Market equilibrium and pareto efficiency

A rough guideline:

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

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

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

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

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

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


Say you have a portfolio with returns described by a random variable X. Call the lowest possible realization of X: xmin. If you take a levered position in that portfolio with leverage A and financing cost r your returns are r(A-1)+AX. There will exist a value of A no larger than 1/xmin where when you get the worst return of X and the levered portfolio returns -100%. That's game over for your wealth.

So if you invest for the long run and reinvest after each period it should be clear that you can't simply arbitrarily maximize leverage to maximize long run returns. That expected log return maximization objective function is exactly the objective function of long run return maximization. This objective treats -100% returns as -infinity utility and so avoids it at all cost.

If instead you maximized expected returns this wouldn't happen. Your preference would be for unboundedly large leverage, in line with your intuition.

Another way to think about this is risk aversion. Log utility in wealth is analogous to a CRRA utility parameter of 1, while the expected return utility function a CRRA parameter of 0 (risk neutral). Since the risk neutral guy doesn't care about risk, he sees an asset with positive expected returns and wants a gigantic levered bet on it. The log utility guy is relatively risk tolerant but still worries about extreme outcomes and so doesn't want too much leverage.

And if the range of values taken by X are mostly in the range of +/- 30%, it turns out that E[log(X)] is nicely approximated by E[X]+0.5Var(X) so can be well approximated with Markowitz mean-variance portfolio optimization where lambda is 1/2. Where as E[X] preferences is analogous to lambda of zero which won't have a bounded solution without an additional constraint.


Gali and van Rens: The Vanishing Procyclicality of Labor Productivity

A napkin theoretical exploration of the issue, to find possible directions to look for explanations, could go like this:

According to the paper, the volatility of output and productivity has fallen. So for the correlation to have gone down, the covariance must have fell steeply.
Modeling output as

$$Y = AK^a(h\cdot L)^{1-a}$$

where $h$ is the "effort per unit" variable, we have

$${\rm Cov}(Y, Y/L) = E(Y^2/L) - E(Y)E(Y/L)$$

Now

$$Y^2/L =   A^2(K/L)^{2a}\cdot L \cdot h^{2(1-a)}$$

$$Y/L =   A(K/L)^{a}\cdot h^{1-a}$$

So

$${\rm Cov}(Y, Y/L) = E\left[A^2(K/L)^{2a}\cdot L \cdot h^{2(1-a)}\right] -E\left[A(K/L)^{a}\cdot L\cdot h^{1-a}\right]E\left[A(K/L)^{a}\cdot h^{1-a}\right]$$

We observe ${\rm Cov}(Y, Y/L) \rightarrow 0$. Now, I am in the mood of assuming that the capital-labor ratio is essentially a constant, as is $A$. More over, that the effort-per-unit variable is independent of the amount of labor hours. Under this scenario, we have

$${\rm Cov}(Y, Y/L) = A^2(K/L)^{2a} \cdot E(L)\cdot \big[ E\left[(h^{1-a})^2\right] -\left[E(h^{1-a})\right]^2\big]$$

$$\implies {\rm Cov}(Y, Y/L) \rightarrow 0 \implies {\rm Var}(h^{1-a}) \rightarrow 0$$

So here, the effort variable should be approximately a constant too.

--

In a more agnostic approach, using lower case letters to denote logarithms, we would have

$${\rm Cov}(y, y-\ell) = Var(y) - {\rm Cov}(y, \ell)$$

If what we observe is

$${\rm Cov}(y, y-\ell) \approx 0 \implies  Var(y) \approx {\rm Cov}(y, \ell)$$

But this means that in a least-squares regression of $\ell$ on a constant $\gamma_0$ and $y$
$$\ell = \gamma_0 + \gamma_1y + u$$
we would obtain $\gamma_1 =1$ and the relation

$$\hat \ell = \hat \gamma_0 + y \implies \approx Y = e^{-\gamma_0}L$$

(with $\gamma_0 <0$ most probably). This is consistent with the previous approach since the constant $e^{-\gamma_0}$, may be seen as containing the $A(K/L)^ah^{1-a}$ terms, which were assumed or emerged as constants for the result of zero-covariance to occur. Here, there is more flexibility since we only require that their product is constant, not each one separately.


Intuition - Why does elasticity vary along a curve?

Think of it this way. A yacht costs $\$0.01$. How much do you demand? Then double the price. Now how many would you demand?

Then consider something closer to a market price. Something in 6 or 7 figures I guess. If you've got some money, you might buy one at the market price. But then if that price doubles, wouldn't that affect your demand a lot more than the 1 to 2 pennies change?

Simply put, as you go up the demand curve, the same percentage change in price is a much bigger absolute movement in price.


You should be fine to use negative interest rates for the risk-free rate when calculating a Sharpe ratio. If rates are negative, they are negative. That does not change the calculation of excess returns.
One point, however: EURIBOR and EONIA are interbank rates and thus are not risk-free but have a small credit spread above risk-free rates. The latest ECB recommendations call for €STR to become a risk-free overnight rate with a transition planned to set EONIA := €STR+8.5 bp going forward.


Saddles and Rational Expectations. Language barrier preventing me to solve this exercise

$\newcommand{\vect}[1]{{\bf #1}}$

Note that you can write the system in the form

$$
\dot{\vect{x}} = A \vect{x} + \vect{b}
$$

where

$$
\vect{x} = \left(\begin{array}{c} x \\ y\end{array}\right), ~~~~ A = \left(\begin{array}{cc} 1 & 1 \\ 2 & -1\end{array}\right), ~~~~~ \vect{b} = \left(\begin{array}{c} 1 \\ 5\end{array}\right)
$$

A fixed point $\vect{x}^*$ is a point for which $A\vect{x}^* + \vect{b} = 0$, which in this case is

$$
\vect{x}^* = \left(\begin{array}{c} -2 \\ 1\end{array}\right)
$$

Now define the variable

$$
\vect{z} = \vect{x} - \vect{x}^*
$$

and note that

$$
\dot{\vect{z}} = \dot{\vect{x}} = A\vect{x} + \vect{b} = A\vect{x} - A\vect{x}^* = A(\vect{x} - \vect{x}^*) = A\vect{z}
$$

That is,

$$
\dot{\vect{z}} = A \vect{z}
$$

What you need to do now is to find the eigen-decomposition of $A$


Can a country survive and prosper without international trade or investment from overseas?

Besides allowing countries to take advantage of comparative advantage, as mentioned, an open economy can engage in intertemporal trade.

This possibility is useful because it allows a country to smooth consumption over time, even when domestic output fluctuates. For instance, in good times the economy can save (consume less than actual output, lending to foreign countries), allowing the economy to consume more than actual output in bad times.
This in fact is welfare improving relative to autarky (in which consumption in any period must equal current output (to simplify assume there is no investment), when we assume that consumer's preferences are convex.

Therefore, I don't know if a country can survive/prosper being a closed economy, but certainly there are many advantages for an economy that engages in international trade.


There are many refinements of pooling equilibria. For starters, you want to make sure all your equilibria are Perfect Bayesian Equilibria. After that, you can also require any equilibrium to survive the intuitive criterion or other forms of forward induction. Lastly, you could also require things like efficiency or symmetry.

The choice of refinements is not always straight-forward, it depends on what you mean by "best" and how well the assumptions implied by each refinement match the story of in your model.


Showing that a preference relation admits a utility function representation

You just need to use the violation of transitivity and proceed by contradiction.

Suppose you have  that $(x_1,y_1)R(x_2,y_2)$ and $(x_2,y_2)R(x_3,y_3)$ and a utility function, $u:[-1,1]^2\rightarrow \mathbb{R}$, exists, then $u(x_1,y_1)\geq u(x_2,y_2)$ (these are two reals) and $u(x_2,y_2)\geq u(x_3,y_3)$. (another two reals). Since the reals are transitive, we conclude  $u(x_1,y_1)\geq u(x_3,y_3)$ which in turn implies that $(x_1,y_1)R(x_3,y_3)$. However this is a contradiction (if you choose carefully your three bunddles).


If I understand correctly we have observed earnings first-differences that we decompose into an "expected" part and an "unexpected" part. But f they are unexpected, it means something like

$$y_{t+1} -y_t = h(I_t) + u_{t+1} , \;\;E(u_{t+1}\mid I_t)=0$$

where $I_t$ is the "information set" (in the broad sense) at time $t$. So by construction, the unexpected part will have conditional mean zero. But , this does not imply necessarily that $u_{t+1}$ will also have a symmetric distribution around zero. If as you conjecture, there are "more shocks downwards (negative) rather than downwards", the distribution will have more probability mass in the negative. Then, in order to maintain a zero-mean the distribution will have to have higher values on the positive: more and smaller negative shocks, fewer and higher positive shocks.  Something like a shifted to the left gamma distribution, so as to have zero-mean,

Is this close to what you have in mind?


The answer, that can be found in greater detail in the following book

Rudolf Richter, Money — Lectures on the Basis of General Equilibrium Theory and the Economics of Institutions, Springer, 1989.

is quite simple. One has to create a positive marginal utility to the user. How one can achieve this aim? One has to create a positive demand for fiat money, which can be accomplished that taxes can only be paid with fiat money. Then it has a positive marginal utility, and the trick is done.


If you would like to go step above Varian's intermediate microeconomics and you like Varian as an author then his book Microeconomic Analysis is on graduate level.
However, I personally prefer more  Mas-Colell,  Whinston, and Green (MWG) Microeconomic theory. That is probably the most comprehensive microeconomics textbook there is on the market.


Multiplicative factorization of stochastic growth time series--solving for an eigenfunction/eigenvector

The progress given so far appears to be correct. Finishing this problem just requires us to argue that $H$ must be chosen so that the equation
$$
(D' + H'(A - I)) x + (F + H'B)'(F + H'B) = \eta
$$
holds for all $x$. Thus, we must have $D' + H'(A - I) = 0$ and, consequently,
$H' = D'(I-A)^{-1}$. This implies that $\eta = (F' + H'B)'(F' + H'B)$.


The official rate is about 10, but the black market rate is about 1000.  Also, yes, apparently the real exchange rate at the 1000 rate is pretty low, in the sense that one dollar buys a lot of local goods, (like boarding). Indeed, that's what it means, 10,000 dollars buys you 1,000,000 VB's, which pays for years and years of rent. The ratio in your numbers is that Venezuela is about 50 times cheaper. In this story, staying in a hotel costs $5, which is also about 50 times cheaper than staying in a hotel in L.A....

(It would be hard to change such a large amount of dollars without getting noticed though...)


Quoting Wikipedia:

  the quantity theory of money (QTM) states that money supply has a
  direct, proportional relationship with the price level.

According to this theory an increase in money in circulation increases price levels.

So an "extremely simple mathematical model" (and extremely simplistic) would be something like:

$$ \pi_t = \dfrac{M_t-M_{t-1}}{M_t} $$ where $M$ stands for money supply and $\pi$ stands for inflation.


What is meant by "Exchange rate expectations are static" (In the Mundell-Fleming model)

The expression "Static Expectations" unfortunately can be found to be used with two different meanings in the economics literature

1) The one apparently less used but used nevertheless is, for some variable $x$ that needs to be predicted while standing at period $t$ (the superscript $e$ denotes "expectation" in the general meaning of the term, without specifying how it is formed)

$$x^e_{t+1 |t} = b,  \;\;\; \forall\, t$$

where $b$ is some constant. This reflects the following attitude: agents are oblivious to any information accrual as time passes, they even ignore possible movements in the realized values of $x$, and if asked, they always think that "tomorrow" the value will be $b$ (perhaps a better way to describe such expectations would be "fixed expectations" rather than "static").

2) The other and most widely used meaning of "Static expectations" is

$$ x^e_{t+k |t} = x_t, k=1,2,...$$

i.e. "what is will continue to be".

In the Mundell-Fleming model, the expression is used with this meaning.

(I note that with this meaning, "Static Expectations" coincide with Rational Expectations when the $x$-variable follows a pure random walk -but again, in Static Expectations we do not specify how we arrive at the value we expect).

Given this, imperfect capital mobility would lead to the domestic interest rate being different in equilibrium from the international interest rate in the sense $i \neq i^*$, since the exchange rate is not expected to change. So the Uncovered Interest Rate Parity (in logarithmic terms) $i = i^*+\Delta s^e$ becomes under Static Expectations $i = i^*$, and stops to hold under imperfect capital mobility.


What is the standard error on quarterly GDP figure?

Good question!

I've never seen standard errors reported alongside official figures (e.g. here for US, here for UK). You would expect they could calculate/report them, since GDP is computed from samples (so randomness is explicitly involved in the design). It is more common to see standard errors reported for GDP forecasts though. For example, check Figure 2.1 of the latest Bank of England Inflation report (data here).

Regarding GDP, what is very common however is the publication of revised figures. That is something both the US and UK include in their quarterly reports (see links above), and also common in many (in not all) countries. Comparing actual and revised figures gives you an idea of the uncertainty involved on preliminary estimates. For example, in the release notes for quarterly GDP of the US economy (see link above), towards the end there is a section called "Updates to GDP", which contains:

  BEA releases three vintages of the current quarterly estimate for GDP:  "Advance" estimates are
  released near the end of the first month following the end of the quarter and are based on source data
  that are incomplete or subject to further revision by the source agency; “second” and “third” estimates
  are released near the end of the second and third months, respectively, and are based on more detailed
  and more comprehensive data as they become available. [...]

  The table below shows the average revisions to the quarterly percent changes in real GDP between
  different estimate vintages, without regard to sign.

The Bank of England also provides its estimates of the past uncertainty (what they call "backcast", in opposition to "forecast"). This is available in the last sheet of this dataset.

Another interesting example comes from the Euro Area. Eurostat produces quarterly "Preliminary GDP flash estimates" for the Euro Area for selection of periods: 30, 45, and 60 days after the end of the quarter. As the page states, there is "a trade-off between timeliness and accuracy". An example of subsequent revisions are:


Edit this answer was written for the original question, before user completely edited whole question to something completely different and unrelated to original question for some reason:

How is it that the coupon rate is higher than the interest rate?
How can the coupon rate be higher than the interest rate?
Doesn't that mean that after one year, you're beating the interest rate. For example, if the coupon rate was 8% and the interest rate was 4%. On a $100 priced bond, I would get 8 dollars in the first year, which is 8% returns, as opposed to 5% returns on the interest rate. Am I missing something? If I'm not missing something, then is this scenario not possible?

I believe that at the heart of this question is confusion between the price of the bond and face value of the bond.
A general bond pricing formula (see here an example) can be expressed as as a discounted stream of future cash flows from the coupon and the value of the bond itself. Hence we have:
$$ P = \sum^T_{t=1} \frac{C_t}{(1+i)^t}+\frac{M}{(1+i)^T}$$
where $P$ is the price of the bond, $C$ is the coupon, $i$ is the interest rate, and $M$ is the value at maturity (i.e. the face value — the value printed out on the bond).
Based on the above formula if the coupon was $8\%$ of the $\\\$100$ face value (i.e. $\\\$8$), the interest rate $4\%$, and we assume $T=2$ the actual price of the bond simply cannot be $\\\$100$.
The correct price would be:
$$ P = \sum^2_{t=1} \frac{\\\$8}{(1+0.04)^t}+\frac{\\\$100}{(1+0.04)^2} \approx \\\$107.54$$
If the price would  be lower than $\\\$107.54$ people would increase their demand for the bond until it would reach its equilibrium at that price and vice versa if the original price would be above.
It is simply not viable for the price to stay at $\\\$100$ in free bond market. In real life there might be some mispricing occurring at some times (although to discuss why is outside the scope of this answer) that can create arbitrage opportunity but as people start taking advantage of it, it will eventually disappear unless the mispricing is deliberately encouraged by some policy.


Why shouldn't governments use Forex reserve to pay their debts?

If you had 12,000 dollars owing on a loan, 10,000 dollars cash in the bank and 1000 dollars monthly expenses, but you weren't sure if incoming cash flow could stop for 6 or 12 months, any time, with no notice, would you deplete your entire cash stock to reduce debt?

At that point, if your next paycheque was a week late, you would go hungry. Or, assuming that you had assured your basic subsistence, you might find yourself with no money to buy some other basic things that make your life a lot easier or more efficient.

Otherwise stated, there's a reason that a common measure of forex reserves is "Total reserves in months of imports".

Basically, if India used all of its forex to pay off external debt, then on a moment's notice it could have zero (or at least highly constrained) ability to access critical inputs into production processes. The highly negative impact that autarky or excessive focus on self sufficiency (beyond subsistence needs) could be highlighted in the context of this explanation.

Also (among other things), while a lot of economists disagree that using forex to manage an exchange rate in a situation of short-term crisis is a good idea, it is logically true that if you have no forex then this option simply does not exist.

For a specific example of how this can be a problem, you could refer to the story of the 1997 Asian Financial Crisis. In response to this crisis, most central banks in Asia undertook to substantially increase their forex holdings in order to not suffer a repeat of such a crisis.


How can the abuses of monopoly power lead to market failure?

First, let's be clear what we mean by abuse of dominance: this is when a firm is a monopoly or near-monopoly, and attempts to use that position in order to perpetuate or enhance its dominance of the market at the expense of competition.

So we need to know: why would a market with a monopoly perform worse than one with competition. Here are the main typical answers:

A monopoly will increase the price above marginal cost by artificially constraining supply. This means that some units that consumers' value above the cost of production don't get traded, resulting in the usual dead weight loss triangle.
Some people speculate that a monopolist has less incentive to innovate (create new products or means of production) than a competitive firm because there is no competitive pressure to be more efficient or to have a better product than other firms. Others, though, have argued that monopolists have a larger incentive to innovate because they can capture more of the gains from invention. This is the subject of a classic debate between Arrow and Schumpeter. The empirical evidence on whether competition is good or bad for innovation is extremely mixed.
Rent-seeking is a concern. If a monopoly makes profit $\pi_m$ and a competitive firm makes profit $\pi_c<\pi_m$ then a firm would be willing to spend $\pi_m-\pi_c$ to become (or remain a monopoly). This is fine if the resources are spend on something socially constructive. But not if it is spent on wasteful advertising, lobbying, engineering products with poor compatibility, creating needless switching costs, etc.
Another concern is managerial slack / x-inefficiency. If a firm has no competitors then there is very little incentive to exert effort to become more efficient, meaning that the firm may end up spending more resources on production than necessary.

If you want to read more about these things, I would recommend the first couple of chapters of Motta's book on competition policy.


Implications of the concepts of combination/expansion of an information structure

Answer to Question 1: Yes. To see this, note that for every combination of elements of $T^1\times T^2$ in the support of $\pi^*$ you get a, possibly different, posterior. To show $\pi^*$ is at least as informative as $\pi^1$ you need these posteriors to be a mean-preserving spread the posteriors induced by $\pi^1$ (there are actually several characterizations of Blackwell's partial order, I'm thinking there is a more elegant proof, but this is the one I can think of right now). You can easily show this is the case if you average up the posteriors induced by $S^*$, over the realizations of $T^2$.

Answer to Question 2: Yes. For any signal $S$, you can find a "trivial" expansion by combining it with $\underline \pi$, so the resulting signal $S^*$ (which is informationally equivalent to $S$) is at least as informative than $\underline S$.

In that sense, Bayesian decision makers can never have less information than their prior.


Why if the economy will produces too much there will be an increase of prices in nearly all goods and services?

The passage from Krugman is quite confusingly written. What Krugman has in mind is not increase in production per se but increase in aggregate demand which is associated with more production.
The short run relationship between real output and price level (change in which gives you inflation) can be visualized on aggregate supply-aggregate demand (AS-AD) diagram (see Blanchard et al Macroeconomics or Mankiw Macroeconomics for more detail).
If production (Y) expands because economy is able to produce more and firms expand their production that will lead to decrease in prices (P) - and hence deflation.  As the picture I below drew shows, when economy produces more because firms on the whole supply more to the market (AS shifts to the right), the price level would be expected to fall ceteris paribus:

However, what Krugman has in mind is change in aggregate consumer preferences that shift demand more (i.e. AD shifts to the right), this will lead to higher prices and more production (in the short run) because this higher demand and prices motivate firms to supply more. However, Krugman makes it sound in the passage above as if there was increase in supply by firms (where the chains of events starts with more production rather than with more demand as in this second case).
In this case aggregate prices increase because people demand too many goods and services and firms can only satisfy that extra demand at higher cost (since production capability of economy given by AS did not change). Consequently, firms will only be willing to supply extra goods and services if they are motivated by higher prices.


I tried this for a=0.5, T=10, w=2 and I get the following real solutions for t12 and t21:
(1) $t_{12}=t_{21}=2.8165$
(2) $t_{12}=t_{21}=1.1835$

For (1) the function is valued at 3.78821 and for (2) the function is valued at 3.68444.

So it seems that (1) is the solution for the maximum.

If it helps, here is the function and the contour sets at those parameter values.

Function

Contour Set ($t_{12}$ on X axis):

By the way, are your FOCs correct? For the FOC with t12, I have:
$-\frac{(1 - a)}{(T - t_{12})} + \frac{a (-1 + t_{21})}{(T - t_{12} + t_{12} t_{21})} + \frac{a t_{21}}{(
  T - t_{21} + t_{12} t_{21})} = 0$


Monopoly pricing under constant elasticity of demand

A constant elasticity demand function has the form $q=p^{-\epsilon}$. Let's check this indeed gives us a constant elasticity...
$$\frac{d q}{dp}=-\epsilon p^{-\epsilon-1}$$
so, as we'd hoped, the elasticity is constant:
$$\frac{d q}{dp}\frac{p}{q}=-\epsilon p^{-\epsilon-1}\frac{p}{p^{-\epsilon}}=-\epsilon p^{-\epsilon-1+1+\epsilon}=-\epsilon p^{0}=-\epsilon.$$

Now, suppose a monopolist has constant marginal cost $c\geqslant0$ (and no fixed cost). It's profit is
$$q(p-c)=p^{-\epsilon}(p-c).$$
The effect on the firm's profits of a small increase in its price is
$$\frac{d}{d p}p^{-\epsilon}(p-c)=p^{-\epsilon -1} [c \epsilon -p\epsilon+p].$$
Now we can answer the question: if $\epsilon<1$ then $c \epsilon -p\epsilon+p$ is definitely positive for every $p$. That means, no matter how high the price is, the firm would still like to increase the price further—the optimal price is $+\infty$!

Intuitively, constant  $\epsilon<1$ means that the firm can always more than double its price by foregoing less than half its current sales. Provided marginal cost isn't very quickly decreasing, this is always an attractive prospect for the firm.


Convention / standard methodology for calculating real prices?

This is not an issue of "established practice", but of methodological and conceptual consistency.

In theoretical models, for "real" magnitudes we write, say $W_R = W_N/P$ where $P$ is the "price index" and we're done.

But in reality, in order to obtain an actual value for this price index (and hence also of inflation), we must specify a base period, against which we will evaluate the price index for the other periods (past and future). This in practice means that we set the value of the price index at the base period equal to $1$ (or $100$).

In the OP's example, the base period is 2010, since they want to express a 2008 amount in "2010 dollars". Let's see then:

The "start period inflation" is an indication of how much prices increased in 2008 compared to 2007. This cannot be relevant to the task at hand. We are examining the relation between 2008 and 2010, the year 2007 is out of the picture.

So conclusion 1: "Start period inflation" is NEVER included in the calculations.

This immediately eliminates the first two possibilities stated by the OP, as being wrong.

Then the issue is, do we include "end-period inflation"? In our example, end period inflation is how much prices increased in 2010 compared to 2009. This is certainly relevant to the task at hand, because we are examining the evolution from 2008 to 2010, and 2009 lies inside.

So conclusion 2: "end period inflation" is ALWAYS included in the calculations.

Combined the two conclusions, tell us that "possibility 3" is what should and is being used.

And it is also evidently logical, because, in order to have the same purchasing power in 2010 as in 2008, we want to cover for the prices increases that happened in 2009 compared to 2008, and the price increases that happened in 2010 compared to 2009.


demand curve shift in a monopolistic competitive market

But my question is, will the slope of the demand curve change as it moves leftward?
The slope does not change because one assumes that relative prices and consumer preferences do not change. This is a ceteris paribus interpreation you want to see what happens when new competitiors enter the market (keeping everything else as it is). There are the same consumers buying the same productons but simply less (in numbers) for that single firm. Graphically the demand curve makes a parallel shift to the left.
But I am wondering, is it possible that the demand curve intersects ATC at a point where MR = MC?
Since graphs are easier to understand than text i made a little drawing, let $D'$ be the competitive demand and $D$ the demand line that intersects the ATC (your question):

The short answer is no. The longer one is, that it is not an optimal outcome. Remember that $(P_m - ATC(Q_m)) \cdot Q_m$ is the potential profit of the firm (This is simply the difference between the ATC and demand curve, the red lines in the graph). If the demand curve interesects the ATC, there will be opportunity for the firm to change its production plan and make a positive profit. This changes the MR curve, new equilibirum will statisfy MR=MC but it will not be at the intersection point of demand and ATC curve (Points A and B). The optimal monopolistic quantity will always be greater A and smaller B because between those two points profit is greater than at the intersection points itself.


How do I represent this indifference curve graphically?

The problem is that there are no indifference "curves" but indifference "areas". Consider the following graph:

For a reference bundle $A$ (equivalent to $\{2,3\}$), the gray regions indicate the areas of indifference, based on your definition of preferences (the black lines are part of the indifference areas).

Thus, by selecting any bundle, you can find the indifference "areas" by plotting the vertical and horizontal lines across that point. Areas to the North-West and South-East will always be the indifference "areas".

For reference, this is the R code to generate the graph:

remove(list = ls())

a=2
b=3

plot(c(0,5), c(0,5), type = "n", xlab = "Sugar", ylab = "Milk", xaxs="i", yaxs="i")
rect(0, b, a, 5, border = NA, col = "grey60")
rect(a, 0, 5, b, border = NA, col = "grey60")
abline(h = b, col = 1, lwd = 2)
abline(v=a, col = 1, lwd = 2)
points(a, b, type="p", pch=19, col="black", bg=NA, cex=2)
text(1, 4, "Indifferent",cex = 1.3)
text(3.5, 1.5, "Indifferent",cex = 1.3)
text(1, 1.5, "Worse off",cex = 1.3)
text(3.5, 4, "Better off",cex = 1.3)
text(2.2, 3.3, "A",cex = 1.3)


Why would companies want to "cancel out" the wage increase? Their goal is to make money, not to keep real wages constant.

Many things factor into the pricing decisions of companies. There are competitive pressures: if rival vendors are decreasing prices the company has to follow suit or lose market share. There are also cost factors: if the price of an input (e.g. labor) rose, than the company may have to increase prices to stay profitable, or just to achieve maximum profit.

If technological improvements, economies of scale, or similar factors enable the company to produce the same quantity of goods at a lower cost, they can afford to decrease prices while keeping wages the same. This enables real wage growth. There are several similar scenarios where prices rose, but nominal wages rose even more.


Should I include country dummies when I combine datasets of 3 countries?

I would advise you to think deeper about your research question first, as this will guide the decision to use country fixed effects. If you would like to exploit cross country variation, for example by studying how the same industry functions differently across countries, then do not use country dummies because it will absorb the variation you want.

If you want to study a causal relation that is independent of country, then using "within-country" variation (i.e. include country dummies) is the most restrictive approach and will sweep out all fixed country characteristics. Your coefficient will then be identified off deviations from within country means.

Are you sure that countries are similar across most characteristics in order to defend not using country dummies? It may be hard to defend solving omitted variable bias on those grounds. If you really want to go this route, then include several country covariates as controls.


Question on the Definition of A Numeraire

numeraire is by definition some unit of measure by which we measure all other units. In your case, let us say that we would decide to use $x$ as a numeraire. In that case we would divide all prices by $p_x$ so the price for $x$ would become $\frac{p_x}{p_x}=1$ (one $x$ can be worth only itself) and for good $y$ the price would be set as $\frac{p_y}{p_x}$.
For example, let us suppose that 1 apple is worth 2 rocks. In such case if we would decide to make apple numeraire everything would be expressed in apples. Now 1 apple would cost 1 apple and 1 rock would cost 0.5 apples (since 1 apple is worth 2 rocks 1 rock would only cost 0.5). If we would choose to make rocks numeraire then 1 apple would now have price 2 rocks and one rock would have price one rock.
If you want to think of it intuitively, you could consider numeraire to be 'currency' used in your particular problem (in real life money is a numeraire - we value all other goods and services in terms of money, let's say dollars, and one dollar itself is just worth one dollar).


Assume for a contradiction that there exists $y$ such that $p\cdot y<w$ and $u(y)\geq u(x^\ast)$. Let
$$\epsilon^\ast= \frac{w-p\cdot y}{ \sum_i p_i}.$$ Then, for all $y'$, if $\ \|y - y'\| < \epsilon^\ast$, we would have $\left|y_i-y_i'\right|<\epsilon^\ast$. Notice that the cost of any bundle $y'$ would satisfy
$$
\begin{eqnarray}
p\cdot y' &=& p_1 y'_1 + p_2 y'_2 + \dots +p_n y'_n\\
&<&p_1 (y_1+\epsilon^\ast) + p_2 (y_2+\epsilon^\ast) + \dots +p_n (y_n+\epsilon^\ast)\\
&=& p\cdot y+\epsilon^\ast\sum_i p_i \\
&=&p\cdot y + \frac{w-p\cdot y}{ \sum_i p_i} \sum_i p_i\\
&=& w
\end{eqnarray}
$$
which shows that for $\epsilon^\ast$, every $y'$ satisfying $\ \|y - y'\| < \epsilon^\ast$ is affordable.

However, since $u$ is locally non satiated, there exists $y^\ast$ with $\ \|y - y^\ast\| < \epsilon^\ast$ such that $y^\ast\succ y$. Since $y^\ast$ is affordable, i.e. $y^\ast\in B(p)$ this leads to a contradiction with the assumption that $x^\ast$ solves the utility maximization problem because transitivity implies that $y^\ast\succ x^\ast.$


Robust predictions in single-agent decision problem with uncertainty

1) Given that $\mathcal{Y}$ is finite, it is obvious that the BCE is not an empty set (or it exists). Remember that the set of BCE's is the set of decisions that are optimal for some belief about $\mathcal{V}$ that could be formed (as a posterior belief) after observing some signal and having a prior $P_V$.

To show there is at least one distribution satisfying the definition if BCE, you can consider an uninformative signal. Given that there is only a finite set of actions, one of them must maximize the expected utility given the prior. If there is more than one, randomize among them as you please. Let the optimal action be $P_Y^*$ (in this notation I capture both situations where the maximizer is unique and when there is more than one maximizer) and define $P_{Y,V}=P_Y^*P_V$, then such a distribution constitutes a BCE.

Now, you should not expect uniqueness because if there was only one distribution that constitutes a, or if BCE was a singleton (since we usually think of BCE as a set), then information is essentially irrelevant. It would mean that there is an action in $\mathcal{Y}$ that is the best action regardless of any extra information you might receive.

2) To characterize this, simply find the optimal strategy when the belief is equal to the prior, $P_V$. If you want to  put it in terms of the two expressions you have, then $P_{Y,V}$ must  be a product distribution, i.e $P_{Y,V}(y,v)=P_Y(y)\cdot P_V(v)$ for some $P_Y(y)\in\Delta(\mathcal{Y})$ (This will capture the fact that the player has no further information beyond the prior and so it cannot correlate his actions with the state of the world). The second inequality will pin down $P_Y$. In fact, using the notation from the previous bullet point, it should be equal to $P_Y^*$.


Your second answer is correct. The problem your first approach has, is that you assume that you may only buy a discrete amount of goods. However, your demand curve is continuous and the usual assumption in microeconomics is that you can consume any amount of this good, for instance $1.478926574$ units. Thus, your first calculation for the surplus is not correct, the correct way to do it is

$$CS = \int_{5}^{10} (10-p) \mathcal{d}p = 10p-\dfrac{p^{2}}{2}|^{10}_{5} = 100-50-50+12.5=12.5$$

This result is equal to the second calculation of yours simply by the fact that it is a triangle. If demand wouldn't be linear, you'd have to use the integral form to calculate CS.


The lender is typically supplying loanable funds in the form of ‘savings’. I.e. money she wouldn’t spend until a future date. But the market for loanable funds allows one to earn interest on these savings rather than laying dormant (think about when people would store money under their mattresses). So, with this surplus of funds, another can make use of these funds (for a price, interest) and increase the amount of spending they themselves do. Hence, the total amount of spending would increase because if that person hadn’t borrowed money to increase their spending, the money would sit in savings.


The equilibrium effects of a change in government purchases on consumption

Generally government purchase would have effect on the price of goods in the market (the "crowding out" effect). The question is just saying that $c$ and $\ell$ are not functions of $g$.


How rate rise by the Fed, higher than market expectation will lead to strengthening of Dollar?

A higher than expected interest rate in the US would result in a higher than expected interest rate differential with other currencies. This will drag more money to the US as investors will try to benefit from this higher interest rate. Moreover, a faster tightening may create the perception that the Fed is more confident about the economic environment. This good news for the US can translate in dollar appreciation.

An appreciated currency makes imports cheaper, which negatively affects inflation.


"Buy a country", expression or possibility?

What do you mean? A country per se cannot be bought. The wealthiest man, Gates, has a net worth of around 67 billion USD (forbes). What could he buy?

GDP

Whatever it the country produces. The poorest country by GDP per capita is Burundi, with a total GDP per capita of around 267 USD, and a population of 10 million. That gives then a total GDP per year of 2.7 billion USD. Gates could easily afford that.

Wealth

We don't have - to my knowledge - reliable wealth data on most developing countries, but the wealth/income ratio is typically around 4-6. Let's assume the maximum, 6. That would give the people of Burundi 13 billion USD, still affordable.

Public Assets

Let's pick a random country, switzerland. They have public assets worth 200 billion USD. Too much for Gates to afford. But for a smaller country, he could surely buy it out.

The answer is somewhat unsatisfactory. Why? Because the question is. In the end, before asking can xyz really happen, you should ask what is xyz  and what do I mean by it?

What else is there?

I'm not aware of a country where the people are still counted as property, so you most likely can't buy them.

Perhaps you could buy all the land? There are many small islands (potentially for sale) that would grant you independence. Some people (mostly rich and eccentric) own them, which means I presume that they bought those countries at some point. So, it turns out,

  You can buy a country

but you can't buy the people. !Viva la revoluçion!


Which database language and database management system should an economics student learn? (Provided that one has to be learned)

First, some clarity. The standard language used for interfacing with a DBMS is SQL. However, its syntax and functions slightly vary according to the DBMS system you are interfacing with.
The most popular ones are MySQL and PostgreSQL. The choice between them is mostly indifferent. In my personal experience, I found that building a MySQL database from scratch is slightly easier, even if PostgreSQL is more flexible.
If you are a complete beginner, first I suggest you to start by learning SQL syntax and best practices, and if you need to build your own database by choosing between one of these two.


Illustrating difficulty to find dominant strategies: series of simple one-shot two-players games

Start with the following matrix.
$$ \begin{matrix}1 && 0 && 0\\
1 && 1 && 0\\
1 && 1 && 1  \end{matrix}$$

You will choose a $1$ in the matrix and change it to a $0$. The other player will choose a $0$ and change it to a $1$. Your goal is for the resulting matrix to have an even determinant. The other player's goal is for the resulting matrix to have an odd determinant.

Hover over the box to find out the dominant strategy:

   Change the $1$ in the bottom left corner to a $0$.


It is correct that quantitative easing (QE) is not exactly the same as just printing money. However, I would hesitate to say equating it to printing money is complete misnomer (although its not accurate analogy either) and I also dont think that your description of QE is completely accurate.
QE actually begins with creation of new money. As explained by Fed St. Louis QE is defined as:

large-scale asset purchases—in the hundreds of billions of dollars range—of, for example, mortgage-backed securities and Treasury securities.

Moreover, this large-scale asset purchase is payed for by Fed creating new reserves. This creation of new reserves is in principle equivalent of actually physically printing out the bills even though it is done electronically.
The reason why the above should not be  equated to just printing new money (as explained here in this VOX article) is that:

asset purchases by the central bank are financed by money creation, but not money in the form of bank notes. The money is in the form of reserves held at the central bank. That form of money is a liability of the public sector – taking the central bank and the government together in an expanded definition of the public sector. It is a liability that pays the interest rate set by the central bank.

The above is why QE can be considered  an asset swap and why you can argue that the asset purchase just by itself is not money creation (although it indirectly encourages it) as it also reduces the money supply since Fed (or other central bank) is taking the money (in form of those bonds that can count as broad money) out of economy.
However, there is a caveat to this. The ultimate purpose of QE is to increase money supply which also happens when you print new money. The QE, if successful, does de facto increase the money supply when it encourages more lending. Even if the original swap of reserves for bonds might leave broad money supply unchanged the additional lending will increase it.
Thus this is essentially an argument about proper nomenclature/semantics. For example, lets say that in certain situation we know that if central bank would lower the interest rate by some percentage the corresponding increase in lending would make money supply expand by $\\\$ 1000$ and we know we could also do that directly by just printing out the $\\\$ 100$ bill which through multiplier (assuming $10\%$ reserve ratio) would also expand the money supply by $\\\$1000$. Hence, is it valid to call cutting interest rate printing money? Well certainly from narrow scientific semantic point of view that would be a misnomer as it would not follow proper economic nomenclature that makes distinction between the two actions. But this being said, since those actions would have equivalent effect I would not say that using such analogy is completely incorrect.
On a certain level this really is just an English language/semantics problem - If you go somewhere and later realize that you have to turn back and backtrack to the same spot would saying; 'well that is the same as staying still,'  be a good analogy or a misnomer? The actions have different implications in terms of effort etc. yet in both cases outcome is the same.  Moreover, before turning to economics I used to study physics which abounds with analogies that are far more outrageous yet commonly used in classroom (see this rant).
Hence to sum up, while calling QE printing money is not completely proper, the analogy has some merit to it.


what percentage of a banks money do they lend

I'll put my comment as an answer and add one or two things. To see how much a bank lends in comparison to deposits, you can look at what is called the loan-to-deposit (LTD)ratio of banks in their annual reports. Usually it is between 60% and 90% but as you mentioned this can vary upon institutions .. And geographic areas as well ! For instance, in 2015 in the EU the average LTD ratio was 108% whereas in the US the average is about 79%. You can check per state the LTD ratio from data gathered by the Fed and the FDIC.

As to factors that determine this amount, I would probably put regulation as the first point. If I had to think about other macro points that could affect the LTD I would probably mention interest rate and in particular LIBOR rates as they tend to give an idea about the financial stress on the banking level and therefore to a certain extend the willingness for a bank to lend money and/or to require more liabilities.


The translog function can be used not only in preferences but also in production and cost functions. I am not very familiar with its implications in consumer theory, but from the production point of view, i've seen it widely used.
The Translog Function doesn't impose additivity and homogeneity, and hence Constant Elasticity of Substitution. This is interesting because it doesn't require a "smooth" substitution among the inputs (in production analysis). I guess that in consumer theory the interpretation would be similar.
So basically, the translog function is less restrictive than a cobb-douglas. If you impose some restrictions while calculating the translog function parameters, you get a cobb-douglas function. That is why it is a "generalization". In other words, the cobb-douglas is a specific case of the Translog function imposing additivity and homogeneity (i.e. imposing constant elasticity of substitution).

Edit: i added more info to reply your comment.
I think that the other reply is more complete than mine. But i'm just going to add something i consider useful for you to have a broader understanding. I assume that you're familiar with indifference curves. I refer you to this site (from where i took the graphs), in case you're not.
An indifference curve is just a mapping of all the combinations of two (or more) goods that give you the same utility, or "make you happy at the same level".
First, see this indifference curve:

Fig 1: source
This setting is known as "complements". Because as you can see, adding one thousand units of good x (that is moving to the right), without adding good y (that is not move upwards) does not make you happier: you move along the indifference curve. Think of this as the left shoe and the right shoe. It is useless to have a thousand additional left shoes without adding a right shoe because they are perfect complements.
Now, look at this one:

Fig 2: source
This one is called "substitutes". It's the opposite case to the complements. You can think of this as beef and chicken. You can cook using only beef, or you can substitute and cook using only chicken. But you can cook with certain combination too, say 150 grams of beef and 100 grams of chicken beccause they are perfect substitutes (Sorry, i couldn't come up with a better example but this one makes the point).
Now, this extreme cases make easier to imagine all the settings that are "in the middle". That is, two types of good that are not perfect complements neither perfect substitutes. Think of food and drinks. They can't be perfect substitutes because you can't have lots of food without drinks. The're not perfect complements either because the mix of food and drinks is not fixed. For this setting the cobb-douglas could be a nice approximation as can be seen in the next figure:

Fig 3: source
Now, the Cobb-Douglas utility function doesn't solve everything, as it imposes certain constraints by construction. For example, the line that goes from the origin through all the curves (the expansion path) is 45° and straight by construction: it cannot be changed. This means that as you get richer (even infintely rich), your preferences over this goods remains constant. The formal name is homotheticity or homothetic preferences. This is empirically false, as it has been shown that the richer you are, you use a smaller share of your income for food. With Cobb-Douglas preferences, this cannot happen. Translog preferences relaxes this assumption.
In the next figure, you have a utility map relaxing the homotheticity assumption:

Fig 4: source
Think of this graph as good y being food and good x being entertainment. As you get richer (or farther from the origin), you will destin more of your income to entertainment.
Finally, i will talk about the Elasticity of Substitution which is known as $\sigma$ (sigma) which can be imagined as being the curvature of the indifference curve. In the Fig 1, the perfect complements $\sigma = 0$: No curvature. In the perfect substitutes, $\sigma = infinity$: straigt line. In Cobb-Douglas, $\sigma = 1$: a slight curvature. Nonetheless, as you get richer (distant to the origin), this Elasticity of substitution remain constant in the three settings. Even in the non homothetic preferences seen in Fig 4, the elasticity of substitution remains constant. These are the **Constant Elasticity of Substitution (CES)**preferences. But what if you allow the curve to have different shapes as you get richer? Look at Fig 5:

source
In this example, the indifference curves get less elastic every time. Hence, they are not CES preferences. The advantage of the Translog preferences is that, since you don't impose neither CES neither homotheticity, you can test this hypothesis with observed data. You can see that the Translog utility function is much less restrictive than the Cobb-Douglas preferences.
As a final remark, i shall say that it can be the case that you do not reject the hypothesis of homotheticity, CES and $\sigma = 1$ in a dataset of observed behavior. This would leave you in a Cobb-Douglas preferences setting. So, by using Translog you're not necessarily ruling out Cobb-Douglas.


You are using it correctly. It means the processes and protocols created by banks or those imposed to them by authorities. If the bank also has some quality certificate, like iSOs, them this imposes new compliance costs and processes.


Would some help me fill in missing steps from a textbook exercise left for the reader

Overall it is almost correct but there are some small mistakes and it is incomplete.
1st Mistake:
in step (1) you write:

$\frac{Y_t}{L_t}=\frac{K^{\alpha}_t}{L_t}\frac{L^{1-\alpha}_t}{L_t}$

But it should actually be
$$\frac{Y_t}{L_t}=\frac{K^{\alpha}_t L^{1-\alpha}_t}{L_t}$$
You cannot divide LHS of the equation by $L$ and RHS by $L L \implies L^2$ as that will generally violate the equality between $ Y= K^{\alpha}L^{1-\alpha}$.
However, this error had no impact on the rest of the problem because in spite of making this mistake you still used the correct definition for output per capita:  $y_t=k^{\alpha}_t$
2nd mistake
you say you divide both sides by $k_t$, but you forgot to do that for the last term. You write:

$0 = \frac{\phi[k^{\alpha}_t\sigma]}{k_t}+\frac{sk^{\alpha}_t}{k_t}+\frac{\lambda[k^{\alpha}_t\sigma]}{k_t}-(\delta+n)k_t$

correct steady state equation should be:
$$0 = \frac{\phi[k^{\alpha}_t\sigma]}{k_t}+\frac{sk^{\alpha}_t}{k_t}+\frac{\lambda[k^{\alpha}_t\sigma]}{k_t}-(\delta+n) \tag{*}$$
3rd 'mistake': It is unfinished
I am using here ‘mistake’ in quotes because this is not a mistake in the same sense as other ones but it is still not good practice to leave problem unfinished. If you would do it on an exam it would cost you some points (some professors might even refuse to give any points for unfinished work).
I get the impulse to just call it a day since you are in home stretch now but proper answer should have solution for $k^*$ (unless the problem specifically states it does not want full solution which I don’t see in this case in your Q).
Solving the equation $*$ for optimal $k^*$ yields:
$$ k^* =  \left( \frac{(\phi+\lambda)\sigma +s}{\delta +n} \right)^{\frac{1}{1-\delta}}$$
The above being said don’t get discouraged most of the attempt was correct. Just be more careful about the arithmetic and don’t stop just before finishing.


The National Bureau of Economic Research is a good source for the latest in economic research: https://nber.org/

You can also go to Brookings institute:  https://www.brookings.edu/

Federal Reserve Board:  https://www.federalreserve.gov/econres.htm


How Does GDP Differ From NNW as an economic measure?

I cannot find any reference to a specific "Net National Wealth". But I assume that you refer to the National Wealth, in general (also called Net National Worth).

The GDP has been explained in another question, so I will not come back to it.

The National Wealth is a measure of the total wealth of a country: assets, liability, etc.

To my understanding, the main difference resides on the fact that the NW is a rather static measure, whereas the GDP focuses on the dynamic part. Some examples

the value of a house will appear in NW, but not in the GDP;
if you buy a product from your neighbour, the NW will stay constant, but the GDP will increase.

If a country has a high GDP, its citizens/companies should accumulate wealth, and thus the NW should increase. But as the GDP also contains movement within the country those are not reflected in the NW. A diamond is in NW, the operations to cut it is in the GDP.


What are some good resources to read on Development Economics from a neoliberal or Austrian perspective?

The reason why you cant find any sources on this is that you seem to be confused about what development economics and those different schools of economics are. I will first address the confusion and then give you references for some of the sources you requested.
Confusion about different terms:
First, development economics is a positive not normative field. Following the
Debraj Ray and Clive Bell  Development Economics, the development economics is defined as follows:

a subject that studies growth, inequality, poverty and institutions in the developing world

An important takeaway is that this definition implies that development economics (as well as a wast majority of economic fields) is a positive field. It is concerned with what is not what ought to be.
On the other hand both Chicago School and Austrian Schools nowadays do not exist as a separate positive fields but they are almost purely normative schools that do interact with positive economics through study of political economy (i.e. the field of economics that, among other things, examines what are implications of different ideologies on economic policy. See the ‘Political Economy’ by Peter Groenewegen or this investopedia entry).
Historically in their respective time periods these schools were associated also with some positive economics. An example of this would be the problem of socialist calculation debate where members of Asutrian school such as Hayek were heavily involved.
However, in present day those positive propositions of Austrian or Chicago school that panned out were incorporated into mainstream neoclassical theory and the schools for most part do not pursue any positive research projects.
Next in you question you seem to equate Neoclassical economics with Chicago school which is simply not correct. You should not confuse Neoclassical economics with Chicago school as Neoclassical economics is a set of tools rather than doctrine.
For example, Sargent is considered to be a representative of 'Chicago school' because of his ideological beliefs but his whole work is Neoclassical. At the same time, Stiglitz who is ideologically as left leaning and anti-free market as reasonably possible within the mainstream of the profession (we are talking about a person who praised Hugo Chavez) is also purely Neoclasical economist and would probably get a stroke if you would try to associate him with Chicago school or Austrian school both of which he criticizes heavily. Next, a firmly New Keynesian economist such as Gregory Mankiw is ideologically pure libertarian/classical liberal clearly ideologically influenced by Milton Friedman and Hayek, yet his whole work could not be further from Neoclassical economics.
When it comes to 'Neo-liberal economics' there is not even such thing within economics. The New Palgrave Dictionary of Economics does not even has entry for 'Neo-liberal economics' only for Neo-liberalism as an ideology, and certainly in political economics you might study how Neo-liberal ideology can lead to different economic policies but there really is not a 'Neo-liberal economics' as a proper sub-field or even school of economic thought.
Hence a big problem with your question is that you keep mixing positive approaches and sets of tools in economics such as the Neoclassical economics with a normative largely ideological and political economic perspectives that Austrian or Chicago school in present day consist of.
Resources:
Now finally after getting all  the above out of a way, if you want to get some resources on Neoclassical perspectives on development economics then any textbook will do. A large part of development economics consist of purely neoclassical models. A good source is a Handbook of development economics by Rodrick & Rosenzweig (2009). A large number of models presented in the book can be safely classified as purely neoclassical models (albeit definitely not all). An excellent example would be chapters in the handbook that focus on economic growth that are for most part purely neoclassical.
Furthermore, you can have a look at sources cited in the handbook to get to see even more Neoclassical models and perspectives on development economics although the handbook is quite exhaustive when it comes to the current research and thinking on economic development.
If you want to get Austrian or Chicago perspective on economic development then you really have to study political economy and not positive fields such as development economics.
For example, you can get some idea about the perspectives of Chicago school on economic development by reading Milton Firedman's Capitalism and Freedom, where he shares among other things also opinions on why Hong Kong or Japan successfully developed.
For an Austrian perspective you can read some of the works of Benjamin Powell who identifies as Austrian economist (although his academic work seems to be more or less neoclassical) who shares his thinking on political economy and development in books such as Out of Poverty: Sweatshops in the Global Economy (Cambridge University Press, 2014) or Making Poor Nations Rich: Entrepreneurship and the Process of Development (Stanford University Press, 2008).


No competitive equilibrium for pooling contracts

As one of the comments points out, there are many models of equilibria in insurance markets. But it sounds to me like you are referring to the Rothschild-Stiglitz (1976) paradigm. I will provide a basic summary of the key takeaways here, but there is a more complete explanation in this set of slides and in those from any second-year graduate course in Public Economics.

In this paradigm, mixed strategy equilibria are ruled out. So when we say that "no equilibrium exists", we are actually just saying that no pure strategy equilibrium exists.
There is never a pooling equilibrium. The basic reason for this is that a firm could profitably come in and offer a cheaper package with less insurance and poach away the low risk types. This leaves the existing firm with only the high risk types and it makes a loss.
There may or may not exist a separating equilibrium. If there does not, it is because - starting from a separating equilibrium - there exist pareto-improving cross-subsidies (from high to low risk types). This means that a firm can profitably come in and offer a package that attracts both types (i.e., a pooling contract).
EDIT: To answer your second question, the separating equilibrium - if one exists - is indeed not pareto efficient but it does benefit both high and low risk types. The pareto efficient separating contract would offer full insurance to both type, but the Rothschild-Stiglitz candidate cannot. To see why, suppose that it did. Then both receive full insurance but the low-risk types get it for less. Clearly the high risk types will pretend to be low types and the incumbent firm will make a loss.

In summary, this particular model offers a theory of pure strategy Nash Equilibria in insurance markets with adverse selection and multiple risk types. An equilibrium may or may not exist. If it does, then it is separating.

Additional information

There are competing paradigms that solve this problem with assumptions about the timing of when offers are made and accepted (e.g., Miyazaki-Wilson-Spence). Another classic model to compare is Akerlof-style adverse selection. Neither alternative suffers from the non-existence problem.


What does "sample period" in this case mean?

It is definitely hard to interpret, and I wish the authors had been more clear.
However I think "sample period" refers to the whole dataset.  There is actually nothing in the identification or data sections that suggests that the authors are using the [-2,+5] window around law implementation in the main regression results. I could be missing something but I only see this window mentioned describing the graphical analysis at the very beginning of the results section.  It is not very clear to have a graphical analysis that doesn't actually correspond to the regression being run, but I think maybe that is what is going on.  If the authors were running a regression based on offsetting the data around the law passed I think they would have described the "t" index in the regression equations as referring to "years since passage of the lenience law" rather than simply "years".
A way to confirm with certainty that the authors are not using the seven year window in the main regression analysis would be to check the sample size under the regression tables. Seven years of data per country will generate a very different N to 22 years of data per country.


Compute variables in perfect competition, in monopoly and duopoly

Under perfect competition, $p=MC$. So you may substitute this price into the demand equation to obtain the quantity.

A monopolist chooses output in a way such that $MR=MC$. So you have $P‘(Q)*Q+P=MC(Q)$, where $P(Q)$ is the inverse demand  equation.

For Cournot, it is also simple. Note that the profit function of firm i is given by $$\pi(i)=P(q_{i}+q_{-i})*q_{i}-MC(q_{i})*q_{i}$$. You just use the first order condition to solve for the price and quantitiy respectively. $q_{-i}$ denotes the output of all players except for $i$.

Finally, for Stackelberg I would like to know how many players/firms you have. Then, I would be able to explain it to you in a suitable way. I hope this helps.


What makes a shareholder a "shareholder of record"?

I typed "shareholders of record" into Google, and I got

Stockholder of Record
Also known as shareholder of record, record holder or owner, or registered holder, or owner. The stockholder whose name is listed on the company's books and records as the owner of shares as of a particular date (the record date). A stockholder of record holds stock in a direct relationship with the company and has direct title to the shares, as opposed to a beneficial owner that owns stock indirectly through a broker, dealer, bank or other entity. Beneficial owners holding through a broker, dealer or bank are also said to hold their shares in street name.

Source

A. You have to have a direct relationship with the company.
B. Seems like the one share you can buy through a brokerage won't cut it. If somehow, before the company went public you made an agreement with them to get a share and hold it in your name - I guess?
C. You can if you are a shareholder:

Shareholders [...] are entitled to review only basic documents, such as the charter, bylaws, minutes of board meetings, and the list of shareholders of record.

Source: Page 14 of Velasco: The Fundamental Rights of the
Shareholder


The answer depends on whether you want accuracy with respect to original (national) sources, more comparability, and how much time you want to spend compiling data.

If you want to focus on a very few countries, or even just one country, I suggest to always go to the primary source, which is the national statistical office, or central bank (depending on the country). There you will have ample documentation about what is being measured. Albeit countries are urged to use the latest UN System of National Accounts, there might be particular issues that are nation-specific, which you will find in the national documentation. Additionally, you will find references to GDP measured in different ways (output, income, or expenditure approaches), which do not always give the same result.

To compare a few countries, you can use primary sources and then decide yourself how to compare them. For example, you can compare growth rates, transform GDP into an index (e.g. GDP=100 in 1990), or into a common currency, where you can decide which exchange rate to use.

This approach on primary sources gives you more accuracy with respect to national data, and also greater transparency in what you are actually measuring or comparing.

Now, if you want to compare many countries, or entire regions (e.g. OECD, EU, NAFTA, etc), using a consistent definition, or perhaps have little time for this comparative exercise, it might be better to use some international database like World Bank or Penn Tables. Here, you depend on the particular arrangements of each institutions (in terms of assumptions regarding exchange rates, price indexes, etc).

For example, the World Bank data states its source as World Bank national accounts data, and OECD National Accounts data files.  And yet, it indicates some limitations of the calculations:

  Limitations and Exceptions: Gross domestic product (GDP), though widely tracked, may not always be the most relevant summary of aggregated economic performance for all economies, especially when production occurs at the expense of consuming capital stock. While GDP estimates based on the production approach are generally more reliable than estimates compiled from the income or expenditure side, different countries use different definitions, methods, and reporting standards. World Bank staff review the quality of national accounts data and sometimes make adjustments to improve consistency with international guidelines. Nevertheless, significant discrepancies remain between international standards and actual practice. Many statistical offices, especially those in developing countries, face severe limitations in the resources, time, training, and budgets required to produce reliable and comprehensive series of national accounts statistics. Among the difficulties faced by compilers of national accounts is the extent of unreported economic activity in the informal or secondary economy. In developing countries a large share of agricultural output is either not exchanged (because it is consumed within the household) or not exchanged for money.

The Penn Tables are also based on national accounts. As the Documentation states:

  One of the major inputs into PWT is National Accounts (NA) data on gross domestic product (GDP)  at  current  and  constant  prices, in  local  currency  units.  [...] We  primarily  source  these  data  from  the  UN  National  Accounts  Main  Aggregates Database,
  see http://unstats.un.org/unsd/snaama/Introduction.asp

There are difference between these two sources. On the one hand, the WB database offer data in local currency units and US dollars (either in current or constant prices), and also in PPP. Additionally, the WB dataset includes all the exchange rates used to calculate its series. Penn Tables only include data in constant USD and PPP.

On the other hand, Penn Tables include data on PPP GDP measured in terms of output and expenditures. With WB, it is not clear which they are using for each country. Additionally, Penn Tables data is usually longer (some series start in 1950).

To check consistency across these databases, I made a few comparisons between the WB and Penn Tables data, for constant USD dollars, and for all countries I tried (France, India, Brazil, Canada), the correlation between the two series was 0.999). So clearly, there might not be much of an issue which one you use (as long as the one you want is available).

Finally, regarding IMF, I would not use these. Their comparative advantage is not to be a database of GDP statistics but of financial and government variables. Its database is not intuitive to use, and GDP documentation is in my opinion of low quality. If you want other alternatives to the above two, you can check UN stats, OECD, Eurostat, or other regional bodies. Finally, you can also use FRED.

(there are other benefits of Penn Tables, not related to GDP itself, which is that you can compare components of GDP (e.g. investment, government spending, etc) or TFP in a consistent fashion across countries)

TL;DR

For single or few country analysis, use primary source (national statistic office or central bank offices). For comparative analysis across many countries or macro-regions, the World Bank is the most comprehensive database. Penn Tables include just a few definitions of GDP. In any case, for common definitions, the two datasets are practically identical.


Set of Pareto efficient allocations is given by the dashed line in the Edgeworth Box. It is the set of feasible allocations satisfying $y_1 = x_1$ and $x_1y_1 \geq 9$
.


Not sure about utility function being $\min$ and it being a case of perfect substitutes...

In general, I would say there are a few different types of utility functions but if you know more or less the theory and can imagine various functions' graphs, then (as mentioned in the comment) you will be fine.

So I think we should start with a Cobb-Douglas preferences which can be represented by utility function of the form: $$u(x_1,x_2) = x_1^ax_2^b \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

Then you have a perfect-substitute preferences which can be represented by utility function of the form: $$u(x_1,x_2) = ax_1 + bx_2 \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

You can also have a perfect-complement preferences which can be represented by utility function of the form: $$u(x_1,x_2) = \min\{ax_1,bx_2\} \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

Finally, you can have a quasi-linear preferences which can be represented by utility function of the form: $$u(x_1,x_2) = f(x_1) +x_2 \:\:\: \textrm{where f is increasing and concave}$$

In all of the above graphs I assumed $a=b=1$ for simplicity. In the last case the concave function was a square root.
Your example is (more or less):

You can now try and argue what kind of preferences these are.

If you are interested in consumer theory, you should find some proper course book or other materials - there's plenty of it on the Internet - just google it.

The last piece of advice I can give you - try to draw as many graphs as you can think of and then try to find a story that the graph is explaining. Or the other way round. I remember my professors trying to "play" with us by giving us different kinds of utility functions - when you drew them, they looked "strange" but when you actually thought about them - they did have some story to tell.

EDIT.
There is also a special case in a way - monotonic transformation is a real-valued function $f: \mathbb{R} \to \mathbb{R}$ which basically transforms numbers into numbers in a way that preserves the order of the numbers, that is, for every two numbers $p$ and $k$ it hods that $p>k \implies f(p)>f(k) $. This means that if a function $f$ is a monotonic transformation of utility $u( \cdot )$, then $f(u(\cdot))$ also represents the consumer’s preferences. $$(x_1,x_2) \succeq (y_1,y_2) \iff f(u(x_1,x_2)) > f (u(y_1,y_2))$$

All this means is that you need to be careful for a case of utility function like: $\ln{x_1} + \ln{x_2}$ which is just a natural logarithm of Cobb-Douglas preferences $x_1x_2$ with $a=b=1$.


When to prefer maximisation of surplus and when instead to prefer minimising differences in price and q to find market equilibrium?

Market Equilibrium occurs when the difference between demand and supply is minimized. Equilibrium price and quantity is determined by solving this problem.

Surplus Maximization (or efficiency) is a property that we would want equilibrium to have. First Welfare Theorem says that in competitive markets (in the absence of externalities) market equilibrium is surplus maximizing. In standard demand-supply framework, solving for equilibrium and efficiency yields identical outcomes, and that is why these methods are used interchangeably.


Why people respond to surveys like that of Purchasing Managers' Index?

For your first question:

Some have noted that surveys make people feel like they are a part of something bigger than themselves and that their opinion matters.

According to Ed Halteman (survey expert) two reasons why people participate in surveys, ,  are that they feel their participation will affect something they care about, and they want to share their opinion with someone who will listen and act on the information.

For your second question:

Once the surveys(or census) are collected there are two stages that typically takes place to ensure that the data products are of high quality.  They are data processing and statistical adjustments.

In data processing, a survey statistician or analysts,   use codes to capture data from their raw state to an edited data file that can be used within the survey organization for quality assessment in preparation for statistical adjustment and dissemination.

They use coding to classify survey responses into categories with associated numeric values. They also decide how coding and data capture will be conducted and monitored. A survey statistician or analysts will edit the data to check for errors throughout the survey lifecycle.

Statistical adjustments:  Consideration will be made for missing data or data values that are obviously wrong using single or multiple imputation methods.

Analysts will use codes to calculate outcome rates based on available data according to the agencies methodology(s).  They will also develop survey weights for each interviewed element on the sampling frame.

Throughout the above processes, documentation is done during both data processing and statistical adjustment steps.


Why is the annual transferred money worldwide quite different?

There is no contradiction here as not every transfer is a remittance.

  remit
  late Middle English: from Latin remittere ‘send back, restore’

It is my understanding that the word remittance is usually used to describe transfers by foreign workers to their home countries. From Wikipedia:

  A remittance is a transfer of money by a foreign worker to an individual in his or her home country.

These transfers may be conducted via SWIFT. As such they make up a small part of all SWIFT traffic, which would also include domestic payments, international business transfers, etc. So it is quite possible that the yearly volume of remittances is $\$5 \cdot 10^{11} $ whereas the yearly volume of SWIFT transfers is $\$1.25 \cdot 10^{15} $.


Treatment of natural disasters and wars in consumption of fixed capital

I am answering this both as a PhD in economics and as a Certified Accountant (I am both).
This quote for the wikipedia page is garbage, because it comes after the article has correctly informed the reader that CFC is an estimate that attempts to capture the economic reality rather than the accounting convention as regards consumption of capital for productive purposes.
If we characterize such estimates as "capricious", then all economic theory and econometric practice is "capricious". Certainly there are people who believe that, but one could imagine that we don't think highly of such opinions in this forum. I don't think such across-the-board unsupported criticism can have, or deserves to have, a "definite answer" from some "respectable source". In other words, I do not have to prove that I am not an elephant. Let the other side prove (not just argue) that I am.
More over, the sentence seems to imply that accounting/tax data are somehow "true to the essence of things". My objection to this is not related to the possibility of falsified records to avoid taxation or boost profits etc. This phenomenon exists but all past experience says that it does not exist to such extend as to make accounting/tax data unreliable on the aggregate.
So assume perfectly "honest" accounting data. In what sense are these data true to the essence of things? What is the essence of things here? Is the historical cost (which is a certain piece of information but referring to the Past), relevant for the Present?
Also, the passage appears to ignore that modern official accounting standards (both FASB and IFRS) advocate more and more for using "fair values"/estimates related to balance sheet assets than historical costs, exactly in an attempt to capture the economic reality of things, and in full knowledge that this will lead us to estimated Financial Statements, with a degree of uncertainty, inaccuracy, etc... So what? The purpose of Financial Statements is not perform or reflect account-settling, but to be the base of financial and economic analysis.
We "settle accounts" with the government through the filing of tax returns forms. We settle accounts with customers, suppliers, personnel, banks, etc by using the accounting ledgers, journals, the detailed bookkeeping and records of financial transactions. But we adjust all these data to produce Financial Statements, to the degree that the transactional reality differs from the economic reality. The fact that we may disagree as to what is the economic reality after all, should not surprise us, nor hinder us in the pursuit of relevant knowledge. It will not be the first "reality" that we disagree on.


What does Adam Smith mean when he says wages are determined by: "advancing, stationary, or declining state of the society."

Taken literary the passages do not hold up, but the passages also have to be read in the context they were written. An important caveat is that Adam Smith actually did not propose any explicit theory of wage growth rather the above quotes are observations he made at the time of the writing (at least I think that the passages when read in the context of the chapter should be interpreted as empirical observations). Take the first quote:

  It is not the actual greatness of national wealth, but its continual increase, which occasions a rise in the wages of labour. It is not, accordingly, in the richest countries, but in the most thriving, or in those which are growing rich the fastest, that the wages of labour are highest.

This observation was made on the basis of him comparing England which was traditionally a backwater - a poor country, with stagnant but 'rich' countries such as Qing dynasty China.  But the observation itself was incorrect. At the time no published real GDP statistics existed and so Smith just thought that China is richer due to the fact that in the West China was often mystified and its wealth was exaggerated.

As the Maddison's estimates show (see graph at the bottom), at the time of publishing of the Wealth of Nations the Britain was already much richer than China.  However, even at that time it was well known China was stagnant and UK was growing rapidly due to its early industrialization and so Smith conflated those.

Moreover, also in recent history we can see that Euro-area (comprised of mostly Rich developed countries) in last decade experienced very slow growth - that although technically not stagnation comes close to it - has higher real wages than China which  experiences rapid growth (between 2011 and 2019 it was always higher than $6\%$), yet real wages are higher in the Euro-zone than in China.

However,  wages tend to raise with economic growth. In fact from macroeconomic perspective there is no meaningful distinction between output and income. Moreover, until very recently average wages tracked productivity very closely and even though in US it is no longer true that they do in many places they still continue to do so. Over long periods of time eventually the output and wages will be higher in countries which are 'thriving' in sense they experience economic growth. This can be shown in the same estimates from the Maddison which shows that although all of the countries in the graph were roughly equally poor in 1500, the countries that experienced high economic growth, are the ones that now have relatively high average wages and standards of living.

So to sum up Adam Smith was not wrong in observing that wages tend to be higher in the 'most thriving' countries, but he was not correct when he said the wealth of the country - at least understood as the output of the country or GDP is not accompanied with high wages.

Of course, converse applies as well. A country which experiences negative economic growth will see its real wages and standards of living falling, and stagnant countries will experience stagnation also with respect to real wages.


Modifying Hotelling's lemma: Is this valid?

You are basically on the right track but might have been carried away by not writing down function arguments. The profit function $\pi$ has arguments $p$ and $w$; don't drop them.

In Hotelling's lemma, $y$ is the output. On the other hand, you write $y$ for the production function. Let's make things clear by writing neutrally $f$ for the production function. The profit-maximizing input choice is $x(p,w)$, so the quantity produced is $f\big(x(p,w)\big)$, the correct meaning of $y$ in Hotelling's lemma.

Hotelling's lemma then becomes
$$\frac{\partial \pi(p,w)}{\partial p}=f\big(x(p,w)\big).$$
The profit function is given by
$$\pi(p,w)=pf\big(x(p,w)\big)-wx(p,w).$$

Taking the derivative with respect to $p$ gives us
$$\frac{\partial \pi(p,w)}{\partial p}=f\big(x(p,w)\big)+pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial p}-w\frac{\partial x(p,w)}{\partial p}.$$
This is pretty much what you got there above.

Now $x(p,w)$ is by definition a maximizer of $pf(x)-wx$. We have the first order condition
$pf'(x^*)=w$. Therefore,
$$pf'\big(x(p,w)\big)-w=0,$$
which implies $$pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial p}-w\frac{\partial x(p,w)}{\partial p}=0$$
This reduces our expression for ${\partial \pi(p,w)}/{\partial p}$ to Hotelling's lemma.

This tell's us also what the important message of Hotelling's lemma is: Indirect effects don't matter. A marginal change in prices has a direct effect on profits since the output gets multiplied by a different number. There is also an indirect effect that comes from firms adjusting how much to produce. Hotelling's lemma just tells us that the latter effect is zero for a profit-maximizing firm, which need not do much adjusting. At a profit maximum, the marginal effect of adjusting production on profits is zero.


Are simultaneous equation models out of fashion? Why?

I think this is already answered in the paper itself and in addition I think this is due to the way how they define 'simultaneous equations' topics.
First as authors of that paper opine:

This presumably reflects
declining use of an orthodox multi-equation framework, especially in macroeconomics. The reduced coverage of Simultaneous equations has made space for modest
attention to Panel data and Causal effects, but the biggest single expansion has been
in the coverage of Functional form (mostly discrete choice and limited dependent
variable models).

In addition I would add that macroeconomic modelling also shifted more towards not just panel data but also time series as macroeconomic data with lengthy $T$ dimension became more readily avaiable.
Second, note how they define simultaneous equation models (in table 1):

Simultaneous equations models -  Discussion of multi-equation models and estimators, including identification of simultaneous equation systems and
system estimators like seemingly unrelated regressions (SUR)
and three-stage least squares (3SLS)

Third, note how they define time series [emphasis mine] (again from table 1):

Time series - Time series issues, including distributed lag models, stochastic processes, autoregressive integrated moving average
(ARIMA) modeling, vector autoregressions, and unit root
tests. This category omits narrow discussions of serial correlation as a violation of classical assumptions

Fourth, instrumental variables have their own category (again from table 1):

Instrumental variables (IV), two-stage least squares (2SLS), and
other single equation IV-estimators like limited information
maximium likelihood (LIML) and k-class estimators, the use of
IV for omitted variables and errors-in-variables problems.

As far as I can see from skimming the paper it is not possible for a topic to be in two categories at once. So in addition to the switch toward panel and time series methods note that they do not seem to define simultaneous equations the same way as they are colloquially known in economics.
For example, I often hear at conferences people colloquially referring to VAR, VECM or 2SLS as a 'simultaneous equations' model, yet authors here created separate categories for them. Yet, these models would be not included in the simultaneous equation category, and we can see that the time series category and IV category expanded relatively to the simultaneous equations category (in comparison to 1970s - table 2 column 1 vs 3).
Lastly, many simultaneous equations approaches (as defined in the paper) like 3SLS, are usually considered advanced topics. The research you cite focuses on undergraduate texts. I don't think any econometrics teacher would go over 3SLS before going over 2SLS and a reasonable writer of undergraduate econometric textbook might want to make the textbook narrow and focused as opposed to forever expanding page count as new techniques emerge. Hence, I think that often topics like 3SLS simply won't make a cut compared to more basic techniques like IV when it comes to undergraduate textbooks.
Response to questions in edit:

Were these models found especially problematic?

I don't think these models are considered generally problematic. At least no more problematic than other strands of models having separate categories in the paper you cite that are used more. I was not also able to find some work that would consider simultaneous equations approach especially problematic (although absence of evidence might not be evidence of absence). I am a graduate student and I visit conferences where I see simultaneous equations being used (although as mentioned above colloquially the term is used far more broader than how the authors of the paper define it, and of course this is just anecdotal evidence - the lowest form of evidence there is).

Were alternative, more attractive models proposed to deal with the same research problems?

Yes, many panel data/time series techniques can be considered alternative to simultaneous equations approach. For example, fixed effects regression that can correct for time invariant omitted variable bias  can be often considered more preferable to SUR.
Moreover, Angrist & Pischke in that paper subset all 'simple' simultaneous equation approaches such as 2SLS/IV in a separate category. Given that this research is about undergraduate texts I am not surprised that this kind of categorization yields this sort of result. 3SLS/Multiple eq. GMM would probably be considered too advanced for most undergraduate texts, and simple SUR is less used nowadays.

Were the research problems requiring structural equation models abandoned?

I am not aware of any such problems. As mentioned above you will see these models used in research, also typing just 3SLS Macroeconomics into google scholar with filter $year \geq 2020$ yields 262 results. Sure some of those will be just references but I would argue that this showcases that such models are still used.

Did changes in data availability (size, dimensionality of samples) play a major role?

I think yes. Currently you see a lot of research that could be done by simultaneous equations as defined by Angrist and Pischke by panel methods/time series methods.  For full disclosure I am fairly young graduate student so I don't have first hand knowledge of this transition, but I see models such as SUR being applied more in past research than in present.
Also, more generally data availability always played role in modeling choices. Even at the present you can see people more and more often talking about 'panel time series' analysis rather than just panel analysis (e.g. see ch 10.6 in Verbeek (2008) Guide to modern econometrics). The idea that people are adopting more 'time-series-esque' models is also supported by the paper you cite which shows that the focus on time series increased over time

Did the increase in computational power play a major role?

It is possible, some of the newer techniques definitely have higher computational requirements but as I am fairly young scholar I don't have much insight into how limiting factor this was when deciding between models mentioned above and I did not managed to find any relevant discussion of it (maybe someone else will provide answer).


Why charge higher interest rates to poorer customers?

First I would not call this problem a circular problem as that term has specific meaning in modeling and it does not really apply here.

Rather you can think about this as a version of a multiplier. For example, consider standard concept of money multiplier. In fractional reserve system if a central bank issues 100€ and reserve ratio is 10% then the original 100€ gets deposited 10€ is kept and 90€ lent which is then again deposited ad infinitum. So you might get puzzled here and think that the original 100€ creates infinite money but that would be incorrect as sum of infinite addition can be still finite. In this case even with infinite banks the original 100€ would crate only max $100(1/0.10)=1000€$.

How this applies here? Well here again even though higher interest rate makes person poorer and hence riskier leading to even higher interest rate it does not necessary make the person twice as riskier. Assuming for simplicity proportional relationship between riskiness and interest rate (relaxing this would not change the intuition) as long as the total effect of this increase measured as a fraction is less then 1 then the problem will converge to some final interest rate $r$ at which it would be possible to lend money to the poor person. If the fraction would be above 1 the series would be divergent meaning the optimum interest rate would be infinity and in that case bank would simply not lend money to this poor person.

More nuanced and complex versions of this kind of questions are studied by development economics. You may want to look into literature discussing borrowing constraints, micro-finance etc. I think good place to start would be: Poor Economics from Banerjee and Duflo


What kind of contractual problem is this?

This looks like a case of incomplete contracts, or what is sometimes known as non-contractibility.

In the principal/agent paradigm, adverse selection arises when an informed agent transacts with an uninformed principal and, in particular, the principal cannot distinguish between desirable and undesirable agent trading partners.

I don't think your example is a good fit for adverse selection: here the principal would be the government and the agent would be the prison company. But the quality uncertainty concerns the guards rather than the agent itself, and the government is not directly interacting with the guards here.

From the world of information asymmetries, this looks more like a situation of moral hazard where the agent is not incentivised to act in the prinipal's interests. Moral hazard is only really a problem when you Cann't write a contract stipulating thet the agent should behave in the principal's interests. This could be for various reasons

the agent's action can't be directly observed;
while the angel't action cab be observed, its badness can't be objectively verified (i.e., it can't be conclusively proven in court that the action was bad, so a court could never enforce the contract);
the number of contingencies is so large that it is impossible to write a contract specifying all of them
there is no relevant authority to enforce the contract (e.g., a drug dealer and his supplier can't write a meaningful contract because they can't exactly expect a court to enforce it)


Why are US equity prices and inflation inversely correlated?

The price of a stock is the discounted value of its future expected payments. The rate for discounting includes inflation. If nothing else related to the stock except the expected inflation rate changed, the discount rate would increase, driving down the price of the stock. That's a theoretical/mechanical reason why prices might drop due to a change in inflation and only inflation.
But inflation doesn't tend to happen all by itself. It accompanies other macroeconomic changes.
If inflation is happening because of economic problems, such as a shortage of materials or trade problems driving up prices, then we might expect the cash flows to decrease at the same time as the discount rate increases.
If prices are moving up in a perfectly orderly and expected way and not because of economic problems, then we could imagine both the cash flows (numerator) and discount rate (denominator) increasing in such a way that they offset each other. In other words, the numeraire could be irrelevant.
If inflation is driven by increased demand, we might expect the cash flows to increase faster than the discount rate, leading to a rise in stock prices. And of course, there's also the mechanical relationship between today's inflation and prices. If suddenly a dollar is worth a lot less, you might expect everything to increase in price, including stocks.
Many aspects of macroeconomics are too complex or unstable to reduce to a few rules of thumb. Macro models help us understand what happens in a fairly specific set of circumstances. More specific than "inflation of unknown origin." Based on the wording of your question, it seems like you have a reasonable handle on this fact. You are correct in thinking that the relationship between inflation and stock prices is not reliably in one direction.


Find the optimal demand functions for capital and labour for this firm

A firm requieres capital $K\in\mathbb{R}_+$ and labor $L\in\mathbb{R}_+$ to produce the final good $Q$. The technology is of Cobb-Douglas type $Q:(K,L)\mapsto K^{\alpha}L^{1-\alpha}$, here $\alpha=0.75$. Denote the capital rent by $r$ and the worker's wage by $w$. Production costs are then given by $C:(K,L)\mapsto rK+wL$. Instead of maximizing profits, the firm wants to minimize costs to produce a given quantity $\bar{Q}>0$ (look up the duality of $\min-\max$ problems in producer theory). We end up with the following programm
\begin{align}
\min_{K,L}~C(K,L)\quad\text{s.t.}~~Q(K,L)\geq\bar{Q}.
\end{align}
Which is
\begin{align}
\min_{K,L}~(rK+wL)\quad\text{s.t.}~~K^{.75}L^{.25}\geq\bar{Q}.
\end{align}
Set up the lagragian
\begin{align}
\mathcal{L}=rK+wL+\lambda(\bar{Q}-K^{.75}L^{.25}).
\end{align}
Note that the constraint is binding (Why? Look that up too!) The FOCs are given by
\begin{align}
\frac{\partial \mathcal{L}}{\partial K}=0\\
\frac{\partial \mathcal{L}}{\partial L}=0\\
\frac{\partial \mathcal{L}}{\partial\lambda}=0
\end{align}
Solve for $K$ and $L$.


Duality of cost minimization and profit maximization

If $F(K,L)$ is a homogeneous function of degree one then so is
$$
\Pi(K,L) = F(K,L) - R \cdot K - w \cdot L.
$$
This follows straight from the definition of homogeneity. (A definition of homogeneous function can be found here.) This means that if a maximal profit exists it is zero. Otherwise you could increase all inputs by say 100%, thereby increasing both revenues and costs and thus profits by 100%. So $\Pi(K^*,L^*) = 0$.

By Euler's Homogeneous Function Theorem we have $\forall K,L$:

\begin{align}
\Pi(K,L) &= \Pi_K(K,L) \cdot K + \Pi_L(K,L) \cdot L  \\
\Pi(K,L) &= (F_K(K,L) - R) \cdot K + (F_L(K,L) - w) \cdot L.
\end{align}

Since $\Pi(K^*,L^*) = 0$, we have
$$
-(F_K(K^*,L^*) - R) \cdot K^* = (F_L(K^*,L^*) - w) \cdot L^*
$$
We know that $K^*,L^* \geq 0$, so if we can show that the signs of
$(F_K(K^*,L^*) - R)$ and $(F_L(K^*,L^*) - w)$ match we will have proven them to be equal to zero. Otherwise one side of the equation would be negative and the other positive. From cost minimization you have
\begin{align}
R + \lambda F_K & = 0\\
w + \lambda F_L & = 0.
\end{align}
If $\lambda >1$ then
\begin{align}
F_K(K^*,L^*) - R & < 0\\
F_L(K^*,L^*) - w & < 0,
\end{align}
if $\lambda =1$ then
\begin{align}
F_K(K^*,L^*) - R & = 0\\
F_L(K^*,L^*) - w & = 0.
\end{align}
and if $\lambda <1$ then
\begin{align}
F_K(K^*,L^*) - R & > 0\\
F_L(K^*,L^*) - w & > 0,
\end{align}

so the signs do indeed match, hence
\begin{align}
F_K(K^*,L^*) - R & = 0\\
F_L(K^*,L^*) - w & = 0.
\end{align}