Was factory beef farming (in the USA) the result of
  government-subsidized corn?

Answer A) The answer is negative, according to

MacDonald, J. M., & McBride, W. D. (2009). The transformation of US livestock agriculture scale, efficiency, and risks. Economic Information Bulletin, (43). USDA.

Although not a historical-statistical account, they write (p. 21, bold my emphasis)

  "Two recent reports argue that Federal commodity programs reduce prices for a particular input, purchased feeds, and that those
  reductions drive structural change (Pew Commission on Industrial Farm
  Animal Production, 2008; Union of Concerned Scientists, 2008).
  Specifically, they assert, drawing on Starmer and Wise (2007), that
  policy encourages increased production of feedgrains, that the
  increased production substantially reduces feedgrain prices, and that
  the buyers of purchased feed benefit from lower feed prices. While
  producers of homegrown feed have been the direct recipients of
  commodity payments, the reports argue that the payments have not been
  large enough to offset lower commodity prices, so that commodity
  programs have largely benefited large-scale animal feeding operations
  at the expense of smaller diversified crop and livestock farms.
  However, the size-related differences in production costs that are
  summarized in figures 6-8 cannot be attributed to the effects of
  commodity programs. In developing cost-of-production estimates, ERS
  prices homegrown feed at its market value—that is, the price that
  homegrown feed would have drawn as feed in regional feed markets—and
  does not attempt to estimate the actual cost of producing the feed. To
  the extent that larger operations realize lower feed costs in ERS
  estimates, it is because they use less feed per cwt of production and
  not because purchased feed costs them less.
  But even if commodity policies reduce purchased feed prices, there’s no reason why that should alter the size structure of livestock farms.
  No technological barrier prevents small farms from replacing homegrown
  feed with purchased feed: if purchased feed prices are lower than the
  costs of growing feed, then small livestock operations can simply buy
  feed and realize the same savings as large farms. A difference
  between the price of purchased feed and the cost of homegrown feed
  does not explain why feeders build large rather than small
  operations."

-__________________________________________________________________

Answer B) The answer is positive according to

Starmer, E., & Wise, T. A. (2007). Living high on the hog: factory farms, federal policy, and the structural transformation of swine production. Tufts University.

They report the following estimations, among others: The feed costs for Hog represent ~$60\%$ of operating costs of CAFO's. That in the period $1997-2005$ feed subsidies meant that feed was sold at a price $26\%$ below production cost. Hence, the savings in operating costs for CAFO's was around $60 \% \times 26\% \approx 15\%$ (of Costs, not of Revenues). For  the top four CAFOs in terms of market share, the authors estimated that this amounted to total cost-savings of $\approx 4.4$ billion USD for the 1997-2005 period.

This paper appears to be an aggregator of previous more detailed studies, so lots of references.


Reduced Form of an econometric model, identification problem and test

The question doesn't make much sense to me as stated. If the problem says that exercise is endogenous (correlated with error term), you can't assume the opposite in the solution. Plus, one usually speaks about reduced vs. structural form in the context of IV estimation. If exercise is endogenous, you need an instrument for it (variable that predicts exercise, but doesn't affect health otherwise) to obtain causal effects. For example, if some people in your sample randomly won gym membership coupons, that could be a valid instrument.

Identification assumptions would then be

coupon really does predict exercise
coupon is orthogonal to $u$

What is called structural form would be two equations, one your original model, the other regression of exercise on coupon and other explanatory variables from the original model (the first stage). Reduced form would be when you substitute first stage into the main equation, so you regress health on age, weight, ... , work and coupon (but not exercise, as that has been substituted out). Reduced form is sometimes used for explaining properties of IV estimation, but AFAIK it's not much used in practice.


In Lemma 1 they say that the support $\left[\underline{p}_t, \bar{p}_t \right]$ is such that $ c < \underline{p}_t $.
So even though the support is connected, it does not extend to $c$, hence the $p \to c$ problem never arises.


Finding the money supply graph for the UK

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

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

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

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

UPDATE
Here is the answer I received from Bank of England

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

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

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

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


What analysis has been done to predict the next big financial crash?

A useful principle to have in mind here is the so called 'volatility paradox´ oft-cited by Marcus Brunnermier (Princeton prof):

The idea is as follows: a) financial institutions are in the business of taking reasonable risks; b) when the financial system is deemed very stable, then it makes sense for financial institutions to engage in risky activities, because, if the system is stable those risky bets are manageable, reasonable bet; c) when all financial institutions jointly engage in risky activities, the system becomes unstable. In other words, safety engenders individual risk-taking, which creates aggregate risk. This works the other way too: riskiness engenders risk averse attitudes, which lead to a stable system.

If you believe this story then you get some insight into why its difficult to predict crises: they happen when risks build up during periods of safety and stability...


What does national debt really mean? and why is it bad to pay it off?

You borrow \$100 from John, at 3% interest per annum. You are indebted to John, to the tune of \$100.
The Wakanda government borrows \$100M from the public, at 3% interest per annum. The Wakanda government is indebted to the public, to the tune of \$100M.

Conceptually there is no difference between (1) and (2).

But in practice, (1) is more likely to be informal. Maybe the agreement between you and John is not even written down and is simply verbal. Or maybe it is casually scrawled on a scrap of paper.

Whereas (2) is more likely to be formal. Indeed, officially, governments don't call it "borrowing". Instead, they say that they "issue bonds". Anyone in the "public" (including Joe on the street or the Chinese government) is free to buy these bonds. If say the bonds are one-year bonds that pay 3% interest per annum, then the government promises to repay the bond-holders $103M in a year's time.

  why is it bad to pay it off and have zero national debt?

I don't think any economist believes it is "bad to pay it off and have zero national debt". Instead, as with most questions in economics, the answer is "It depends." You can have too much debt or too little debt. Having zero national debt is neither always ideal nor always bad.

But in practice (i.e. in the real world), many governments tend to err on the side of having too much debt.


Is special case of utility function in "Intertemporal Substitution in Macroeconomics" also a special case of a Ramsey growth model utility function?

Let's call your first utility function [1], and the third utility function [3].

If [3] is a special case of [1], then for some [1] we can find parameter values such that the first derivatives of [1] and [3] will be equal for all values of $c$ and $l$.

So the following must hold:

$$c^{1-\sigma} = -(1+\beta)l^{\beta}/\omega_l(l)$$

$$c^{\sigma -\alpha} = (1-\sigma)(1+\omega(l))$$

Solving both for $c$, and the setting the equations equal to one another, you'll find a functional equation which we need to solve for $\omega$ given its restrictions.

$$ (-(1+\beta)l^{\beta}/\omega_l(l))^{1/(1-\sigma)} = ((1-\sigma)(1+\omega(l)))^{1/(\sigma-\alpha)}$$

Now check this equation to see if you can satisfy the sign restrictions on $\omega$'s derivatives, and you fill find that you cannot for any meaningful specification.


Finding the Pareto optimal set with preferences for 3 individuals

I am somewhat unsure where your exact difficulty lies (and hence what your question is). As you write the two Pareto impromevents are $(a,b),(a,c)$. Meaning $b$ and $c$ can be improved upon and are not efficient. Also $(a,d)$ is an improvement, notice that individual 2 is indifferent while others prefer $a$ to $d$. Alternative $a$ cannot be improved upon, and hence it is Pareto-efficient. So the Pareto-set would be $\left\{a\right\}$.


The paper is a couple of years old by now, but I believe the first people to look seriously at the issues you're thinking of are Akbarpour, et al.

It might be useful to look at that paper and the handful of others that cite it. The paper was originally called 'Dynamic Matching Market Design', so it may have more citations under that name.


Why did the pound sterling lose its reserve status to USD?

As written, this question has an extremely wide scope, and an answer would be the length of a book. I will attempt a narrow answer.
People use “reserve currency” at present in a very loose manner. In the context of this question, it had a very specific meaning. Currencies were pegged, ultimately to gold. Currencies that had inadequate gold reserves used “strong” gold-backed currencies as extra backing for their pegs. Hence, the strong currencies were used as reserves.
Britain lost almost all of its gold reserves in World War II, in order to finance the war effort. (The process started earlier, with large losses in World War I.) Without adequate gold reserves of its own, the pound was not a credible backing for another country’s gold peg. Hence, it lost reserve status.
The Bretton Woods agreement enshrined the financial reality of the post-war system: the U.S. has accumulated almost all of the official gold reserves as a result of supplying the warring countries. The system was set so that the US dollar was gold-backed, and other currencies were backed by the USD.
Anything beyond this outline turns into a financial history of 1910-1950, for which there are many choices.


In which market structure does the consumer has almost zero knowledge about the good?

The market for lemons. Paper is here.

The example most commonly given is used cars. The result is market failure.


Sustainability of Collusion in a finite Bertrand competition with N > 2 firms

You're already on the right track. Just need to reorganize your argument a little bit.
Hint: Prove by contradiction. Suppose that collusion is sustainable in an equilibrium. Show that the collusive strategy (i.e. choosing the monopoly price in every stage) is not optimal given that everybody else is doing the same. Then conclude that collusion cannot be an equilibrium, hence a contradiction.


I agree that cryptocurrency can be used to launder illicit gains, but I think this question needs to consider a broader societal context. It is estimated that there is at least $20 trillion hidden within tax havens around the world. Much of this is laundered by the very institutions (banks) now claiming that cryptocurrency is a potential space for criminal activity.

I think a good example of this is China's recent clampdown on Bitcoin within the context of capital outflows. Although it was speculated that Bitcoin trading within China could be used to avoid capital controls, the comparative amount of Bitcoin was a fraction of the overall amount. Bitcoins entire market cap is about USD18 billion, whereas Chinas capital outflows were USD220 billion in 2016.

I suppose my answer is that cryptocurrency will only ever be a cog within a broader illicit machine. Reducing criminal activities like laundering will require systemic and governance changes within our societies, starting tax avoidance and effective regulation on those who are most likely to carry it out - the super rich.


get statistics about the research intensity of an industry

Perhaps someone with more expertise on the matter can answer your question more directly, but in the mean time:

In the literature on innovation, it is fairly common to use patent data to measure the intensity of innovation activity.

Here are Aghion et al (2005) on their approach:

  "There is a large literature on measuring innovation intensity, with the most commonly used measures being R&D expenditure and patenting activity. We use the average number of patents taken out by firms in an industry, and to reflect the
  heterogeneous value of patents, we weight each patent by the number of times it has been cited by another patent. These data are generated by matching the NBER patents database to accounting data on firms listed on the London Stock Exchange
  (from Datastream). Our sample includes all firms with names beginning "A" to "L" plus all large R&D firms. After removing firms involved in large mergers or acquisitions and those with missing data, we have an unbalanced panel of 311 firms spanning seventeen two-digit SIC codes over the period 1973-1994. We also have information on citations to and from each patent, which enables us to construct a count of citation-weighted patents. We take the average value of citation-weighted patents of firms within each industry (SIC code) in each year. We do not observe a sufficient number of firms in all industries in all years, so our resulting industry level panel is also unbalanced with 354 industry-year observations."

The latest version of the patent database to which they refer can be viewed here. Aghion et al.'s data is available here.


Why is infinite recursion on the common knowledge assumption necessary?

Ariel Rubinstein has a fabulous paper in which he illustrates that Common Knowledge and Almost Common Knowledge are very different!

http://www.cs.cornell.edu/courses/cs6764/2018fa/The_Electronic_Mail_Game.pdf


You can find a didactic exposition of Mirrlees (1971) -- including derivations of optimal tax schedule formulas for different social welfare and utility function -- in section 4.2 of "The Economics of Taxation" by Bernard Salanié (2003, First Edition).

I bet it is still somewhere in the second edition too, but I do not have a copy of it so I don't know in which section that would be.


How do international organizations collect statistics regarding a country?

I don't know how CIA gets its information from other countries on their economic performance, but I can tell you that IMF and WB have residence representative offices in member countries. Central Banks, various ministries and other government agencies supply periodic performance records to these offices.


Worth of collectables - predatory pricing

The way you describe this object (I will call it a coin) it exists in some finite quantity with no reasonable substitute and no new coins can be made. Suppose before you go on the proposed endeavor the world starts with a Pareto efficient allocation of these coins. Additionally, assume the act of destroying a coin has no impact on how much other owning these coins value their collection. This implies a few things:

Each individual who has a coin values the last coin in their collection more than anyone who does not have a coin, otherwise it would be Pareto-improving for someone with a coin to sell it to someone with a higher valuation.
The market-clearing price of coins is the lowest valuation of anyone who owns a coin.
Every individual values their first or an additional coin at something equal to or less than the market clearing price, otherwise they would purchase the coin.
You will never pay less than the market-clearing price for a coin, otherwise someone with a higher valuation would have bought it already.

These facts imply that you can never make money in your endeavor. When you buy a coin you will pay a price equal to the lowest valuation of anyone that already has a coin. But after destroying the coin you haven’t changed the price because the marginal buyer is still the same and has the same valuation.  You can repeat the process and pay a higher price for the next coin, but you still haven’t changed the valuation of the marginal buyer, so the market-clearing price will not change.
This conclusion relies on two assumptions, which may not be true in the real world. If the initial allocation isn’t Pareto efficient, you could make more money simply acting as a broker, buying coins from people with low valuations and selling them to people with higher valuations (since the coins are clearly less valuable after you destroy them). This could even pay for your travel.

It’s more complicated if valuations are impacted by your act of destroying the coins.  In this case, you could make money, but it would be difficult to determine ahead of time.  As a buyer/destroyer of coins you can observe the total number of coins, the number of people who own them, and the market-clearing price of coins. That price tells you nothing of how the preferences of people with valuations higher than the market-clearing price change with each coin you destroy.  For example, if you destroy a coin and the market-clearing price goes up by $100 it may be that everyone increased their valuations or only the marginal buyers increased their valuation. Predicting how prices would change after each coin destruction given what you observe would require assuming the highest-valuation buyers behave similarly to the marginal buyers, which can be a stretch.


What is an example of a utility function where one good is inferior?

A good cannot be inferior over the entire income range.

The paper A Convenient Utility Function with Giffen Behaviour shows that for a person with utility of the form:

$$U(x,y) = \alpha_1 \ln(x-\gamma_x)- \alpha_2 \ln(\gamma_y - y) $$

X is inferior if $\gamma_x$ and $\gamma_y$ are positive, $0<\alpha_1<\alpha_2$, and in the domain $x>\gamma_x$ and $0\leq y<\gamma_y$.

Update:
$$U(x,v) = x + \ln(v)$$
If the budget is $w$, $v^* = \min(P_x / P_V, w)$ so for $w>P_x / P_V$ $v$ is inferior sticky good. Realized this is actually a zero income elasticity not a negative one so it is not inferior.

I found another funky functional form for a utility function where one good is inferior but it too has increasing marginal utility of the other good: An Inferior Good and a Novel Indifference Map

$$U = A_1 \ln(x) + y^2 /2 $$ That function gives a crazy indifference map.

The classic example to me of inferior goods are things like cheap food, where delicious food which is much more expensive crowds it out because there is an additional constraint (stomach capacity) which eventually binds. It should be readily possible to make an example where inferiority is a consequence of this second constraint rather than the utility function.

Update with another example:

The paper The Case of a “Giffen Good” (Spiegel (2014)) shows that for a person with utility of the form:
$$ U = \begin{Bmatrix} \alpha X - \beta X^2 / 2 + \lambda Y + \delta Y^2 / 2 & for & 0\leq X\leq \alpha/\beta \\ \alpha^2 /2 \beta + \lambda Y + \delta Y^2 /2 & for& X > \alpha/\beta\end{Bmatrix} $$
where $\alpha, \beta, \lambda,$, and $\delta$ are constant and positive values.

But as in the above functions, this utility function has increasing MU in one good (Y). This is apparently common in Giffen settings:

  In the case of an additive utility function where the marginal
  utilities of all goods are diminishing with the consumption of the
  goods, that is, the marginal utility of income is diminishing, all
  goods are normal and net-substitutes for each other. However, if for
  some good (in our case, good Y) the marginal utility is positive and
  increasing and for the other good(s) the marginal utility(ies) is
  (are) diminishing (in our case, good X), then the marginal utility of
  income is increasing. The good that exhibits increasing marginal
  utility is a luxury good, whereas the good that exhibits diminishing
  marginal utility is an inferior good. These characteristics were
  proved by Liebhafsky (1969) and Silberberg (1972) and wen: used to
  develop the utility function above that illustrates the case of a
  Giffen good.


Effects of finding infinite gold reserve

The gold  will be free. (price 0 not anymore rare).

banks and others institutions will found others stuff for having reserve. ( platinum, silver, grains, maybe one day water ).


How do you stop Hyperinflation in a Nation?

You can generally stop hyperinflation in very similar way as any inflation. It is generally accepted by economic profession that price level $P$ (change in which is by definition inflation) in an economy is determined by money market equilibrium. In turn a simplified model of money market equilibrium can be given by the following equation of exchange (see Mankiw; Principles of Macroeconomics pp 87):
$$P= MV/Y $$
where $M$ is the money supply, $V$ velocity of money (i.e. how much one unit of currency is used on average) and $Y$ real output (also note this is a simplified version - more complex ones depend also on expectations of these quantities, but given that you are non-economist I think it is better to stick to this simple one - in case you would be really interested in the more complex versions Romer Advanced Macroeconomics is a leading macro handbook).
Next, hyperinflation does not have exact definition, but typically people consider inflation higher than $50\%-100\%$ per month a hyperinflation (e.g. see Sachs, 1987). Consequently, to stop hyperinflation you just need to make sure that the (expected) change in the right hand side $\Delta(MV/Y)$ is less than $50\%$ per month.
Government issuing its own fiat currency has complete control over $M$ so one option would to decrease the rate at which new money is created. Government can also to a certain degree affect velocity of money $V$ since it does depend on nominal interest rate (increasing nominal interest rates will lower $V$). In addition government has also some control over real output to the extend it can make some structural reforms that help economy grow or various policies that can expand $Y$. In addition as mentioned in the second paragraph this model is simplification and in real life expectations matter as well. This is why often countries that try to combat hyperinflation resort to monetary reform where they replace their currency with new one (this is meant to signal to the public government is serious about the issue thereby lowering their inflation expectation). Naturally, some combination of the above is always an option. Of course, this is just very broad and brief explanation. One could write whole tome on this issue, but exhaustive review would be too long for SE (for more in depth treatment you can have look at above mentioned Romer and Mankiw textbooks as well as  Sachs (1987, Kiguel (1989), Reinhart & Savastano (2003)   - in addition Cagan (1989) has whole book on hyperinflation - and sources cited therein).
Regarding whether there are any benefits of hyperinflation, there are generally only very few. One big benefit is that it can help country to inflate away public debt. Also, hyperinflation actually does not help economy to grow.  Regular inflation can help economy grow, but when it comes to hyperinflation, it is generally agreed that its effect on output is on net negative as any benefit that it has in terms of boosting export through exchange rate, or through removing nominal friction are smaller than the negative effects on output which are caused by uncertainty it creates, and due to other costs of inflation, as well as due to the fact that hyperinflation often leads to people abandoning the currency altogether, in worst cases resorting to barter which is very inefficient form of exchange (see the sources recommended in previous paragraph for discussion on this).


Were there any contemporary economists (or other observers) who considered the 1920s a "New Era" of greater growth and stability?

There are several other economists/observers that share view that this was time of growth and relative stability. For example, according to  Smiley writing for the Economic History Association

1920s are a period of vigorous, vital economic growth.

Smiley is drawing upon quite wide list of research on economic history that generally shares that view. I mean in this case you could say that data speak for themselves. As you can see from the figure produced by Smiley, shown below, based on Romer (1988) estimates of the real GNP, we can see that real national output increased significantly during the 20s (excluding Great Depression of course).

More recently, Gordon (2016) in his The Rise and Fall of American Growth considers this period to be a period of significant economic growth and prosperity/stability.
This being said, I think Eichengreen was just referring to how historians call 20's rather than how economists call them. According to The American Yawp
A Massively Collaborative Open U.S. History Textbook by Sandford University Press [emphasis mine]:

The 1920s, of course, would be anything but “normal.” The decade so reshaped American life that it came to be called by many names: the New Era, the Jazz Age, the Age of the Flapper, the Prosperity Decade, and, most commonly, the Roaring Twenties.

The use of quotation marks seems to indicate that Eichengreen was just referring to the commonly used historic term for that period in American history. The way how I read the passage is that Eichengreen is saying this was 'the New Era' - a pun/reference to the historic term, and then just saying that it was a period of economic growth and stability - which is supported by data  and, as mentioned above, also argued by multiple other economists/observers.


Price Changes in Factor and Product Markets

Let's

distinguish exogeneous from endogenous factors,
distinguish partial versus general equilibrium analysis,
consider here the labor market of one product, i.e., a partial analysis,
consider that this labor market is perfectly competitive.

In your reasoning, the decrease in labor supply, which typically leads to an increase in the wage rate, would come from an exogenous shock. This shock causes an endogenous process that stabilizes the wage rate at a new partial equilibrium. This is a situation where the product of marginal labor income equals the wage rate.

In a competitive labor market, the wage has two properties:

It adjusts to make supply and demand for labor equal.
It equals the value of the marginal product of labor.

Suppose the labor supply decreases in our market. The supply of labor shifts to the left. At the initial wage, say $w_1$, the quantity of labor supplied falls behind the quantity of labor demanded. The wage increases from $w_1$ to $w_2$, and the firm reduces its demand for labor. As the firm lays off workers, the marginal product of a worker increases, so the value of marginal product increases. In the new (partial) equilibrium, the wage is equal to the value of the marginal product of labor and both are higher than before.

Regarding your comment and the stability of this new equilibrium

  I think where I am confused is how does the exogenous shock to the labor market cause the wage rate to stabilize? Because the firm will now get higher prices in the product market, shouldn't the demand for labor increase and wouldn't the wage rate go up again and the process would repeat again and keep on cycling upwards?

The above equilibrium is a stable partial equilibrium, but general equilibrium forces exist. As a result of the shock, the firm will initially get higher prices in the product market, but we can assume that if it is a competitive market, demand for its product will decrease. This will lower the prices of its products. Demand for labour will shift to the left ... wages will go down (not up). However, we will need a general equilibrium model to better understand these changes.


At market both price (in this case exchange rate) and quantity are endogenous ('dependent') variables.

You are confusing two completely distinct phenomena. Movement along demand curve and shifts in demand curve.

When exchange rate increases demand will be lower because we are moving along given demand curve - but increase in exchange rate does not cause shift in a demand curve.
In your second example when you talk about 'demand rising' you must be implicitly talking about shift in demand curve otherwise the rest of the paragraph would not make sense.
So  you are actually talking about two distinct concepts. When we talk about movements along demand curve even if you reverse the relationship and 'make the exchange rate dependent variable' there will still be inverse relationship between the two.  For example, if the relationship between demand for currency $(D)$ and exchange rate $(E)$ would be given by:
$$D= 100 - E$$
Then 'making exchange rate the dependent variable' would mean you have to solve $E$ which would give you:
$$E=100-D$$
As you can see the relationship is still inverse.
However, you are not doing the above when you talk about 'making exchange rate dependent variable' and then asking what would happen when demand raises - you are really talking about what happens when demand shifts right which in this case would happen if in the equation $D=\alpha-E$ the parameter $\alpha$ which used to be in the above example  $100$ would increase.
See the graph below from Mankiw's Principles of Economics that showcases the distinction between the shifts in demand and movement along demand. The graph is not about exchange rate market but you could as well just change y-axis label to exchange rate and x-axis label to quantity of currency and the point would be the same.


Income Elasticity and Convexity of Demand in Income

This may not be the answer you are looking for.

What you want may be technically feasible, but just barely. What I mean is that for any $\delta > 0$, you cannot have the income elasticity always be larger than $1 + \delta$. The simple reason is that it would mean that the amount spent on the good always increases faster than the income. Then eventually the amount spent on the good would just overtake the income, but this is, of course, a contradiction.

Income elasticity being exactly equal to one would be easy, e.g. demand derived from Cobb-Douglas.

You should also be able to construct a function $z(y)$ for which
$$
\epsilon(y) = 1 + \frac{1}{y},
$$
so that $\epsilon(y) > 1$, although
$$
\lim_{y\to \infty} \epsilon(y) = 1.
$$
You would need to solve a differential equation:
\begin{align*}
1 + \frac{1}{y} & = \frac{dz(y)}{dy}/\frac{z(y)}{y} \\
\\
z(y) \left(\frac{1}{y} + \frac{1}{y^2}\right) & = \frac{dz(y)}{dy}.
\end{align*}
Unfortunately I am not good at these, so I don't know what $z$ yields a solution, but I don't think this is what you had in mind anyway.


What's the term for an economic policy that has negligible effect for most, but a significant effect for a few?

Utility Trade-offs: Pareto Efficiency vs Equality

While I tend to agree with the notion of Tragedy of the Commons as per the comments, I will provide an additional economic framework to answer your question. Perhaps this is something you already know, but is nonetheless useful for answering your question as per the welfare-economics tag. As to the exact name of the policy, I'm not sure there is a name for the policy, but the underlying economic concept can be characterized as a utility trade-off and that's likely how a policy maker would approach this.

Consider the following social indifference curve, I just made it myself, don't scrutinize it too much:

Assuming the First Theorem of Welfare Economics held before the tariff, then that point would have been on the pareto efficient curve which is the gray line in my graph. This curve is also known as the utility possibilities curve, which shows the highest utility one party can have without reducing another party's utility.

In your case, since the aluminum producers were worse off, then this is a utility trade-off. And we recall that society may be willing to trade off one group's utility for another's utility, sometimes on the grounds of equality, but for other reasons too.


Dominated Strategies in an Infinitely vs Finitely Repeated Game

A usual refinement concept used to deal with weakly dominated strategies is the trembling hand perfect equilibrium. (I do not know others but this one works quite well.

The strategy in question is indeed weakly dominated by the following strategy

In the first period, player plays $B$.
In all other periods, player plays $A$.


Do I need to log transform these two kinds of variables?

You don't need to log transform these variables. Consider the following model

  $\log y = \beta x.$

Then, you can interpret $\beta$ as a semi-elasticity, i.e., the percentage change in $y$ when $x$ increases by 1 unit.

To see this you can transform the previous equation as:
$y = \exp(\beta x),$ in which case $dy/dx = \beta \exp(\beta x)$. Thus, the percentage change in $y$ when $x$ changes is $\frac{dy/dx}{y} = \frac{\beta\exp(\beta x)}{\exp(\beta x)} = \beta$.

Note also that

  $\Delta \log y = \beta \Delta x$ so $100.\Delta \log y = (100.\beta) \Delta x$

It follows that

  $\% \Delta y \approx (100.\beta) \Delta x$


Why don't we transact with securities rather than fiat money?

You've just reinvented bearer bonds. From that page:

  Bearer bonds are bonds that are owned by whoever is holding them, rather than having registered owners like most other securities. Like most other bonds, they have a stated maturity date and interest rate, but coupons representing interest payments are generally physically attached to the security and must be submitted to the company for payment.

It's been done. Like cryptocurrencies, the one class of transactions they turned out to be well-suited for was the class of illegal transactions; money laundering, payment for criminal services, prohibited capital movements, etc.

And having a fiat currency which is simply backed by a country's economy, has succeeded against all other kinds of currency as the most viable.


Help understanding this inclusive capitalism proposal

Robert Ashford has written about his proposal for a better, inclusive
  capitalism, but I don't understand the method. For example, take a
  look at this article. All is very obscure.

  Can't see how this would work.

The article is altogether too sketchy, idealistic, contradictory, very flawed, with too many unfilled gaps and motley elements. Much of it sounds like campaign promises, but a "broader distribution of capital acquisition and income" is not the panacea the article purports.

Initially, the authors admit that "production generally becomes increasingly more capital intensive" (p.70) and that "capital [keeps] doing an ever-increasing portion of the total work" (p.71). Despite that, the authors "predict" that a broadening of capital acquisition will lead to "fuller employment of labor" and that "the present demand [...] [of] employment would increase" (p.71) (even by virtue of the market's anticipation of that broadening!), that it "should be reflected in increased employment of labor" (p.73).

That twist is unrealistic and makes no sense. Simply put, it would be absurd for a profit-maximizing agent to invest on labor even though (or despite that) he knows that labor keeps being increasingly replaced by capital.

The principle of decreasing marginal utility strikes the authors' presumption that higher levels of income will consistently lead to greater demand of goods and services. The authors purport that, unlike the "Keynesian analysis", binary growth would continue materializing in the long run. But they do not (and cannot) articulate how their proposal would overcome the decreasing marginal utility in the long run, or why this would be any different in the aggregate. In short, the authors appear to ignore the notion of oversupply.

The article posits that its so-called "binary economic approach" can also enhance "the prospects for achieving environmental sustainability" (p.70). But then it admits that it is optimistic and doubtful that "binary growth would in the aggregate be greener growth[,] because [...] too many people may also prefer to spend their enhanced capital income on brown rather than greener products and services" (p.73). The article ends up requiring an "active role" from the government "with respect to [...] environmental sustainability issues" (p.73).

The article also goes from the initial premise that its proposal would require "no government command" (p.70) to proposing that the government embrace an "active role" in defining where to promote a "qualified" binary financing"  as well as to determine which producers shall be subjected to "more stringent [standards] than [the] mandatory standards imposed on all producers" (p.73). That governmental active role also includes the implementation of a "credit reinsurance entity modeled after the FHA home loan reinsurance program" (p.74).

The article is so poorly written and so incongruous that it significantly hurts the "prestige" and value of the authors' credentials (listed at the end: J.D. & Ph.D. holders). The only thing for which I would praise the article is the call for a reduction of tax rates. Putting aside the terrible "quality" of the article, it appears as if one team of authors wrote pages 70-71 and their opponents wrote pages 73-74.


Why do we need to "get the economy moving again"?

Trade creates value.

Previously, Kate preferred spending \$50 on food at Alice's restaurant (rather than cook her own food). And Alice preferred spending \$50 getting her hair cut at Kate's (rather than cut her own hair). That Kate must now cook her own food and Alice cut her own hair means that value has fallen (where value is broadly defined as the degree to which individuals' desires are satisfied).

(Note though that the fall in value in the above example is probably not very large. In most other examples and cases, the fall in value will be much larger—consider for example the children who no longer go to school and are now supposed to be learning at home.)

Money is not the source of value.

You write:

  It seems to me that all the money that hasn't been spent in restaurants and hairdressers during the COVID-19 pandemic hasn't gone anywhere, it's still sitting in the bank accounts of the people who would otherwise have spent it.

This expresses a common misconception among non-economists. Money is not the source of value.†

Value is the satisfaction of people's desires—and such value arises through production and consumption. If everyone is sitting at home and not producing anything, then no desires are satisfied, no value is produced, and all the money sitting in banks is worthless.

†The mercantilists of the 17th and 18th centuries likewise believed that gold and silver were the sources of value and that one's country should try to accumulate as much gold and silver as possible. Adam Smith pointed out that this was mistaken--value comes from production and consumption, not precious metals, pieces of paper, or numbers in a bank account.


[...] where does the newly created money go?

This money is used by central banks to "buy" commercial banks' bonds $\iff$ commercial banks are refinancing themselves $\iff$ this money goes to commercial banks.

  Does some entity get free money?

Yes, when central banks buy commercial banks' bonds for free, i.e. at a 0 interest rate.

  Why is that allowed?

The main reason is that as counter-cyclical agents, central banks have to change reference rates to stimulate their national economies, even if sometimes doing so implies to center them on 0, or even on a negative value. Note that in the case of negative interest rates, we are talking about entities that are paid to get money.

E.g. when refinancing themselves at a lower-than-previously cost, commercial banks are theoretically more prompt to be, say, flexible, and thus to stimulate demands (here and there) in the national economy by lending money (here and there). If supplies stay constant, markets will reach a new equilibrium via prices rising (inflation) due to these stimulated demands. Persistent inflation means new business opportunities, which in turn means growth if capacities to cut these opportunities exist domestically, or thanks to foreign direct investments.


How do I calculate the services sector as a percentage of GDP?

The table you are looking at is a GDP expenditure measure.  But this ignores the fact that services are involved in the costs of goods purchased, and that goods are involved in the cost of services purchased.  For example, if you buy a cake in a supermarket, the price reflects not just the food manufacturing industry, but also agriculture, transportation, retail trade and many more.

To consider the importance of different industries in the economy, better analysis comes from the GDP income measure considering the value added (essentially wages and profits) for each industry. So looking at table 5 of another BEA release on GDP by industry, and stripping out duplicated values for sub-industries, you get something like

                                                                     2018
                                                                $ billion
Gross domestic product                                           20,580.2

--Private industries--
Agriculture, forestry, fishing, and hunting                         166.5
Mining                                                              346.6
Utilities                                                           325.9
Construction                                                        839.1
Manufacturing                                                     2,321.2
Wholesale trade                                                   1,212.2
Retail trade                                                      1,126.9
Transportation and warehousing                                      658.1
Information                                                       1,067.7
Finance, insurance, real estate, rental, and leasing              4,301.6
Professional and business services                                2,579.4
Educational services, health care, and social assistance          1,792.5
Arts, entertainment, recreation, accommodation, and food services   860.6
Other services, except government                                   437.2
--Government--
Federal                                                             790.8
State and local                                                   1,753.8

If you add up the first five rows (Agriculture through to Manufacturing) you get \$3,993.3bn (19.4% of total GDP), while if you add up the other rows you get \$16,580.8bn (80.6% of total GDP)


Are there strictly dominated strategies?

I agree with Herr, the payoff matrix looks right. Also, there are no strictly dominated strategies because a strictly dominated strategy cannot be a best response for any possible belief. However, If any player believes that the other player is choosing 19, then every strategy (both pure and mixed) is a best response.


Do I have to bootstrap the dummy variables used for event dates in an event study?

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


Is there a way to adjust trade data between China and EU member countries for the Rotterdam effect?

This seems relevant: China and the Dutch Economy

I have only a limited experience in this, but here are some approaches you could look at.

Re-export data

UN Comtrade has re-exports data (products that the Netherlands import and export in the same form to other countries). Using this, you could make an assumption that x% of goods imported from China are re-exported to the EU. Assuming that this share of re-export actual source countries are the same for Dutch exports to the EU, you could get a rough estimate of how much goods from China goes to each EU country.

Using aggregate data, this could be a very far-off estimate. Belgium could import a lot from Netherlands, but all it imports are tulip flowers. To account for this, you can look at this on a HS-code level and you should get a reasonable estimate.

TiVA

Another approach is to look at the TiVA (trade in value-added) concept. This should tell you each country's value-added into an import of a country. OECD has this data, but I don't believe it covers all EU-28. Here's a link.


Smaller variance of estimators in matrix sense

Relative efficiency between two unbiased estimators $\widehat{\theta }_{A}$
and $\widehat{\theta }_{B}$ of an unknown parameter vector $\theta _{0}\in
\mathbb{R}^{K}$ is usually defined as follow (see for instance Ruud, 2001).
The estimator $\widehat{\theta }_{A}$ is said to be efficient relative to $
\widehat{\theta }_{B}$ if we have:
$$ \mathrm{V}\left[ \widehat{\theta }_{A}\right] \equiv \Omega _{A}<<\Omega
_{B}\equiv \mathrm{V}\left[ \widehat{\theta }_{B}\right] .$$

If $\Omega _{B}-\Omega _{A}$ is positive definite, then the diagonal terms
of $\Omega _{B}$ and $\Omega _{A}$ are necessarily such that $\sigma
_{Bii}^{2}>\sigma _{Aii}^{2}.$ Indeed if $v^{T}\left( \Omega _{B}-\Omega
_{A}\right) v>0$ for any $v\neq 0$ then for $v=e_{i}$ we catch the diagonal
terms out of $\Omega _{B}-\Omega _{A}$ and find that $\sigma
_{Bii}^{2}>\sigma _{Aii}^{2}.$ The converse, however, it is not true: the
mere conditions $\sigma _{Bii}^{2}>\sigma _{Aii}^{2}$ do not garantee that $
\Omega _{B}-\Omega _{A}$ be positive definite. Covariances matter.

The answer to your second question is: no it never happens that $\sigma _{Bii}^{2}<\sigma _{Aii}^{2}$ when $\Omega _{B}>>\Omega _{A}$.
The answer to your first question is: it is important to consider all covariance terms. If we want that the confidence ellipse (at any threshold) of $\widehat{\theta }_{A}$ be nested within the confidence ellipse of $\widehat{\theta }_{B}$ then we need $\Omega _{A}<<\Omega _{B}$ and not just $\sigma _{Aii}^{2}<\sigma _{Bii}^{2}$.

See Ruud, (2001, chapter 9) for a proof and detailed explanations. An example is provided here, illustrating that the confidence ellipses are nested when $\Omega _{B}-\Omega _{A}$ positive definite, and not nested if $\Omega _{B}-\Omega _{A}$ is not positive definite.

Example:
\begin{eqnarray*}
\widehat{\theta }_{A}\left( a\right) &\sim &\mathcal{N}\left( 0,\Omega
_{A}\left( a\right) \right) ,\qquad \widehat{\theta }_{B}\sim \mathcal{N}%
\left( 0,\Omega _{B}\right) \\
\Omega _{A}\left( a\right) &=&\left(
\begin{array}{cc}
4 & a \\
a & 9%
\end{array}%
\right) ,\qquad \Omega _{B}=\left(
\begin{array}{cc}
5 & 0 \\
0 & 10%
\end{array}%
\right).
\end{eqnarray*}

For $a=0$ the matrix $\Omega _{B}-\Omega _{A}\left( 0\right) $ is positive
definite and the $95\%$ threshold confidence ellipses are nested. The Figure
below (left panel) represents the iso-curves centered on $\theta _{0}=0$ and
whose equations are given by $v^{T}\left( \Omega _{A}\left( 0\right) \right)
^{-1}v=5.99$ and $x^{T}\Omega _{B}^{-1}x=5.99$ and illustrates that for $a=0$
the later is nested within the former. This is no longer true for $a=-5$
(right panel) in which case $\Omega _{B}-\Omega _{A}\left( -5\right) $ is no
longer positive definite. In this case the probability that $\widehat{\theta
}_{A}$ is further away than $\widehat{\theta }_{B}$ from the true value $%
\theta _{0}=0$ is positive, and $\widehat{\theta }_{A}$ is no longer
efficient (case with $a=-5$) relatively to $\widehat{\theta }_{B}$. Note
that the variances always satisfy $\sigma _{Bii}^{2}>\sigma _{Aii}^{2}$ but
this is not sufficient for the ellipses to be nested, the covariances must
also satisfy $\left( \sigma _{B11}^{2}-\sigma _{A11}^{2}\right) \left(
\sigma _{B22}^{2}-\sigma _{A22}^{2}\right) -\left( \sigma _{B12}-\sigma
_{A12}\right) ^{2}>0$, which is not the case in this example for $a=-5$.


What are the advantages using Otree to progam economic experiments?

One irreducible advantage of oTree over z-Tree is that you can easily reuse/imply the entire trajectories of results per entity (session/group/player) when doing repeated games. With z-Tree, it is rather complicated (not to say impossible?) to define dynamically temporal- or individual-based autocorrelated/interdependent/recursive games.
Another notable advantage is that you can also easily run (fully automated) functional tests with oTree, be them headless or not, with random numbers following the distribution(s) or mechanism(s) you like, up to the generation of the final results, which is at best complicated if I remember well with z-Tree. Moreover, keeping in mind that it is increasingly mandatory to provide (in appendix) all the screenshots of the steps of an experiment, you can easily do so with oTree, using libraries such as Selenium as soon as something changes.I remember me doing all that manually, with many zLeafs opened, which was cumbersome to say the least. Having 3 screens was not enough...
Finally, learning oTree will strongly initiate your familiarity with a notorious web-framework, Django (not the CMS), which means that learning oTree means learning it as well. Entirely class-based and PEP8-compliant, it also means that via oTree, you will learn coding in Python the "right way". Moreover, if you wanna tweak your frontend, you will also start learning things such as CSS, HTML and even JavaScript. oTree's dividends are numerous.All that with a concrete goal in mind, which (to me) is the best manner to acquire new skills.
That being explained, there is now a new version of z-Tree which I do not know, i.e. z-Tree unleashed which may be far better than its ancestor.


How is inheritance classified in Adam Smiths profit/wage/rent classification?

Profit, wage and rent are rewards for inputs used in production, they constitute a decomposition of Income (or the Income-decomposition of output produced)

Inheritance is a transfer, it does not involve production. So it is invalid to try to fit it into the mentioned classification.

Consider now any gift: say I work, earn a wage in the form of consumer goods, and give them to you. The consumer goods I got as a wage is a reward for my labor. Giving them to you is a transfer of already produced output.

In inheritance, this output has already taken the form of assets rather than consumer goods.


Decreasing and increasing returns to scale question

From your production function: $f(K,L)= 1.8(K^{0.6} + L^{0.6})^{2} \Rightarrow f(K,L) = 1.8(K^{1.2} + 2(KL)^{1.2} + L^{1.2})$

You are able to see that $f(\lambda K, \lambda L) \Rightarrow \lambda^{1.2} f(K,L)$. Say you double both your inputs. Increasing returns to scale tells you that your output will be more than doubled, which is precisely what $\lambda^{1.2}$ represents. The level of $K$ and $L$ does not affect this conclusion.  The point where the level of $K$ and $L$ play a "role" is that for levels such that $K + L >1$ each unit of input results in more than one unit of output (i.e.$f(K,L) > 1$) (again note that we are talking about levels here and not changes, which would regard the returns to scale). You can easily see this by plugging in different values into your formula. But, again, this is not to do with the returns to scale of the function so it doesn't pertain the question you are asked.


How do government purchases crowd out the private sector?

Just to add some more besides the answers above:
Short answer: by influencing interest rate.
Explanation:
In the post Keynesian literature what has come to be appreciated a lot is that government purchases, backed by increased government debt, increases interest rate. Why? Well, that's in itself an interesting explanation and perhaps should be a separate question (interesting, because contrary to what people think that "more govt borrowing means more govt bonds in the market, means lower bond prices and hence higher interest rates" is not at all a satisfactory explanation).
Private investments are assumed to be a function of interest rate. So some of the investment projects (particularly, those at the margin) that were earlier profitable may not be anymore due to increase in interest rate.
This is what is essentially, crowding out of private investment.
This does not require the assumption that Savings have to be fixed. Instead, in Keynesian framework, what happens is that $Y, S, C$ all goes up (because crowding out is not necessarily full, depending on the interest elasticity of money demand - slope of LM curve). It's just the share of $G$ increases and that of $I$ decreases.
Coming to your assessment that historically, at least during some periods, $I$ and $G$ have been found to be positively correlated. Indeed, investment is not just a function of interest rate. It is also a function of sentiments of future of the economy. When govt's start building infrastructure or take measures to improve supply constraints it boosts the confidence in the economy and encourages investment.
That doesn't make concept of crowding out incorrect. Because the concept is applicable when, ceteris paribus, $G$ increases.
Lastly, your own idea that "the capital and labor in the economy is limited, and if the government is spending more money to use capital goods and labor, then there are less capital goods to go around for the investment." is the somewhat closer to the classical view. This assumes that economy is at its full potential and there are no rigidities in the labour market. Without going into details, this view over the year has been seriously challenged. However, it should still be considered as an alternative explanation.


German economic growth in the 20th century

$1.$ My guess is that your videographic's data were simply from the World Bank's GDP (current US\$). This is nominal GDP in US\$ and does not take into account (a) changes in the exchange rate; or (b) inflation.

$\color{red}{\text{2.}}$ And it turns out that the chief culprit here is (a). To correct for (a), we look instead at GDP (current LCU), where Germany's LCU = Local Currency Unit is the euro.

$\color{blue}{\text{3.}}$ We can also further correct for (b) by looking at GDP (constant LCU).

P.S. I would not trust the website VisualCapitalist.com. Here for example, it simply took this videographic from a tweet, which in turn gives no sources.


Why does price drop as a result of increase in supply

You are describing a special (although very important) market where the short-run market supply function is vertical : farmers will attempt to sell whatever they have produced, irrespective of what that will do to the unit price. I am not saying this is not reasonable: many agricultural products are not really storable (or require special infrastructure in order to be stored), and there is certainly increased pressure to sell, compared to other products.

And indeed, in such a case, one can see a "good crop" bringing in lower total revenues, due to relatively inelastic demand. This is after all the core reason why government price subsidies in the agricultural sector (i.e. supplementing the farmer's income to compensate for low market price) is the rule rather than the exception, in virtually all market economies around the world, where very few other final products enjoy price subsidies (and only temporarily, while for the agricultural sector, it is essentially a permanent situation, of course not for all agricultural products).

Also, as another real-world remark,  it is not uncommon for farmers to withhold and even destroy part of their crop in order to safeguard prices (and thus end up with higher total revenues compared to what they would have get if they have sold all the crop at a lower price).

I am not sure I understand the last sentence/question in your post. In what sense farmers "can" or "cannot", "sell at the same or higher price"? And "same" compared to what?


Finding optimum market value of vintage car

Note that, from your SOC:
\begin{align}
P''(t) &= V''(t)e^{-rt} -  rV'(t)e^{-rt}-  rV'(t)e^{-rt} + r^2V(t)e^{-rt} + rme^{-rt}\\
&=e^{-rt}\Bigl[\underbrace{\color{red}{V''(t)-rV'(t)}}_{=D}-r\underbrace{\bigl[V'(t)-rV(t)-m\bigr]}_{=0\text{ from FOC}}\Bigr]
\end{align}


Prove the budget correspondence is upper hemi-continuous

I am assuming that the following facts do not require proofs for the purposes of this question.

Fact 1: Let $h_n$ be a sequence in $\mathbb{R}^K$ such that $\lim_{n\rightarrow \infty} h_n =h\in \mathbb{R}^K$. Then, for each $i\in \{1,2,\ldots,K\}$, we have $h^i_n\rightarrow h^i$.

Fact 2: Let $z_n$ and $q_n$ be sequence in $\mathbb{R}$ such that $\lim_{n\rightarrow \infty} z_n =z\in \mathbb{R}$ and $\lim_{n\rightarrow \infty} q_n =q\in \mathbb{R}$. Then,

i) $\lim_{n\rightarrow \infty}(z_n\pm q_n)=z\pm q$

ii) $\lim_{n\rightarrow \infty}(z_n \times q_n)=z\times q$

Fact 3: Let $z_n$ in $\mathbb{R}$ such that for each $n$, $z_n\leq a\in \mathbb{R}$ and $\lim_{n\rightarrow \infty} z_n =z$. Then, $z\leq a$.

Definition: A correspondence $G:X\rightrightarrows Y $ is upper hemicontinuous at $x\in X$ if for any open neighborhood of $V$ of $G(x)$. There exists a neighborhood $U$ of $x$ such that for all $x'$ in $U$, $G(x')\subseteq V$.

Unlike the previous remarks, the following fact requires proof.

Fact 4: Let $G:X\rightrightarrows Y $ be a correspondence. If for every $x_n\rightarrow x\in X$ and $y_n\in G(x_n)$ there exists a subsequence $y_{n_k}$ of $y_n$ with $y_{n_k}\rightarrow y$ and $y\in G(x)$, then $G$ is upper hemicontinuous. Moreover, if $G$ is compact valued, then the converse is also true.

Now, let's go back to your question. Let $(p_n,w_n)$ be a sequence in $\mathbb{R}^{L+1}_{++}$ such that $\lim_{n\rightarrow \infty} (p_n,w_n) =(p,w)\in \mathbb{R}^{L+1}_{++}$. Moreover, let $x_n\in B(p_n,w_n)$ for all $n$.

Let $w^\ast = \sup_n w_n$. Since $w_n\rightarrow w\in \mathbb{R}_{++}$, it must be true that $w^\ast \in \mathbb{R}_{++}$ because otherwise we could have created a subsequence $w_{n_k}$ of $w_n$ for which $w_{n_k}\rightarrow\infty$ which would contradict with $w_n\rightarrow w$. Similar arguments would imply that $p^{\ast} = \min_i( \inf_n p^i_n)>0$ since $p^n\rightarrow p\in\mathbb{R}_{++}^L$.

The previous observations imply that for all $n$ and for all $i$ we have $x_n^i\leq w^\ast/p^\ast$. Therefore, the sequence $x_n$ is bounded. By Bolzano-Weierstrass Theorem, $x_n$ has a convergent subsequence $x_{n_k}$ with $x_{n_k}\rightarrow x$. From here onwards, we will suppress the subscript $k$.

Since $x_n\in B(p_n,w_n)$, we have $\sum_i p_n^i\times x_n^i-w_n\leq 0$. Let $c_n = \sum_i p_n^i\times x_n^i$. By fact 1, we have $p_n^i\rightarrow p_i$ and $x_n^i\rightarrow x_i$ for each $i$, and by fact 2, we have $c_n=\sum_i p_n^i\times x_n^i\rightarrow p\cdot  x$.

By fact 2, $c_n-w_n\rightarrow (p\cdot  x-w)$. Fact 3 implies that $(p\cdot  x-w)\leq 0$, which in turn implies that $x\in B(p,w)$. This concludes the proof that the budget correspondence is upper hemicontinuous.

Original (incorrect) Answer

The following answer has been posted earlier. However, I realized that there is a mistake with this one. I am keeping this here since the original question had been wondering specifically about how Bolzano-Weierstrass theorem is relevant to the question and comparison of this incorrect answer and the correct version shows why we would need this theorem.

Definition: A correspondence $G:X\rightrightarrows Y $ is upper hemicontinuous at $x\in X$ if for all sequences $x_n\in X$ with $x_n\rightarrow x$, for all $y_n\in G(x_n)$ with $y_n \rightarrow y \in Y$, we have $y\in G(x)$.

Now, let's go back to your question. Let $(p_n,w_n)$ be a sequence in $\mathbb{R}^{L+1}$ such that $\lim_{n\rightarrow \infty} (p_n,w_n) =(p,w)$. Moreover, let $x_n\in B(p_n,w_n)$ for all $n$, with $x_n\rightarrow x$.

Since $x_n\in B(p_n,w_n)$, we have $\sum_i p_n^i\times x_n^i-w_n\leq 0$. Let $c_n = \sum_i p_n^i\times x_n^i$. By fact 1, we have $p_n^i\rightarrow p_i$ and $x_n^i\rightarrow x_i$ for each $i$, and by fact 2, we have $c_n=\sum_i p_n^i\times x_n^i\rightarrow p\cdot  x$.

By fact 2, $c_n-w_n\rightarrow (p\cdot  x-w)$. Fact 3 implies that $(p\cdot  x-w)\leq 0$, which in turn implies that $x\in B(p,w)$. This concludes the proof that the budget correspondence is upper hemicontinuous.


How does Google price the items on Google Play?

The simple answer is they estimate the demand curves for each product and, using their cost structure and market characteristics (competition structure, etc.) set price to maximize profits. This is standard for any firm, though.

How Google in particular and these big firms in general (Amazon, Microsoft, etc.) estimate demand curves is somewhat different than the usual economist might do it. For usual demand estimation, a researcher would have to make use of market idiosyncrasies to identify demand. For example, using supply shifters with 2SLS for basic demand estimation, BLP for discrete choice with heterogeneous products, etc. Identification is such a big issue for demand estimation because a researcher generally just observes equilibrium (p, q) combinations, not the actual demand curve. We are also often constrained purely by the amount of data available.

For a big firm like Google, however, they 1) have the ability to enact exogenous perturbation in price to see how sales change and 2) have access to tons and tons of data. Using 1) they are constantly running little experiments to see how consumer behavior changes. They can then use the results to actually trace out the demand curve. In these experiments, the firm could easily take into account things like movie popularity, genre, etc. With respect to 2), Pat Bajari, chief economist at Amazon and one of the biggest names in modern empirical IO, has a (at this point) working paper with Nekipelov, Ryan, and Yang on how to use machine learning to estimate demand curves across products with lots of sample points bunches of characteristics (think thousands of product characteristics). As a "fledgling computer science researcher," you'd probably be in to this. This approach is especially relevant for people/firms with access to tons of data (like Google, Amazon, etc.)


The utility function is $U(C,R)=CR$ and the time is restricted: $48=R+L$. Now we know that $C=\sqrt L$. $C$ can be replaced by $\sqrt L$. Therefore the langrarian is

$\mathcal L=\sqrt L\cdot R+\lambda (48-L-R)$

The (partial) derivatives are the following. They have to be set equal to zero.

$\frac{\partial \mathcal L}{\partial L}=\frac12 L^{-0.5} R-\lambda=0$

$\frac{\partial \mathcal L}{\partial R}= L^{0.5} -\lambda=0$

Putting $\lambda$ on the RHS

$\frac12 L^{-0.5} R=\lambda \quad (1)$

$L^{0.5} =\lambda \quad (2)$

Dividing (1) by (2):

$\frac12\cdot \frac{R}{L} =1 \Rightarrow R=2L$

The expression for R can be insert in the time restriction

$48=2L+L$


Why can't I find masks for the corona virus?

The supply of masks can't keep up with soaring demand because mask producers have capacity constraints in the short run. See this recent article.


Field experiments on labour market discrimination: do sizes of randomly assigned groups matter?

No, this should not be a major issue. We will very carefully take into account the different sample sizes. Allow me to continue your example:
Assume "Resume A" is the the treatment resume and "Resume B" is the control resume, where the treatment has a less advantaged ethnic name (Jamal, Beyonce) and the control resume contains a anglicized name (James, Sophia). One should be careful about gender here as well.
We then compare the mean acceptance rate between the two resumes.
Suppose $N_a = 300$ and $N_b = 310$.
The accepted resumes are: $Accepted_a = 30$ and $Accepted_b = 62$.
For shortcuts, the mean rates of acceptance are $r_a = 30/300 = 0.1$ and $r_b = 62/310 = 0.2$. Notice the mean rate of acceptance normalizes each group, so size is partially accounted for here.
The variances are then:
$\sigma_a^2 = (1-r_a)*r_a = 0.090$
$\sigma_b^2 = (1-r_b)*r_b = 0.160$
And more importantly, the variance of the means of each distribution are then $\sigma_a^2/N_a$ and $\sigma_b^2/N_b$, respectively.  We are comparing the means of the two distributions with a simple mean comparison test. Notice this step explicitly uses the sample size to account for expected variation between the two distributions.
So we now compare the two to see if there is a difference, under the null hypotheses there is no difference. (In statistics, you nearly always always assume no difference/no relevance/no effect as the baseline.)
$t=\frac{r_a-r_b}{\sqrt{\sigma_a^2/N_a+\sigma_b^2/N_b}} = 3.572$
The reasoning can be very explicitly laid out as follows:

The difference between the two samples is 3.572 standard deviations
away from 0.
Under the assumptions we have made above, it seems very
unlikely that this would occur by chance if we repeated the process
(looking this up on the t-table as a one-tailed test has a p-value of less than 0.001). Bayesians will point out there are some other silent assumptions one may have made, so language is particularly important at this step, but this generally passes the intuition.
Therefore, I reject the idea that these two samples came from the same distribution. It seems implausible.
So, I am left to conclude that there is a difference between the two samples. It seems that indeed people are more accepting of the Anglicized names rather than the ethnic names.


I think at the outset you have to make the distinction between the real interest rate and the nominal interest rate. The nominal interest rate is the real interest rate less inflation. Assuming the money supple curve is a vertical line in M-r space, then a shift of the line to the right (as indicated by a monetary expansion) implies a decrease in the nominal interest rate. Some books make the distinction between money demand being dependent on nominal interest rates, but it makes intuitively more sense that money demand be a function of real interest rate rather than nominal. The real interest rate decreases as the nominal return on assets stays constant but the price level increases (increase in inflation as a result of the monetary expansion). Remember that the price of a stock in a perfect world is the net present value of all future dividends, discounted by real interest rate. As this interest rate decreases, we discount stocks less, and hence price increases. This is the intuition behind the inverse relationship between asset prices and interest rates. This however, is a short run effect, so stock prices increase in the short run. When the real interest rate returns to its original level in the medium run, the stock prices decrease again. In short, the fluctuations in stock prices reflect fluctuations in interest rate. (the illustration is simplest for a fixed coupon bond that pays C every period). Basically, it all depends when you examine the price.


Does diminishing government spending lead to a reduction in trade deficit?

I will give an overview based on the analysis of Chapter 6 of “Monetary Economics: An Integrated Approach to Credit, Money Income, Production and Wealth” by Wynne Godley and Marc Lavoie. This is a Post-Keynesian textbook; neoclassical economists might debate the exact model structure, but their models probably give qualitatively similar results.

There’s a lot of issues buried in your question. One immediate problem is that we should not really look at just the government deficit: what the government is doing also matters. For example, the government could decrease spending in aggregate, yet greatly increase its imports (e.g. defense spending) and so the trade deficit could get worse. We will assume that something like that is not happening.

Arguing that government spending “will not impact savings” (as the question is now stated) is incorrect. Savings desires might well be unchanged, but changing government spending changes incomes, and so actual saving will change. Furthermore, cutting spending presumably has some multiplier effect (the precise level of the multiplier is hotly debated), and this will affect investment plans.

In the model in “Monetary Economics,” cutting government spending reduces the trade deficit. Since it is a multi-equation system that is solved simultaneously, one needs to be careful about verbal descriptions of mechanisms. However, I would describe the mechanism as follows: reduced government spending reduces national GDP, and this implies less imports (since imports are a fixed propensity to import out of all spending). As a result, the trade deficit falls, but it is not 1:1 with the reduction in spending - as your question phrasing (at the time of writing this answer) suggests.

One could debate the model structure, but I cannot think of many model structures that would suggest that the trade deficit would widen in response to government spending cuts. The implication is that the real debate is about the coefficient that relates the size of the cut in government spending to the trade deficit reduction. In almost any model, that coefficient would be parameter dependent, and so one would need to look at observed data to come up with a prediction for the real world.


Marginal Revenue Interpretation of Auctions

This analogy is due to Bulow and Roberts.

Think of a diagram from Econ 101 with price on the vertical axis and quantity on the horizontal axis and consider a monopolist with constant marginal cost of zero. Given a demand curve, you can draw the marginal revenue and find the monopoly quantity where this function intersects the axis.

Now take that same diagram and interpret "quantity" as "probability that a buyer accepts price $p$", i.e., $q = 1- F(p)$. Now, maximize revenue $pq=q F^{-1}(1-q)$ wrt $q$ to reach some form of the well known marginal revenue = marginal cost. The derivative of the revenue above is
$$\frac{\partial pq}{\partial q} = F^{-1}(1-q) - \frac{q}{f(F^{-1}(1-q)} = v - \frac{1-F(v)}{f(v)}, $$
where I replaced $q=1- F(p)$ and set $p=v$. This is Myerson's virtual value. The expected revenue of an optimal auction is the expectation of the highest virtual value. The part $\frac{1-F(v)}{f(v)}$ is the information rent, which takes care of the fact that you don't bid your true value in a first-price auction.


Ricardo's theory of comparative advantage

The solution concept used in Ricardo's modell is the competitive equilibrium. Let the set of countries $N$ be defined as $N = \left\{E,P\right\}.$ (England, Portugal) Then the competitive equilibrium is a vector
$$
\left(p,\left(q_{x,i},q_{y,i}\right)_{i\in N},\left(c_{x,i},c_{y,i}\right)_{i\in N}\right),
$$
where $p$ is the equilibrium price ratio of the goods $x$ and $y$, so $p = \frac{p_x}{p_y}$, and $\left(q_{x,i},q_{y,i}\right)$ and $\left(c_{x,i},c_{y,i}\right)$ are the production and consumption vectors of country $i$. The equilibrium vector has the following properties:

Profit maximization:
Each country choose a production that maximizes her profit given her production capabilites and the equilibrium price ratio. The set of possible production vectors $T_i$ is defined by the size of the countries labor pool $L_i$ and the labor requirements $a_{x,i},a_{y,i}$ as defined in the question
$$
q_{x,i} \cdot a_{x,i} + q_{y,i} \cdot a_{y,i} \leq L_i.
$$
A production vector $\left(q_x^i,q_y^i\right)$ is profit maximizing if
$$
\max\limits_{(x,y)\in T_i} p \cdot x + y = p \cdot q_{x,i} + q_{y,i}.
$$
Utility (welfare) maximization:
The vector $(c_{x,i},c_{y,i})$ maximizes country $i$'s utility if
$$
\max\limits_{p \cdot x + y \leq p \cdot q_{x,i} + q_{y,i}} U_i(x,y) = U_i(c_{x,i},c_{y,i}).
$$
The good markets are in equilibrium, i.e. in the market of each good demand equals supply (technically only the values are equal, but here, i.e. with Cobb-Douglas preferences, prices are always positive in equilibrium, so there is no difference). The equations for these are
$$
\sum\limits_{i\in N} q_x^i = \sum\limits_{i\in N} c_x^i \hskip 20pt \sum\limits_{i\in N} q_y^i = \sum\limits_{i\in N} c_y^i.
$$

Let us examine what these properties imply. The set $T_i$ is a triangle. As goods have positive value in equilibrium so all the labor is used up and the production vector is chosen from the production possibility frontier. Which industry ($x$ or $y$) can employ labor more lucratively? In industry $x$ a unit of labor produces value $\frac{p_x}{a_{x,i}}$. Similarly the value produced in industry $y$ is $\frac{p_y}{a_{y,i}}$. If $\frac{p_x}{a_{x,i}} > \frac{p_y}{a_{y,i}}$ only good $x$ is produced, if $\frac{p_x}{a_{x,i}} < \frac{p_y}{a_{y,i}}$ only good $y$ is produced, if $\frac{p_x}{a_{x,i}} = \frac{p_y}{a_{y,i}}$ it does not matter how labor is allocated among the industries as long as all labor is used. So the profit maximizating quantities are
$$
\left(q_{x,i},q_{y,i}\right) = \left\{
\begin{array}{cc}
\left(\frac{L_i}{a_{x,i}},0\right) & \frac{a_{x,i}}{a_{y,i}} < p \\
\alpha \cdot \left(\frac{L_i}{a_{x,i}},0\right) + (1 - \alpha)  \cdot \left(0,\frac{L_i}{a_{y,i}}\right)
 & \frac{a_{x,i}}{a_{y,i}} = p \\
\left(0,\frac{L_i}{a_{y,i}}\right) & \frac{a_{x,i}}{a_{y,i}} > p.
\end{array}
\right.
$$
The optimum condition for the utility maximization problem is
$$
MRS_i(c_{x,i},c_{y,i}) = \frac{c_{y,i}}{c_{x,i}} = p.
$$
Because the utility functions in England and Portugal have the same form we can take this further. From
$$
\frac{c_y^A}{c_x^A} = p = \frac{c_y^P}{c_x^P}.
$$
we get
$$
\frac{c_{y,A}}{c_{x,A}} = p = \frac{c_{y,P}}{c_{x,P}}.
$$
we get
$$
c_{y,P}  = \frac{c_{y,A}}{c_{x,A}} \cdot c_{x,P}.
$$
Using this
$$
\frac{c_{y,A}+c_{y,P}}{c_{x,A}+c_{x,P}} = \frac{c_{y,A}+\frac{c_{y,A}}{c_{x,A}} \cdot c_{x,P}}{c_{x,A}+c_{x,P}} =  \frac{c_{x,A}}{c_{x,A}} \cdot \frac{c_{y,A}+\frac{c_{y,A}}{c_{x,A}} \cdot c_{x,P}}{c_{x,A}+c_{x,P}} =
\frac{c_{y,A} \cdot c_{x,A} + c_{y,A} \cdot c_{x,P}}{c_{x,A} \cdot \left(c_{x,A}+c_{x,P}\right)}.
$$
so
$$
\frac{c_{y,A}+c_{y,P}}{c_{x,A}+c_{x,P}} = \frac{c_{y,A} \cdot c_{x,A} + c_{y,A} \cdot c_{x,P}}{c_{x,A} \cdot \left(c_{x,A}+c_{x,P}\right)} = \frac{c_{y,A}}{c_{x,A}} = p.
$$
What this says is that the relative demand ($\frac{c_{y,i}}{c_{x,i}}$) is not only equal to the price ratio for individual countries but also the relative aggregate world demand is equal to the price ratio. (Again, this is only true if the individual countries have Cobb-Douglas utility functions with identical parameters.) We now have a relatively easy way to find the equilibrium price ratio: we calculate relative aggregate supply. As aggregate supply equals aggregate demand in equilibrium, relative aggregate supply will equal relative aggregate demand, and as we have just shown it will also equal $p$. We get relative supply from the profit maximizing productions of the individual countries. Let us first discuss the aggregate of the profit maximizing productions, which I will denote by $(q_x,q_y)$. So $(q_x,q_y) = (q_{x,E} + q_{x,P},q_{y,E} + q_{y,P})$ which means
$$
(q_x,q_y) = \left\{
\begin{array}{cc}
\left(\frac{L_E}{a_{x,E}} + \frac{L_P}{a_{x,P}},0\right) & \frac{a_{x,E}}{a_{y,E}} < \frac{a_{x,P}}{a_{y,P}} < p \\
\left(\frac{L_E}{a_{x,E}} + \alpha \cdot \frac{L_P}{a_{x,P}} , (1 - \alpha) \cdot \frac{L_P}{a_{y,P}} \right) & \frac{a_{x,E}}{a_{y,E}} < p = \frac{a_{x,P}}{a_{y,P}} \\
\left(\frac{L_E}{a_{x,E}} , \frac{L_P}{a_{y,P}} \right) & \frac{a_{x,E}}{a_{y,E}} < p < \frac{a_{x,P}}{a_{y,P}} \\
\left(\alpha \cdot \frac{L_E}{a_{x,E}}, (1 - \alpha) \cdot \frac{L_E}{a_{y,E}} + \frac{L_P}{a_{y,P}} \right) & \frac{a_{x,E}}{a_{y,E}} = p < \frac{a_{x,P}}{a_{y,P}}  \\
\left(0, \frac{L_E}{a_{y,E}} + \frac{L_P}{a_{y,P}} \right) & p < \frac{a_{x,E}}{a_{y,E}} < \frac{a_{x,P}}{a_{y,P}}  .
\end{array}
\right.
$$
The relative aggregate supply is the ratio $\frac{q_x}{q_y}$. It is perhaps best described by this image:

The relative aggregate demand is the ratio $\frac{c_x}{c_y}$. As we have discussed $\frac{c_y}{c_x} = p$ so
$$
\frac{c_x}{c_y} = \frac{1}{p}.
$$
As a result one can draw the relative aggregate demand in the previous graph as  hyperbole. The intersection with the relative aggregate supply curve will give yield the equilibrium price ratio and will also yield information about the production of individual countries. Where this intersection occurs depends on the parameters $L_E,L_P,a_{x,E},a_{y,E},a_{x,P},a_{y,P}$. I will distinguish between three types of equilibria, each represented in the following figure:

In the 1. equilibrium the price ratio is $p = \frac{a_{x,E}}{a_{y,E}} < \frac{a_{x,P}}{a_{y,P}}$. Thus Portugal specializes and only produces good $y$, but England does not specialize but produces both good $x$ and $y$. Producing either good gives her the same value. The exact equilibrium quantities England produces are determined by the value the aggregate demand curve takes at price $p$, because
$$
\frac{q_x}{q_y} = \frac{c_x}{c_y} = \frac{1}{p}
$$
and
$$
q_x = q_{x,E} + q_{x,P} = q_{x,E} + 0 \hskip 20pt q_y = q_{y,E} + q_{y,P} = q_{y,E} + \frac{L_P}{a_{y,P}}.
$$
In this case England will still not achieve a net export of good $y$. The preferences tell us that Portugal will consume both goods $x$ and $y$. But the only way it can pay for the goods $x$ consumed is by trading some of its goods $y$, so Portugal, not England, will be a net exporter of good $y$ while England is a net exporter of good $x$.
In the 2. equilibrium both countries specialize: England produces only good $x$, Portugal produces only good $y$. This is usually presented as the textbook case.
The 3. equilibrium is like the 1. equilibrium, but here England specializes and produces only good $x$ while Portugal does not specialize and produces both.

So to answer my original questions:
Given that England has a comparative advantage in producing good $x$,

England will always produce good $x$.
England may also produce good $y$.
England will never export good $y$.


Is "normal distribution" a good choice for modeling future inflation rates?

With many financial variables you can deduce the distribution from the options market , but there isn’t a sufficiently developed inflation options market (at least in the US).  However, experience shows that market participants are unwilling to sell options with strikes either very low (say 0%) or very high (say 5%) at anywhere near theoretical prices, suggesting that the right distribution might be more fat tailed than a normal distribution.  This wouldn’t be shocking , since many financial variables have similar characteristics.
It also matters what measurement period you are working with. The inflation rate for a given 12month period in the future could have a wider distribution than the average inflation over a 20year period, which might be better behaved.
In the US , the center of the distribution can be estimated from the inflation bond and derivatives markets , and is in the area of 2% for long term forecasts, in line with the target of the Federal Reserve.


Here is the Lagrangian function for the stated optimization problem:
$$\mathcal{L}(x_i, t) = v(x_i, t) - cx_i + \mu x_i$$

Necessary conditions for optimality:

$$\frac{\partial\mathcal{L}}{\partial x_i} =\frac{\partial v}{\partial x_i} - c + \mu = 0 $$
and
$$x_i\geq 0, \ \mu \geq 0, \ \mu x_i = 0$$

To solve it, consider the following cases :

$x_i > 0$

$x_i > 0 \rightarrow \mu = 0 \rightarrow \dfrac{\partial v}{\partial x_i} - c = 0$

If there exists $x_i^* > 0$ such that $\dfrac{\partial v}{\partial x_i}\Big\vert_{x_i=x_i^*} - c = 0$, then $x_i = x_i^*$ solves the problem.
$x_i = 0$

$x_i = 0 \rightarrow \mu = c - \dfrac{\partial v}{\partial x_i}$

If $c - \dfrac{\partial v}{\partial x_i}\Big\vert_{x_i=0} \geq 0$, then $x_i = 0$ solves the problem.


Is lowered probability of spending equals savings?

Not going into Risk-Adverse case as this is sensitive to how we define the Government's utility function

$\mathbb{E}[xpnse] = 0.05*(0.07*1000)+0.95*(0.6*(0.07*1000))$

Follows from Law of Large Numbers


Monopolist engaging in international trade

Because the monopolist is only a monopolist in its own country, it is basically in perfect competition in the international market and cannot influence the world price. Therefore, it will have to use the condition that price is equal to marginal cost. His marginal cost (MC) is the first derivative of total cost, i.e. TC'=2Q=MC. If MC=P, then 2Q=10. Therefore, Q=5.

Is that the correct answer from your solutions?


How can two different utility functions represent the same preferences?

First of all, an utility function $u: X \rightarrow \mathbb{R}$ represents the preference relation $\succsim$ if: $$\forall a, b \in X, \; u(a) \geq u(b) \iff a \succsim b.$$

Well, if another function $v: X \rightarrow \mathbb{R}$ represents $\succsim$, then: $$\forall a, b \in X, \; v(a) \geq v(b) \iff a \succsim b.$$

But the above implies the following:

$$\forall a, b \in X, \; u(a) \geq u(b) \iff v(a) \geq v(b).$$

Which means that both functions preserves the order relation, just as Herr K. said it. Another important part of Wikipedia's article is this one. You see, any monotonically increasing (and it must be increasing, not nondecreasing) function preserves the preference relation.

So now I think you will be able to answer both questions. As a tip for the first one, think about the $\ln(x)$ function: is it a monotonic increasing function?


Literature request: Any articles on the signalling theory / game theory / asymmetric information of the firms' recruiting process?

It sounds like you have a background in game theory. Here is a Game Theory text available for free through RAND if you (or anyone else) wants a refreasher: http://www.rand.org/pubs/commercial_books/CB113-1.html

Below are some articles from which you should be able to draw information. Most are not specific to recruiting, but the underlying theory is certainly applicable. If you continue to be interested in this topic, you will find a great deal of information in the literature on social capital, product value signaling (one article included below), and value signaling as it relates to credentials. Denesp recommended a great paper; plug it into google scholar and see which papers cite that one. Doing this you may find a ton of information.

Here is a paper published that takes a game theoretic look at the legal profession. It doesn't directly relate to your scenario, but the concepts and framing certainly do (this is on google scholar): Rasmusen, E., Raghav, M., & Ramseyer, M. (2009). Convictions versus conviction rates: the prosecutor's choice. American Law and Economics Review, 11(1), 47-78.

Here is an article, also available on Google Scholar, that contends with imperfect information: Rothschild, M., & Stiglitz, J. (1976). Equilibrium in competitive insurance markets: An essay on the economics of imperfect information. The quarterly journal of economics, 629-649.

Here is one specific to Signaling Games: Cho, I. K., & Kreps, D. M. (1987). Signaling games and stable equilibria. The Quarterly Journal of Economics, 102(2), 179-221.

Also about signaling: Kirmani, A., & Rao, A. R. (2000). No pain, no gain: A critical review of the literature on signaling unobservable product quality. Journal of marketing, 64(2), 66-79.

Related to employment: Leana, C. R., & Van Buren, H. J. (1999). Organizational social capital and employment practices. Academy of management review, 24(3), 538-555.


Another solution can arise if you allow for uncertainty and for the planner to provide private information to each player. See Bergemann and Morris, 2018 for a survey in what is called information design. In that way, it is not hard to see how each player can be convinced that the social allocation is its most preferred outcome with high probability such that from the ex-ante perspective, the outcome is Pareto-optimal.

Of course, I'm throwing a lot of details under the rug, but having asymmetric information seems like another solution to this conundrum.


Why politicians choose Mega Projects in developing countries?

You are looking for the political economy literature on government expenditures. That is a massive literature. A starting point may be given here.

A summary of all possible reasons for so-called white elephants is difficult without a more specific question. Are you interested in corruption, asymmetric information, institutional explanations, political preferences...?


Restricting "money creation" to the central bank

The proposal is to prohibit fractional reserve banking. So if enacted, Icelandic banks would have to have at least as much in reserves as they have in checking deposits. To do any lending, the bank would need to raise funds either from it's equity holders or by borrowing from investors. A number of US economists, such as Chicago economist John Cochrane, have endorsed this idea.

The main reason for abolishing fractional reserve banking is to eliminate the potential for bank runs. Standard banks don't keep enough money in reserve to pay out everyone's deposits, but they are obligated to let anyone withdraw their deposits at any time. Thus, if too many people withdraw at once the bank can go bankrupt, and that possibility of loosing their deposit in turn causes everyone to try to withdraw at once. It's a self-fulfilling panic that can happen at anytime, and doesn't necessarily even need to be caused by anything in particular. In economics terms, fractional reserve banking has two equilibria at all times: the normal one with deposits and lending, and the bank run equilibrium.

The US and most countries have tried to elimintate the bank-run equilibrium by insuring deposits--this is what the FDIC does. The problem, as countries like Cyprus found, is that if depositors don't believe the government really has the resources to pay out the full insured deposit amount for everyone, then you can still get a bankrun, this time not merely wiping out the banks and everyone's savings, but also the government's finances as well. Small countries that can't borrow and lend in their own currency are especially vulnerable here. Iceland does have its own currency, but often does business in US dollars or Euros anyway.

Additionally, there is concern that fractional reserve banking could lead to increased risk-taking by banks, since part of the risk is born by the depositors who (it is assumed) aren't as sophisticated as equity investors at measuring the banks' risks.

The drawback, of course, is that this likely makes checking more expensive to depositors since the bank can no longer lend the money out to make interest. Presumably it won't lead to higher borrowing costs because the central bank would offset the inability to lend out deposits with a larger base money stock, but it could exacerbate inequality as depositors would no longer share the returns to banks' investments (think Mr. Potter vs Bailey Savings and Loan from the movie "It's a Wonderful Life").


Diminishing mariginal utility and risk preferences

Mas-Colell, Whinston, and Green Proposition 6.C.1 (p. 187) says that for an expected utility maximizer with a Bernoulli utility function $u(\cdot)$ on amounts of money, saying that "the decision maker is risk averse" is equivalent to "$u(\cdot)$ is concave".

However, these other assumptions matter. A concave utility function isn't enough. Diminishing marginal utility in amounts of money doesn't have to imply risk aversion. For example, with gamble $(F)$ taking two values $W_L,\: P(W_L) = \theta$  and $W_H>W_L,\:P(W_H) = 1 - \theta$, and a utility function $u{}$ with diminishing marginal utility in wealth, consider this person's welfare function over risky lotteries $(V)$:
$$V(F, u(), \theta) = u(W_H)$$
This person is not risk averse. They love risk (for example, they prefer a risky lottery to the expected value of that lottery). But, in situations without risk, they have diminishing marginal utility of wealth.


Is it possible to imagine humanity without a modern understanding of money?

The rather famous political economist Marx claims that the wage labour relationship contains within itself the transcendence of that relationship. This is in part as the only socially productive property a worker has consists in the giving away of themselves for a mess of postage.  This suggests a non-engagement with owning socially productive property and instead an emphasis on usufruct. Similarly the commodities of the wage are used (consumed) rather than hoarded.

In Marx’s suggestion that the proletariat can abolish itself, the wage, profit, Capital—in general “the value form”—the abolition of money as we understand it in Marx or contemporary use would necessarily be abolished. Movements of socially useful stuff and services would somehow work on the basis of capacity and need. There might be a mediator, but it wouldn’t have the current functions of money.

Apart from a few very short term instances during revolutionary crises, and generally without ordinary productive activity, there’s no empirical evidence for these relationships; and, there isn’t sufficient material to move beyond large weak concepts like “love,” or “solidarity.”


on the FRED website where do i find definitions of the variables they give charts of?

The bottom of these indicator/series pages on FRED always lists the source. In this case, it's the OECD. If you dig around a bit on their site, and find where they host the data, there is always some meta-data attached to these that explains pretty much everything you would be looking for normally. Otherwise, they also list an email and you can ask them directly--which I've done on occasion.  They're slow to respond, but get back.

For your specific question, for example, if you go to https://stats.oecd.org/, click on the 'Finance', 'Monthly Financial Statistics', 'Monthly Monetary and Financial Statistics' and the select 'Monetary Aggregates - Broad Money M3', it pulls up all the M3 data. On the right-hand side, there is a window you can expand that has all the metadata that answers your question.


Saddle path equilibrium on financial market with rational expectations

Suppose you have a dynamic system
$$ x_{t+1} = Ax_{t} $$
with a stationary point (or steady state as used in growth or RBC literature), say, $x^*$, i.e. $x^{*} = Ax^{*}$.

Now, consider the following question. Starting from an initial value $x_0$, how many paths are there leading to the stationary point $x^*$? If there is an unique path going from $x_{0}$ to $x^{*}$, then your model is well behaved in the sense that you can track the vector of variables, $x_t$, along the transition without worrying which path you are actually on. This is the saddle-path stability case that you are referring to. On the other hand, if the answer is not affirmative, it means that you have at least two routes from $x_0$ to $x^{*}$. There is another case: no matter where you start from you eventually end up at the stationary point $x^{*}$, in which case your model is said to be indeterminate.

So you can think of saddle-path stability as a desirable feature that you would like that your model to manifest in order to analyze the problem at hand. For example, standard RBC models all posses this property.

There are some mathematical conditions that ensure the saddle-path stability.

For the details and more, check out the section 7.8 (The q-Theory Investment and Saddle-Path Stability) in Introduction to Modern Economic Growth (Acemoglu, 2009).

Ok, I'll try to explain intuitively for the Ramsey model (but it's not rigorous). Suppose that from the first order conditions, you can conclude that the equilibrium is unique and furthermore you can express consumption as a function of capital stock, that is $c_{t} = g(k_t)$, for some (smooth) function $g$. If that is the case, for a given initial capital stock $k_0$, you know how the economy is going to evolve. That is, if we denote the vector of variables by $x_t = (c_t, k_t)$, you know $\{ x_{t}\}_{t=0}^{\infty}$; because

$t=0: \quad$ $k_0$ is given, and $c_0 = g(k_0)$,

$t=1: \quad$ $k_1 = (1-\delta)k_0 + f(k_0) - c_0$, and $c_1 = g(k_1)$,
(here $f$ is the production function and $\delta$ is depreciation)

$\vdots $

$t=\tau: \quad$ $k_{\tau} = (1-\delta)k_{\tau -1} + f(k_{\tau -1}) - c_{\tau -1}$, and $c_{\tau} = g(k_{\tau})$,

so on ....

Now, suppose the economy loses some of its capital stock for some reason while it was on $x^*=(k^*, c^*)$ and the new capital stock is, say, $k_{\tau}$.

Where will the economy be with this new capital stock?

Well, we have just claimed that there is a corresponding consumption $c_{\tau}=g(k_{\tau})$, and therefore the economy will jump to the point $x_{\tau}=(k_{\tau}, c_{\tau})$ on impact the shock and then move toward back to steady state $x^*$. You can think of $x_{\tau}$ as the red point in the figure.

Finally, a few remarks on the intuition behind the mathematical conditions ensuring saddle-path stability.

1) Note that, in the model we have actually two  variables $c_t$ and $k_t$, but we have assumed  that one of them can be  expressed in terms of the other, so the number of free variables is reduced to one, which is $k_t$ here. Basically, in order to be able to do this, you want to have a smooth curve mapping capital stock to consumption, i.e. $g: \mathcal{R} \to \mathcal{R}$ is a smooth curve (1-dimensional manifold).

2) Once the economy is at $x_{1}$ after the shock, you want it go back to $x^*$ and this is related to the number of eigenvalues that are less than one in absolute value.

For precise statements (in a general environment) see Acemoglu as I mentioned before.

Regarding rational expectations(RE): it is a solution concept and RE alone does not imply saddle-path stability.


Why are second price auctions preferred if they don't maximize expected revenue?

I don't think sellers would prefer SPA over FPA. In fact, the SPA is riskier than the FPA, if we look at it from a seller's perspective. The reason is because the distribution of prices in case of the SPA is a mean preserving spread of the distribution of prices in FPA(you can refer to Vijay Krishna's Auction Theory for the proof).

However, the VCG mechanism finds a lot of applications in real life scenarios(like in case of network routing algorithms). It just so happens that the VCG mechanism can turn out to be the second price auction, given that certain conditions are met.

The Generalized SPA is another such example, that draws its basis from the standard SPA. Please see https://en.wikipedia.org/wiki/Generalized_second-price_auction .

As far as the English Auction is considered, it is equivalent to the SPA only in the case of independent bidders. If the values realized by the bidders depend upon the the values realized by others(meaning the valuations are now interdependent), the English Auction no longer remains equivalent to the SPA(interdependence is something that on can expect to be more prevalent in real life cases).

Another thing, the RET states that the 'expected revenue' is same for standard auctions with symmetric bidders. The payment received by the seller is just the highest bid. Thus, receiving a higher payment in either auction(FPA or SPA) depends on the values realized by the bidders(and the bidding strategy).


Why doesn't the San Francisco Bay Area have two-story houses?

Areas such as San Francisco Bay Area have very strict zoning laws. In fact as this source shows in San Francisco Bay Area many houses are restricted to be single family houses. The source focuses mainly on Berkeley and Oakland but discusses also the situation of the wider area.
Adding an extra story just for your own family might not be economical even if there is housing shortage. Furthermore, there might be further restrictions on any such buildings (e.g. neighbors complaining to the zoning board that second story blocks their view/sun). Lastly, the housing shortage and exorbitant housing prices in cities such as San Francisco can actually itself be attributed (among other things) to the zoning laws themselves in the first place (see for example The Impact of Zoning on Housing Affordability Edward L. Glaeser, Joseph Gyourko).


Yahoo Finance is a site thats pretty useful for getting historical stock data.

You can get some pretty big datasets (i.e. you can get daily stock market data for Microsoft going back to 1989 till now).

Hope this helps.


Exercise 4.7 in SLP (dynamic programming)

It follows from Assumption 4.10.
Let $\lambda y \in \Gamma(\lambda x)$ be the solution.
Let $\delta = \frac{1}{\lambda}$.
By Assumption 4.10,
$$
\lambda y \in\Gamma(\lambda x)
$$
implies
$$
\delta \lambda y \in \Gamma(\delta\lambda x)
$$
in other words
$$
 y \in \Gamma(x)
$$