Why does quadratic utility function imply $\mu-\sigma$ preference? Why does investors having quadratic utility function mean that their optimal portfolios can be chosen by only considering mean and variance of returns i.e. imply $\mu-\sigma$ preference?

If you have quadratic preferences then your utility function is:
$$ U(W) =  W - \lambda W^2$$
this implies your expected utility function looks like:
$$ E[U(W)] =  E[W - \lambda W^2] =  E[W] - \lambda E[W^2]$$
$$ =  E[W] - \lambda E[W^2 - E[W]^2 + E[W]^2]$$
$$ =  E[W] - \lambda E[W^2 - E[W]^2] + \lambda E[E[W]^2]]$$
$$ =  \mu_w - \lambda \sigma_w^2 + \lambda \mu_w^2$$

Therefore, we have established that expected utility depends only only the mean $(\mu_W)$ and variance $(\sigma^2_W)$ of wealth and the risk parameter $(\lambda)$.


How does one calculate an optimal consumption bundle if the MRS only contains one variable? For example, say the MRS has simplified to the form $MRS(x,y) = -y$, given a utility function
\begin{equation*}
U(x, y) = e^{(x + ln(y))^{1/3}}
\end{equation*}

With a budget line of, say, $2x + y = 10$ which would yield a budget line slope of -2, equating the MRS to the BLS would simply yield $y = 2$, but there is no x to solve for in the equating of the two slopes.

Is the last step, in cases like this, to plug back into the budget constraint?  In which case, one would get $x = 4$, yielding an optimal consumption bundle of $(4, 2)$?

I wanted to add this as a reply to denesp's comment, but I do not have enough reps.

MRS and a binding BC gives a system of two equations from which we can solve the optimum bundle. In case of income  = 10, these two equations have positive solutions, in case of income = 1, these two equations do not have positive solutions. See this:

Income = 1 just makes the choice set small enough that there is no MRS = price ratio, hence the corner solution.

I completely agree that income =1 is really the more fun question.


Ito's Lemma derivation I'm getting into asset pricing and was looking at Ito's Lemma, but cannot understand a few steps that are given.

Ito's Lemma states that given

$$dx_t = \mu dt + \sigma dz_t \\
y_t = f(t, x_t)$$

then

$$(1) \quad dy_t = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dx_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} dx^2_t \\$$

I understand this part using chain rule and a second order Taylor expansion, of the second equation. I don't understand why then the following holds:

$$(*) \quad dy_t = \left[\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} \mu + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \sigma^2 \right] dt + \left[  \frac{\partial f}{\partial x} \sigma \right] dz_t \\$$

When I substitute in $dx_t$ into $(1)$ and use the fact that $dz^2_t = dt$, it isn't enough to arrive at $(*)$. I think $dx^2_t$ can literally be interpreted as $(dx_t)^2$, but if there is a better way of handling that term, any guidance there would be appreciated.

$$dx_t = \mu dt + \sigma dz_t \\
y_t = f(t, x_t)$$

A key idea here is that $\left( dx_t \right)^2=\left( \ldots \right)dt^2 + \left(\ldots\right) dzdt + \sigma^2 dz_t^2 = \sigma^2 dt$. The loose reasoning is that $\left( dz_t\right)^2 = dt$ and all the other terms (i.e. $dt^2$ and $dz\, dt$) are infinitely smaller than $dt$.

Horribly loose intuition for $dz_t^2 = dt$ is that $dz_t$ is normally distributed with variance $dt$, and hence the expectation of the square of $dz_t$ is $dt$.

In any case, we then have:

\begin{align*}
\quad dy_t &= \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dx_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} dx^2_t\\
&= \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} \left( \mu dt + \sigma dz_t  \right)+ \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \sigma^2 dt \\
&= \left(\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}\mu + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \sigma^2 \right)  dt + \left( \frac{\partial f}{\partial x}\right)\sigma dz_t
\end{align*}

Which is Ito's lemma.


Why do Reserve Banks only move in 0.25 basis point 'leaps' It seems to me that a lot of uncertainty is created by Reserve Banks spending 5 months telegraphing "I'm gonna do it, I'm gonna raise rates", as is currently seen in the USA.

This seems to be the case where everyone knows the rate is wrong, but changing it quickly will cause shocks.

Instead of waiting 5 months and then increasing by 0.25%, why not increase by 0.05% every month for 5 months? Or smaller amounts if that is required.

The Fed normally only change rates during their regular monetary policy meetings, of which there are 8 per year (roughly six weeks between meetings). Therefore, they cannot change rates every month.

The amount 1/4% comes from interest rate quote conventions; historically they also used 1/8% (12.5 basis points). If they were to go to smaller increments, it would probably be 1/16%.

The first reason why they do not announce finer increments is that the Fed Funds rate that they target is a market interbank rate; they do not have complete control over it. The Fed publishes the "effective fed funds rate" in the H.15 report, which is their estimate of where the market is trading; it does not exactly match the target. Therefore, they cannot credibly deliver such small incremental changes to the overnight rate that matters.

The second reason is that such a small increment would probably make the Fed look silly. How do they have enough confidence in their ability to fine tune the economy so that moving the policy rate by 1/16% is now the right level?

Realistically, the current level of interest rates does not matter, so much as they expected path. At 25 bps per meeting, the total change over one year is 2%. That should be enough to match changes in economic conditions. An increment smaller than 1/8% would not be enough to keep up with conditions.

As for the current situation, it is unusual. Interest rates are very close to 0%, and it is difficult for them to go negative (the "zero lower bound"). Everyone knows that interest rates can only go one way (assuming no negative rates), so it is just a question of timing. This creates a different mood for interest rate watchers than when interest rates can go either way. Fed Reserve officials have been cautious about raising rates, since they have no leeway to cut rates if the economy rolls over. For this reason, they would not have considered any rate hikes until they were certain that rates needed to be well above 0%. (Even when the rate hikes occurred, they have been quite tentative.) Furthermore, if you want by news stories, "everyone" knew that the policy rate was "unsustainably low" every year after 2010. In other words, we need to be careful about what "everyone" knows.


Why do firm output decreases when fixed cost decreases in Long Run? So in long run if fixed cost is lowered, more firms enter the industry and increase the supply, bringing up industrial output. According to the firm diagram, Average Cost lowered so that the firm output is lower. But why is that? Shouldn't the firm produce more when the fixed cost lowers?

The quantity produced does not depend on the fixed costs. Fixed costs only influence the decision of whether or not to produce, not how much to produce. The individual firm's level of production is determined by MR=MC, independent of fixed costs. Firms will produce less when fixed cost decrease, because, as you point out, more firms will produce (entry in the market). On the aggregate market supply shifts out, so that price declines. For each individual pricetaking firm, the exogenously determined price has declined. Put differently, MR has shift down, resulting in an optimum (MR=MC) at a lower level of production.


Does the H-1B visa program have the effect of reducing wages in the U.S In either the short or long term, does the H-1B visa program have the effect of reducing wages in the U.S. for U.S. citizens?

The program would increase the supply of labor to those specific sectors. As a result more foreign people would get into the sectors bidding the price down (lowering the wage). In addition, some Americans would not be able to get hired, that otherwise would have.

In addition, companies would become more profitable so more companies would be started and the reduced labor costs would be visible in other sectors, increasing labor demand elsewhere.

In the very long run free movement of labor is at least in theory more efficient than restricted movement, as people will more efficiently be able to navigate to the most efficient job locations available.


How does a budget deficit help grow a trade deficit? What are the mechanism that explains this, and does this help explain the U.S.-China trade deficits?

It all boils down to the macroeconomic identity

$$ (G -T) \equiv (S – I) +(M-X), $$

where $X$ stands for exports, $M$ for imports, $G$ is government spending, $T$ are taxes, $S$ private savings and $I$ private investment.

Given $S-I$, the larger is the budget déficit, the larger the trade deficit is. $S-I$ can be thought of as the private sector excess financing ability. With this amount, it needs to finance both the budget deficit and pay for the difference between what it is able to sell abroad and what it buys abroad. If, as in the US, $S-I$ is not too large (due to low $S$), as soon as the government incurs in a budget deficit, a trade deficit will appear.

This does not tell us with which specific country the trade deficit will happen. However, it does tell us that those countries with high $S-I$ and low budget deficits (or even surpluses) need to be running trade surpluses, and this is the case of China (or Germany).


Renminbi demand and CNY rates I read something similar to here, http://www.businessinsider.com/onshore-and-offshore-renminbi-2014-2?IR=T, paragraph 6, that demand for renminbi drives down cny bond market?

I can understand how demand for bonds can drive the currency markets (because to buy local denominated bonds mean you have to purchase the currency) but I cannot understand how demand for the currency can drive the bond markets and to what extent this happens.

What the paragraph actually said is

  "Fervent demand for renminbi from international investors has driven down rates in Hong Kong and thereby created incentives for companies considering using the renminbi for trade or financing," explains Financial Times correspondent Robert Cookson.

This means that international investors were buying offshore Renminbi (CNH) in Hong Kong and using that to buy offshore bonds denominated in Renminbi. That pushed the price of those bonds up, corresponding to reduction in their effective interest rates ("driving down rates").


Does the growth rate of a neoclassical production function converge as all input factors grow with constant, but different growth rates? Suppose you have a neoclassical production function with N-inputs

$F(x_t^1,...,x_t^N)$

All input factors grow in continuous time with constant, but not identical growth rates $g^j$. Assume $g^1 \leq g^2 \leq ... \leq g^N$. The growth rate of $F$ is then

$\hat{F}=\sum_{j=1}^N \varepsilon_{F,x^j} g^j$

with $\varepsilon_{F,x^j}$ being the elasticity of $F$ with respect to $x^j$. Since $F$ is linear homogenous I know that $\sum_{j=1}^N \varepsilon_{F,x^j}=1$ holds. $\frac{\partial F}{\partial x^j}>0$ and $\frac{\partial^2 F}{\partial {x^j}^2}<0$ imply $\varepsilon_{F,x^j}>0$. Hence

$g^1 \leq \hat{F} \leq g^N$ $\forall t$

My question is: Is $\hat{F}$ going to converge as $t \to \infty$? I find it hard to imagine that the elasticities fluctuate (periodically) around some value as all input factor ratios go to zero or infinity (or stay constant the whole time in case of equal growth rates).

I tried to show that the elasticities all converge. I suspect that the result might hinge on the propety that $\frac{\partial^2 F}{\partial x_j \partial x_k}>0$ for $k \neq j$ which always holds in the two goods case, but I'm not sure about that.

Thank you all in advance for your help! If my English seems a little awkward, it's because I'm German. But I hope you understand the problem anyway. :)

I am not sure that elasticities converge for arbitrary growth rates.

Here is a treatment of the issue. Subscripts in $F$ denote partial derivatives. I will omit the time subscript. By definition, the elasticity of the $i$-th input is the ratio of marginal product ($F_i$) over Average Product $(F/x_i)$. So for an elasticity to stabilize intertemporally, it must be the case that the growth rate of the marginal product  is equal to (or equalizes a the limit with) the growth rate of the average product.

For input $x_{1}$ we have

$$\frac {d}{dt} F_1 = F_{11}\dot x_{1} +F_{12}\dot x_{2} +...+F_{1n}\dot x_{n}$$

All inputs are strictly positive so we can multiply and divide, and also divide by $F_1$ to obtain the growth rate (denoted by a tilde)

$$\tilde {F_1} = \frac {dF_1/dt}{F_1}  = \frac{F_{11} x_1}{F_1}\frac{\dot x_{1}}{x_1} + ... +\frac{F_{1n} x_n}{F_1}\frac{\dot x_{n}}{x_n}$$

The first ratio in each term is the elasticity of the $F_1$ with respect to each input, say $\eta_{1k}$. So,

$$\tilde {F_1} = \sum_{k=1}^n\eta_{1k}g_k $$

and analogously for the other inputs.

Turning to the average product of the first input, we have

$$\frac {d}{dt} (F/x_1 ) = \frac {F_1x_1 - F}{x_1^2}\dot x_1 + \frac {F_2}{x_1}\dot x_2+...+\frac {F_n}{x_1}\dot x_n$$

Manipulating,

$$\frac {d}{dt} (F/x_1 ) = \left[F_1-\frac {F}{x_1} \right]g_1 + \frac{F}{x_1}\frac{F_2x_2}{F} \frac {\dot x_2}{x_2}+...+\frac{F}{x_1}\frac{F_nx_n}{F} \frac {\dot x_n}{x_n}$$

$$= \frac {F}{x_1}\cdot \Big( (e_1-1)g_1 + e_2g_2 + ...+ e_ng_n\Big)$$

$$\implies \frac {d(F/x_1)/dt}{(F/x)} = \tilde F - g_1 = \sum_{k=1}^ne_kg_k - g_1$$

So for all elasticities to stabilize (and so for $\tilde F$ to stabilize) we want, at least eventually, that

$$ \forall i=1,...,n\;\;\;  \sum_{k=1}^n\eta_{ik}g_k = \sum_{k=1}^ne_kg_k - g_i$$

Denoting $\mathbf H$ the $n\times n$ the matrix of the elasticities of the marginal products, $\mathbf I_n$ the identity matrix, $\mathbf e$  an $n \times 1$ vector containing the output elasticities $e_i$, $\mathbf i$ a column vector of ones, and $\mathbf g$ the $n \times 1$ column vector of growth rates of inputs, we can write the conditions for all elasticities to be stable compactly , as

$$\big(\mathbf H + \mathbf I_n - (\mathbf i \oplus \mathbf e')\big)\mathbf g =0$$

I note that the sum-of-matrices term does not depend at all on the growth rates vector. So the above orthogonality condition cannot hold for arbitrary $\mathbf g$ (except if I am neglecting some property stemming from the homogeneity of $F$).


Why do some researchers include oil price in a VAR model? Why do some researchers include international oil prices in a VAR model for small open economies? My intuition is that international oil prices cannot be a endogenous variable in the context of a small open economy. For example, I don't see how one would justify a two-variable VAR model (say, for international oil price and GDP) for a country like Ghana. Because oil prices are not endogenous for small open economies, right? I mean, Ghana's GDP or its lags do not affect the international price of oil. However, I see in the literature that people still use a VAR model with oil prices as one of the endogenous variables in the model. I am finding it hard to understand how they can do that, and these papers do not state how.
Any help is appreciated.

Output and oil price can be endogenous even for relatively small and open country if the country is oil producer. Actually, in international trade whether economy is small depends on whether in particular case country can affect world price by varying its output. For example Kuwait is generally small open economy, but when it comes to oil production they are (at the time of writing) ranked 10th in the world so in oil market they are not small economy. For Kuwait there would be definitely an endogeneity between oil prices and output.
Ghana (at the time of writing) it is ranked 49th in world oil production which is somewhere in the middle between being significant and trivial so I can imagine someone arguing that there could be some endogeneity between oil prices and output. I suppose it is possible that if Ghana would shut down its oil production world price would move slightly.

VAR model actually does not require for all variables to be endogenous. It is completely fine to estimate VAR model where there are also exogenous control variables. Doing so does not violate any assumption of VAR. VAR allows for endogeneity to be present, it does not require it. However, if you are completely sure those variables are exogenous, and you are only using two variables, running VAR would be equivalent to just running two ARDL models where one of those ARDL models will be superfluous. It would not be the most efficient modeling choice but I also can't think of reason why it would introduce bias.


Is there a journal for replications in experimental economics? Is there a journal that publishes replications (failed and successful) of (published) experiments? And, if not, why not?
And what about failed experiments (i.e. experiments with insignificant results)?

I think replications of experiments are very important, as experiments in economics depend on many different things and most of them we don't really understand. Having replicated an experiment helps to make sure that the results obtained are not by pure chance (or worse). And, more importantly, failed attempts to replicate a study shows that there was something wrong in one of the studies.

As this was not clear: I am talking about lab experiments, not field experiments.

Experimental Economics states: "The journal will also consider
articles with a primary focus on methodology or replication of
controversial findings."
The Economics e-Journal takes replications from any field.

The Behavioral Economics Replication Project could be interesting for you. They presented their research at the 2016 American Economic Association Annual Meeting.


Why is 10% the necessary upper bound for a negative interest rate? Source: p 710, Economics, 3 Ed, 2014, by N G Mankiw, M P Taylor

  ... In that same article, Professor Mankiw recounted a
  discussion with one of his graduate students at Harvard about a scheme, put forward by the student,
  whereby the central bank announces that in one year’s time it would pick a digit from one to nine out of
  a hat and any currency ending in that number would cease to be legal tender. People would thus know
  that in one year’s time 10 per cent of the cash would cease to be legal tender, so what would they do? The
  logic is to spend it. The additional spending would increase aggregate demand and act as a boost to the
  economy. Such a policy might enable central banks to set negative interest rates provided the rate was
  less than 10% because there would then be an incentive to lend at (say) −4% rather than
  potentially losing 10%.

Please explain the last sentence (that I bolded). I don't understand the significance of the 10% and −4%. The foregoing discusses NEGATIVE interest rates, the maximum of which must be 0%.
So why bound the interest rate above, with a POSITIVE number (10%)?

Assuming your home is safe, would you keep your money in a bank that only gave 96% of it back? (-4% interest rate)

What if your home has mice which eat exactly 10% of your money? Would you then keep your money in the bank that only returns 96%?

And finally, what if every bill has a 10% chance of being eaten? This means that in expected value you lose 10% of your money.

In the above situations, what would be the lowest interest rate you would accept from the bank?


Why is there so little automation at McDonald's? I'm currently at a McDonald's in Norway (so, not a country where labour is dirt cheap, even for relatively unqualified jobs), and I'm struck by how manual every process is, from cooking the meat to assembling burgers to serving.
The fries-making process is particularly baffling. It goes like this:

An employee pours frozen fries in a basket and dunks it in oil
After a while, an alarm rings and an employee has to come to manually raise the basket from the oil
An employee put the fries in a bin and salt them with a handheld salt shaker
The employee shovels the fries into little bags that are brought to the trays

It seems like it would be trivial to automate away the second step here (for example by having the basket rise by itself after a set time), and not that hard to have the rest done by a machine either.

Other industries employing unqualified labour, like groceries stores, have spent considerably more money and effort into automating their checkout lines, for example. Why hasn't the fast food industry followed suit?

Automation would increase the upfront cost of opening a McDonald restaurant. Which wouldn't be a good thing for its franchise business.
Also, you have to take into account that by automating its store McDonald's will drastically increase its electricity consumption (which will not only increase its costs overall but might impact its brand image in terms of global warming). So in other words, automation wouldn't be such a cheap alternative.


Explaining the development of income inequality I'm a syndicated writer working on an explainer about income inequality and I realized I didn't really understand how it can possibly develop in an economy where all dollars are spent. That is, let's take a hypothetical: I understand how it's possible where one person creates a product that everyone else purchases (creating a concentration of wealth). But, then, that person invests or saves the money, which ultimately gets circulated back into the economy. If total money is zero-sum (all saved money is spent, all received money is spent), how does any one actor in the system end up with more money than others?

As I see it, the only mechanisms by which income inequality could increase within a group of people is if the system included globalization, where wealthy people spend money outside of the system (i.e. sending the money to other countries which were previously poor)

I'm open to answers that include a closed hypothetical economy (with 3 actors) or some other explanation. Thank you and let me know if I can clarify this question.

Update: Some have suggested its a duplicate of "how wealth is created". This is different because im asking about income (money supply and distribution), not the distribution of outputs. I get how some people might produce more. I dont understand how money concentrates in a system where all savings is invested or spent.

First off, dollars are not the sole determination of wealth.  People can be wealthy by having a large house and a nice car and still have \$0 in the bank.  Dollars hold wealth only because they can be exchanged for goods or services that actually do have value.

With that being said, let us take a look at what you have asked:

  I understand how it's possible where one person creates a product that everyone else purchases (creating a concentration of wealth). But, then, that person invests or saves the money, which ultimately gets circulated back into the economy. If total money is zero-sum (all saved money is spent, all received money is spent), how does any one actor in the system end up with more money than others?

The two phrases that I have emphasized above, hold the answer to your question.  In fact, you have almost answered your question in the first phrase.

The only reason that wealth has concentrated in Person A (who creates a product that everyone else purchases) is because everyone else has determined that they would rather have that product than the money that they have.  The benefits of having that product outweigh the benefits of keeping the money.  Many economists would argue that everyone is better off because the seller has improved his life by getting more money and the buyers have improved their lives by getting the better product.

Now, what does Person A do with the money?  Person A does the exact same thing that everyone else did originally: he looks for products that he would rather have than money.
In the marketplace, Person B, Person C, and Person D are all selling unique products of their own.  Person A sees value in purchasing the products from Person B and Person D, but values his money more than the product from Person C.  So, Person A purchases from Persons B and D but not from Person C.  This means that the money flows from Person A to Person B and Person D.

This cycle will continue to repeat indefinitely.
As long as Persons A, B, and D continue to produce products that people value more than money, they will continue to gain wealth.
On the flip side, as long as Person C does not produce a product that people value more than money, he will not gain wealth.


Log linearization and Conditional Expectation How to log linearize an equation that has a conditional expectation? Any reference or example would be appreciated.

You need to remember two properties of derivatives and expectations. Everything else stays the same.

$E(f(x))$ is not equal to $f(E(x))$ unless f is linear.
$\partial E(f(x)) / \partial x$  is equal to $E(\partial f(x) / \partial x)$.

Also remember that as a result, linearizing models with uncertainty reduces them to their certainty equivalent version. This may or may not matter for what you are doing.


About studying econometrics in undergraduate I’m currently undergraduate student studying economics and mathematics. I took econometric class and mathematical statistics class last semester. So during the vacation, I decided to study econometrics more because I’m planning to do Phd in economics in the future and I thought it will be helpful. But, come to think of it, I feel like I made a wrong choice. I don’t know what the exact reason is, but whenever I study econometrics, I find that I’m just ‘reading’ a book. And I think that’s because of my lack of statistics and other math ability. So now I’m considering to study mathematical statistics more deeply and other maths, and postpone studying econometrics. One of my professor said that it’s fine not to study econometrics now deeply because I will study during Phd course, but it’s crucial to fully study and understand maths, statistics before Phd. What do you think?

If you are looking to understand how Econometrics are and should be grounded in Statistics, read Aris Spanos "Probability Theory and Statistical Inference: Econometric Modeling with Observational Data". Be prepared for a bit of a shock.

As for Mathematical Statistics, they are a difficult subject, and there is no magical textbook that has ever succeeded in making them "accessible".


Connection between negative interest rates and negative bond yields In an answer to one of my other questions: Why does falling global bond yields signal coming deflation, the answerer states:

  Whereas central banks set rates by policy, long term bond yields are determined by the market (i.e., supply-demand equilibrium). An inverted yield curve signals recession and implies rates are set too high relative to market expectations. If central banks set rates in positive territory when long term bond yields are negative, they will de facto force the yield curve to invert.

I cannot see the mechanism as to why if central banks set rates in positive territory when long term bonds yields are negative how this will cause the yield curve to invert.

I cannot see the mechanism as to why if central banks set rates in positive territory when long term bonds yields are negative how this will cause the yield curve to invert.

No need to seek a mechanism because the inverted yield curve occurs by definition in the scenario you describe. This Investopedia video defines inverted yield curve as follows:

  An inverted yield curve occurs when short term interest rates are higher than long term interest rates.

If short term rates are positive and long term rates are negative then short term rates must be higher than long term rates. Therefore, the yield curve will be considered inverted by definition.

Remember, central banks set the interbank overnight lending rate as a matter of policy. In the U.S., this is called the Federal Reserve Funds Rate. This is the shortest possible maturity. So this is the leftmost point on the yield curve.

By contrast, supply-demand equilibrium (i.e., the market) sets long-term bond prices. And, as we know, bond prices are inversely correlated to their yields. So when the market demand increases for long term bonds, their prices rise and their yields fall.


Is the consumer surplus on a Giffen/Veblen good negative? When drawing the demand and supply curves on a quantity/product space for an upwards sloping demand, assuming the two curves intersect, I noticed that the traditional consumer surplus region lies below the equilibrium price, and possibly even under the supply curve. What happens to consumer surplus in this situation? Does it become negative (as an integral definition would suggest), or do we simply say it defaults to zero?

Actually, neither demand for Veblen good nor for Giffen good is strictly increasing in price.

In case of Giffen good the demand actually looks as shown below in picture 1. The reason for this is that you can only increase demand for the Giffen good up until you consume your entire budget. Once the price gets higher then that you still get normal downward sloping demand.

Consequently if you would apply the definition of CS as the area below the demand and above equilibrium price (excluding the corner solution when Q=0) you would still get positive consumer surplus, as no matter what the equilibrium price is there still will be some finite positive area under the demand and above equilibrium price.

In case of Veblen good the demand actually looks like at the picture below. The reason for it looking like this is that the Veblen demand cannot start already at zero price, as for example if diamonds would cost 1 euro they could not be considered status symbol. Only at a certain equilibrium price when the good is not accessible to 'ordinary' people does the demand become upward sloping above the 'snob'line.

Hence, in this case as well the CS defined as area below demand curve and above equilibrium price will be positive.

This being said, I actually looked at all microeconomic textbooks I own, and I do own a lot of them both at undergraduate and graduate level, and none actually shows calculation of CS, or mentions how CS behaves for either of these cases even when they mention them. So to be honest I am not 100% confident that the CS idea would extend to these cases, but in principle blind application of CS definition to the above cases would still give you positive CS.


Why are high/rising property values considered a good thing? Increasing prices for housing are pretty much universally treated as positive in the news and in political discourse: newspapers run headlines like "Good news: Home values are up" and talk about "housing recoveries" when home prices go up and housing's "remaining depressed" (as opposed to "remaining moderate" or "remaining affordable") when they don't. (See here for several more examples.) But as with any other good, rising home prices are good for sellers and bad for buyers. I can't see any reason why everyone seems to treat the sellers' status as more important.

In fact, when people move between (owned) homes, they usually sell one house and buy another in quick succession. Any overall change in housing prices is a wash for them. Many people buy a home and then die before selling it - high housing prices don't seem to help them much. By the time people sell a home to move to a rental retirement home, they are usually receiving Social Security, Medicare, and a pension, and don't have a long life expectancy, so it seems to me that they would prioritize the money they receive from a sale less than a young first-time homebuyer would prioritize the amount that they lose. And of course, most people don't sell their homes very often, so higher prices don't seem to do them much good over the vast majority of their lifetimes. The most concrete impact I can think of that changes in the average homeowner's house price have on their day-to-day life is through their property taxes, in which case rising house prices are a bad thing for the homeowner.

(It's true that it's good for you if your neighbors' house prices rise, because then they pay higher property taxes which go to local services that you can take advantage of. But people seem to focus more on their own house's price then on their neighbors'.)

It seems to me that for anyone other than the tiny percentage of the population who are professional real estate agents, roughly speaking, rising housing prices are a bad thing until you buy your first house, and then after that are fairly neutral, at least until you're very old and about to move into a retirement home. Housing is a notoriously illiquid good, so an increase in the fraction of total wealth tied up in housing would seem to decrease overall economic flexibility. Am I missing an obvious reason why rising prices are a good thing for the economy as a whole?

Considered a good thing by whom?

Financial institutions and speculators: yes, since houses for them are just another asset. Higher asset prices is a key mechanism of profit making for financial institutions (capital gains). In the recent financial crisis house prices collapsed and people defaulted on their mortgage payments. Thus,  the fall in the value of houses repossessed by banks meant the return on financial assets associated to housing translated into massive losses for financial institutions and government agencies associated with the housing market like Fannie Mae and Freddie Mac in the US.
Home owners: partly yes. A price increase is good, particularly if you have a mortgage to pay (so if you sell the house you can pay the loan back). Rising house prices are also good for pensioners, which main source of wealth is their house. Rising prices also give people the feeling they are richer (and hence spend more). Some argue that the positive appreciation of the media of rising house prices is partly due to the fact that newspapers are written and read mainly by older people, not by youngsters i.e. renters.
Renters: no, because rents are associated with house prices. If incomes are not growing at the pace of house prices, renters will be squeezed, to the benefit of landlords and homeowners. This is true in the UK (and in many other advanced economies, see here):

Central banks, and perhaps the overall economy and population: not necessarily, because rising house prices fueled by speculation and indebtedness (e.g. speculators and investors entering into the buy-to-let market) mean possibilities of a bubble bursting and a subsequent financial crisis. For example, in the UK, Mark Carney has expressed such concerns in the past.
Estate Agents: yes, as their business as intermediaries depends partly on fees from sales. People is more willing to sell when the market is on the up, fostering the property chain. If prices are falling, people prefer to wait instead of selling their house, breaking the property chain.


Interpretation of utility function I am reading Lucas (1980) and I am a bit confused about the way he formulates the utility function.

So there is one non-storable good that comes in $n$ colours and one unit of labor produces $y$ units of any color. He says:

  Consumption is now a vector $(c_{1t},...c_{nt})$ where $c_{it}$ is the consumption of color $i$ in period $t$.

So far I have no problems. But then he defines current period utility as:

$V(c_{1},...c_{n}) = U[\pi_{i=1}^{n}(\frac{c_{i}}{\alpha_{i}})^{\alpha_{i}}]$ where U is a standard utility function. $\sum _{i} \alpha_{i}=1$ and $\sum _{i} c_{i}=c$

What seems to confuse me is first of all the use of lower case pi. Surely Lucas is referring to the multiplication operator? I.e $V(c_{1},...c_{n}) = U[(\frac{c_{1}}{\alpha_{1}})^{\alpha_{1}}...(\frac{c_{n}}{\alpha_{n}})^{\alpha_{n}}]$

Then he says that the relative prices of the goods must be unity which I understand. However, the next part is where my main source of confusion lies, he says:

  With these prices, consumers will select color proportions $c_{i}/c=\alpha_{i}$ and given this mix   $V(c_{1},...c_{n}) = U(c)$. Without altering the example one can think of all agents having the same $\alpha$-weights, of agents distributed by a c.d.f. $F(\alpha)$ of weights. (...) In each of these cases, the equilibrium output mix (per capita) is $(\bar{\alpha_{1}}y_{1},...,\bar{\alpha_{n}}y_{n})$ each period, where  $\bar{\alpha_{i}} = \int\alpha_{i}dF(\alpha)$

I tried equating MRS to the relative prices (i.e 1) but since I'm unsure about the form of the utility function I don't know if my working is correct. Also, I am very confused about the definition of $\alpha$. What does it mean for agents to have weights? Is it just a way of stating that they are equally distributed?

Yes, $\Pi^n_{i=1}$ refers to the product of a sequence.

Regarding Lucas' solution, the Lagrangian of the problem is:

$$ L (c_1,...,c_n) = U(c_1,...,c_n) - (c-\sum c_i) $$

The FOC with respect to $c_i$ and $c_j$ are:

$$ \frac{\partial L}{\partial c_i} = \frac{\partial U}{\partial c_i} +1 = 0 $$

$$ \frac{\partial L}{\partial c_j} = \frac{\partial U}{\partial c_j} +1 = 0$$

From here you get (the MRS):

$$\frac{\partial U}{\partial c_i} = \frac{\partial U}{\partial c_j} $$

Since

$$ \frac{\partial U}{\partial c_i} = \left(\frac{c_i}{\alpha_i}\right)^{-1} \Pi^n_{i=1} \left(\frac{c_i}{\alpha_i}\right)^{\alpha_i} = \left(\frac{c_i}{\alpha_i}\right)^{-1} U(c_1,...,c_n) $$

from the MRS you get that:

$$ \frac{c_i}{\alpha_i} = \frac{c_j}{\alpha_j} $$

Using these, the definition of $c=\sum c_i$, and the fact that $\sum \alpha_i =1$, you can obtain Lucas's solution (I leave the last bit of the solution to you).

Finally, regarding weights, I think that is just to make the model interesting, as it means consumers might prefer some colors more than others (they spend $\alpha_i$% of their income in colour $i$). If not, they would spend $1/n$ of their income on each, to which any loyal disciple of William of Ockham would rightly observe "they why to bother with multiple colours at all"?


Intermediate Case of Bertrand and Cournot I am wondering whether there is a model of oligopolies in which we have some intermediate case of Bertrand and Cournot competition. What I do not mean by "intermediate" is the mixed Bertrand - Cournot game in which one firm sets a price and the other sets a quantity. What I also do not mean (I think) is monopolistic competition in which goods are not perfect substitutes.

Rather, I mean a model in which there are degrees of commitment to prices and quantities. For example, a cookie producer can softly commit to a cookie amount by only purchasing a certain number of ovens and softly commit to a price by printing the price on the cookie boxes. At some extra cost, it is however possible to later scramble to rent an extra oven or to reprint price labels. I am wondering whether this has been modeled anywhere. Ideal would be a parametric version with the parameter indicating whether we are in the pure Bertrand or in the Cournot case.

These papers could be interesting to you. First, a classical contribution:
Singh Nirvikar and Xavier Vives, 1984. "Price and Quantity Competition in a Differentiated Duopoly," RAND Journal of Economics, vol. 15(4), pages 546-554.
And these two interesting papers, using the concept of competition toughness to reconcile Cournot and Bertrand:
d'Aspremont, Claude & Dos Santos Ferreira, Rodolphe, 2009. "Price-quantity competition with varying toughness," Games and Economic Behavior, 65, 62-82.
d'Aspremont Claude and Rodolphe Dos Santos Ferreira & Louis-André Gérard-Varet, 2007. "Competition For Market Share Or For Market Size: Oligopolistic Equilibria With Varying Competitive Toughness," International Economic Review, 48, 761-784.


Can machines create value? Marx wrote that only humans can create value. But why is that so? Ok humans can, with their work, create more value than they need to consume in food and so on, that's obvious. But why is this so important? There has to be a physical explaination as to why they can create much more value than they need to supply themselves. How efficient is the human body in energy creation?
Another thing can comes to my mind is that everyone talks about that in a few years, everything will be automated and there will be no jobs left.
But if everything is done by machines, then there can't be profit. Because machines can only create things that have less value than the machine itself., otherwise you would have an perpetuum mobile, which is according to the laws of thermodynanics impossible. Ok it depends of course on demand. If demand is very high, then you can make profit, but assuming that there is equilibrium of supply and demand, then you always make a loss if you only produce with machines.
Has anyone every recognized that fact? If everything is automated, then capitalism will abolish itself. How can there be profit if everything is automated?  Why does no one see this?

Modern economists believe that machines can create value. The reason why Marx thought that machines cannot create value is that he subscribed to classical notion of labor theory of value (LTV). Labor theory of value is an objective theory of value that postulates that all value is determined by labor inputs (see the Durlauf & Blume 2008 entry in the Palgrave dictionary of economics for more on LTV).
However, current mainstream economic thinking rejects LTV in favor of the subjective theory of value precisely because LTV is empirically inconsistent with valuations that we observe. Subjective theory of value postulates that value is subjective and in the eye of the beholder as in subjective theory of value value depends on marginal utility of particular person. Under subjective theory of value even object with no labor content can have value (see the Durlauf and Blume entry (2008) on utility, and on utility and decision theory and sources cited therein  - this concept is also discussed in virtually any contemporary textbook so you can learn more also from there).
Marx also thought that profit comes from exploitation of labor but modern economists do not accept that view so, yes economists do realize that profit can be made from capital alone. Profit by definition total revenue minus total cost $\Pi = TR-TC$ and there is no reason to assume that fully automated firm could not have higher revenue than cost.
Lastly, capitalism in economics is poorly defined. In most economic textbooks you wont even find the word capitalism printed. Since the term lacks proper definition there is not much to discuss about it without first defining it.


Do Chinese tariffs on American goods make them more expensive to American consumers? This CNN article seems to imply that Chinese tariffs on American goods will make them more expensive to American consumers. Is this true? I did think that tariffs by your own country on imported foreign goods made those imported goods more expensive, but I also thought tariffs by a foreign country on goods your country exports actually made those goods cheaper for you.

Example statement in the article:

  The cost of the new tariffs can be found in the roughly 1,300 individual products that have suddenly become more expensive.

This statement is followed by a list of "US goods hit by Chinese tariffs
". Clearly the implication is these goods are among the products that have become more expensive.

Well, due to Lerner's Simmetry Theorem, an import tax has the same effect of an export tax. When China makes its imports from the US more expensive, its exports get more expensive too.


Market equilibrium and pareto efficiency How can I prove that the equilibrum point $D(p)=S(p)$ is pareto efficient?

The definition of pareto efficient: there is no way to make any person better off without hurting anybody else

$D(p)$ is the market demand curve and $S(p)$ is the market supply curve

I´m working with Hal Varian intermediate microeconomics and he proves that any amount less than the equilibrium amount cannot be Pareto efficient :

At any amount of output less than the competitive amount $q^*$ there is someone who is willing to supply an extra unit of the good at a price that is less than the price that someone is willing to pay for an extra unit of the good. If the good were produced and exchanged between these two people at any price between the demand price and the supply price, they would both be made better off. Thus any amount less than the equilibrum amount cannot be Pareto efficient, since there will be at least two people who could be better off

But I´m having trouble proving that the equilibrium point is pareto efficient

I would really appreciate if you can help me with this problem. Any comments, or suggestions would be highly appreciated

A rough guideline:

You already know the first case. At $q < q_c$ (competitive quantity), we have

$$S(p) > S(p_c) = D(p_c) > D(p)$$

This is assuming you have "normal" (monotonic) assumptions on the supply and demand curves, which are pretty important to this proof. Notice that $S(p) = q_s; D(p) = q_d$ and in this case $q = q_s < q_d$, that is, there is a shortage. At any higher price $p_0 \in (p, p_c]$, there is a Pareto improvement. A new seller will sell at this higher price and there will exist some buyer willing to still buy it.

The other case is analogous. At $q > q_c$, now

$$D(p) > D(p_c) = S(p_c) > S(p)$$

You can use a similar argument as Varian's. At any lower price, there will be a buyer willing to buy the good and a seller still willing to supply the good.


Finding the money supply graph for the UK I have been trying to find a reliable source for the UK money supply over the past couple of decades, but it seems that in each different place I look I get completely different looking answers. Now I am aware that there are different measures of the money supply (M1, M2 etc) and I am also aware that these are defined differently in different countries, I am also aware that these definitions change from time to time.

What I would really like is the sum of instantly spendable British money that is currently possessed by all individuals and businesses. So I think this is the sum of notes&coins held outside banks plus the sum total of all sterling denominated demand deposits.

Certainly notes&coins on their own is not what I'm after because there is much more spendable money than that. This measure would be too small.

Also certainly I do not want to include the value of bonds or shares owned by people because that has to be sold (taking money from someone else) in order to convert their value into money. This measure would be too large.

I suspect that the data is probably available on this page on the Bank of England website, but there are so many choices and so much jargon that I am not at all confident of choosing the right data. Can anyone give me any pointers?

"What I would really like is the sum of instantly spendable British money that is currently possessed by all individuals and businesses"

The closest to what you want is $M1$ measure (see the informative table in the wikipedia article), but it seems that Bank of England does not provide it in its webpage (see the "explanatory notes" subpage if you want to learn what the different measures include).

This site provides the $M1$ measure and it appears you can export the data too.

On the same site, consider also the $M2$ measure. It includes "near-money" (i.e. assets that are quickly and easily converted into cash) and many economists consider it to be more representative of "money supply".

UPDATE
Here is the answer I received from Bank of England

  Our Bankstats tables can be found here -
  www.bankofengland.co.uk/statistics/pages/bankstats/current/default.aspx

  We publish M1 and M2 data in Bankstats table A2.3. Data is monthly and
  split into Seasonally adjusted and Not seasonally adjusted and then
  split again into Amounts outstanding, Changes and Growth rates. These
  are all shown on different tabs within the table.

  For more details about the components and construction of these money
  measures, please see the M3 explanatory note which can be found here
  www.bankofengland.co.uk/statistics/Pages/iadb/notesiadb/m3.aspx

The table they refer to has the title "$M3$" but if you download the spreadsheet you will see that it contains also $M1$ and $M2$ measures.


How much more can the US Federal Reserve spend on COVID-19? On Mar. 26 2020, the Federal Reserve’s Balance Sheet Tops $5 Trillion for First Time. Is the Fed's spending capped at some maximum?

I studied in Canada and am ignorant of U.S. I'm asking about just Fed, not US govt. I quote MarketWatch

  In an NBC News interview aired Thursday [Mar. 26 2020] on “The Today Show,” Fed Chairman Jerome Powell said there was essentially no limit to the Fed’s emergency lending ability.

  Advertisement

  “We can continue to make loans, and the point of all that is to support the flow of credit in the economy to households and businesses,” he said.

  The Fed is only limited by how much backstop it gets from the Treasury Department.

There’s no real dollar cap on what the Federal government can do, rather the limit is on what is available to buy, and how much it is willing to spend. (There are various institutional limits on spending, but those limits can be raised. This is what allows the size of government to grow in line with increasing nominal GDP.)

Examples

It can only buy as much medical equipment as is available for sale. (As noted in a comment, it can use emergency powers to increase the supply, but purchases are still limited by what is in inventory and what can be produced in a given period of time.)
If it lends to a corporation (a big part of announced packages), there are limits to how much they want to lend to them, based on the size of the firm, etc.

One can point to limits that are created by the structure of the bond market. This limit is fuzzy, and there is no well-defined “cap.” One can point to the experience of World War II government finance as an example where the government modified how borrowing operations worked to eliminate restrictions created by the bond market.


In the real world, how does a firm know their marginal costs and revenue? How do they mathematically calculate MC=MR? I understand the theory behind it but in practice how is this done?

According to the orthodox economist, in the real world, firms do not consciously try to calculate $MC$ or $MR$. Nor do they consciously try to produce/sell at the point where $MC=MR$.
Instead, the theory you have learnt is simply theory. This theory argues that:
If firms are maximizing profits, then they must be producing/selling at the point where $MC=MR$.
This theory does not claim that firms consciously seek to produce/sell at the point where $MC=MR$. Indeed, most firms do not.
Nonetheless, the orthodox economist claims that thanks to the discipline of competitive markets, firms are forced to maximize profits and thus forced to produce/sell at the point where $MC=MR$. And so, as a simplifying assumption, we may assume that firms act as if they are seeking to produce/sell at the point where $MC=MR$.

Here is one such orthodox economist (Milton Friedman, 1953) explaining this at length with the use of the analogy of the billiards player:

Consider the problem of predicting the shots made by an expert billiard player. It seems not at all unreasonable that excellent predictions would be yielded by the hypothesis that the billiard player made his shots as if he knew the complicated mathematical formulas that would give the optimum directions of travel, could estimate accurately by eye the angles, etc., describing the location of the balls, could make lightning calculations from the formulas, and could then make the balls travel in the direction indicated by the formulas. Our confidence in this hypothesis is not based on the belief that billiard players, even expert ones, can or do go through the process described; it derives rather from the belief that, unless in some way or other they were capable of reaching essentially the same result, they would not in fact be expert billiard players.
It is only a short step from these examples to the economic hypothesis that under a wide range of circumstances individual firms behave as if they were seeking rationally to maximize their expected returns (generally if misleadingly called "profits") and had full knowledge of the data needed to succeed in this attempt; as if, that is, they knew the relevant cost and demand functions, calculated marginal cost and marginal revenue from all actions open to them, and pushed each line of action to the point at which the relevant marginal cost and marginal revenue were equal. Now, of course, businessmen do not actually and literally solve the system of simultaneous equations in terms of which the mathematical economist finds it convenient to express this hypothesis, any more than leaves or billiard players explicitly go through complicated mathematical calculations or falling bodies decide to create a vacuum. The billiard player, if asked how he decides where to hit the ball, may say that he "just figures it out" but then also rubs a rabbit's foot just to make sure; and the businessman may well say that he prices at average cost, with of course some minor deviations when the market makes it necessary. The one statement is about as helpful as the other, and neither is a relevant test of the associated hypothesis.
Confidence in the maximization-of-returns hypothesis is justified by evidence of a very different character. This evidence is in part similar to that adduced on behalf of the billiard-player hypothesis-unless the behavior of businessmen in some way or other approximated behavior consistent with the maximization of returns, it seems unlikely that they would remain in business for long. Let the apparent immediate determinant of business behavior be anything at all-habitual reaction, random chance, or whatnot. Whenever this determinant happens to lead to behavior consistent with rational and informed maximization of returns, the business will prosper and acquire resources with which to expand; whenever it does not, the business will tend to lose resources and can be kept in existence only by the addition of resources from outside. The process of "natural selection" thus helps to validate the hypothesis-or, rather, given natural selection, acceptance of the hypothesis can be based largely on the judgment that it summarizes appropriately the conditions for survival.
An even more important body of evidence for the maximization-of-returns hypothesis is experience from countless applications of the hypothesis to specific problems and the repeated failure of its implications to be contradicted. This evidence is extremely hard to document; it is scattered in numerous memorandums, articles, and monographs concerned primarily with specific concrete problems rather than with submitting the hypothesis to test. Yet the continued use and acceptance of the hypothesis over a long period, and the failure of any coherent, self-consistent alternative to be developed and be widely accepted, is strong indirect testimony to its worth. The evidence for a hypothesis always consists of its repeated failure to be contradicted, continues to accumulate so long as the hypothesis is used, and by its very nature is difficult to document at all comprehensively. It tends to become part of the tradition and folklore of a science revealed in the tenacity with which hypotheses are held rather than in any textbook list of instances in which the hypothesis has failed to be contradicted.


Do leverage pools force net negative outcomes? How are leveraged futures profits payed? Pre-note: Although I am trying to make this question general, in the case that there are any ambiguities, I would like to direct this question toward cryptocurrency futures trading platforms that offer leverage

In general, when it comes to trading, one can make the statement "for every person that gains money, someone else loses money".
What this means is that the net outcome of an entire system is zero, i.e the profits and losses sum to zero.
This is indeed the case between two time periods if the opening and closing price of a stock happens to be the same. However, if the closing price is greater than the opening price, then the net outcome in that time window is positive/profit, and vice versa.
So in simple terms, if the general trajectory of a price is upwards then there are more people making profit than making losses.
In general, when the economy is going well, you do expect more people to be making profits than making losses.
However, this begs the question, how is this net positive outcome achieved when trading with leverage pools?
Those that put money into these pools are expected to be able to eventually take that money out (let's forget for a moment that they also intend to make interest).
However, if more people are making profit than making losses, doesn't money have to be taken from the pools to be paid out to gainers, which results in the available money in the pools shrinking? Is this the case, and if so how does this not pose problematic? What solutions are used to prevent this in these pools?
The only solution I can think of is to force the system to be a net zero outcome (where there are equal gainers and losers) or a net negative outcome (where there are more losers than gainers). This solution could be achieved through transaction fees, funding fees, lending fees, etc. Is this the case in these pools, is the net outcome forced to not be positive? If so, wouldn't people be less likely to use these leverage pools?
One of the things that makes me confident when investing money is that long-term (assuming the economy is going well), there are more gainers than losers. Is it possible to still have this confidence in leverage pools? Or are there more losers than gainers? Because it seems to be that you can't have this confidence, because it seems to me that leverage pools have to force a net zero or net negative outcome.
It would be great to get clarity on this, please let me know if my question is too complicated and needs improvement.

Edit: This question seems to apply more to future/perpetual markets, where you are in a closed system. When working with the actual underlying asset, profits are also gained from selling to people outside the leverage exchange you are using, so this problem doesn't seem to exist here.

Warning
I am currently editing this answer, as I feel like I might have come to incorrect conclusions. (Maybe I didn't, but I am putting this warning in place in case I did). I hope to update this answer with more information soon.

Answer

It appears that a futures market is a zero-sum game.

Futures markets
the sum of all the long positions must be equal to the sum of all the short positions. In other words, risk is transferred from one party to another is a type of a zero sum game

Zero-sum game
a zero-sum game is a mathematical representation of a situation in which each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero

See also another interesting question related to this topic: Where does the profit come from in short selling futures?
Conclusion
When it comes to futures markets, whenever you gain, someone else is losing. This differs from spot markets where, if the general price trajectory is upwards, you can gain without someone else losing. Of course a market cannot have infinite growth so eventually it will stagnate or come down, but as long as it doesn't drop to zero, there is a net-profit outcome. But a futures market always has a net-zero outcome.
With futures markets it is more important than ever to ask yourself why you think you'll be the one gaining and not the one losing. with spot markets, it is possible for more people to gain than to lose, and may be considered less risky if this is the case (long-term investments often match this critera).
In futures markets, funding fees appear to be an example of how gains and losses are redistributed to ensure a zero-sum outcome.

Interesting follow up question: Can futures markets exist that are not zero-sum?


Total factor productivity and revenue On page 116 of The British Industrial Revolution in Global Perspective Robert Allen writes

  "...the total factor productivity of enclosed farms was 18 per cent
  higher than that of open fields. If rent accounted for one-third of
  revenues, then enclosure boosted rent by 64 per cent, for example a
  rise from 12 shillings to 20 shillings per acre."

Is the 64% rent increase derived from the 18% total factor productivity increase here?  If so, what is the working in detail?

Rent was one third of revenue. Revenue increases by 18%. Assuming this all goes to rent
$$
\frac{18}{100/3} = 54.
$$
Perhaps the 6 instead of 5 is a typo?


Do I have to bootstrap the dummy variables used for event dates in an event study? I am doing a project where i am trying to estimate the effect of the inclusion of a stock in an ETF on its returns, meaning that i am trying to see how the inclusion of a stock in a given ETF affects that stocks returns, I have collected return data on more than 150 equities, their event dates (when they were included in the ETF and so on).
my design is this, i have divided the sample into the estimation periot starting 130 days before the inclusion and the forecast window which inclused 10 days.
I am using the Multivariable Regression Model (MVRM) technique and the market model with dummy variables, the market model's dependant variable is the firms returns on the left hand side of the equation and on the right hand side of the equation qe have the market returns represented by the returns of the S&P500 index and 10 dummy variables where every dummy takes the value of 1 for the observations within the forecast interval.
More can be read on the model in Karafiath 1988.
I want to do determine the distribution of the sum of the dummy estimators which pretty much represents the cummulative abnormal returns in this case by bootstrapping.
Now my question is this:
Do i have to use a bootstrap method by simply resampling with replacement from the Dependant Variable (firm i's returns) and the market returns (S&P500's returns) and once i have drawn a sample I run a regression with the resampled values of those two and estimate the the coefficients of the dummies and save them, then draw another sample from the DepVar and IndVar and again run a regression and estimate the coefficients of the dummy variables and save them and repeat this process 10 000 times and then construct the empirical distribution of the dummy variables?
OR
Should i draw 10 000 samples and run the regression for all the variables in the regression including the dummies?
will this second case imply that when resampling from the dummy variable which is basically a bunch of zeros with only one 1 at the position of the date in the forecast interval, would this mean it will randomly assign the number 1 to some other non forecast interval because that would not make sense.
Thanks in Advance.

One way is to rely on a simple block bootstrap procedure. That is, you can draw $n$ contiguous blocks of length $t^{bs}$ (both LHS and RHS variables) which result in an overall bootstrapped sample of length $T$, which should be identical to the length of your original sample.
Then repeat your estimation for each bootstrap sample and save your results, e.g. 10000 times. Thus, inference is based on the distribution of your coefficients.
From my personal experience, dummy variables often result in singular (or nearly singular) matrices within your estimation procedure. However, you can either manually add these observations to your bootstrap sample or alternatively discard all samples resulting in numerical errors.


Comparing Nash equilibria Suppose two players play the following game:

\begin{array}{cc}
  & L & R  \\
U & 1,1 & 0,0 \\
D & 0,0 & 4,4
\end{array}

Is there any way to compare the top-left Nash equilibrium with the bottom-right one? Is there any way to distinguish between two equilibria and to define which one is "better"?

Yes there absolutely is. It is generally agreed upon that utility is (at least) ordinal. That means I can compare utility levels for a single person (not necessarily across people) and the numbers have meaning in this sense. So for a single person:

$U(down, right) = 4 > 1 = U(Up, Left)$.

This means (Down,Right) is strictly better for each agent than (Up, Left). You will notice this satisfies the definition of a pareto improvement. So (Down,Right) pareto dominates (Up, Left).

It is also generally agreed that a good way to compare outcomes in economics is pareto efficiency (no person can be made better off without someone being made worse off).

Hence we can safely say that the equilibrium with (Down,Right) each is comparable and better than (Up, Left). This is generally accepted, not controversial and doesn't require any assumptions we dislike.

Note that it is not necessary to assume here that utility is cardinal. That would mean we can compare utility levels across agents. E.g. for the statement: if U(agent 1)=4, U(agent 2)= 1, then U(agent 1)>(agent 2). Another example: saying the sum of utilities for everyone is larger under policy A than under policy B and hence A is better than B requires cardinal utility. This is a wrong assumption, but sometimes used (for example in public and welfare economics) to make comparisons. It is however controversial to use, since it is actually wrong. Public economists see it as a necessary evil. What everyone accepts is that utility is ordinal and that is all we need to compare the two equilibria.


Which part of the market value a thief can expect to get for the stolen item? If a thief steals an item to make money from it, which percent of the legal market value the thief typically gets?

For instance, if the thief steals a bike that costs $1000 new in the shop, how much money that thief typically makes from this operation?

Most likely it should be much less than $1000, because why someone would buy a used item from the questionable sources for the same price. Also, if a thief would use some black market handler, that handler also needs to earn.

If this differs a lot for different goods, please address the differences.

I would even say, you might be able to make more profit on the black market, and sell it for more than the market price. the fact that you do not pay any taxes on the black market gives you the opportunity to make profit in the range of what the tax would be.

Surely there are cases where that might differ.
Depending on supply and demand. Still i think it is not far fetched to assume, you can sometimes make a lot more money by selling something on the black market. Whether it is stolen or not.

  Most likely it should be much less than $1000, because why someone would buy a used item from the questionable sources for the same price.

That would then depend on the "kind" of black market. Surely there are black markets considered "safe" und "trustworthy" in the eyes of some dealers and thieves.

  Also, if a thief would use some black market handler, that handler also needs to earn.

Here you could split the profit made due to a lack of taxes and national regulations between all the people involved in that trading process.


The core and competitive equilibrium Can someone please explain why the concept of a core is equivalent to a competitive equilibrium?
I am just writing a term paper for my matching theory class and I'm having a hard time following along with the literature.
In particular I'm referring to Gale and Shapley (1962), Shapley and Shubik (1971), and Crawford and Knoer (1981).
Thanks

tldr: The core is a very general concept that can be used in a vast amount of models. Applying it to the setting of a general equilibrium, you can show that every competitive equilibrium is in the core.
The core
The core is a concept that can be defined for very abstract environments.
Consider a population of agents $N$ and an space $\Omega$ of outcomes. Assume that every agent has nice (transitive and complete) preference relation $\succeq_i$ over the elements in $\Omega$.
In addition, agents can form coalitions $A \subseteq N$ and for every state $x \in \Omega$ and coalition $A$ there is a correspondence $\Gamma_A: \Omega \rightrightarrows \Omega$ that determines to which states the coalition $A$ can deviate to when starting at $x$.
For example if $y \in \Gamma_A(x)$ this means that at state $x$ the coalition $A$ has the power to move to state $y$.
We say that a state $x$ dominates the state $y$ by coalition $A$ if:

$y \in \Gamma_A(x)$
for all $i \in A$: $y \succeq_i x$ and there is at least one $j \in A$ such that $y \succ_j x$.

Condition 1 states that $A$ has the power to move from $x$ to $y$. Condition 2 requires that every member of $A$ (weakly) prefers $y$ over $x$ and at least one member strictly prefers $y$ over $x$. If both conditions hold, then you can say that $x$ is not a stable situation as $A$ can find a better state $y$.
Now given such structure $(N, \Omega, (\succeq_i|i \in N), (\gamma_A|A \subseteq N))$ we can define a the core as the set of all states $x \in \Omega$ that are not dominated by some other state. In other words, no coalition can find a profitable deviation from $x$.
Exchange economies and perfect equilibria
Now give this, let us show that for an exchange economy, the competitive equilibrium is in the core. An exchange economy consists of a set of agents, $N$, a set of goods $J$. Each agent has a locally non-satiated utility function $u_i: \mathbb{R}^J_+ \to \mathbb{R}$ and an endowment $\omega_i \in \mathbb{R}^J_+$. A price vector $p \in \mathbb{R}^J_+$ together with an allocation $(x_1, \ldots, x_N)$ for the agents in the economy is a competitive equilibrium if for all agents $i \in N$:
$$
x_i \in \arg \max_{q} u(q) \text{ subject to } p' q = p' \omega_i
$$
(equality of the budget constraint holds due to locally non-satiation) and:
$$
\sum_{i \in N} x_i = \sum_{i \in N} \omega_i
$$
This second condition requires demand to be equal to supply.
To show that every allocation of a competitive equilibrium is in the core, we first need to get a mapping from our exchange economy to the structure given above:

The set of agents is simply the set $N$

The set of states $\Omega$ can be any allocation $(x_1, \ldots, x_N)$ such that $\sum_{i \in N} x_i = \sum_{i \in N} \omega_i$.

The preferences $\succeq_i$ are the ones that correspond to the utility functions.
$$
   (x_1, \ldots, x_n) \succeq_i (y_1, \ldots, y_N) \iff u(x_i) \ge u(y_i).
   $$

Now for the possible deviations of a coalition $A$, we assume that each coalition can move to an allocation by simply reallocating their own endowments. This means that they can move to a new allocation for their members as long as the total consumption within the coalition does not exceed the total endowment of the members in the coalition. (There is an issue here as I am not specifying what happens to the consumption of the members outside the coalition once a coalition deviates, but we will see that this does not really matter for the argument).

Given this, let us show that any competitive equilibrium is in the core.  Towards a contradiction, assume that $(x_1, \ldots, x_N)$ is a competitive equilibrium with price vector $p$, and assume that coalition $A$ has found a profitable deviation $(y_i| i \in A)$. This means that:

for all members $i \in A$:
$$
   u_i(y_i) \ge u_i(x_i),
   $$

for at least one member $j \in A$
$$
   u_j(y_j) > u_j(x_j).
   $$

The total consumption in $A$ must not exceed the total endowment of members in $A$
$$
   \sum_{i \in A} y_i \le \sum_{i \in A} \omega_i
   $$

Let us show that these three conditions lead to a contradiction. First, using a simple revealed preference argument it follows from  $u_i(y_i) \ge u_i(x_i)$  that:
$$
p' y_i \ge p' x_i.
$$
Indeed, otherwise $p' y_i < p' x_i \le p' \omega_i$ which means that $y_i$ was cheaper than $x_i$. This however contradicts the assumption that $x_i$ was utility maximising for $i$ at prices $p$. Also using a similar revealed preference argument, we can show that $u_j(y_j) > u_j(x_j)$ implies:
$$
p' y_j > p' x_j.
$$
So adding these inequalities across $i \in A$, we get:
$$
\sum_{i \in A} p' y_i > \sum_{i \in A} p' x_i.
$$
On the other hand from the budget constraint conditions and 3. we also have that:
$$
\sum_{i \in A} p' x_i = \sum_{i \in A} p' \omega_i = p' \sum_{i \in A} \omega_i \ge p' \sum_{i \in A} y_i = \sum_{i \in A} p' y_i
$$
a contradiction.


Are FedWire transfers and volume tracking exchanges of reserves between banks that have accounts at the Federal Reserve? Looking at Fedwire® Funds Service - Monthly Statistics it appears that these transactions are between banks but I'm trying to confirm is whether the stuff they are exchanging is classified as reserves. That is, FR liabilities shown in "Deposits with F.R. Banks, other than reserve balances" in H.4.1?

When sending a payment order to the Fedwire Funds Service, a Fedwire participant authorizes its Federal Reserve Bank to debit its master account for the amount of the transfer. If the payment order is accepted, the Federal Reserve Bank holding the master account of the Fedwire participant that is
  to receive the transfer will credit the same amount to that master account. Each payment order sent through the Fedwire Funds Service is processed individually upon receipt through an encrypted electronic network.

The above came from the Fedwire product sheet.

So yes, it is a transfer of a balance held at a Federal Reserve bank.


How is reducing government expenditure supposed to reduce aggregate demand? In my coursebook it says that in a deflationary fiscal policy governments may reduce government expenditure (such as building hospitals and road constructions) to reduce aggregate demand but I don't see how is this supposed to affect aggregate demand.

Aggregate demand is the total domestic demand for goods and services in an economy. Cutting government spending could reduce aggregate two ways.

Direct. Government demand is obviously lower, and is a component of domestic demand.
Indirect. The recipients of income from government spending have less money to spend.

It is theoretically possible that other sectors can raise spending to offset those effects. (If taxes were cut at the same time, this is obviously easier. In that case, there can just be a shrinkage of the government’s share of the economy.) There is a large debate about the empirical effects of spendng cuts. However, since the question just asked for a mechanism, that literature is somewhat out of scope.


Nominal Values or Real Value & stationary I'm going to estimate a time series regression which has consumption as a dependent variable.
My data source give it in nominal form and unfortunately there are no other sources. It is
Question :
Can I run a TS model with nominal values ?
If not, what should I do ?
I think there are 2 possible ways :
First: Change variable form to Real form by divide it to CPI(consumer price index). Then change it to a stationary form (e.i Log form) and enter it in the model.
Second: Change it suitably and correctly to become a stationary variable. e.i Log form
My question clearly is ; It is possible to use a nominal variable in a time series regression? (regardless of stationary issues)
Please explain which one is right to be done?

My question clearly is ; It is possible to use a nominal variable in a time series regression? (regardless of stationary issues)

Is it possible? Yes.
Should you actually do it? Probably not.
Nominal variables are combination of real variables and prices. For example, nominal consumption $C_n$ will be given by:
$$C_n = P C_r$$
where $C_r$ is real consumption. If you will run regression (changes are included to get rid of non-stationarity) with
$$ y = \beta_1 + \beta_2 \Delta C_n + \epsilon \Longleftrightarrow  y = \beta_1 + \beta_2 \Delta P C_r + \epsilon $$
$\beta_2$ will not tell you what is the effect of change in consumption on change in dependent variable. This is because you cannot know, without actually examining real consumption, whether variation in $C_n$ is due to $P$ or also due to $C_r$. It is very well possible that all variation in  $C_n$ is just due to prices where you would be estimating effect of prices on your dependent variable and not consumption. Of course, in reality it is even worse because likely $C_r$ and $P$ both change at the same time, meaning $\beta_2$ will be combined effect of both of those and will not have any stright forward interpretation.
This being said if you are using AR model for forecasting and you just want to forecast nominal consumption based on past nominal consumption where you don't care about above issue it would be fine to use nominal consumption.


Maximum price a consumer is willing to offer? In microeconomics it is standard to maximize utility subject to a constraint. For this problem income and prices are normally and the solution is the Marshall which plugged into direct utility function gives the indirect utility function.
Similarly, taking instead the utility to be achieved and prices as given, the expenditure minimization problem finds the expenditure (income) necessary to achieve the given level of utility at given prices and the solution is the Hicksian demand and the expenditure function.
Is there in a similar fashion a problem where utility and income are taken for granted but where prices are solved for?
The idea would be to solve for the maximal price that a consumer is willing to pay for a good if the consumer is to achieve a certain level of utility given a certain income.

There is indeed and it is called a bid function. Consider a standard set up where preferences of the agent are given by the utility function $$u(c,h) = \bar u$$ with $c$ being a composite good and where the budget constraint is given as $$c + ph = I.$$ In this case the price $p$ must satisfy $$p = \frac{I-c}{h}$$ and the maximal price $\phi(\bar u,I)$ that the consumer can offer for a unit of $h$ conditional on attaining a utility level of at least $\bar u$ given the income $I$ can be defined as
$$\phi(\bar u,I):=\max_{c,h} \left\{\ \frac{I-c}{h} \ \Bigg\lvert \ u(c,h) \geq \bar u \right\}, $$
under standard assumptions on $u(c,h)$ the constraint is binding and therefore using that $h = u^{-1}(c,\bar u)$ the optimization problem can be reformulated to
$$\phi(\bar u,I):=\max_{c,h} \ \frac{I-u^{-1}(c,\bar u)}{h}.$$
The bid function complements duality by satisfying
$$V(\phi(\bar u,I),I) = \bar u \phantom{xxx} \wedge \phantom{xxx}\ \phi(V(p,I),I) = p,$$
in relation to the indirect utility function $V(p,I) :=\max_{c,h}\{u(c,h)\lvert c+ph=I\}$ as well as the relations
$$E(\phi(\bar u,I),\bar u) = I \phantom{xxx} \wedge \phantom{xxx}\ \phi(\bar u,E(p,\bar u)) = p,$$
in relation to the expenditure function $E(p,\bar u) :=\min_{c,h}\{c + ph\lvert u(c,h) = \bar u\}$.
This implies that the bid-function can be found by inverting the indirect utility function or the expenditure function. So with Cobb-Douglas preferences
$$u(c,h) = \left( \frac{c}{\alpha}\right)^\alpha\left(\frac{h}{1-\alpha} \right)^{1-\alpha},$$
the indirect utility function is given as $V(p,I) = I/p^\alpha$ implying that the bid function is
$$\phi(\bar u,I) = \left(\frac{I}{\bar u} \right)^{1/\alpha}.$$
For an overview of the more well-known duality relations see this post duality and for the perhaps most important application of the bid function see this post monocentric city dealing with the monocentric city model.


Is the Nash product really maximised ex post? In my game theory class this term, we studied Nash bargaining. It is only now when starting to prepare for the exam that I have come to realise there is something I fundamentally don't understand, and hope that somebody here might be able to help. :)
This is a question highlighting my issue:

The Nash bargaining solution to this is (confirmed with the lecturer) the payoff pair (3/2,3/2). This is achieved by player 1 playing T and player 2 mixing L and C with 1/4, 3/4 probability. The perhaps simplest way to find this solution is to find the midpoint of a tangent to the convex set of possible utilities.

My issue is, that I am not convinced that this maximises the Nash product s_1*s_2 (the disagreement pair is (0,0)).

IF one assumes that the payoff of player 1 and player 2 vary independently, then the above is indeed the solution because E[s_1] = 1/4*3 + 3/4*1 = 3/2 and E[s_2] = 1/4*0 + 3/4*2 = 3/2, which gives E[s_1*s_2] = 9/4 (which can be shown to be maximum).

However, the payoffs do not vary independently! (T,L) corresponds to Nash product 0 and (T,C) corresponds to 2. Hence the true expected value for the Nash product under this randomisation is 1/4*0+3/4*2=3/2. This is not optimal, as purely playing (T,C) gives a higher Nash product.

Hence, how comes that the Nash bargaining solution isn't actually maximising the Nash product? Where do I go wrong in my thinking?

Very many thanks in advance! :)

The Nash bargaining solution DOES maximize the Nash product. You have to separate the playing of the game from the bargaining problem. If the players negotiate a binding agreement they will realize that their maximal total payoff from playing the game is $3$. This can be achieved by playing $(T,L)$ or $(T,C)$, or any mixture of those two profiles.

The bargaining solution just asks: Given a total payoff of $3$, how will the players divide this payoff among them, given a threat point of $(0,0)$? So the efficient frontier is (a part of) the line $u_1+u_2=3$, and by symmetry (or by maximizing the Nash product $u_1(3-u_1)$) you find $u_1=u_2=3/2$. The only remaining question is how to play the game (under a binding agreement) in such a way that these are the expected payoffs. But you already solved this part.


Census Population per County Data Looking for the simple time series of the Census data, population per county thru the years available.

Thank you much for anyone who has this data, or can link to the right source

The US Census Bureau American Community Survey: https://www.census.gov/acs/www/data/data-tables-and-tools/


Why do stock exchanges not operate at a fixed frequency? Would this not be more fair towards traders who don't trade at high frequency?

And would it not be possible to distribute this fixed frequency to other exchanges, modulo relativity?

The reason I'm asking is because I recently learned that high-frequency traders are beating inter-market sweep orders.

As long as trading faster is not explicitly forbidden, there will be people who want to trade faster if they can take advantage of it.

If you are able to trade fast enough, you can take advantage of arbitrage situations.
For instance, if you can detect a fall in the price of soy on the New-York exchange, buy soy on the New-York Exchange, and sell it on the Chicago exchange before the price in Chicago adapts, you will profit from it.

So, on the positive side, the answer to your question in the title "Why do stock exchanges not operate at a fixed frequency?" seems to be "because it's not impossible/forbidden to operate faster and people benefit from it".

Now, on the normative side, there are certainly a fair amount of people who believe that stock exchanges would perform ``better" if they operated at a fixed frequency.

Roughly, increasing the frequency of trades is good if it allows prices to adapt faster to new economic information. Some people argue, however, that past a certain threshold, increasing the speed of trade cannot foster more effective price adjustments. The argument is that relevant economic information only arrives every so often, and there is no point in trading faster than the fastest stream of relevant information.
Past this threshold -- so the argument goes, increasing the speed of trade only allows for increased high-frequency speculation in which traders benefits from arbitrage situations that are unrelated with ``actual" economic fundamentals.

I don't think they are the first to propose it, but Budish, Crampton and John have a recent pair of papers in which they advocate for a cap on the speed of transactions for similar reasons : The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response, and Implementation Details for Frequent Batch Auctions: Slowing Down Markets to the Blink of an Eye.


Deriving intertemporal budget constraint from flow constraint I am trying to learn how to solve difference equations in order to derive intertemporal budget constraints. Consider the government's flow budget constraint:

$$\frac{B_{t}-B_{t-1}}{p_{t}} + \frac{M_{t}-M_{t-1}}{p_{t}} + s_{t} -i_{t-1}\frac{B_{t-1}}{p_{t}} = 0$$

where $s_{t}$ is real taxes net of government consumption. I re-write the above as

$$\frac{R_{t}B_{t-1} + M_{t-1}}{p_{t}} = \frac{B_{t} + M_{t}}{p_{t}} + s_{t}$$

where $R_{t}$ is the gross nominal interest rate.

Now I am quite lost. I think what is confusing me is that I have $t-1$ and $t$ terms on the left-hand side. The examples I have tried before are usually on the form $a_{t} = a_{t+1} + b_{t} +(...)$ which I then solve "forward" by substituting for $a_{t+1}$ over and over again until I see a pattern developing. But in this case I can't really re-write the LHS in a proper manner.

Any thoughts on how I should proceed? And if there are any books that go through difference equations I would be very interested in some names as I am self teaching myself so far.

EDIT: Thanks to @luchonacho I have obtained the following:

$B_{t} = \frac{B_{t+T}}{R_{t+1}R_{t+2}...R_{t+T}} + \frac{D_{t+T}}{R_{t+1}R_{t+2}...R_{t+T}} ...\frac{D_{t+1}}{R_{t+1}}$ since we are in an inifite period world I write this as"

$B_{t} = \lim_{T\to\infty}  \frac{B_{t+T}}{R_{t+1}R_{t+2}...R_{t+T}} + \sum\limits_{j=1}^{\infty} \frac{D_{t+j}}{\prod_{s=1}^{j} R_{t+s}}$

I might have messed up the summation/product notation but this is one of the expressions I have seen for the intertemporal constraint. However, another common way I have seen it written is as follows:

Ignore the right-hand side as it assumes a functional form of the utility function. I am interested in obtaining an expression with both $B_{t-1}$ and $M_{t-1}$ on the LHS. Is there a way I can go from the expression I obtained above to the one with both $B_{t-1}, M_{t-1}$? I am interested in deriving this because I want an expression for the valuation of all government liabilities.

Take your second equation, move it forward one period, and rearrange. You get:

$$ B_t = \frac{p_{t+1} s_{t+1} + \Delta M_{t+1}}{R_{t+1}}  + \frac{B_{t+1}}{R_{t+1}} $$

Then, define the nominal primary deficit as $D_{t+1} = -(p_{t+1} s_{t+1} + \Delta M_{t+1})$. The above transforms into:

$$ B_t = -\frac{D_{t+1}}{R_{t+1}} + \frac{B_{t+1}}{R_{t+1}} $$

which has the format you are looking for. To find the infinite-period intertemporal budget constraint, simply proceed with forward substitution.

The notation I've used (regarding the deficit) is taken from Wickens, Chapter 5. The above is roughly equal to equation (5.11) in the book. There, the author has a more general expression, which includes inflation, GDP growth, and debt-to-GDP ratio. Your example is a simplification of his. I suggest you go through Section 5.4 of the book, which it gives a very detailed analysis of the Government budget constraint (it also gives the solution you are looking for, in case you want to compare).

To find the final expression you are looking for (after edit), take your second equation and replace $B_t$ with the infinite-period intertemporal budget constraint:

$$\frac{R_{t}B_{t-1} + M_{t-1}}{p_{t}} = \frac{B^*_{t} + M_{t}}{p_{t}} + s_{t}$$

where $B^*_t$ is the result of the forward substitution exercise.


interpretation: linear regressions with both unit dummies and time dummies Suppose I have a panel data with N units and T time periods.

For model 1 with only unit dummies:
$$y_{it} = \text{intercept} + \beta_1 x_{it} + \sum_{j = 2}^{N}\delta_j I\left(i = j\right) + \text{error},$$
least squares estimation of $\beta_1$ only uses the within-unit across-time variation in $x$.
For model 2 with only time dummies:
$$y_{it} = \text{intercept} + \beta_2 x_{it} + \sum_{k = 2}^{T}\gamma_k I\left(t = k\right) + \text{error},$$
least squares estimation of $\beta_2$ only uses the within-time across-unit variation in $x$.
For model 3 with both unit and time dummies:
$$y_{it} = \text{intercept} + \beta_3 x_{it} + \sum_{j = 2}^{N}\delta_j I\left(i = j\right) + \sum_{k = 2}^{T}\gamma_k I\left(t = k\right) + \text{error},$$
least squares estimation of $\beta_3$ apparently does not use the within-unit within-time variation in $x$, because no such variation exists in the dataset.

My question is: what exactly is the variation used in model 3?

I understand that for model 3 we are essentially de-meaning $x$ and $y$ in forms like
$$\tilde{x}_{it} = \left(x_{it} - \bar{x}_i \right) - \left(\bar{x}_t - \bar x \right),$$
where $\bar {x} _i$, $\bar {x} _t$, and $\bar {x}$ are within-unit mean, within-time mean, and total mean of $x$.

Economists using model 3 often loosely say they "have controlled for unit fixed effects and time fixed effects", but "controlling for x" usually have the ceteris paribus interpretation, meaning we are comparing within groups with same values of x. See this answer for a nice presentation. I'm looking for intuitive and verbose interpretations that are more precise than "controlling for unit fixed effects and time fixed effects".

what exactly is the variation used in model 3 with both unit and time
  dummies?

The variation that is being used to identify $\beta_3$ is basically the individual level deviations away from both the individual mean and average across individuals for the year. So to the degree that your variable of interest is varying over time but does not vary in a differential manner across individuals, you will fail to detect its effect.

  Economists using model 3 often loosely say they "have controlled for unit fixed effects and time fixed effects", but "controlling for x" usually have the ceteris paribus interpretation, meaning we are comparing within groups with same values of x. See this answer for a nice presentation. I'm looking for intuitive and verbose interpretations that are more precise than "controlling for unit fixed effects and time fixed effects".

To be clear, when we say a unit fixed effect in econometrics, we are referencing any time invariant observed or unobserved determinant of the dependent variable. It's easy to show that all of these are "wiped out" by the individual level demeaning. The individual level demeaning also controls for the average differences in observed and unobserved independent variables across individuals. Individual fixed effects in the model mean that any source of bias must be time varying. Thus, if someone argues that your variable is endogenous on account of some  variable that is constant in your sample, you have already controlled for that by only using variation over time to identify your point estimate.

So with the inclusion of these individual fixed effects you can focus on determining time varying covariates that determine your independent variable. Time fixed effects will strip away any changes in variables that are the same for all individuals in a given period of time. For instance, if "individuals" are grouped together in the same state or municipality, and there are some changes in state or municipal policies for all individuals in that year,then the time fixed effects could strip away these effects without a need for measuring them. This only leaves concern over variables that have different time paths within different individuals.

So with both time and panel fixed effects, to identify the effect of your variable of interest, exogeneity concerns aside, that variable of interest must

be time varying

and

have variation in within-individual time paths across individuals (i.e. individual heterogeneity, must be varying at $it$ not just homogeneous across $i$ within $t$.)

So we are indeed "controlling for" those time invariant unobserved confounding factors, as well as average differences in observed factors across individuals with panel fixed effects. We are indeed "controlling for" covariates that have common variation across individuals within a given year with time fixed effects.

Let $y_{it}=\beta_i+\lambda_t+X_{it}\beta+\epsilon_{it}$ be the specification with $i=1\dots N$ and $t=1\dots T$.

Initially you do the panel demeaning, which creates the transformed variable

$\overset{..}{y}=(y_{it}-\overline{y_i})=\underbrace{(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)}_\text{panel demeaned time dummies}+\text{other demeaned terms unrelated to year FE}$.

This removes the $\beta_i$, along with any other time invariant confounding factors.

Then you demean with year FE, doing the transformation $\overset{...}{y}=\overset{..}{y_i}-\overline{y_t}$ where $\overline{y_t}=\frac{1}{N}\sum_{j=1}^{N}y_{jt}$.

$=\underbrace{(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)-\frac{1}{N}\sum_{i=1}^{N}(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)}_\text{want to show this = 0}+\text{other demeaned terms unrelated to year FE}$

.
.

since $\lambda_t$ is common to all $i$ in a given year, $\sum_{i=1}^{N} \lambda_t=N\lambda_t$ and similarly for $\sum_{i=1}^{N}\frac{1}{T}\sum_{j=1}^{T}\lambda_j=N \frac{1}{T}\sum_{j=1}^{T}\lambda_j$

.
.

$=(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)-\frac{1}{N}(N \lambda_t -N \frac{1}{T}\sum_{j=1}^{T}\lambda_j)+\text{other demeaned terms unrelated to year FE}$

.
.

distribute the $\frac{1}{N}$

.
.

$=\underbrace{(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)-(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)}_\text{=0}+\text{other demeaned terms unrelated to year FE}$

$=\text{other demeaned terms unrelated to year FE}$


Augmented Filtrations and Martingales in the Martingale Representation Theorem Note: This question is related to the following question about complete markets in continuous time. In the linked question, the answer mentions that complete markets in this setting is a result of the Martingale Representation theorem.

I'm trying to understand the statement of the theorem as given in its Wikipedia article:

  Let $B_t$ be a Brownian motion on a standard filtered probability space $(\Omega, \mathcal F, \mathcal F_t, P)$, and let $\mathcal G_t$ be the augmentation of the filtration generated by $B$.
  If $X$ is a square integrable random variable measurable with respect to $\mathcal G_\infty$ then there exists a predictable process $C$ which is adapted with respect to $\mathcal G_t$, such that
  $$
X = E[X] + \int_0^\infty C_s dB_s.
$$
  Consequently,
  $$
E[X \mid G_t] = E[X] + \int_0^t C_s dB_s.
$$

In this definition, where is the relationship to Martingales? I see that it is assumed that $X$ is square integrable with respect to $\mathcal G_\infty$ and I assume that it has something to do with the fact that $\mathcal G_t$ is an "augmentation of the filtration generated by $b$." Also, what is an "augmentation of a filtration?"

The augmentation of the filtration generated by $B$ is typically just called the augmented filtration and is defined using the below construction. It is more commonly referred to simply as the standard Brownian filtration.

The collection $\mathcal{C}$ of all sets of probability $0$ in the sigma-field $\sigma\{B_{s} \: : \: s \leq t\}$
The collection of null-sets, $\mathcal{N}$, of all $A$ such that $A \subset B$ for $B \in \mathcal{C}$. It is assumed $\mathbb{P}(A)=0$ for all such $A \in \mathcal{N}$.

Then the augmented filtration for Brownian motion to be the filtration given by $\{\mathcal{F}_{t}\}$, where for each $t$ we take $\mathcal{F}_{t}$ to be the smallest sigma-field containing $\sigma\{B_{s} \: : \: s \leq t\}$ and $\mathcal{N}$.

The point of this is to endow the filtration with some properties which are considered helpful.

The relationship to martingales is simply that $X_{t}:=\mathbb{E}(X\:|\:\mathcal{G}_{t})$ is a martingale.


Can there be too little unemployment for the health of the economy? The cyclical rate of unemployment can be negative, which implies that the business cycle sometimes produces too little unemployment. How can there be “too little” unemployment for the health of the economy?

Let me answer a simpler question first. Is it possible for there to be too little orange juice in the grocery store? Definitely.

Let's say that the price of OJ is set too low (\$1 per liter) and customers who arrive early in the morning buy all the OJ. Assume that those customers think buying all the OJ is a good deal and they figure they'll enjoy drinking it, but that at regular price (\$4 per liter) they'd buy much less, they wouldn't want any more at those prices, and there would be lots left on the shelves.

People who arrive later in the day want some OJ at \$4 per liter. If they could find them, the early OJ buyers would happily sell them the OJ at that price and pocket the profit (that follows by the demand schedule I sketched out in the second sentence of the last paragraph). That means that there are deals to be done that would improve the allocation of OJ (there are welfare increasing, Pareto-improving, trades to be made). But the early customers and the late customers have trouble finding each other and so the OJ stays in the early shoppers' refrigerators.

Just about everything I just said about OJ holds for worker labor. Surprise! If the price of labor is set below the equilibrium price then early buyers purchase an inefficient quantity. Later buyers can't find the workers they need even though the marginal value of those workers is higher than the price at which the early employers bought them. Because (or perhaps I should say if) it is less efficient to look for a job while you have a job, when unemployment is too low workers ended up employed by firms that do not employ them at their highest marginal product. If only alternative employers could be found output and welfare could be improved. In fact, using a Kaldor-Hicks mechanism it is possible to make the new employer, the old employer, and the employee all better off under such circumstances. Thus it is not efficient under production efficiency or welfare efficiency criteria.