that they never one have worked, and I find a lot of current events documents arguing why particular tariffs wont work, but I'm having it hard finding any economist pointing to a tariff that worked as intended.

Bonus points if the tariff worked as intended and was not considered to be a net loss for the country that imposed it.

I interpret your question more broadly as one about whether protectionism has ever "worked". Two economists that think it has are Chang Ha-Joon and Dani Rodrik. You can therefore look up their work.

Two arguments they use are:

The infant industry argument.

In a 2007 article, Chang gives various examples of protectionism "working", including the US in the 19th century:

  the US shifted to protectionism after the Anglo-American War of 1812. By the 1830s, its industrial tariff rate, at 40-50 per cent, was the highest in the world, and remained so until the Second World War.

The other oft-cited examples are East Asian economies like Japan, Korea, and Taiwan in the postwar era.

The second-best argument.

The world is a second-best place and so movements towards seemingly-freer trade may not always be a good thing. Classic example as told by Rodrik (2015):

  imagine that beef is supplied by the United States to Germany at a price of \$100. Assume that Germany imposes a tariff of 20 percent, raising the consumer price of US beef in the German market to \$120. France, meanwhile, can supply beef of equivalent quality only at a price of \$119. Prior to the preferential agreement between France and Germany, French suppliers, facing the same tariff rate as US producers, were outcompeted. Now consider what happens when Germany eliminates its tariffs on imports from France but keeps in place those on the United States. French-supplied beef suddenly becomes cheaper in Germany (\$119 versus \$120), and imports from the United States collapse. German consumers are better off by \$1, but the German government forfeits \$20 of tariff revenue previously collected on US beef (which could have been handed back to consumers or used to reduce other taxes in Germany). On balance, Germany gets a raw deal.

Two footnotes.

Evaluating whether something has ever "worked" is fiendishly difficult in economics.

First we have to decide what the goals of the policies were and what would count as "working". Then we have to do the empirical analysis and evaluate quantitatively the degree to which the policies "worked".

So while Chang and Rodrik might give examples of protectionism "working" (19th-century US, Japan, Korea, Taiwan), other economists might disagree that protectionism really "worked" in these cases. Indeed, I am not aware that Chang or Rodrik has actually conducted any empirical analyses on this matter.

Chang and Rodrik merely try to argue against the extreme position that free trade is always a good thing for everybody and in favor of the modest position that protectionism has sometimes "worked".

Most (all?) economists, including Chang and Rodrik, would agree with the following statements:

Free trade is generally a good thing for most people.
Tariffs are generally a bad thing for most people.

In contrast, a significant portion of the general public would disagree.


Scaling Adjustment Costs Take a cont. diff, convex and increasing cost function $c(X)$. Say you start with a stock $K$, and want to (dis)invest $I$. Many adjustment cost functions (for example, the first example on page 2 here) then are of the type

$$ K c\left(\frac{I}{K}\right)$$

What is the purpose of scaling investment costs with $K$ here? Marginal cost of investment is

$$ c'\left(\frac{I}{K}\right)$$

It seems that this gives the property that marginal cost of investment decreases in existing capital stock. Is this observed in the real world, or does it rather simplify computations? Are there (popular) alternative formulations of adjustment costs?

The New Palgrave Dictionary of Economics Article on Adjustment Costs gives a nice overview of the use of adjustment costs in economic models.

What is the purpose of scaling investment costs with K here?

This function form implies that adjustment costs of size X% of capital $I/k$ cost the same fraction of capital ($c(I/k)$) regardless of the amount of capital($k$) involved. It is a constant returns to scale form of adjustment cost.

It seems that this gives the property that marginal cost of investment decreases in existing capital stock.

I think you have the marginal cost of adjustment wrong here.

$$ \frac{\partial}{\partial K} K \cdot c\left(\frac{I}{K}\right)$$
$$ = c(\frac{I}{K})  + K \cdot c'(\frac{I}{K})$$

The first term is decreasing in $K$ for fixed $I$ but the second need not be.