Your notation is a bit all over the place, so I'm going to try and standardize it for a general case.

Let $X$ be the regression matrix (which includes a column of 1s for the intercept term) and $\beta$ be the vector of coefficients to be estimated via OLS (which includes the intercept term). $Var(u_i|X) = 10 = \sigma^2$, for all $i$. Thus the errors are homoscedastic (constant variance) and the variance-covariance matrix of the errors $\Omega$ has diagonal entries that are all equal to $\sigma^2$. Furthermore, if you assume that the error terms are not serially correlated, then the off-diagonal covariance terms $Cov(u_i, u_j|X)$ for all $i \neq j$ in the variance-covariance matrix $\Omega$ are zero.

Those are two of the standard Gauss-Markov assumptions used to establish the BLUEness of the OLS estimator. Under these assumptions, $\Omega$ is

$$
E[uu'|X] =
    \left[
    \begin{array}{ccccc}
    \sigma^2                                    \\
      & \sigma^2    &   & \huge0        \\
      &               &   \ddots                  \\
      & \huge0 &   & \sigma^2        \\
      &               &   &   & \sigma^2
    \end{array}
    \right] = \sigma^2I.
$$

To get the variance of the OLS estimates $b = \hat{\beta}$, first note that

$$
\begin{align}
b &= (X'X)^{-1}X'y \\
&= (X'X)^{-1}X'(X\beta + u) \\
&= \beta + (X'X)^{-1}X'u \\
\implies b - \beta &= (X'X)^{-1}X'u,
\end{align}
$$

using $y = X\beta + u$. Then,

$$
\begin{align}
Var(b|X) &= E[(b-\beta)(b-\beta)'|X] \\
&= E[(X'X)^{-1}X'u((X'X)^{-1}X'u)'|X] \\
&= E[(X'X)^{-1}X'uu'X(X'X)^{-1}|X] \\
&= (X'X)^{-1}X'E[uu'|X]X(X'X)^{-1}.
\end{align}
$$

But recall that we derived $E[uu'|X] = \sigma^2 I$ via the Gauss-Markov assumption of spherical errors. Thus, by substitution,

$$
\begin{align}
Var(b|X) &= (X'X)^{-1}X'E[uu'|X]X(X'X)^{-1} \\
&= (X'X)^{-1}X'(\sigma^2 I)X(X'X)^{-1} \\
&= \sigma^2 (X'X)^{-1} X'X(X'X)^{-1} \\
\implies Var(b) &= \sigma^2 (X'X)^{-1}.
\end{align}
$$

The standard deviation of $b$ is just the square root of the variance, or

$$
sd(b) = \sqrt{\sigma^2 (X'X)^{-1}}.
$$

To find the standard deviation of the $k^{th}$ estimated coefficient in $b$, $b_{k}$, we simply extract the $k^{th}$ diagonal element of $(X'X)^{-1}$, denoted as $(X'X)_{kk}^{-1}$:

$$
sd(b_k) = \sqrt{\sigma^2 (X'X)_{kk}^{-1}}.
$$

$\sigma^2$ is the (common) variance of the errors, and unfortunately, this value is unobserved in our sample. In your question, however, it appears that this value is just given to you directly — it's $10$. In general, however, $\sigma^2$ must be estimated using the data. It turns out that given homoscedastic errors, an unbiased estimator of $\sigma^2$ is

$$
s^2 = \frac{e'e}{n-P},
$$

where $n$ is the number of observations, $P$ is the number of columns in $X$, and $e = \hat{u}$; that is, the vector of residuals $y - Xb$.

Thus, the standard error of $\mathbf{b_k}$ is estimated as

$$
\boldsymbol{se(b_k) = \sqrt{s^2 (X'X)_{kk}^{-1}}},
$$

which is Hayashi's result.


Find Change in output from marginal products of labor/capital

I think your solution is golden.

Taking the integral of the marginal products could only work if the marginal products were constant everywhere. This is not mentioned in the text, and in fact if you assume it to be constant then production given this level of $K$ and $L$ would not be 231.

Another interesting tidbit:
 If you knew the marginal product function of $L$ and also that of $K$ taking the integrals and summing them up would only work if $MPL$ is constant in $K$ and vice-versa.


Dec 2015: What would cause natural gas and crude oil to go up?

Two things would cause the price to go up.

The supply curve could move to the left: some suppliers could leave the market, some wells might switch off.

The demand curve could move to the right: new sources of demand; e.g. a burst of extremely cold weather.

This price drop is a step towards economic efficiency. You'll know when it's more efficient, because almost the whole oil & gas industry will have closed apart from a few niche applications.


Why can't we control for squares and quadratics of predicted values in regression?

For prediction, yes you can consider the models
$$
y = \beta_0 + \beta_1 x_1 + \cdots + \beta_p x_p + \gamma_2 \hat{y}^2 + \cdots + \gamma_m \hat{y}^m + error,
$$
where $\hat{y}$ represents the first-step OLS fitted values and $m$ is chosen by something like cross validation.

I haven't seen this approach used before. My personal guess is that this approach is not as useful as other commonly used ones (SVM, splines, GAM, etc.). For example, if $p$ is large (in comparison to the number of observations $n$), the first-step OLS may already be overfitting so including $\hat{y}$ is not practical. (Yes, you can use lasso residuals but that's a different story.) If $p$ is small, nonlinearity can perhaps be better handled by splines or even by simply augmenting the equation with quadratic and cubic terms of the features. Some generalized additive models (GAM) are already there too.

My personal experience is that nonlinearity is not so important (for prediction using economic data). It is usually a lot more important to avoid overfitting nicely. To me, your suggestion seems to be useful in some cases but not in many.

That said, I do not want to dissuade you from pursuing this issue, although there is a (high) chance of ending up with the conclusion that it is not very useful given the availability of other methods. BTW, you would already know it, but just in case, Hastie, Tibshirani and Friedman's book (The Elements of Statistical Learning) is helpful.


This is some dynamic supply-demand model (I am not aware of it having some special name).

The first equation gives you the evolution of prices. It says that there will be inflation if there is excess demand $d_t>s_t$ and deflation if there is excess supply (that’s why the first equation has ($d_t-s_t$).

The second equation tells you how supply changes in response to price in time. It is basically a dynamic version of basic supply curve. If the prices increase supply increases as well

The third equation is a demand function. The $y$ is most likely income - hence higher income the higher demand, and then the demand is decreasing in price


How is "opportunity cost" an useful concept? Even when it does exist?

Probably the most useful aspect of the concept of "opportunity cost" is to distinguish between choice problems where a particular choice is available, but the alternatives differ.  The concept of opportunity cost stresses that when we make a choice of some action, we forego alternatives, and so the gain from that action is the difference between the benefit of that action, and the benefit of the next-most-preferred alternative.

For example, you might have two choice problems (e.g., two nights of the week) where you have the opportunity to go to the movies on your own.  You get a certain amount of enjoyment from this, and so you might consider making this decision by thinking about how much you enjoy this activity.  The concept of opportunity cost would tell you to bear in mind the available alternatives, so that you remember you are foregoing other possible activities if you choose to go to the movies.  In one of these cases (e.g., the first night) you might not have any exciting alternatives, and so going to the movies is pretty attractive (the opportunity cost is low).  However, on another night, you might have been invited to an exciting party, with lots of close friends, and so now going to the movies and missing the party seems like a relatively bad idea (the opportunity cost is high).

All that "opportunity cost" is really doing is reminding you of the fact that decision problems must be considered on the basis of all the available actions, and the relative value of the optimal action is affected by the value of the best foregone alternative.  Most people understand this implicitly (for the most part), but economists are particularly aware of this, and have developed a concept around it.

One consequence of considering opportunity cost is to highlight the sunk cost fallacy, where a person makes a decision to pursue some action even though it is less desirable than the available alternatives, because they have already committed resources to that course of action, and they do not wish to accrue losses from a change of action (i.e., they have loss aversion).  In the example above, suppose you had already pre-paid for your movie-ticket before you find out about the party, and you can't get a refund.  The payment you made for the ticket is a "sunk cost".  You might be inclined to go to the movies, and miss the party, just so that you don't "waste" the money on the ticket.  However, doing this would just compound one mistake (buying the ticket) with another mistake (choosing to see the movie over a party you would have enjoyed more).  The concept of opportunity cost tells you to compare the available actions against the alternatives, noting that the sunk cost occurs under every case (and is therefore not relevant to a particular action).


CV, EV for additive utility; confirm or deny

Questions with numbers are usually not as good as questions without numbers. If you had written down the formula for CV and EV you would probably have noticed that your premise is false.

CV and EV are not supposed to have opposing signs. You can see this from their definitions where
$$
CV = e(p_1,u_1) - e(p_1,u_0), \hskip 20pt EV = e(p_0,u_1) - e(p_0,u_0).
$$
Either $u_1 > u_0$ and then a larger income is needed to reach $u_1$ given any price or the opposite is true.

In your final comment you also note that good 2 is inferior. As can be seen from your demand function

$$
x^*(p, w) = \left(\frac{p_2 w}{p_1(4p_1 + p_2)} , \frac{4p_1 w}{p_2(4p_1 + p_2)}\right)
$$

this is not true. If income were to increase $x_2^*(p, w)$ would increase as well.


What's wrong with have a 'one second tick' on stock market trades, to prevent high frequency trading?

There is nothing wrong with your idea. Some people may say that it would reduce liquidity - but there is way more than enough liquidity in the system already. HFT is just parasitical - it serves no useful purpose.

A small transaction tax would have a similar effect.


If I understand the question correctly, I believe the neatest way to put it is to simply call you solution concept "individual rationality".

Individual rationality can easily be seen as a cooperative game theory solution concept in its own right. I am not aware of any other name for a solution concept consisting only of an individual rationality requirement.


Weighted voting: Vote of representative weighted by number of election votes

You ask an excellent question, as it has been partly but not fully discussed in the scientific literature.

Your exact proposal (weighting representatives by the amount of votes they got in the election) has been discussed on a blog post (in french) by Jean-François Laslier see here.

In practice, representatives are often elected on a geographical basis, and their weight is determined by the local area they represent (typically: its population). There is a whole literature on that subject. Balinski and Young [BY] focus on how to design (integer) voting weights that are as close as possible to proportional to the populations. A consequent literature has questioned the desirability of the idea of proportionality, a representative paper in this area is Barbera and Jackson [BJ].

Besides, your proposal seems to be one feature (among other) of the liquid democracy system advocated by democracy activists. In that system, each one can delegate or vote on each decision, and delegates' weight equals the number of received delegations. The system has been experimented by the German Pirate party. It has been studied recently in computer science. One issue is that some influent members may get too many votes [GKMP].

[BY]: Balinski and Young. Fair Representation. 1982

[BJ]: Barbera and Jackson. On the Weights of Nations: Assigning Voting
Weights in a Heterogeneous Union. 2006

[GKMP]: Gölz, P., Kahng, A., Mackenzie, S., & Procaccia, A. D.. The Fluid Mechanics of Liquid Democracy. 2018


What you have is basically the Fisher Two-Period Optimization Problem. (Fisher)

For iii)
You first need to find the Euler equation, which tells you how to optimally trade off first-period and second-period consumption.

For starters, your optimization problem is set up incorrectly:
$$\frac{C_{1}^{1-\theta }-1}{1-\theta }+\frac{1}{1+\rho}\frac{C_{2}^{1-\theta }-1}{1-\theta }+\lambda (w-C_{1}-\frac{C_{2}}{1+r})$$

You find the Euler equation by deriving and equating the first-order conditions of both consumption terms.

$$\frac{1}{1+r} C_{1}^{-\theta }= \frac{1}{1+\rho}C_{2}^{-\theta }$$

Then just solve for $C_{1}$, substitute $C_{2}=s(1+r)$, and plug into $s=w-C_{1}$, which gives:

$$
s=w-(1+r)^{\frac{\theta-1}{\theta}}(1+\rho)^{\frac{1}{\theta}}s \\
s=\frac{w}{1+(1+r)^{\frac{\theta-1}{\theta}}(1+\rho)^{\frac{1}{\theta}}}
$$

The derivative, giving the response to a change in $r$, is:
$$
\frac{\partial s}{\partial r}= -\frac{(\theta -1) w ( 1+\rho)^{1/\theta } (1+r)^{1/\theta }}{\theta  \left((\rho +1)^{1/\theta }+r (\rho +1)^{1/\theta }+(r+1)^{1/\theta }\right)^2}
$$

The derivative for $0<\theta<1$ is positive, indicating that a higher return on saving will lead to an increase in the amount saved. Preference for first period consumption is still low enough, so that the consumer is willing to save more to consume it later. Here the substitution effect dominates the income effect.

When $\theta>1$, the derivative is negative, meaning that preference for first period consumption is so high that the consumer is willing to lower savings for current period consumption. Here the income effect dominates the substitution effect.


How do stock index futures help an economy?

Actually empirically there is causality going both ways there (i.e. economic growth helps stock index futures to grow but also stock index futures help economy to grow). However, focusing only on the effect going from futures to the economy, the reason for this relationship is the fact that capital markets (including the futures market), help to ensure efficient allocation of capital and this empirically has first order effect on economic growth (e.g. see Levine, 2005).
As summarized by Şendeniz-Yüncü, Akdeniz, & Aydoğan (2018):

... well-functioning futures markets
allow for greater and more efficient risk sharing, thereby making it possible for firms to
undertake relatively riskier projects and, hence, promote growth. On the other hand,
futures markets widely distribute equilibrium prices that reflect demand and supply
conditions, and knowledge of those prices allows investors, consumers, and producers
to make informed decisions. Consequently, the amelioration of information and
transaction costs fosters efficient resource allocation, thus leading to economic growth.

In addition note that literature overall suggest that the effect from the direction of stock futures gets weaker for high income economies. This is thought to be due to the fact that capital in high income economies is already more or less efficiently allocated.


How can I adjust the base year of this price index?

No, you want to re-scale proportionally, so you want to solve the equation $115.9x=100$, so that $x=100/115.9$ and then multiply each value by $x$. The value for 1997, for example, should be 86.3 (after rounding).


How should the general public evaluate economic arguments?

For most purposes, economics is not effectively understood in 10-second segments or catchy phrases used by politicians.

Generally, a willingness to openly acknowledge that something is complex, and to acknowledge merit to empirical evidence and/or ethical argumentation on multiple sides of a debate, is a good indication that what follows has at least some merit. Presentation of an issue which is preoccupied with fun-sounding insults, or promoting black/white thinking on a question, is unlikely to be based on quality analysis.

Differentiating between acknowledgment of complexity, as opposed to obfuscation, is also important. If complexity is acknowledged and some other analysis claims "ah HA, you admit you don't know, stooopid, because you said it's complex", then don't pay any attention to the people producing the second 'analysis', unless wanting to understand the nature of their manipulation.

Another strong indicator for a non-expert to guess at the quality of arguments is the transparency in who funds the research. If there is no disclosure of the funding, the research output should be treated as worthless unless you have specific abilities to understand the analytical methods and data inputs. An independent researcher may explicitly declare that no external funding was provided, as compared with someone paid to produce a pre-determined result who may not disclose anything. Disclosing funding sources at least allows a non-expert to observe that the conclusion does or does not align with whoever paid for it. This can be gamed, but a couple minutes of online research should most often be sufficient to determine if it's likely to be BS -- compare a) a web search ending with unnamed corporations or funding organizations with no detail about who/what/when/why of some front organization, and no ability to contact any person from the front organization, as opposed to b) a web search that leads you to a named organization registered in a specific jurisdiction with a named board of directors and an explicit mission statement.

Also, this is fundamentally not an economics question. The thematic area is more "how can a non-expert evaluate any expert claim?"


Walk through for calculating impulse response function, given VAR equation & coefficients

I ended up figuring it out. I was including the estimated constant in my IRF equation/calculation in Excel, which shouldn't have been there and was throwing my calculation off slightly. This obviously compounded with each subsequent period.

I found this resource provided the eureka moment for me: https://www3.nd.edu/~esims1/arp_companion.pdf


Where can I find research on the impact of informal/unregulated suppliers on prices and externalities?

A quick way to do this would be through google scholar? Papers are then available via libgen or scilib
If you're part of an institution such as a university then you should have access to EBSCO Econlit which allows you to scan every economics paper using Boolean search strings to filter down to what you're looking for.


Why are interpersonal utility comparisons not possible

Whether `interpersonal comparisons of utility' are possible depends on what you mean by 'utility':

If you mean something like welfare, i.e. the extent to which a
person's life goes well, then interpersonal comparisons are
obviously possible. Example: if Anna is tortured on a daily
   basis, whereas Bob lives a life of luxury, then we would (given standard background assumptions) be justified in believing that Bob has a higher
   level of welfare than Anna. (If you don't like the example, feel
   free to change the details. The general point is that, in some cases
   at least, it is reasonable to believe that one person is better off
   than another person.)
In 'standard' microeconomics, however, the word 'utility' has come to
mean something rather different to welfare. Instead, utility is defined as whatever people maximise. To be more precise, a utility
function is simply a way of representing individual rankings over
outcomes; and the outputs of that function are termed utilities. Now,
if all utilities are supposed to do is represent rankings, it is
clear that there are many possible ways of assigning utility values.
For example, if I go prefer Paris to London, we could say that
$$u(\text{Paris}) = 2 > u(\text{London}) = 1;$$or instead that
$$u(\text{Paris}) = 7 > u(\text{London}) = 3;$$or more generally assign
any utility values such that $$u(\text{Paris}) > u(\text{London}).$$ And as a result, it is clearly meaningless to say that your utility
level is higher than mine. For utility levels are purely arbitrary: we can always make your utility higher than mine, but also rescale the utility functions so that my utility is higher than yours.
So if you are wondering whether utility levels can be compared across
individuals, you need to ask yourself a simple question: what do you
mean by 'utility'?


Very basic question about confidence intervals

Yes your vague recollection is correct. You can verify it with a trivial example.
Lets take $\bar{X}=10$, with $\sigma(X)=2$ and $n=30$. So the $95\%$ confidence interval (using student's t distribution) is approximately given by  $10 ± 0.75$.
Now suppose $\bar{Y}=11$ with $\sigma(Y)=1$ and $n=30$ so the $95\%$ confidence interval is given by approximately $11 ± 0.37$.
Confidence intervals above clearly overlap since for $\bar{X}$ the upper bound is 10.75, but for $\bar{Y}$ the lower bound is $10.63$ but when we actually do the t-test for difference in their means we get:
$$ t=\frac{10-11}{\sqrt{\left(\frac{2^2}{30}+\frac{1^2}{30}\right)}} \\
 \approx -2.5 $$
Hence in this case we clearly can reject the null hypothesis that the means are same even at $5\%$ (in fact the $t$-stat is very close to being rejected at $1\%$ even) despite that their confidence levels happen to overlap.


why is aggregate demand equal to aggregate expenditure?

When we say that AD=AE we don’t mean that aggregate demand function is equal to aggregate expenditure - we mean by that the actual total quantity (Q) demanded on the market given the aggregate prices at a specific period of time is equal to aggregate expenditure. Aggregate expenditure is equal to the sum of all spending on output (Q) sold in the economy at the given time.

They must be equal because the total quantity demanded is actually quantity that’s being purchased on the market - it’s not some hypothetical demanded quantity that’s not purchased.


Have there been attempts to measure the value of specific taught skills?

Have there been any empirical attempts to estimate the value of being taught specific skills - for example, phonics or solving algebraic equations?

If I may be brazen enough to challenge the basis of this question. We first need to have an economic criteria to evaluate a subject in a curriculum before we start analysing its relevance to the labour market .

I got all this information from the University of Mumbai- Economics of Education for M.A Education
. (free download and an easy read).

In terms of how an economist should view a curriculum, the criteria is twofold, as stated in the course kit:

  One needs to asses the value of the subject and its
  existence in the system.
   There are two criteria for a particular
  subject being studied or taught in a given course of study.

  1) Theoretical consideration:
  The subject should be a body of fundamental knowledge
  which is a prerequisite for the understanding of other subjects.

  2) Practical consideration:
   The subject should have a practical significance in the sense
  that its understanding would be useful in taking efficient decisions
  in some of the practical problems. (pg.108)

This being said, its inappropriate to measure productivity on the basis of knowledge of phonics and algebra as these are studies which are only prerequisites of more prerequisites for another subject with labour market implications.

Hope this helps


Why was the price of gold lower when gold confiscation act was issued but raised afterwards?

Answer to Q1:

Wouldn't the effect on deflation be the same if they fixed the price of gold at 35$ in the beginning?

Yes it should. The value of currency under gold standard is given by the price of gold so if one ounce of gold is set to be worth $\\\$100$ then $\\\$1$ will be worth $1/100$ ounce of gold. If government would set the price for gold to $\\\$200$ per one ounce then the value of $\\\$1$ would be only $1/200$. Hence increasing the price at which dollar exchanges for given amount gold is inflationary under the gold standard.
The rationale for confiscating gold is actually already hinted at on that Wikipedia page:

The main rationale behind the order was actually to remove the constraint on the Federal Reserve preventing it from increasing the money supply during the depression. The Federal Reserve Act (1913) required 40% gold backing of Federal Reserve Notes that were issued. By the late 1920s, the Federal Reserve had almost hit the limit of allowable credit, in the form of Federal Reserve demand notes, which could be backed by the gold in its possession.

Wikipedia does not mention source for this statement,  but it is a statement that you will find across literature (see below). Furthermore,  Wikipedia is not clear about this but the reason why Fed was running out of gold reserve was that people expected that there will be devaluation of dollar so people wanted to earn profit by first converting dollar for gold at the Fed at the old price $\\\$20.67$, then wait for the devaluation of dollar to happen and then convert gold back to dollars at higher price (they could not know exactly what the higher price will be but it was reasonably expected that there will have to be some devaluation to combat deflationary pressures).
For example, according to Bordo 1989:

that rumors of devaluation appearing weeks before the banking holiday-events such as bills in Congress proposing to devalue the dollar,
statements by leading financial figures, and FDR’s unwillingness to
commit himself to the current exchange rate-triggered the run on the
dollar. The run manifested itself in both an internal and foreign demand
for gold by individuals and central banks. l9 Furthermore, he argues
that though the increase in currency was three times the amount of
gold reserves lost by the New York Fed, the fact that gold losses
threatened to reduce the Fed’s reserves below the legal limit-while
at the same time it had a virtually unlimited ability to meet demands
for domestic currency-was crucial.
Wigmore also attributes the calm which immediately followed the
banking holiday to the Roosevelt administration’s international policies: the embargo on gold ownership and export, and restrictions on
foreign exchange dealings. The former cut off the domestic channels
for a speculative run on the dollar, and the 60 percent devaluation of
the dollar in the ensuing three months removed the source of the speculative pressure

Answer to Q2:

It is stated that the governement used the profit to fund the Exchange Stabilization Fund Is this the only reason the government set the price of gold AFTER all the gold has been collected?

Well we can only make conjectures about what was going on in the heads of government officials at the unprecedented times of Great Depression, but I would guess their main objective was to prevent the speculative run on and  fall of gold reserves. Getting funds for the program was probably secondary issue. There might have been further non-economic reasons (e.g. political reasons).


By neutral, the author probably refers to something that can hardly be influenced by external forces. Money can be affected by market forces and by central banks, and the consequences are variations on inflation and exchange rates. However, if you have gold or land, these will hardly lose their relative value, at least that was the idea in the 50s-60s.

Coming back to the negative interest rates, for a rational investor that would need to decide between investing his money in return for a negative interest rate, or use the money to buy gold or land, which yield a neutral or positive interest rate, he/she would take the latter.


You are correct. A monopoly is a price maker. Not a taker.

A monopoly has the power to influence the price it charges as the good it produces does not have perfect substitutes. A price maker within monopolistic competition produces goods that are differentiated in some way from its competitors' products.

I suppose a monopolistic firm could be a price taker via matching random re-sellers prices on singular items, however that would be operating at a loss a majority of the time. The idea with that though is to retain the customer business for the future, not to make a single small profit on one item, one time.


Cross-section experiment with Differences-in-Diffrences estimation

Dif-in-dif (DiD) strategy relies on the identifying assumption of parallel trend. This essentially means that in the absence of the treatment, the control group and the supposedly treatment group would have evolved similarly (ideally both in the pre-treatment and post-treatment period). The information you provided did not mention anything specific to this underlying assumption. If the systemic difference you mentioned introduces some differences in levels between the control and treatment group but nothing else, i.e. they evolve similarly notwithstanding the difference in levels, then you can use the dif-in-dif. More generally, the dif-in-dif strategy assumes there is no other time-varying factor affecting the groups other than the introduction of the treatment.
There are other things that should also be taken into account when doing a dif-in-dif study. Is the assignment of the treatment affect the control group? Does the composition of the treatment and control groups change as a result of the treatment?
For a more comprehensive yet approachable analysis of the DiD strategy, you can have a look at chapter 5 of Mastering Metrics: The path from cause to effect by Angrist and Pischke (2014).


The CIA assumption is equivalent to $E\{ \nu_i D_i \vert X_i \} = 0$. However, it is not the case that the unconditional expectation is equal to zero. The whole reason we need the CIA is that we don't have random assignment, but potentially we have random assignment conditional on the covariates we can observe.

I think it may help to think of $\nu_i$ as an error term that is composed of other variables $Z_i$ with effect $\gamma Z_i$. In this case, $\nu_i \perp D_i \vert X_i$ means that $Z_i$ does not predict treatment after conditioning on the observed covariates $X_i$.

To answer your IV question, this is indeed similar to the exclusion restriction in that you are making assumptions about the unobserved variables $Z_i$ in your data generating process and your treatment/instrumental variable. But I wouldn't try to connect the ideas too much or else you may come to wrong conclusions.


Use Lag Operator to find Lifetime Budget Constraint

@T. G.: I think I obtained the expression for $s_{t+1}$ as a function of the other variables. I don't know what lifetime budget really means so I'll write the answer here and hope it's useful.

$s_{t+1} = \sum_{t=0}^{\infty} \lambda_{t} (w_{t}(1-l_{t}) -c_{t} - \tau_{t})$

where $\lambda_{t} = \prod_{i=0}^{t-1} (1 + r_{t-i})$.

I'm not certain if it's correct but intuitively, it looks like exponential smoothing of some kind of expression each period, except that the smoothing constant is not constant and is a function of the past interest rates ??


A type-1 seller will trade her car only if the car's measure of quality $x$ (privately known) is less than or equal to the average car quality in the market, $\mu$. $x \in [0,2]$ since no buyer values a car more than $2$ and a car cannot be sold for less than $0$. To find the probability of a car having quality less than or equal to $\mu$, we consider the CDF of $x$, $F(x)$, assuming $x$ is uniformly distributed over the support $[0,2]$:

$$
F(x)=
\begin{cases}
0 && x \leq 0 \\
\frac{x}{2} && x \in  [0,2] \\
1 && x \geq 2.
\end{cases}
$$

So the probability that a car has $x \leq \mu$ is $P(x \leq \mu) = F(\mu) = \frac{\mu}{2}$. A seller facing a price $p$ for her car becomes indifferent towards selling at $\mu = p \implies F = \frac{p}{2}$. Scaling up by a total of $N$ cars possessed by type-1 sellers implies that the market supply from type-1 sellers is $S_1(p) = \frac{p}{2}N$.


Alex Schwartz's "Housing Policy in the United States"
https://www.amazon.com/s?k=housing+economics&i=stripbooks&ref=nb_sb_noss

https://www.researchgate.net/post/Is_there_any_classic_theory_or_book_for_housing_policy_and_housing_market

Lund Brian ' Understanding Housing Policy' 2017

Housing Economics
1st Edition
Authors: George Fallis
https://www.elsevier.com/books/housing-economics/fallis/978-0-409-82940-2

Housing and Mortgage Markets in Historical Perspective (National Bureau of Economic Research Conference Report)
by Eugene N. White, Kenneth Snowden, et al.

That's what I found.  Good Luck.  There are some forums that may have a better answer for you.


Most profitable conversion of stock into money

Yes.  There is a good chapter on this general question in The Valuation Handbook book by Ashok Abbott.  It is a whole chapter so it is too long reproduce here.

Shortened down, he shows that there is a half-life effect given the amount of volume that a dealer normally carries and how long it normally takes them to liquidate a position.

ISBN-10: 0470385790


How do tech giants manage turnover costs?

Companies like Google and FB are talent focused. Turnover is high for a few reasons:
 1). Because the employees tend to be VERY good at what they do, thus their opportunities are vast; and people tend to gravitate towards who pays the most.
 2). Stress.
 3). Contracts. Many employees in these big firms tend to only have short term contracts for specific projects.
 4). These companies demand some of the best talent and work, and if an employee isn't reaching par consistently, no reason to keep them.
I'm sure there are more reasons, but those are researchable. =-)

Hope this helps.


To calculate cross elasticity of demand, does percentage change in quantity demanded have to be market demand instead of individual demand?

If you calculate XED you have two relative   terms.

$XED=\frac{\Delta q_i}{q_i}\cdot \frac{p_j}{\Delta p_j}$

It is right that in your case $\frac{\Delta q_i}{q_i}=\frac{2-1}{1}=1$

Now it depends on the current price of good j and the change of the price how large XED is. From the relative change of the demand you cannot conlcude to the level of XED.


Logically inconsistent optimality condition

There is $C_t$ the number and $C_t()$ the function. This is already so in your first equation. Your second equation says that the function is constant over $[0,1]$. In which case it is true that $C_t$ the number equals $C_t()$ the function's value times 1.

The same thing with a different notation:
$$ C_t \triangleq \int_0^1 f_{C,t}(i) di.$$
If there exists a constant $c_t$, where for all $i$
$$f_{C,t}(i) = c_t.$$
then
$$ C_t \triangleq \int_0^1 f_{C,t}(i) di = \int_0^1 c_t di = c_t. $$


If a rational preference relation over simple lotteries $\succsim$ are convex then they satisfy independence?

It's well known that if $\succsim$ satisfies independence, then it is also convex.

Since $\succsim$ satisfies independence,
$L\succsim L^{'} \iff \alpha L+(1-\alpha)L^{''}\succsim \alpha L^{'}+(1-\alpha)L^{''}$ for all $\alpha \in [0,1]$ and $ L, L^{'}, L^{''}\in \mathfrak{L} $
Convexity requires:
$L\succsim L^{''}$ and  $L^{'}\succsim L^{''} \Longrightarrow \alpha L$ + $(1-\alpha)L^{'} \succsim L^{''}$ for all $\alpha \in \left[0,1\right]$  and $ L, L^{'}, L^{''}\in \mathfrak{L} $
Pick any $ L, L^{'}, L^{''}\in \mathfrak{L} $  with $L \succsim L^{''}$  and $L^{'} \succsim L^{''}$. By completeness, $L \succsim L^{'}$ or $L^{'} \succsim L$ or both. Without loss of generality, assume $L \succsim L^{'}$. Then, by independence, for all $\alpha \in [0,1]:$
$ \alpha L$ + $( 1- \alpha) L^{'} \succsim \alpha L^{'} + (1-\alpha)L^{'}  = L^{'} \succsim L^{''} $, which we wanted to show.

Is it true if $\succsim$ are convex then they satisfy independence?

On the other side, convexity does not imply independence. To see this, take a look
at Figure (see below). Triangles are indifference curves and arrows show the direction in which
utility increases. It is convex, but not independent.


Why doesn't the price of gold drop as mining occurs that produces more gold?

You cannot reason directly from a price change. Prices may increase because supply has fallen or demand has changed. In general we can't distinguish between the two unless we know something about the demand and supply curve in particular and not just prices and quantities.

For example, when demand increases (the demand curve shifts to the right but supply curves are held fixed) this increases demand for gold at all prices. Which means that equilibrium quantity increases (more gold is produced) as well.

.

But that gold doesn't decrease prices. The only reason more gold was produced was because prices were higher. If prices fell in response to higher quantity produced the producers (who have the same supply function by assumption) would not produce more. So in response to a demand shock for gold prices increase in order to clear the market for gold. Otherwise there is not enough gold at the old prices.


You intuition about the assumption is correct.

Given the usual assumptions of downward sloping demand curve ($p'(q)<0$) and non-decreasing supply curve ($c''(q)\ge0$), if $p(0)<c'(0)$, then demand would lie entirely below supply. This means the equilibrium quantity must be zero---as if the market did not exist. Therefore, for the analysis to be non-trivial, the assumption of $p(0)\ge c'(0)$ is maintained.

With the use of calculus, we are assuming implicitly that quantities are infinitely divisible. Thus $c'(0)$ can be roughly interpreted as the cost of producing the "first infinitesimal unit" of the good. Mathematically,
\begin{equation}
c'(0)=\frac{\mathrm d\, TC(q)}{\mathrm d\,q}\Bigg\vert_{q=0}=\frac{\mathrm d\,(FC+VC(q))}{\mathrm d\,q}\Bigg\vert_{q=0}=\frac{\mathrm d\,VC(q)}{\mathrm d\,q}\Bigg\vert_{q=0}.
\end{equation}
Thus $c'(0)$ is the derivative of the variable cost function, evaluated at $q=0$.


Relative risk aversion and lotteries exercise

You want the household to be ex-ante indifferent between taking the lottery and paying some $p$ and voiding it.

With van-Neumann-Morgenstein utility, the utility of taking the lottery is given by $$U_L(w) \equiv 0.5 U(w-1000) + 0.5U(w+1000)$$

Now, you're looking for a payment $p(w)$ such that $u(w - p(w)) = U_L(w)$.


There is no commonly-accepted definition. Different countries and different organizations within a country will have different definitions.

For example, the African Development Bank defines the rich to be those with "per capita consumption levels" of >\$20 per day (2005 PPP).

That would be around or below the poverty line in most developed countries. For example, the 2018 US federal poverty line for an individual is \$12,140 per annum or \$33.26 per day.


I know that this is an old question, but I thought I'd add an answer in case it's helpful to others with a similar question. My interpretation of the original question was that the question asker was looking for a concrete example of how a production function and a transformation function might be related. So I am providing a simple example.

Suppose we have a firm that produces a single output, $ q$, using a Cobb-Douglas production function of the following form:

$$ q\le f(x_1,x_2)=x_1^\alpha x_2^\beta $$

That is, $ f(x_1,x_2) $ describes the maximum amount of $ q$ that can be produced by a firm with this technology. So the production possibilities set, $ \mathbf Y$, is the set of all vectors $ \mathbf y=(-x_1,-x_2,q), \mathbf y\subseteq \mathbb R^3 $ (in this case - clearly could be extended to $ \mathbb R^n $), such that the inequality above holds.

$$ \mathbf Y=\{\mathbf y|q\le f(x_1,x_2)\} $$

Note that the vector has negative signs on $ x_1$ and $ x_2$ because they are inputs, while the output, $ q$, is positive.

An equivalent way to represent this is as a transformation function, $ F: \mathbb R^3 \to \mathbb R$, where

$$ F(\mathbf y) = q-x_1^\alpha x_2^\beta $$

Using this notation,

$$ \mathbf Y=\{\mathbf y|F(\mathbf y) \le 0 \} $$

Where $ F(\mathbf y) \le 0 $ implies that the first inequality is satisfied.

This is fairly simple to interpret when there is only one output; however the concept of a transformation function is useful because it allows us to think about production technologies that might use multiple inputs and produce multiple outputs.

I hope this is useful!


Why do cities with rich areas still contain poor areas?

Regarding NYC, it has mostly to do with public housing and rent stabilization.

Here's a map of public housing in NYC. Notice the large public housing developments in lower Manhattan, The Bronx, and Bushwick.

About half of all apartments in NYC are rent stabilized, and these apartments tend to be priced at a fraction of the market value and inhabited by tenants who never move (because then they'd lose their precious deals). Thus rents in NYC have a bimodal distribution.


On the (rather strange) notion of strategies in repeated games

Your proposed notion of what a strategy should be in repeated game can be adapted to all extensive form games has been called a plan of action by Ariel Rubinstein, who discusses this and related issues in the paper

Rubinstein, Ariel. "Comments on the interpretation of game theory."
Econometrica: Journal of the Econometric Society (1991): 909-924.

Some concepts become harder to define, such as backward induction in games of perfect information. However, one can still do pretty much all of game theory. The book " Rationality in extensive form games" by Andres Perea uses the definition as a plan of action throughout.


Yes, the same authors Berry, Levinsohn, Pakes have written a second paper that uses both macro and micro data to estimate demand for automobiles as a function of the characteristics of the car.

"Differentiated Products Demand Systems from a Combination of Micro and Macro Data: The New Car Market"

http://dash.harvard.edu/bitstream/handle/1/3436404/Pakes_DifferentiatedProducts.pdf


Proving there exists no arbitrage opportunities given 3 states and 2 assets

First, you are in fact given $p$. You can think of return as a security that costs 1 in current period and pays off $R$ in the next period. The price vector and the payoff matrix are thus
$$
p = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \  X = \begin{bmatrix} 1.1 & 1.0 & 0.9 \\ 1.0 & 1.0 & 1.0 \end{bmatrix}
$$
and we need to find positive vector $q$ of state prices such that $p = Xq$. Because the system is undetermined, there can be many such vectors, but as long as some of them are positive, we can be sure there's no arbitrage. To actually do the computation, one way would be to treat $q_3$ as a parameter, solve for $q_1, q_2$ as functions of $q_3$, then try to find $q_3 > 0$ such that implied values of $q_1,q_2$ are positive. But in this case it's easy to see that $q = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ does the job.


"Durable goods" are a form of utility-generating capital. But they are capital, and what is actually generating utility is the flow of services from them, not them directly. So when we buy a durable good, this is not consumption, it is investment. The phenomenon of uneven intertemporal allocation of purchasing expenses in durable goods is not related to "saturation", in the sense the word is used in microeconomic theory: saturation and non-saturation is a static concept. Intertemporally, the capital stock may very well decline, as an optimal choice : we pile up capital early on, and then, the rate of usage is greater than the rate of further accumulation: we use what we have bought in the past, wearing down the stock.

The right way to model durable goods (and it has been done of course), is to specify a utility function like

$$U = U(C, h(D))$$

where $C$ are non-durable goods and $h(D)$ is the service-flow function related to durables $D$. Sometimes we simply write $C_D$ (consumption of the service flow). The difference is that the stock of durable goods is a state variable, alongside any other state variables the model has.

As a recent example of a theoretical model (not a canonical one), see the paper
Mansoorian, A., & Michelis, L. (2010). Monetary policy in a small open economy with durable goods and differing cash-in-advance constraints. Economics Letters, 107(2), 246-248.

Public Capital also enters sometimes the utility function, to represent the services flow that come from public infrastructure (contrasted to current public expenses). See for example

Chatterjee, S., & Ghosh, S. (2009). Public goods, congestion, and fiscal policy: Do consumption-based instruments matter?


second order stochastic dominance without the same mean

Let $u(x) = x$ which is increasing and concave. Then the defining condition of SOSD reads

$$\int x\mathrm dF(x)\ge \int x\mathrm dG(x) \implies E_F(X) \geq E_G(X) \tag{1}$$

..which would contradict the case  $E_F(X) < E_G(X)$ that would be permissible under a general "different means" postulate. On the other hand, we see that the "same mean" condition can be violated by the defining condition for SOSD itself. What does that tell us?

1) That $E_F(X) \geq E_G(X)$ is a necessary condition for $F$ to SOSD $G$.

2) ...And so that the requirement "$F$ and $G$ have the same mean" wrongly restricts the application of the concept of SOSD.


Contest game: second order condition satisfied, but negative profits?

Well it says in the paper on p. 423:

"Thus the sufficiency assumption implies a unique pure strategy equilibrium. And if the sufficiency assumption is violated, at least one player will receive a negative payoff; that player can improve by reducing his or her effort to zero, which will destroy the equilibrium." [emphasis mine]

So that player will not make any effort because he/she cannot win. I leave it to you to figure out what the best reply from the other player is in that case.

I guess to be completely mathematically correct one would have to formulate these kind of games with non-negativity constraints, but because it complicates the analysis they're often left out and an interior equilibrium is either assumed or, as here, checked afterwards.


Conceptual explanation/prediction of utility function. Extraction of utility function from Big Data

I believe what you are looking for are hedonic regressions:

  In economics, hedonic regression or hedonic demand theory is a
  revealed preference method of estimating demand or value. It
  decomposes the item being researched into its constituent
  characteristics, and obtains estimates of the contributory value of
  each characteristic. This requires that the composite good being
  valued can be reduced to its constituent parts and that the market
  values those constituent parts. Hedonic models are most commonly
  estimated using regression analysis, although more generalized models,
  such as sales adjustment grids, are special cases of hedonic models.

  An attribute vector, which may be a dummy or panel variable, is
  assigned to each characteristic or group of characteristics. Hedonic
  models can accommodate non-linearity, variable interaction, or other
  complex valuation situations.

  Hedonic models are commonly used in real estate appraisal, real estate
  economics, and Consumer Price Index (CPI) calculations. In CPI
  calculations hedonic regression is used to control the effect of
  changes in product quality. Price changes that are due to substitution
  effects are subject to hedonic quality adjustments.

Hedonic regression on Wikipedia

The BLS has a nice fully worked example about using this method to price changes in the quality of camcorders. They explore how brand, storage format, image stabilization, weight, and other features influence the price in order to capture how changes in the value of camcorders influences substitution in consumption decisions.


The first equality, as you've already found, equates $A$'s and $B$'s MRSs. The last equality comes from setting the marginal rate of transformation $MRT_{XY}$ between $X$ and $Y$ to the common MRS. For a production function with a single input and a single output, $MRT_{XY}$ is equal to $MP_X$, the marginal product of the input good $X$. That is:

$$ MRT_{XY} = \frac{1}{2}\sqrt\frac{5}{X}. $$

For details, see Mas-Colell, Whinston and Green (1995), section 16.F.


Market Behavior under expected and unexpected news

Short answer: the market has a forecast of the government numbers but the market considers the actual government numbers to contain information not available to the market. Therefore when unemployment  is lower than the market forecasts this indicates that on average things will be better than than the market forecast before the release. This is not to say that the forecast has no use, nor that the government number is perfect. Instead, it merely requires that there be some information in the release not in the forecast such that differences from the forecast are informative.


Is there a problem with Inflation-corrected real price?

You are right that if the inflation rate is calcualted using PPI, then PPI-adjusted price for copper didn't not change, i.e. the real prie for copper did not change. However, we would also consider the real income and real prices of the necessities in determinging wether copper has increased in value reative to income and daily necessities. Since this time when we compute the real income, we would be using the PPI calculated inflation rate as well. If nomial income rose by $5\%$ by the end of the fifth year while the price of copper along with other PPI goods rose by $15\%$ at that time, we would get a real income that is lower than the real income five years ago, which then would be apparent that the value of copper, or the relative value of copper, has increased.


Is a financial bubble harmless if rich people choose to hoard their wealth?

an economic bubble happens when too much money was chasing after too little goods

This is not correct statement. When too much money chases too little goods you get inflation including asset prices inflation but asset prices inflation or inflation in general does not necessarily leads to bubbles. Bubbles are often thought to be behavioral in origin, see for example Minsky's Stabilizing an Unstable Economy or Shiller’s  Irrational Exuberance. For example Minsky argues that bubbles ultimately start whenever investors get excited about some new paradigm which can be some new technology or it could also be financial in nature such as shift to permanently more loose monetary policy but it is not the increase in money itself that creates bubble but rather the exuberance investors feel. There is no guarantee that just having too much money will do this. The Shiller's work has similar message.

There can also be rational bubbles where the expectation of an asset price deviates progressively from the path dictated by its economic fundamentals  (see Nelson Mark's textbook international Macroeconomics and Finance) which could be caused for any number of reasons unrelated to the amount of money in circulation.

Hence already the whole premise of your question is not generally valid as bubbles are not necessarily linked to the amount of money in economy.

Furthermore, whether there is an bubble or not, just hoarding money can be harmless or detrimental to the economy but it is definitely not a positive. If rich people would just hoard money by not spending them or not reinvesting them then the money would become 'sterile' - it would be an equivalent of removing the money from the economy which would create deflationary pressure for the economy.

Arguably, during boom such deflationary pressure is more or less harmless, although even then it would be better for an economy if the money would be invested into some productive activities instead of just hoarded. If an economy is in recession creating additional deflationary pressure is just doing extra harm to the economy as due to wage stickiness it will lead to increase in real wages precisely at times when they should be cut.  In addition, if there already is bubble on the market then the deflationary pressure would just make it burst not keep going (although arguably it might be better to burst bubble sooner than latter). Deflationary pressure at right times could help prevent asset bubbles but that would be only in cases where it is actually exuberance about inflation that causes the bubble and besides your question implies that the bubble already exists. It would also be unlikely that rich people would just 'hoard' money to create the optimum level of monetary contraction to prevent it so any such task should be preferably done by central bank.

To sum up, hoarding money does not make bubbles harmless and it is not positive for economy. Hoarding money itself might be in some situation harmless to the economy but in most cases it would actually be detrimental in itself. Furthermore, by creating deflationary pressure it would actually make peoples real incomes raise faster than the productivity in presence of 'sticky wages' (at least in the short run until the deflationary pressure would become part of peoples expectations).


How can there be an electronic bank run?

What you describe is the "traditional" bank-run.

The 2008 incident you refer to, was called an "electronic bank run", because the problem was not caused by people trying to get tangible paper money in their hands in excess of the actual paper money available to the banks.

The problem was that through electronic banking and electronic wire-transfer orders "funds were redirected out of the U.S. money markets" as one account of the incident puts it.

So why is this a problem?

Because under "fractional banking", electronic money functions exactly as paper money. You work, and your employer pays your salary through a wire transfer. You go buy something and you pay with your debit card, where you transfer electronic money from your account to the shop's account.

In other words, electronic money, once created, is as much actual purchasing power as paper money is. It is wealth.

But this means that this "electronic bank run" was impoverishing the US banking system. But since the transfers were electronic, by necessity they went out of the US economy also -to some bank accounts in other countries.  It was wealth going out of the US economy, not just paper money going out of the banks but still in the economy -and at a speed and volume unimaginable in the old days, due to the wonders of the internet and digital technology.

So essentially, it is a totally different kind of problem (and more serious) than the "traditional" bank run, so it should perhaps have been characterized by using another term.


what is $b(p)$ in the Almost Ideal Demand System?

In the original article by Deaton and Muellbauer (1980), An Almost Ideal Demand System, they characterize the expenditure (cost) function for the class of PIGLOG preferences as:

$$\log c(u, p) = (1-u) \log \{a(p)\} + u \log \{b(p)\}\tag{1}$$

Where they go on to describe in intuitive terms:

  " With some exceptions (see the Appendix), $u$ lies between $0$ (subsistence) and $1$ (bliss) so that the positive linearly homogeneous functions $a(p)$ and $b(p)$ can be regarded as the costs of subsistence and bliss, respectively. " (p. 313)

Some algebra gets you equations $(2)$ and $(3)$ in the paper, which are analogous forms of the equations you have above.


Why do we have autarky price of mud pies equal to 0.5 of cheese, while it must be zero?

In Ricardian model prices depend also on consumer utility and demand. Without that it would be impossible to solve the model no matter what the goods are. However, this is usually not explicitly mentioned in bachelor level textbooks where all models are for better or worse described in overly simplified form as not to load too much on most students. In fact you always have to make an assumption on demand in Ricardian model. Standard undergraduate model assumes that utility is identical for everyone home and abroad and such that the demand derived from utility function is homothetic across countries. Of course, you could also make some own 'unrealistic' assumption on utility and demand by assuming lets say country only wants one good or just some fixed quantity of the good etc. Under some assumptions it could be even impossible to actually solve for the price. However, if you for some didactic purposes want to assume there will always be 2 industries no matter what the demand is then price can be still determined even without any explicit demand.

The reason why you can say that relative prices are equal to the ratio of productivity is that assuming perfect market, then the prices will have to be equal to the marginal cost of production which has to be in equilibrium equal to marginal product of the factor of production (in this case only factor is labor). As long as you impose restriction that both industries exist prices have to be such that they equal the ratio of productivity. If this would be violated then all workers would just go to the sector where they can get higher wage  for the same effort in terms of labor (here there is no economic profit all accounting profits are paid out as wages). For example, if productivity is such that 1 cheese = 1 labor and 1 mud pie = 2L then if the relative prices $\frac{p_{ch}}{p_m}$ are anything else than ratio of the productivity $1L/2L$ the other sector would completely collapse and could not exist anymore, violating the assumption both sectors exist.

But again all of above is extremely simplified version of Ricardian model. In actual Ricardian model you dont have just 2 goods but $n$ number of goods, you have wages utility and much more. Following PhD level textbook by Feenstra, the set up of Ricardian model includes:

Continuum of industries indexed by $z\in [0,1]$ with productivity in each industry being $a(z)$
The utility is given by $U = \int_0^1 \ln c(z) dz$, and since consumption has to be equal to the income this utility function is subject to the constraint $c(z)= wL \implies c(z)= p(z)z$, where $z$ would be some good consumed (remember these are all distributions, its not just 1 good but potentially infinite amount of goods since $z \in [0,1]$ and there are infinite numbers between 0 and 1). The demand is derived from utility function by solving the consumer optimization problem (that is how to spend resources in a way to maximize utility).
To clear markets in autarky prices are set $p(z) =wa(z)$ where $w$ is the wage.
Next you would assume the all above would also hold for foreign country and then let them trade (I might have also skipped some minor auxiliary assumptions - for full treetment see Feenstra (2015) chapter 3.).
Once countries open up to trade, balance of trade is assumed so: $\frac{w}{w^*} = \frac{z}{1-z}\left(\frac{L^*}{L}\right)$, where star * indexes the variables from the second country. Once the trade opens up the prices will be given not by national supply and demands but by world supply and demand which could be derived from the above assumptions.

Now, I wont post the full solution to this model but due to the budget constraint prices also depend on what the utility, and by extension what demand is (since demand is always derived from utility). Hence, in Ricardian model there is absolutely nothing strange about price determination - its a part of the whole system and each part makes quite a lot of sense if you think about it individually.

To sum it up, the full treatment of Ricardian model does and not only that but also has to include both utility, demands, consumption, wages etc. So your description of Ricardian model in your question does not make it justice. In a proper Ricardian model the question does not make sense because the prices actually depend also on utility and demand - you cant really make models in economics without including utility somewhere in one way or another, explicitly or implicitly. However, dont expect to learn all this at once. The purpose of the simplified version of Ricardian model that does not include all of the above explicitly is to make it easy for students to follow, but as a result the simplified Ricardian model will have many 'plot holes' and if you just try to rigorously examine the simplified Ricardian model it will never hold up - it misses too many important parts but again thats just for didactic purposes - not because the 'full' Ricardian model does not cover these.

References:
Feenstra, R. C. (2015). Advanced international trade: theory and evidence. Princeton university press.


Solving a Cournot oligopoly game with N firms

Your FOC is wrong.

So $$\max_{q_i} \quad q_i(a − \sum_{j=1}^nq_j) - q_i^2$$

$$\implies \max_{q_i} \quad aq_i − q_i\sum_{j=1}^nq_j - q_i^2$$

The real FOC is

$$a - \sum_{j\neq i}q_j - 2q_i - 2q_i = 0.$$


How to determine covariant stationary values?

Hi: Here's an attempt at a heuristic solution.
Dividing both sides by $(1 - \rho L)z_{t}$ ( using $\rho$ instead of B and $\epsilon_t$ instead of $u_{t}$ because that notation is easier for me ),
gives
$z_{t} = \frac{a}{1-\rho L} +  \frac{\epsilon_t}{1 - \rho L} + \frac{\epsilon_{t-1}}{ 1 - \rho L} $
= $\frac{a}{1-\rho L}  + \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i} + \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i-1}$
This means that we have a similiar expression for $z_{t+1}$.
$ z_{t+1} = \frac{a}{1-\rho L}  + \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i+1} + \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i}$
To calculate $cov(z_{t}, z_{t+1})$, the first term on the RHS doesn't play a role since it's not random.  Now,  since the $\epsilon_{t}$ are independent, one only needs to find the terms in the two expressions where the same $\epsilon_{t}$ occurs because that will result in a non-zero covariance term. The second infinite sum in $z_{t+1}$ coincides exactly with the first infinite sum in $z_{t}$ so all of those terms coincide without any shifting.  Next,  the terms in the second infinite sum of $z_{t}$, shifted over two time periods, coincide exactly with the terms in the first infinite sum of $z_{t+1}$ so all of those terms coincide.
So, now it should be clear that $cov(y_{t}, y_{t+1})$ will be equal to $cov(y_{t+k}, y_{t+k+1})$ because, with the terms in each expression be shifted  over by k time units, the term terms that coincide won't change. Of course, this is a heuristic argument but hopefully you can work out the details.


The assumptions of Rational Expectations Models

The assumptions are that the agents understand the situation, model or game they are in, that they are rational - in economic sense of the word and try to maximize their utility.

I think that one reason why you might have trouble finding a list of assumptions behind rational expectations is that rational expectations don’t really rely on their own assumptions rather they are application of rationality (with its set of assumptions) and utility maximization to uncertainty. Rationality and utility maximization under uncertainty simply give you rational expectations as rational expectations are just the mathematical/statistical expectations of any uncertain situation.

To illustrate this if you have a gamble $G$ where you toss a fair coin with payoffs $T=100$ and $H=-200$ the statistical expectations are given by:

$$E[G]=0.5(100)+0.5(-200)=-50$$

Now rational expectations are also -50 as if you understand the game, and want to maximize your utility the best possible expectation of the outcome of gamble is just to calculate it statistically.

You can find assumptions behind utility maximization and rationality in almost any economic textbook, but the most detailed explanations is in game theory textbooks. For example, Tadelis game theory: an introduction has clear treatment of the assumptions behind rationality.

It’s not really possible to say how restrictive rational expectations are without comparing them to some other model of expectations and considering the specifics model you want to apply them.

The critique of rational expectations is that the predictions of rational expectations don’t always hold in empirical testing. However, they are still being widely used even though we know they are not completely correct as there is no other consistent general theory of expectation formation that would provide better predictions than rational expectations even though in some isolated individual instances different expectation theories may actually provide better predictions.


A monopolist maximizes profit. For me, it is usually easier to do this in the quantity space. So you rearange the demand and maximize
$$\max_{Q_p,Q_r} \quad P_p(Q_p)Q_p + P_r(Q_r)Q_r - TC(Q_p+Q_r)$$
$$\max_{Q_p,Q_r} \quad (Q_p-10)Q_p + (28-2Q_r)Q_r - 5-2(Q_r+Q_p) -\frac{(Q_r+Q_p)^2}{8}$$
The FOC gives you two equations with two unknowns, $Q_r$ and $Q_p$, which you solve for. Both FOC equations can be written in the form $MR=MC$. You then plug these $Q$s into the demand equations to get the prices. Next, you calculate consumer surplus and producer surplus and add them to get social welfare.


Cournot oligopoly - first-order condition

Note :$Q = \sum_{i=1}^n q_i$.
Thus the optimisation problem of firm $i$ is:
\begin{align}
max_{x_i\in\mathbb{R}_+}\pi_i(q_i,Q)
\end{align}
where $\pi_i(q_i,Q) = \big(Q^{-1}(q_i;q_{-i}) -c_i\big)q_i$. Assuming interior solution, the first order condition is
\begin{align}
\frac{\partial\pi_i}{\partial q_i} + \frac{\partial \pi_i}{\partial Q}\frac{\partial Q}{\partial q_i} &= 0\\
\implies (Q^{-1} - c_i) + (-1)Q^{-2}q^*_i* 1 &= 0\\
\implies q^*_i &= Q(1-Qc_i)
\end{align}


Are Credit Card Points, Airline Miles, and Store Credit representations of M2 money supply?

It seems unlikely that any authoritative source would consider credit card points, airline miles or store credit as apart of M2 money supply.

As you have probably read on investopedia:

  What is 'M2'
  M2 is a measure of the money supply that includes all elements of M1 as well as "near money." M1 includes cash and checking deposits, while near money refers to savings deposits, money market securities, mutual funds and other time deposits. These assets are less liquid than M1 and not as suitable as exchange mediums, but they can be quickly converted into cash or checking deposits.

According to this definition, credit card points are not counted to as apart of M2 money because they are not directly tied back to a cash exchange directly,(though there are some firms who will purchase them,) but are tied to specific services provided by the authorizing firms.

Since Credit card points are only usable for a specific services and are viewed  as  "in kind" transfers  rather than actual money.


Update of value function in continuous time - HJB

You iterate towards a fixed point, so you want to reach a situation where plugging in your current iterated value produces itself. Now using your notation, we are told that we should calculate

$$V_{n+1}(a) = V_n(a) + \Delta$$

where

$$\Delta = \ u(c(a^*)) + \dfrac{\partial V_n(a)}{\partial a}da_t(a^*) - \rho V_n(a)$$

Insert the second into the first to see what the iteration rule is:

$$V_{n+1}(a) = V_n(a) + \ u(c(a^*)) + \dfrac{\partial V_n(a)}{\partial a}da_t(a^*) - \rho V_n(a)$$

When you reach a point where

$$V_{n+1}(a) = V_n(a) $$

(or $\epsilon$-so)

It will mean

$$\rho V_n(a)= \ u(c(a^*)) + \dfrac{\partial V_n(a)}{\partial a}da_t(a^*) $$

which is what you have to satisfy.

Some stars etc may have to be adjusted in the above, for a fully consistent notation.


First-price auction with discrete values

The suggested pure strategies do not create an equilibrium, because if $X_2=X_1=2$, there is a tie. Whichever tie-breaking rule is employed, one of the players would want to deviate to $\beta_i=1+\epsilon$ to avoid the tie and win with certainty. If tie breaking is always in favor of the second player, player 1 deviates to $0+\epsilon$.

Discrete values in this setting do suggest a mixed strategy equilibrium (non-degenerate), because the probability of a tie in valuations is non-zero. This is why the pure strategy equilibrium from the continuous case is missing here.

The more general case of this problem is in Doni & Menicucci (2013).


The U-3 unemployment rates is the total unemployed, as a percent of the civilian labor force (official unemployment rate). By definition it excludes the discouraged workers, which, as Varun and Kent understand, is not true of all unemployed statistics. I liked what Karabell (2014) says about this statistic:

  ...[T]here is nothing wrong per se with the [U-3] unemployment figure,
  provided you know that the definition of employment is a construct,
  not a counting exercise....to be unemployed, according to the
  Bureau of Labor Statistics, you have to be in the workforce. To be in
  the workforce, you have to have a job or be looking for a job. To be
  looking for a job, you have to answer when surveyed that you have
  actively sought employment at some point in the past four weeks. If
  you haven't, you aren't actually unemployed. You are a discouraged
  worker, are marginally attached to the workforce, and are not counted
  as part of said workforce. And if you haven't been looking for a year,
  you're not, statistically speaking, even discouraged. You simply,
  statistically speaking, don't exist. You do, of course, actually exist
  (though that is a whole other topic), but not in the statistical
  universe of employment data.

  The result of such methods is that it is possible, and not just
  theoretically, for the unemployment rate to drop not because more jobs
  were created or more people were hired but simply because more people
  gave up looking for work and dropped out of the workforce. That is, in
  fact, What happened in the United States and swaths of Europe after
  2009, though in America, there were also real job gains in 2010 and
  after. Nonetheless, the unemployment rate, which receives so much
  public scrutiny, went down in part not because the economic system was
  creating so many new jobs but because it was failing to do just that.
  To be fair, the BLS keeps multiple unemployment statistics and
  different unemployment rates, including ones that include
  underemployed, marginally attached, and part-time workers, but there
  is only one headline number. That is what the BLS points to in its
  monthly release; that is what the media pick up; and the other numbers
  are at best footnotes.

The Leading Indicators: A SHort History of the Numbers That Rule Our World
Zachary Karabell (2014)

Therefore, if your question is why does U-3 exclude discouraged workers, the answer is that's an essential part of the definition of discouraged.

If your question is why did they define U-3 this way in the first place, I am not sure. I suspect the following. The definition of the labor force, which is workers and those actively looking for work or waiting to be recalled to work. Detached and marginally attached workers are not in the labor force. Once you defined the labor force, a natural question is what fraction of the labor force is not working, and that statistic is U-3.

If your question is why is this statistic more widely discussed than other economic statistics, the answer may simply be that it is older. The U-3 statistic goes back to 1948 while U-4, U-5, and U-6 start in 1994. People may be used to talking about the better established statistic and enjoy the longer time series, and see insufficient benefit to switching to the newer statistics, despite the benefits. A sort of Dvorak vs. Qwerty keyboard issue for economic statistics.

In a way, this is like an issue of GDP measurement. Home production is not in GDP and Robert Barro has argued that investment is double counted in GDP. If you want to use GDP as a measure of something else, these may be big problems and other measures may serve better. However, if you want to compare GDP across countries or time, these issues matter less than having something useful and consistent to compare them with. Commenter ABC at Marginal revolution has a nice quote about this:

  The algebra is fine as it is, but I take strong issue with the idea
  that GDP "mismeasures" something. As any decent intro macro course
  will tells its students, "GDP is not a measure of welfare." Indeed,
  the Kuznets quotes that Barro points to are exactly of this nature; we
  have this high profile GDP number, but it doesn't do what you want it
  to do - or at least not ALL the things you want it to.

  What Barro does here is construct a new measure - present discounted
  value of consumption - and shown how it relates to GDP (and GDP's
  present discounted value). In addition, he provides a capital income /
  labor income decomposition for both measures (GDP having such an exact
  decomposition because of constant returns to scale, Euler's identity
  for homogenous functions, and marginal-value input prices). He then
  says present discounted value of consumption, and its associated
  decompositions, are "right" while the other the construction GDP is
  "wrong" and thus gives "misstated" decompositions. Of course,
  "right/wrong" and "correct/incorrect" begs the question - right about
  WHAT? Correct for WHAT?

  Barro says that in present discounted value terms GDP "double counts"
  investment. The "double counting" is only relative to how things are
  calculated for the present discounted value of consumption. This just
  reflects the fact that GDP counts production, while consumption only
  counts consumption.

ABC's comment on Marginal Revolution Post: Robert Barro says we double-count investment


In classical macroeconomics, why Labor supply responds to real wages and not money wages?

To begin with, economics is all about assumptions. Because of classical dichotomy, money level doesn't impact the growth of an economy in the long run. Money has no effect on GDP in classical macroeconomics. If you want to model the effect of money, you have to introduce nominal rigidities in your model. That means, at least the goods market or the labour market is NOT perfect. If labour responds to money wages, the result would be a prefect market situation.

According to Keynes (General Theory, 1936),  the nominal wage (money wages) is completely unresponsive to current-period developments (at least over some range):$$W=\bar{W}$$
As you can see, because nominal wages are sticky, the result is an imperfect labour market. Because the labour market is imperfect, there is unemployment.  The
labor market has some non-Walrasian feature that causes the equilibrium
real wage to be above the market-clearing level.
In Keynes's model, aggregated labour supply is given. Firms are competitive and their prices are
flexible, and so they hire labor up to the point where the marginal product
of labor equals the real wage: $$F'(L)=\frac{\bar{W}}{P}$$

The main conclusion of Keynes' model is that GDP and real wages are countercyclical. Later, it was turned out to be moderately procyclical by empirical research.

Other more sophisticated models:
 - sticky prices, flexible wages, perfect labour market
 - sticky prices, flexible wages, imperfect labour market
 - flexible prices, sticky wages, imperfect labour market

If you want to model nominal effect, you should take note of nominal wages. If you are modelling fiscal shocks without the effect of the money, then considering real wages are maybe enough. It depends on one's research philosophy.


Is there any nation which doesn't have any external debt?

The simplest answer is that yes, there are a few countries without external debt, all small.  Wikipedia has a list of the external debt of countries.  For instance Niue finished paying off its debt in October 2016.


Relationship between Growth in Purchasing Power and Real Interest Rate equations

$Growth\ in\ Purchasing\ Power=1+r_r = \displaystyle\frac{1+r}{1+i}$

Also it is given that $real\ interest \ rate=r_r$

So :

$$real\ interest \ rate=(1+r_r)-1=\frac{1+r}{1+i}-1=\frac{1+r}{1+i}-\frac{1+i}{1+i}=\frac{r-i}{1+i}$$


Incentive compatibilty conditions for multi-item auctions

Yes, it is essentially the same idea as with just one unit.
For example, see the text leading to Proposition 14.1 in the book "Auction Theory" by Vijay Krishna.
Let $x$ be the vector of willingness-to-pay.
Define $U(x) = \max_z \{ q(z)x - m(z)\}$ as the equilibrium utility of type $x$. The incentive compatibility (I am quoting Krishna now) "implies for all $x$,
the probability vector $q(x)$ is a subgradient of the payoff function $U$, which
is convex, at the point $x$. In other words, the vector $q(x)$ is perpendicular to
the hyperplane that supports the function $U$ at $x$ — the graph of the function $U$
lies above the hyperplane." He goes on to show that
$$U(x) = U(0) + \int_0^1 q (t \cdot x) x d t,$$
where instead of directly integrating over the type $x$ you integrate  over $t\in (0,1)$.


In the standard model, all workers have the same productivity. So if J < 0, your expected value of a vacancy would also be negative. That the reason why, once a match is made, the only variable to be set is the wage. In the real world, they have to deal with different expected productivities.

Pissarides 94 distinguished between two kinds of applicants (but in this case, the firm is passive in the decisions).


Acemoglu - Introduction to Economic Growth - Existence of a one-to-one relation between human and physical capital

I think this needs little bit context to be answered, because in your question you missed a whole host of background assumptions - this is not a result that will hold for arbitrary function.
The equations actually describe are derived from first order optimality condition for a steady state from a Hamiltonian
$$f_k(k(t), h(t)) − f_h(k(t), h(t)) = \delta_k − \delta_h,$$
where $f$ is production function, $k$ per capita capital and $h$ per capita human capital and $\delta_k$ and $\delta_h$ are the depreciations respectively. Furthermore, as stated in the first paragraph you omit host of many important assumptions about the production function.
These assumptions are too many to be listed here (the assumptions take several pages to explain in the textbook itself in ch 3.3 on pp 85 and following pages), but the main important assumptions (and their implications) are:

$f$ has constant returns to scale
$f$ is strictly concave in $k$ such that: $f (k^∗, h^∗)>f_k(k^∗, h^∗)k^∗ + f (0, h^∗) \implies f (k^∗, h^∗)>f_k(k^∗, h^∗)k^∗$
$f_k(k(t), h(t))>0, f_h(k(t), h(t))>0$ and $f_{kh}(k(t), h(t))> 0  $ implies function is monotonically increasing.
Inada conditions.

These imply that if you double the factors of production output will double  and that one would always want to use human capital and capital together. This means that one would always want to increase usage of capital together with human capital rather than just usage of one factor.
Since whenever we increase use of $k$ we want to also increase use of $h$ and since difference between marginal productivity of both of them will be always constant, there should be some one-to-one mapping between $k$ and $h$ described by some function $k=\xi(h)$. Also this is why the textbook assumes that $\xi(\cdot)$ is strictly increasing, unique and differentiable. It has to be strictly increasing because the more human capital $h$ we use the more we will want to use regular capital $k$. It is unique given that from all conditions we impose on the model there will always be some unique equilibrium $(k^*,h^*)$ combination, and differentiable simply because this is obviously continuous function.    Also this result would not hold for any arbitrary function $f$.


Can stimulus packages lead to hyperinflation?

No chance. The economy is heading for a deep recession, because of an exogenous fall in demand (think of a supply and demand graph, where demand falls drastically, and so Q and P falls). Recessions most of the time mean less inflation (firms sell less). Exceptions are called stagflation and are fairly uncommon, except in totally destroyed economies where inflation is a deep cause of the recession, like in Venezuela.

numbers are most likely calculated by looking at needs, size of market, etc. In most cases these stimulus are funds. They are money to be put in a box and taken as needed. Not all might be used.
a lot of funds are targeted and not given directly as cash to people. In any case, people won't be spending the money as crazy, partly because there is little on which to spend it (no eat out, no holidays). Plus all the uncertainty about how long this will last make people averse to spend it. Wage costs are low and partly subsidized too so firms won't increase prices.


Meaning of UK Employment Change Index - 3 Months Average

Let's say the raw series is $x_1, x_2, \ldots$. The three-month moving average is given by $y_n = \frac 1 3 \cdot (x_{n-2}+x_{n-1}+x_n)$. What you have, though is $z_n = y_n - y_{n-1} = \frac 1 3 \cdot(x_n - x_{n-3})$.

Without knowing any of the $x_n$'s, you can't back out $x_n$'s from $z_n$'s. Knowing only $x_1$, however, would give you $x_1, x_4, x_7, \ldots$. It's likely that you would know this, since the series we're interested in is an index, which usually gives the base year.

In all, you need to know at least $x_1, x_2, x_3$ (or, to be more precise, at least one $x_i$ where $\mod(i, 3) = j$ for $j \in \{1, 2, 3\}$ to be able to back out the entire series of $x_n$'s.

Usually, people are interested in three-month moving average because the value for each month is highly volatile and can't be used with much precision.


Let me focus on the use of duality theory in demand analysis (as this is what I'm most familiar with).
Direct approach
Usually, in order to obtain a demand system (which you can then estimate) you need to take the following steps.

specify a utility function $u(x_1, \ldots, x_n)$.
maximize this with respect to a budget constraint $\sum_i p_i x_i \le m$. .
Obtain a closed form solution for $x_i$ as a function of all prices $(p_1,\ldots, p_n)$ and income $m$.
Estimate these functions.

Step 3 is the trickiest part as usually no closed form solutions are possible. To see this, notice that the first order conditions give the following system:
$$
\frac{\partial u(x_1, \ldots, x_n)}{\partial x_n} = p_i\,\, \lambda,\\
\sum_i p_i x_i = m.
$$
Solving for $\lambda$ we have:
$$
\dfrac{\dfrac{\partial u}{\partial x_i}}{\sum_j \dfrac{\partial u}{\partial p_j}} = \frac{p_i}{m}
$$
This gives the (normalized) prices on the right hand side as a function of the quantities (so the inverse demand functions). To obtain a closed form solution of the quantities in terms of the functions, this system needs to be inverted. Except in some particular cases (e.g. Cobb-Douglas or CES) we do not know how to do this.
Dual approach
The dual approach starts from a specification of the expenditure function and uses known duality results to obtain a closed form expression for the demand functions. It takes the following steps:

Specify an expenditure function $e(p_1, \ldots, p_n,u)$.
Use Shephard's lemma to obtain the Hicksian demand function $h_i(p_1,\ldots, p_n, u)$.
Invert $e(p_1, \ldots, p_n, u)$ with respect to $u$ to get the indirect utility function $v(p_1, \ldots, p_n, u)$.
substitute $v(p_1, \ldots, p_n, u)$ in $h(p_1, \ldots, p_n, u)$ to get the Marshallian demands $x_i(p_1, \ldots, p_n, m)$.

As an exemple, consider the widely used Almost Ideal Demand System specification of Deaton & Muellbauer
Start from the following (flexible) specification for the expenditure function.
$$
\ln e(p,u) = \alpha_0 + \sum_i \alpha_i \ln(p_i) + \sum_{i,j}\frac{1}{2}\gamma_{i,j} \ln(p_i) \ln(p_j) + u \prod_i p_i^{\beta_i}
$$
We impose $\gamma_{i,j} = \gamma_{j,i}$, and by homogeneity of degree 1 we have that $\sum_i \alpha_i = 1$ and $\sum_{i = 1}^n \gamma_{i,j} = 1$ and $\sum_{i} \beta_i = 0$.
First we can take the derivative $\frac{\partial \ln e(p,u)}{\partial \ln p_i}$ to get the demand share equations. Shephard's lemma gives:
$$
\frac{\partial \ln e(p,u)}{\partial \ln p_i} = \frac{p_i h_i(p,u)}{e(p,u)} = \alpha_i + \sum_j \frac{1}{2}\gamma_{i,j} \ln(p_j) + u \beta_0 \beta_i \prod_i p_i^{\beta_i}
$$
Here $h_i(p,u)$ is the Hicksian demand for good $i$.
Writing by $v(p,m)$ the indirect utility function, we know by duality between utility maximisation and expenditure minimisation that:
$$
m = e(p,v(p,m)) \text{ or } u = v(p,e(p,u)).
$$
Using this, we can invert the expenditure function to obtain the indirect utility function.
$$
v(p,m) = \frac{\ln m - \alpha_0 - \sum_j \alpha_j \ln(p_j) - \sum_{i,j}\frac{1}{2} \gamma_{i,j} \ln(p_i) \ln(p_j)}{\beta_0 \prod_i p_i^{\beta_i}}
$$
Also by duality, if $x(p,m)$ is the Marshallian demand function then:
$$
x_i(p,m) = h_i(p,v(p,m)).
$$
So we can substitute $v(p,m)$ into $h_i(p,u)$ to get the shares as a function of prices and income.
$$
\begin{align*}
s_i(p,m) &= \frac{p_i x(p,m)}{m} = \alpha_i + \sum_j \frac{1}{2}\gamma_{i,j} \ln(p_j) + \beta_i \left(\ln m - \alpha_0 - \sum_j \alpha_j \ln(p_j) - \frac{1}{2}\sum_{i,j} \gamma_{i,j} \ln(p_i) \ln(p_j)\right),\\
&= \alpha_i + \sum_j \frac{1}{2}\gamma_{i,j} \ln(p_j) + \beta_i \ln(m/P).
\end{align*}
$$
where:
$$
P = \alpha_0 + \sum_j \alpha_j \ln(p_j) + \sum_{i,j} \frac{1}{2}\gamma_{i,j} \ln(p_i)\ln(p_j).
$$
This is the Almost Ideal Demand System. Conditional on $P$, shares are linear in the log of income and log of prices. The symmetry and homogeneity restrictions immediately translate to conditions on the coefficients of this system.


This simply tells us that the mathematical object $f(x) = 1/x$ is not friendly to all the economic concepts we want to associate/obtain from its use as a demand function.
So, it is not the right tool to represent a demand function, we should use something else.

Mathematically, this relates to the harmonic series which is divergent (think of the integral as a discrete sum).


Marx: Relative and equivalent-form. Difference?

It has nothing to do with new products in the market, nor with confusion of use-value and exchange-value. To make it a bit simpler:

1) Commodities in capitalism have a two-fold character. The use-value (they are useful products for the buyer to consume) and the exchange-value (they can be exchanged to another products in a given way).

2) The exchange-value of the product depends on the embodied labor of the product, which is named as Value by Marx. So in a sense, exchange-value = Value.

3) Products can only be compared in relation to one another. You cannot understand anything by saying ex. 2 apples = 2 apples.

Thus, we reach to the Elementary Form: 20 yards of linen = 1 coat
In this equation, linen plays the active role or Relative form and coat plays the passive role, or Equivalent form. In other terms, you define the Value of linen comparing it to the Value of coat.

Now, here is the core of the analysis.  Marx claims that since there is a comparison (20 linen=1 coat), there must be a common element to compare. That common element is not the use-value (linen is not the same as coat). It is the exchange-value, or Value, or the human labor embodied in the two products. In other words, this equation states that labor needed to develop 20 yards of linen is equal to labor needed to develop 1 coat (given constant technological advancement, resources, etc.) Use-value, on the other hand, just acts as a carrier of Value.

The importance of this realization, is to understand the essence of exchange in capitalistic world and connect it to the money form (20 yards of linen = 1 coat = 100 Euros), and understand socio-economically how money was created and what does it really mean.

Karl Marx. Capital Volume One, Part I: Commodities and Money, Chapter One: Commodities


Interpeting equivalent annual cash flows

You're missing the distinction between costs and benefits. In your first probem you have positive NPVs and therefore the equivalent annual value is the equivalent annual cashflows that comes in.

In your second problem you have costs only. Converting that that to an EAV means converting it to an annual cash flow that goes out. Obviously you want the former to be as large as possible and the latter to be as small as possible.


how to find Cournot equilibrium for 2 firms having different MC?

Let the profit functions of the two firms be
$$
\begin{aligned}
\pi_1(q_1,q_2)&=p(q_1,q_2)q_1-C_1(q_1)\\
\pi_2(q_1,q_2)&=p(q_1,q_2)q_2-C_2(q_2)
\end{aligned}
$$
where $p(q_1,q_2)$ is the inverse demand depending on the total output of the two firms $q_1+q_2$, and $C_i$ is the total cost function of firm $i\in\{1,2\}$. Thus $C_i'$ is the marginal cost (MC) of firm $i$, and we can impose the condition that $C_1'\ne C_2'$ on the cost functions so that the firms have different MCs.

Now we can solve for the Nash equilibrium (NE). Profit maximization would yield the following first-order conditions for the two firms:
$$
\begin{aligned}
p'_1(q_1^*,q_2)q_1^*+p(q_1^*,q_2)-C_1'(q_1^*)&=0\qquad\qquad(1)\\
p'_2(q_1,q_2^*)q_2^*+p(q_1,q_2^*)-C_2'(q_2^*)&=0\qquad\qquad(2)
\end{aligned}
$$
These are the usual MR=MC condition (for each firm). Solving for $q_1^*(q_2)$ in $(1)$, we get the best response of Firm 1 as a function of Firm 2's output $q_2$. Likewise, from $(2)$ we get Firm 2's best response to Firm 1's output, $q_2^*(q_1)$.

Recall the a NE in a two-player game is a pair of mutual best responses. Therefore the NE of this two-firm Cournot game with different MCs is the pair $(q_1^*(q_2^*),q_2^*(q_1^*))$, where each firm is best responding to the other firm's best response (to the former firm). Practically, this means to plug $q_2^*$ into $q_1^*(q_2)$ and solve for $q_1^*$, and then plug $q_1^*$ back into $q_2^*(q_1)$ and solve for $q_2^*$.


How likely is a total economic collapse due to the potential coronavirus impact?

As a disclaimer, this is not an answer, rather a long-form explanation of the difficulties of answering.

The first leg is uncertainty about the danger of the epidemic. Economists are not medical experts. For example, one could imagine that there is an amazing breakthrough where a common medicine saves patients. The economic effects of that scenario are very different than the scenarios presented by governments. More realistically, we do not know how strong activity restrictions have to be to control the epidemic.

The next leg of uncertainty is the government response. Governments have the capacity to bail out entities (firms, households) that get in trouble. We do not know what they will do.

Even if we knew all that, there is uncertainty about the best model that takes all this in. It is very unusual for governments to restrict economic activity in such a broad-based manner, and so there are no standard models. (There is a flood of research.)

Finally, “conventional” economics does not really spend much time looking into societal collapse scenarios. There are areas of academia that do this, and there is presumably studies in economic history. Experts in those subject areas are not normally visiting this website.


Why do Higher Interest Rates attract Foreign Investors?

The answer lies in this part of your posted quote:

"The higher interest rates that can be earned tend to attract foreign investment, increasing the demand for and value of the home country's currency."

Foreign investors would be attracted to the higher interest rate if they are able to receive a better return on their investment than they would get from their local market.

It sounds like your confusion might arise from the fact that a higher interest rate would mean the borrower has to pay out more in interest. Your quote refers to investors, the party receiving the higher interest payments.

A common way in which foreign investors would use higher interest rates to their advantage is by borrowing money locally at a lower rate and investing it in foreign markets at a higher rate. Profits would be calculated based on the difference in interest on the money (in a simplified situation). A more realistic scenario would also  include exchange rate fluctuations and other variables.


Deriving aggregate output from labor demand and supply

Use $W_{i}=V \cdot \frac{P_{i}^{\beta}}{Z_{i}}=VQ_i$, then $$L_{i}=\left(\frac{\alpha^{1-\eta} \eta^{\eta}}{R^{\eta}} \cdot \frac{A_{i}}{(V_{i} Q_{i})^{1-\eta}}\right)^{\frac{1}{1-\alpha-\eta}} \cdot T_{i}$$ and $$\sum L_i = {\left({\frac{\alpha}{V}}\right)}^{\frac{1-\eta}{1-\alpha-\eta}} {\left({\frac{\eta}{R}}\right)}^{\frac{\eta}{1-\alpha-\eta}} \sum\left(\frac{A_i}{Q_i^{1-\eta}}\right)^{\frac{1}{1-\alpha-\eta}}T_i = 1$$ $$\frac{V}{\alpha} = {\left({\frac{\eta}{R}}\right)}^{\frac{\eta}{1-\eta}} \left(\sum\left(\frac{A_i}{Q_i^{1-\eta}}\right)^{\frac{1}{1-\alpha-\eta}}T_i\right)^{\frac{1-\alpha-\eta}{1-\eta}}  $$.
Use the FOC on labor, $W_i=\alpha \frac{Y_i}{L_i}$, then $$\sum Y_i = \frac{V}{\alpha}\sum L_iQ_i = \frac{V}{\alpha} \bar{Q}$$, and replace $\frac{V}{\alpha}$ with above equation then you get equation (7).


Behavioral economics and utility function

I don't know the paper, but I can make a general point.

In behavioral economics, agents do not arrive at their optimal choice by maximizing a standard expected utility function. They may maximize some function with other characteristics or, alternatively, they use heuristics to make choices. That is, they may end up with a choice that is not optimal in terms of "objective" expected utility. This paper uses this objective expected utility as welfare criterion - with the true probabilities and true outcomes, not the perceived probability weights or perceived outcomes that led to the choice of the agent. The argument behind this approach is that although the agent makes choices according to a perceived world, he experiences utility in the true world. Therefore, instead of using the expected utility perceived when making the choice as a welfare measure, it makes sense to use the "objective" expectation of the objectively experienced utility.

I interpret the footnote without having read the paper: The agent uses subjective probability weights to make choices and incurs 'thinking costs'. The welfare measure ignores these thinking costs.


Bertrand Duopoly Equilibrium for Discrete Prices

The problem formulation admits the following Normal Form representation. We can reject any strategy involving price greater than 2, as demand falls to zero and such strategies are strictly dominated by those for which prices are either 1 or 2.

    0         1         2

0 [0,0]     [0,0]     [0,0]

1 [0,0]   [0.5,0.5]   [1,0]

2 [0,0]     [0,1]     [1,1]

With this setup you can see that there are three Nash equilibria: (0,0), (1,1) and (2,2). At (0,0), firms are indifferent to the outcome of a higher price, and will not deviate. At (1,1), Firm 1 will not deviate to 2, nor will Firm 2 deviate to 2, as this cedes the market to the opponent. At (2,2) firms are indifferent to reducing price to 1.

(0,1), (0,2), (1,0), and (2,0) are strictly dominated by deviating to a price-matching strategy, splitting the market between firms.

Similarly, (1,2) and (2,1) fail to be Nash equilibria because whichever firm opens with a price of 2 can capture half the market by deviating to a price of 1.


How do you utilize event studies in policy research?

It sounds like you are really looking at differences in differences, not event studies.
Diff in diff compares outcomes before/after a policy intervention (as does an event study) (the first diff) and between treated and untreated units (the second diff).
The underlying assumption is that, in the absence of the policy, treated and untreated units (here: states) would have experienced the same changes over time.
A good reference is Ch 5 of Mastering Metrics (by Angrist and Pischke), but the previously suggested "Harmless Econometrics" by the same authors is also very good.


Arrow's Impossibility Theorem merely states that (with a ranked voting method) we cannot simultaneously satisfy IIA, Pareto Efficiency, Non-dictatorship and unrestricted domain.  You can satisfy any subset of those conditions, though.  So Non-dictatorship with IIA is fine — so long as you're willing to drop Pareto Efficiency.  Similarly, Non-dictatorship and Pareto Efficiency can be satisfied — if you drop IIA.

You can also use a system that is not ranked, such as Cardinal Voting although there is some debate as to  how to adapt the IIA criterion in such cases, and some such systems nonetheless fail different criteria not addressed by Arrow, such as the Majority Criterion.

(Unrestricted domain has more to do with how to decide which candidates are eligible to appear on the ballot, and which voters are allowed to vote in general, and so isn't usually considered as one of the criteria that can be relaxed... it's more a formality in Arrow's theorem, than the other criteria.)


Cholesky Decomposition in R package "vars"

According to page 6 of the documentation for vars, the Cholesky decomposition matrix is lower triangular:

  The long-run impact matrix is the lower-triangular Choleski decomposition of the above matrix and the contemporaneous impact
  matrix is equal to:

  $$(I_K − A_1 − \dots − A_p)Q$$

  where $Q$ assigns the lower-triangular Cholesky decomposition.

In the theoretical literature, Cholesky factorization is traditionally defined as lower-triangular; that is, as $A = L L^{T}$. There is a function is base R, chol(), that returns the upper-triangular Cholesky decomposition of a matrix $X = R^{T}R$.


Suppose Alex, Bryan, and Chris run in a race. Alex is the fastest and Chris is the slowest. So far I have only given you ordinal information about where they finished.

I guess you'd be okay with me taking that ordinal information and saying "Alex finished first, Bryan finished second, and Chris finished third". But then I just defined a function that assigns a number to each of the ordinal places:
$$f(x)=\begin{cases}1\text{ if }x=\text{Alex}\\2\text{ if }x=\text{Bryan}\\3\text{ if }x=\text{Chris}.\end{cases}$$
Note that the numbers 1, 2, 3 don't tell us how much faster Alex was than Bryan, only that he was somewhat faster.

Utility functions are just like this: they give bigger numbers to the best alternatives. But the size of those numbers don't tell us how much better the best alternative is, only that it is better. That is the sense in which the function is ordinal.


Appropriate estimator for FDI "gravity" model

An alternative is to use stocks rather than flows, but this does not
  seem to fit very well with the theory of gravity equations which seem
  to me to inherently deal with flow concepts.

Using stocks rather than flows and avoiding the disinvestment issue is a good alternative. Gravity equations have been shown to do a good job fitting stocks of FDI (see the Handbook of International Economics, Gravity Equations: Workhorse, Toolkit, and Cookbook, page 149).

This paper considers a model in which FDI takes the form of acquisitions. Using the discrete choice framework in a way that resembles Eaton and Kortum (2002), the authors develop a gravity equation for FDI which fits the data well. This paper extends the model to greenfield investment by imagining that instead of bidding for assets, each corporation selects the best "investment project" across all host countries.