Mathematical Micro/Macro Economics Textbook Recommendation

You should also try Advanced Microeconomic Theory (3rd Edition) by (Jehle, Reny, 2011).
Note that, imho, Mas-colell-Whinston-Green (1995!) is the best choice for those with initial background in Math switching to Economics. When it comes to Economics majors the former seems to be more appropriate (and more modern).
For modern Macro with strong math see Introduction to Modern Economic Growth (Acemoglu, 2009).
To integrate the micro/macro problems with econometrics and math see Economic Modeling and Inference (Christensen & Kiefer, 2009). Hope, that helps.


I don't know if Amazon breaks out downloads such as Kindle or movies from the non-AWS part of income statement. You might look at the annual filing to see if there is a breakdown. You are interested in "retail". If we take "retail" to mean non-AWS then we can use the table I posted. It looks like in 9 recent months retail made 5.7B USD in North America and AWS made 9.9B USD. This is only an operating profit so expect actual profit to be less. If they had overestimated depreciation rates and surged retail investment then it might be that reducing retail investment will increase the retail profit margin. The retail operating margin looks like 3.5% in North America and 0.5% international. At Walmart it was 4.0%. So Amazon is delivering and they make a similar amount on a margin basis in at least in North America albeit the nature of the goods is different. The international segment is the laggard at Amazon.

For comparison is this Walmart statement that follows. Note that at Amazon the recent data column is to the right of the "year ago" column and at Walmart the order of the columns is the opposite...


Who is Jacob? And where did he say this?

As we discussed in the comments, in  William Jacob's An Historical Inquiry Into the Production, and Consumption of the Precious Metals, on page 101 it says

  It is probable that in all ages those metals have cost more in their
  production than their value ever repaid.

This is the likely source of the quote.


The indifference curves are not "parallel", as they are not straight lines. They are however shifted, that is they are supposed to maintain vertical distance regardless of the value of $x$.

The curves you map maintain horizontal distance regardless of $y$. That is because the non-linear variable here is $y$, not $x$. The curves are still shifted, but along the other axis.

Taking the logarithm of the utility function you get
$$
x + \ln y
$$
which is more clearly quasi linear.


Something Terrible is Happening - But When Is it Likely to have Happened?

I assume that $r(t)$ is continuous.

The idea is that, this Poisson process with time-varying parameter $r(t)$, as the limit of Bernoulli trial with time-varying probability of success, is memoryless:

For a generic finite partition $\mathscr{P}$ of $[0, T]$ as $\{[0,t_1),[t_1,t_2),[t_2,t_3) \ldots,[t_{n-1},T]\}$, let the $n$th cell following a Poisson distribution with a fixed parameter $r(t_n)$, so
 $$P_{\mathscr{P}} = 1 - \underbrace{e^{\sum_{k=1}^n-r(t_k)(t_k-t_{k-1})}}_{\text{Probability of no accidents}}$$. So  $P = 1 - e^{-\int_0^Tr(t)dt}$. Notice that the Riemann integral $\int_0^Tr(t)dt$ is the limit of a net indexed by the set of all finite partitions of $[0,T]$ with vanishing maximal length of cell with respect to number of cells. $f(x)=1-e^{-x}$ is continuous, so this limit preserves.


In this demand curve, $a=6$ means that the autonomous demand for this good is of 6 units. That is, if the market price of that good was $0$, then people would demand 6 units.

It is reasonable to think that a positive income shock, or a shift in preferences making this good more desirable would make $a$ higher. I don't think there's any deeper meaning here, just that a downward sloping demand curve requires a positive intercept or else quantity demanded would be negative at any price $P$.


Which measure to check in analysis (p-values or $R^2$)?

R-squareds and p-values give different information. For example, suppose that you regress birth weight on mother's smoking. The p-value tells us whether the association is indeed nonzero (in the statistical sense), and the R-squared tells us how much of variation in birth weight is explained by mother's smoking.

p-value > .15 and R-squared 40% would mean that (1) we can't say that the association is nonzero (2) but the explanatory power for the given sample is quite large. The two neither reinforce each other nor are contradictory.


Can the US government selectively cancel debt held by the Federal Reserve?

The total federal debt equals intragovernmental holdings plus debt held by the public. Technically the federal debt held by the Federal Reserve System (Fed) is considered to be debt held by the public. However by law the Fed covers its operating costs and then returns to the federal government almost all of the interest payments it earns from holding the debt of the federal government.
If Fed is treated as the monetary authority of the federal government, despite its political independence, then the Fed balance sheet can be consolidated with the Treasury balance sheet for the federal government. Note the Treasury keeps an incomplete balance sheet because the value of land and natural resources is not considered in the national accounts.
If one consolidates the Fed and Treasury balance sheet positions then this reduces the debt held by the public by the amount of debt held by Fed. Also it eliminates the Fed liabilities due to Treasury. But it would not reduce the Fed liabilities held by the public. So the public holds a mix of Fed and Treasury liabilities issued to manage fiscal policy and monetary policy on behalf of the federal government.
The federal government issues debt to implement fiscal policy and authorizes Fed to implement monetary policy in large measure by the purchase and sale of Treasury debt in the open market. So there is nothing to be gained in terms of fiscal or monetary policy objectives even if the government tried to cancel debt owed to Fed.


What is the exact difference between the $\hat{u}$ in PRF vs SRF?

Population Regression Function (PRF) is the relation between conditional expectation of population $Y_{i}$ conditional on population $X_{i}$, versus the corresponding $X_{i}$, which is also a random variable (to be precise, a matrix of random variables) with some specific given values, say $x$, that is, $E(Y_{i}|X_{i}=x)= f(X_{i})$, which is PRF. For linear regression, $f(X_{i})=\beta_{0}+\beta_{1}X_{i}$.

For a specific value $X_{i}=x$ in population, there could be multiple corresponding $Y_{i}$. (Similarly, imagine when we have a sample dataset and we sort by $X$, we can find multiple corresponding $Y$). Thus, using the relation we get above, we have: $Y_{i}=E(Y_{i}|X_{i}=x)+\mu_{i}$ (Note: it's $\mu$, the deviation of population $Y_{i}$ from its conditional expectation value, rather than $\widehat{\mu }$, which is an estimated value from a specific sample). In other words, PRF captures the perfect, or ideal situation where if $X=x$ happens, we get $Y=y$. But remember both $Y_{i}$ and $X_{i}$  are random variables that may deviate from the perfect situation. Given $X_{i}=x$, the model uses an error term $\mu_{i}$ to capture the deviation caused by other random factors other than $X_{i}$.

Sample Regression Function (SRF), on the other hand, is the estimated relation between estimated $Y_{i}$, denoted as $\widehat{Y_{i}}$, meaning it is calculated by SRF, that is, $\widehat f(X_{i})=\widehat\beta_{0}+\widehat\beta_{1}X_{i}$, and $X_{i}$. (Note that $X_{i}$ and $Y_{i}$ here are actual values from sample, rather than from population, although we usually use the same denotations). $\widehat\beta_{0}$ and $\widehat\beta_{1}$ mean they are calculated or estimated from sample, rather than the actual relation between population $Y_{i}$ and $X_{i}$. Also, $\widehat f(X_{i})$, $\widehat\beta_{0}$ and $\widehat\beta_{1}$ are estimated parameters/relation based on sample, in order to "guess" the true parameters/relation in population. The estimated error term $\widehat{\mu}$, is the difference between the sample $Y_{i}$ and the calculated value $\widehat{Y_{i}}$ using the model we specify, and is the estimator of population ${\mu}$.


What is the difference between the demand schedule and the demand curve?

Demand schedule is a table that gives you the quantity demanded at different prices.

An example of demand schedule that I found on Wikipedia is shown below:

Demand curve is a curve that plots the demand at different prices in the 2D space defined by $Q$ and $P$ (see example picture I took from investopedia below).

Demand curve is essentially a plot of demand schedule.


I've seen them summed somewhere but I cannot exactly remember where. Ultimately I don't think that it makes much difference. The quarterly sum is just the average multiplied by three. Since local projections are just a bunch of OLS - one for each horizon, this is how you can think about the issue:
If $y_{t+h} = \alpha^h + \beta_h news\_shock_t + \sum_{j=1}^{L}\delta_j^hx_{t-j} + \dots + \sum_{j=1}^{L}\gamma_j^hy_{t-j} + \epsilon^h$ is the OLS for the h-th horizon, adding the shocks up is going to yield a $\beta_h$ that is exactly three times smaller than the $\beta_h$ you will obtain by averaging. The statistical significance will not be affected.
Since the units of the news shock most likely don't have any particular meaning both regressions yield exactly the same information.
A different approach instead of aggregating the news shocks up is to use industrial production data, which is usually available on monthly basis to disaggregate the GDP series down to monthly levels using Chow-Lin interpolation. A recent paper using this method would be Anaya et. al..


Would directing more money to science and research and development be an effective response to recessions?

tl;dr:
R&D spending cannot stimulate economy during just by increasing long-run aggregate supply because recessions are fluctuations around the long-run aggregate supply and not necessary affected by the long-run aggregate supply.
A in which spending on R&D in principle could be more effective in fighting recessions than other spending. The effectiveness of spending on increasing output through the multiplier depends on marginal propensity to consume. Hence, a redistribution of resources from people with low marginal propensity to consume to people with high marginal propensity to consume is helpful in increasing the effect government spending has on the economy. If you could argue that people who work in R&D have higher marginal propensity to consume than other people you could argue spending on R&D is more effective. However, I would be extremely surprised if this would be the case, because it is established that marginal propensity to consume decreases with people's incomes and people engaged in R&D tend to be one of the better payed workers (see For example above mentioned Blanchard et al).

Full Answer:
Not in general, even though spending on R&D might be beneficial for long run growth, it is not, in general, any more effective in stimulating economy in recessions or more effective response than some other spending. There are several reasons for this.
First, a recession (especially demand driven one) is, from a perspective of macroeconomics, a short-run phenomenon.  As opposed to microeconomics in macroeconomics long run has slightly more vague definition but scholars generally agree on that it is a very long time period.
For example, Blanchard et. al. in Macroeconomics a modern perspective offer probably most vague definition of long-run:

long run A period of time extending over decades.

More advanced texts such as Romer's Advanced Macroeconomics use the term long-run to define a period in which all relevant variables (wages, prices etc.) adjust to the state of the economy.
Depending on the book/research paper the definition might slightly vary but these definitions imply that in general recession is a short-run phenomenon, not a long-run one and this hold especially true for demand driven recession.
To clarify, recessions is defined as defined as a period of negative growth of economic output (and for statistical/research purposes often as negative growth of output for at least two consecutive quarters - again see the above cited Blanchard et al.).
Under either of the above definitions of long-run recession would generally not qualify as a long run phenomenon. I cant think of any historical example of recession spanning decades, and even if there were such example it would be an exception rather than the rule. Under the second more narrow definition recession could only occur in long-run if the economy's own long-run growth path would itself be negative which is very implausible.
Moreover, under both Real Business Cycle theory and New Keynesian theory of business cycles recessions are considered transitory phenomena. In both of these it does not really matter what is the long-run aggregate supply for the recession - in fact the business cycle occurs as economic output fluctuates along the long-run aggregate supply (See Romer's Advanced Macroeconomcis for more detailed treatment of business cycle theories).
Consequently, the long-run changes in aggregate supply do not affect recessions. Of course, it is desirable to increase the long-run AS as higher output will lead to higher material welfare but there is no reason why increasing  economy's productive capacity should have any effect on reducing the size of fluctuations themselves.
In addition, the idea that spending on R&D even affects rate of economic growth and hence the rate at wich long-run aggregate supply expands is somewhat contentious one. In a standard Solow-Swan growth model a economy will growth at an exogenously given rate no matter what is it's R&D spending as technological progress in such models is completely exogenous. In an endogenous growth theories (e.g. Romer model) spending on R&D actually increases long-run economic growth (and thus also rate at which long-run aggregate supply expands). Unfortunately, an empirical evidence is not yet settled on this issue and Solow model is the more accepted one (again see Romer's aforementioned book).
However, there is a way in which spending on R&D in principle could be more effective in fighting recessions than other spending. The effectiveness of spending on increasing output through the multiplier depends on marginal propensity to consume. Hence, a redistribution of resources from people with low marginal propensity to consume to people with high marginal propensity to consume is helpful in increasing the effect government spending has on the economy. If you could argue that people who work in R&D have higher marginal propensity to consume than other people you could argue spending on R&D is more effective. However, I would be extremely surprised if this would be the case, because it is established that marginal propensity to consume decreases with people's incomes and people engaged in R&D tend to be one of the better payed workers (see For example above mentioned Blanchard et al).
Thus to sum up, generally you cant claim that R&D spending is more effective in stimulating the economy and aggregate demand in the short-run than lets say social or infrastructure spending. It also does not make sense to talk about 'long-run solutions'  to recessions as they are by definition short-run phenomena (where I take the solution to mean response to a recession and not a policies that might try to lets say reduce likelihood of recession occurring in the first place).


What is the public spending / government expenditure share of GDP in Europe?

The data provided by Statista are estimates of public expenditure to GDP ratio for year 2017 while the Eurostat data are for year 2019. Consequently I do not see any a priori reason why the two series would be expected to match. They are not necessarily contradictory as they are for different years.
Furthermore, I can't see the information on Statista on the methodology/sources as it is hidden behind a paywall. It is quite possible that Statista's definition of public spending does not match the Eurostat definition of government expenditure. The definition provided by Eurostat is:

Final consumption expenditure by general government includes the value of goods and services purchased or produced by general government and directly supplied to private households for consumption purposes.

Where general government is defined thusly:

of institutional units which are non-market producers whose output is intended for individual and collective consumption, and are financed by compulsory payments made by units belonging to other sectors, and institutional units principally engaged in the redistribution of national income and wealth.

If you happen to know the Statista definition you can compare them.


Is trading in the commodity futures much more riskier than trading in stock market futures?

Let's start with the statement -

'Consider the commodity exchange where the futures price of a commodity is determined by supply demand during trading hours and is not directly determined by the real price'

first, this statement is not accurate. Not everyone that buy a commodity future ever intend to take delivery.

Second, you need to look at financial future vs. Physical future. Most of the time they move in tandem (correlation nearing 1). However, near delivery, they become less correlated. In fact, physical commodity carries delivery risk. If you have not disposed of the future in time, you'll have to take the delivery. In that case, you have to deal with storage cost.

This leads to the third point - convenience yield. Hard to quantify, hard to hedge (unlike stocks or other financial instruments) - where would you store 1MM barrels of oil today? At what cost?

Last point is - what is 'real price'? The cost of producing a barrel of oil? 1 ounce of gold? If so, how would companies (and in turn their stock) profit?

Now, to the question itself - to truly assess whether commodities are more volatile then other financial instruments you have to define several parameters including observation period, asset classes etc. If you don't you cannot make a proper comparison.

Finally, Commodities post 2009 - part of the drop in commodity prices could be attributed to the fact that due to costs and regulation banks are not such big participants in the assets as they used to be. Here, you could say that 'lack' of interference is simply bringing prices to where supply and demand dictates.

Now, compare all of the above to recent price drops in Twitter and Linked in stocks. With that in mind, Commodities look 'stable'...


Difference between Sequential and Weak Sequential (Weak Perfect Bayesian) Equilibria?

Let's review the definitions of the two concepts. Let $\sigma$ be a profile of strategies and $\mu$ a system of beliefs.

  A pair $(\sigma,\mu)$ is a weak perfect Bayesian equilibrium (WPBE)
  if

  $\sigma$ is sequentially rational given $\mu$, and
  $\mu$ is derived from $\sigma$ using Bayes rule whenever applicable.

and

  A pair $(\sigma,\mu)$ is a sequential equilibrium (SE) if

  $\sigma$ is sequentially rational given $\mu$, and
  $\mu$ and $\sigma$ are consistent in the following sense:   There exists a sequence of completely mixed strategies $\sigma^k\to\sigma$
  as $k\to\infty$ such that, for each $k$, $\mu^k$ is derived from
  $\sigma^k$ using Bayes rule (which is always applicable), and that
  $\mu$ is the limit of $\{\mu^k\}_k$.

From the definitions, we can see that the two concepts differ in their restrictions of beliefs: WPBE requires beliefs to be consistent with Bayes rule whenever it is applicable, and places no restriction on cases where Bayes rule does not apply (as when an information set is reached with probability zero). In contrast, the consistency requirement imposed by SE forces Bayes rule to apply everywhere in the tree by demanding that the SE-belief is the limit of a sequence of beliefs derived from a profile of completely mixed strategies.

To illustrate, consider the following example of market entry game. The entrant first decides to enter ($I$) or stay out ($O$), and if entered, the entrant (E) and the resident (R) simultaneously choose whether to fight ($F$) or abstain ($A$).

The equilibrium where entrant chooses $(O,\text{$A$ if $I$})$ and the resident chooses $F$, together with the resident believing that the entrant will choose $F$ when entered constitute a WPBE. Clearly the resident's belief is wrong, but that is allowed in WPBE because the information set at which the resident makes his decision is reached with probability zero. Thus Bayes rule does not apply, and he is free to have any belief.

SE will not allow this to happen, because as soon as $I$ is played with a positive (albeit tiny) probability, Bayes rule applies at the resident's information set, and given the entrant's strategy, the only consistent belief is to assign probability $1$ to the node after $A$. Thus the strategy profile $((O,\text{$A$ if $I$}),F)$ is not a SE.

Therefore we have the following theorem relating SE and SPE (subgame perfect equilibrium):

  If $(\sigma,\mu)$ is a SE, then $\sigma$ is a SPE.


How does a bank create new money when it pays its own staff?

If an employee performed work there is a liability called "current liabilities". Paying into the employee's account is just changing one kind of liability into another kind of  liability of the same size so total liabilities is unchanged and total assets is unchanged. The new liability is a recorded as a deposit.

Paying the employee with an amount of cash reduces assets and reduces liabilities.

@IñakiViggers is correct, of course, that part of the assets were the result of interest earned and fees charged. My answer addresses only what happens at the instant in time the employee is paid.


How does reducing government spending increase the money supply?

The government cuts public spending but continues to "run a deficit by borrowing from the central bank."
This is the polite way to say "increase yearly the money supply by an amount equal to the government's deficit".

So the money supply will also increase in the second year, albeit by a smaller amount than before. But increase it will.


It could simply be conspicuous consumption, a case of signaling. He does not buy the more expensive item for quality's sake, but to show that he has the income to shop there. In this model you could tweak his utilities in such a way that he would not spend all his money on needless expenses, but he would spend some of it.


Fill out Normal-Form Game to obtain exactly one mixed equilibrium

To ensure that the game has a single, completely-mixed, Nash equilibrium there needs to be "cyclic unique best responses", i.e., either

BR(L) = O, BR(O) = R, BR(R) = U, BR(U) = L,

or

BR(L) = U,  BR(U) = R, BR(R) = O, BR(O) = L.

The strict cyclic preferences both rule out pure equilibria and allow a (unique) completely mixed equilibrium.

There are infinitely many ways to achieve either of the two cases above. Aiming for the first case with BR(L) = O, given that the payoff to the row player for (O,L) is 1, to ensure that BR(L) = O, i.e., the unique best response against L is O, one can set the payoff to the row player for (U,L) as any number strictly less than 1, i.e. so that against L, the row player strictly prefers O with payoff 1 to U with payoff strictly less than 1. The rest of the payoffs can be filled in similarly.


"Intelligent money" or "smart money" is not an economic term per se, but a finance term.

It is colloquially used in the finance industry to describe the collective group of individuals who are paid to actively manage funds, rather than passively manage funds. Active management differs from passive management in that the former chooses how to invest money based on individual perceptions of expected return, whereas the latter follows a predetermined simple rule, such as "buy equities in a weighting equal to their individual market valuations amongst the largest x stocks".

More generally, 'smart money' refers to the subset of active managers that act as if they possessed an informational advantage, and/or the ability to adapt a fund's investment strategy to market conditions. In this sense, smart money consists of hedge funds and stock-picking mutual funds and 'dumb money' consists of passive mutual funds. (Passive funds are more likely to be ETFs, though it's not mutually exclusive).

However few economists would consider investing with active managers to be the intelligent choice (given equal risk, passive investment outperforms active investment on average, net of fees).

Since the term is colloquial, it is hard to find an example which provides a definition. Notwithstanding this, the following article's title lines up with this explanation, and further, explains why "smart money" isn't ordinarily a smart place for money.

http://time.com/money/3657009/indexing-investing-inflows-vanguard-blackrock/


I think this question is too open-ended and unqualified. One can point to two economists in particular who highlight the importance of debt in the cycle - Hyman Minsky and Steve Keen. Both of these economists are not considered to be “mainstream” (although Minsky has been more heavily cited since the Financial Crisis).

If we look at the data, we see that credit (debt outstanding) grows during the cycle, and this can cause financial distress (not always). However, one could argue that the debt growth is a side effect of growth, and not the causal factor. Even though I am highly sympathetic to Minsky’s work, that criticism appears valid. We need to qualify exactly what we mean, but the question has no such qualifications.


Finding distributions of wealth from utility functions

From what it looks like, you are literally trying to subtract $(2)$ from $(1)$, incorrectly as well (You didn't distribute the minus out correctly.) It is saying to use equations 1 through 2 to solve this problem.

You have to set up a Lagrangian with the maximization problem:

$$\max_{x_1, x_2} \ (x_1 + 1)x_2 \\ \text{s.t.} \ 36 - (x_1 - 4)^2 - (x_2 - 6)^2 = c
$$

So we have

$$\mathcal{L} = (x_1 + 1)x_2 - \lambda[36 - (x_1 - 4)^2 - (x_2 - 6)^2 - c]$$

and you should be able to move from there.


Why are there fewer high schools than elementary schools?

Two relevant economic principles are:
Economies of scale: These are a more significant issue for middle and especially high schools which have a greater need for specialist teachers, as others have pointed out, and also for specialist facilities such as science labs with supporting technicians.  Such teachers and facilities would tend to be under-utilised in small schools, resulting in higher costs of provision per child. Elementary schools, on the other hand, rely more on generalist teachers so can operate efficiently on a smaller scale.
Based on a survey of literature on scale economies in education, Andrews, Duncombe & Yinger (2002) found “some evidence” that the optimal size of elementary schools was 300-500 pupils while that of high schools was 600-900 students.  The reason for identifying an optimal size (rather than concluding that larger is always better) was that large schools can also have disadvantages, such as more bureaucracy leading to lower student and staff motivation and less parental involvement.  Note that these optimal ranges were based on studies focusing on the US: different ranges might be optimal elsewhere given different school curricula and different staff salary levels.
Travel costs: Older children are more likely to make their own way to school, while younger children are more likely to be accompanied by a parent. Hence the costs of travel to school, for a given journey, and including not only fares and/or fuel costs (if any) but also and especially the opportunity cost of a parent's time, tend to be greater for younger children.
More formally, family behaviour can usefully be considered within the framework of a household production model in which households derive utility $U$ from commodities $Z_i$ which they “produce” using inputs consisting of market goods $x_i$ and/or time periods $T_i$.  Becker (1965) writes this as (p 495):
$$U = U(Z_1,…,Z_m) ≡ U(f_1,…,f_m) ≡ U(x_1,…,x_m; T_1,…,T_m)$$
The $f_i$ are the “production” functions, although the term production is used here in a somewhat specialised sense: Becker gives the example of the commodity sleep being “produced” via input of a home, a bed and time. Households aim to maximise utility subject to constraints on both income and time (albeit there may be scope to trade off income against time by adjusting hours spent in paid work).  Within this framework a child’s education can be regarded as a commodity produced using time spent in school, doing homework and travelling to and from school, and perhaps market goods (school fees if applicable, fares or fuel costs if required for travel).  A longer journey to school will imply a greater input of time, with a consequent lesser availability of time for other commodities and therefore less overall utility.  Crucially, this effect will tend to be larger for younger children who are more likely to be accompanied by a parent, whose time should also be counted as an input.  Moreover in some cases a parent’s time on a twice-daily school trip will reduce their hours of availability for paid work, again resulting in lower overall family utility.  Hence families, seeking to maximise their utility, will have a strong incentive to try to ensure (via voting or campaigning) that public schools for younger children are small and numerous so that travel times to school are generally short, and less incentive to do so for schools for older children.


Does the price level in a competitive market have to be at the intersection of the Average Cost curve and the Marginal Cost curve?

One of the assumptions of perfect competition is that firms are price takers. Ultimately price is determined by the quantity of goods supplied, and with perfect competition, there are infinite (or an arbitrarily large) number of firms, so a firm that changes their price by itself will simply have no business, since there are cheaper places to buy from. The quantity supplied will also not change, and will leave the prices unchanged.

So what happens if all infinite of these firms tried to unilaterally raise their prices to the same level? This is maybe in some cases a Nash equilibrium, but not a coalition-proof Nash equilibrium. So firms could also unilaterally deviate from the higher price to take the whole market between themselves.

If price (marginal benefit), was suppose, forced to be higher, so that you moved along the marginal cost curve where MC $\neq$ ATC, then the supply curve would shift, because the optimal production quantity would increase for all firms at this price control. And the market would then not clear.


How does "buy low, sell high" relate to supply and demand?

In the comments @henry already provided you with the correct conceptual answer, I will try to offer some extra intuition and way how this could be modeled. Currency exchange is a combination of a retail market with service market so here if you would want to visualize it with supply and demand you would have to use two diagrams for both parts of the market.
If that still sounds too exotic just imagine classical retail. A supermarkets will first purchase its produce (let us say carrots) from carrot farmers. This will be separate market where supermarkets are the buyers and where they jointly together form demand and where farmers are suppliers. On this 'market for carrots from farms' there will be supply and demand and price given by the equilibrium.
However, once supermarkets acquire carrots then they will sell them on second retail market to regular consumers. On this second market, let us call it 'market for retail carrots' supermarkets are the suppliers and regular consumers are part of the demand. This market will have its separate price that might be different from the 'market for carrots from farms'. This however does not mean that market is not competitive because supermarket has additional marginal costs for warehousing those carrots, if they sell them just for price that covers these additional cost the market would be perfect (of course, in real life the market might not be perfect, but the point is  you can't just say it from the fact that there is a price differential).
In the same way the exchange window at an airport can be thought of operating at two distinct markets. At a market where it buys currency from individuals where the exchange windows are the customers and part of demand and people are the suppliers, and then second market where the exchange window operates as a retail and act as a supplier and people who buy currency are the buyers.
Again this is all consistent with perfect competition. Consider the following simple model:
Exchange window sells USD at price $P_H$, but buys currency at $P_L$ and exchange window additionally has to pay the person sitting at the window wage $w$ for facilitating intermediation (let us assume that wage is paid per unit of currency exchanged to simplify calculation).  Let the quantity here be the number of USD exchanged ($Q_{USD}$). Now this can be fully consistent with perfect competition. Let us suppose market is perfectly competitive so $P_H$ is treated by firms as being exogenously set (e.g. see any introductory economics text such as Mankiw Principles of Economics). This means that the profit function of this exchange window will be given by:
$$\Pi = P_H Q_{USD} - P_L Q_{USD} - w Q_{USD}$$
Now take the FOCs and we get:
$$ P_H - P_L - w =  0 \implies \underbrace{P_H}_{\text{price}} = \underbrace{P_L + w}_{\text{Marginal Cost}} $$
Hence, here we have literally a textbook example of  perfectly competitive market, where $P_H>P_L$ (e.g. a situation where firm buys low sells high). Yet there is no additional markup as price that the firm provides is exactly equal to marginal cost so the markup $\mu=1$ and we have perfectly competitive outcome.
The intuition here is that for the exchange window buying currency from people is part of their cost, but then they have other costs (e.g. wage in example above and in real life also capital costs etc) since the exchange window is in business of providing intermediation service. In perfectly competitive market price at which firms supplies product will be equal to marginal cost, but here the marginal cost equal not just the cost of accruing the currency from people in the first place but also marginal cost of intermediation.
Lastly note, that this is of course not enough to prove that exchanges are competitive, rather this is proving the opposite. Namely, a difference between price at which firm buys product and price at which it sells product in itself does not tell you anything about whether market is competitive or not. $P_H>P_L$ on both competitive and non-competitive markets. Here you would have to add more reasoning to argue it is competitive, like in a question you mention that when it comes to exchanges there are usually many firms offering the same service and so on. Or you could actually try to empirically estimate marginal costs of the firm and see if indeed firm just sells its product at marginal costs or if the prices are higher indicating both presence of markup and market power.


Why are the formulas for price elasticity of supply and demand the same in intro microeconomics?

The elasticity is a very general concept. Consider an arbitrary function

$$y=f(x_1,x_2, \cdots , x_n) $$

The elasticity of $y$ with respect to $x_i$ is

$$ e_{y,x_i} = \frac{\partial y}{\partial x_i}\frac{x_i}{y} $$

It measures the percentage change of $y$ when variable $x_i$ changes by 1% - a measure of responsiveness, as you say. It can be applied to anything, including the demand and the supply, where you get the common concepts like own-price, cross-price, income elasticity, etc. The sign of the elasticity is given by the sign of the derivative, which depends on the function $f(\cdot)$.

Elasticities applied to functions other than standard demand and supply abound. For example, individual factors of production, relative factors of production, investment, taxation, preferences, city size, et cetera.


Your example is the classic Allais paradox.

I think the best way to see how the preference pattern $L_1\succ L_2$ and $L_3\succ L_4$ violates independence is to visualize it geometrically. Consider the following figure:

Observe that
\begin{align}
L_2&=L_1+(0.01,-0.11,0.1)\\
L_3&=L_4+(0.01,-0.11,0.1).
\end{align}
Therefore, the indifference curves passing through $L_1$ and $L_4$ should be parallel. Since $L_1\succ L_2$, the latter lottery should lie below the IC through $L_1$. By implication of the parallelism of the ICs, $L_3$ should also lie below the IC through $L_4$. But we're given that $L_3\succ L_4$. This means that $L_4$ lies below the IC through $L_3$ (the blue dashed line). Since parallelism of ICs is a necessary condition of independence, the agent's preference is inconsistent with independence, because it can only be justified by a set of non-parallel ICs.

Alternatively, you can think of the inconsistency between the agent's preference and expected utility theory in the following way.
Suppose expected utility theory holds. Then $L_1\succ L_2$ is equivalent to the inequality
\begin{equation}
u(\$1)>0.01u(\$0)+0.89u(\$1)+0.1u(\$5).
\end{equation}
Adding $0.89u(\$0)-0.89u(\$1)$ on both sides, we get
\begin{equation}
0.89u(\$0)+0.11u(\$1)>0.9u(\$0)+0.1u(\$5).
\end{equation}
But this second inequality is simply saying $L_4\succ L_3$, which is contradictory to the agent's preference.


How large is the Underground Economy in the U.S

There has been some research on this. For example, see the papers by Cagan (1958), Tanzi (1983) and Schneider (2007), Schneider et. al (2010).

If you would like a survey kind of thing, there is a staff paper from St. Louis Fed: Restrepo-Echavarria (2015). Link: https://www.stlouisfed.org/publications/regional-economist/january-2015/underground-economy
and also Schneider and Enste (2000).

Based on Table 3.3.3 (page 24) of Schneider et. al (2010) and the discussion (page 17-19 of same paper) which makes me believe that the numbers on the table are percentage of GDP, USA moved from 8.8% in 1999 to 8.4% in 2007 with a country average of 8.6% in this time period.

Note that (as these papers would mention) there is significant debate about defining and estimating "shadow economy".

References

Cagan, Phillip. "The Demand for Currency Relative to the Total Money Supply." Journal of Political Economy, August 1958, Vol. 66, No. 4, pp. 302-28.
Restrepo-Echavarria. "Measuring Underground Economy Can Be Done, but It Is Difficult"
Federal Reserve Bank of St. Louis The Regional Economist, First Quarter 2015, Vol. 23, No. 1
Schneider, Friedrich. "Shadow Economies and Corruption All Over the World: New Estimates for 145 Countries." Economics: The Open-Access, Open-Assessment E-Journal, July 2007, Vol. 1, No. 2007-9, pp. 1-66.
Schneider, Friedrich & Buehn, Andreas & Montenegro, Claudio E. "Shadow economies all over the world : new estimates for 162 countries from 1999 to 2007," , 2010, Policy Research Working Paper Series 5356, The World Bank.
Schneider, F. and Enste, D. Shadow Economies: Size, Causes, and Consequences. The Journal of Economic Literature, 2000 , 38(1): 77–114.
Tanzi, Vito. "The Underground Economy in the United States: Estimates and Implications." Staff Papers—International Monetary Fund, 1983, Vol. 30, No. 2, pp. 283-305.


What's the relationship between IS-LM model and loanable funds model and what's the differences?

The LM-curve is completely independent from the loanable funds market. This is because the loanable funds market is entirely determined by the real side of the economy. Therefore, it is linked to the IS curve, not the LM curve. Even more, equilibrium in the loanable funds market for given $Y$ allows you to derive the IS curve, just as the keynesian cross allows you to do.

To see this, consider both relationships in the loanable funds market. The demand side of loanable funds is given by the demand for investment (negatively related to the real interest rate $r$) and the supply side is given by all savings available (positiveley related to $r$). For equilibrium it is required that $S=I$ holds. Which is exactly the equilibrium condition represented by the IS curve: what are the combinations of $r$ and $Y$ for which savings equals investment.

To derive the negative IS-relationship from the loanable funds market, just go through the following reasoning. Suppose that the loanable funds market is in equilibrium. This equilibrium holds for a given $Y$. If $Y$ increases, and the marginal propensity to consume is smaller than unity, the level of savings will increase for each $r$, so that the savings curve shifts to the right. More income does not affect investment, so that a new equilibrium requires a lower $r$. Conclusion: if $Y$ increases, a lower $r$ is needed for $S=I$ to hold. This is the IS-curve.


What Level of Government Debt to GDP Ratio is Sustainable?

As you have pointed out: where it comes from is very important. As to the Japanese situation it is quiet different from the US position from example. In fact most of the Japanese debt is owned by Japanese people (90% of the current debt). More specifically the BoJ plays a big role as a buyer, and puts pressure on Japanese yield, which makes it cheaper for the government to issue bonds !

Another interesting point that is usually left by anlysts: Japan is the biggest creditor in the world. The country holds a net amount of about 3 trillion USD (367 trillion yen) of financial assets through the world, which makes Japan the first creditor worldwide (before China !).

An interesting further reading which gives other information on the Japanese debt under stress tests of the IMF (P.40): https://www.imf.org/external/pubs/ft/scr/2015/cr15197.pdf

You also might want to read this paper of Rogoff and Reinhart called "Growth in a time of debt": http://www.nber.org/papers/w15639 . It was really criticized but is a good first glimpse... Afterwards you might want to go a little bit deeper into the debt sustainability analysis of the IMF...

Edit

The reference of Rogoff is more an introduction to the comprehension of debt dynamic than an argument for the existence of a threshold than would reduce growth. On the controversy of the paper, like pointed out by @dv_bn, you can check this intersting article: http://www.newyorker.com/news/john-cassidy/the-reinhart-and-rogoff-controversy-a-summing-up
Thanks to @denesp: Whenever you see "first creditor" this usually takes public and private assets into account. However, if you want to analyze the public debt you should only take financial assets owned by the government (mostly T-Bills regarding Japan).
I add also an interesting article I have read about it, which was written by economists from the NY FED, check it out: http://libertystreeteconomics.newyorkfed.org/2016/06/the-rapidly-changing-nature-of-japans-public-debt.html#.V2qJW2r2aM8


What mathematical structure or formalism can be used for modelling strategic investment decisions of agents in the context of competition?

Disclaimer: I am NOT presuming to have an answer to this question but given the lack of any sort of input I am throwing some ideas out there so hold your horses with your downvotes.

It sounds to me like this is something you can model as a game of incomplete information. As a very very basic set up (see for example Moulin for a detail on games), in a Bayesian form a game specifies:

The set of $N$ playes
The set of pure strategies $X_{i}$ for each player $i$
The set of types $T_{i}$ for each player $i$
The set of beliefs of each player $i$, represented by a probability distribution $\pi_{i}(.|t_{i})$ over $T_{N \text\i}$: one distribution for each possible type of player $i$
The payoff function $u_{i}(x;t)$ for each player $i$, where $x \in X_{N}$, and $t \in T_{N}$

A Bayesian equilibrium is described by a mixed strategy (combination of strategies) for each player, conditional on his type: $s_{i}(t_{i}) \in \Delta(X_{i})$. The equilibrium property then is $$ \forall i,t_{i} \in T_{i}, \forall s^{'}_{i} \in \Delta(X_{i}): $$
$$ \Sigma_{t_{-i}\in T_{N \text\i}} \pi_{i}(t_{-i}|t_{i})u_{i}(s(t);t) \geq \Sigma_{t_{-i}\in T_{N \text\i}} \pi_{i}(t_{-i}|t_{i})u_{i}((s^{'}_{i},s_{-i}(t_{-i}));t) $$

were the following notation is used $$ s_{j}(t)=s_{j}(t_{j}); s(t) \in \Pi_{j \in N \text\i} \Delta(X_{j})$$

Now, if the sets $X_{i}$ and $T_{i}$ are finite, the game possesses at least one Bayesian equilibrium. (This is a direct consequence of Nash's theorem and you can easily find a prove online so I will not get into this).

Now, this is all very abstract, sort to speak. But applied in a simple way to the example you provided your components are quite straightforward: you have some set of agents who will make an investment decision with some goal in mind (obviously here is to maximize some payoff function), and a stochastic component in the returns that each investment offers. Types may refer to some heterogenity that you might want to impose on your players, and beliefs may refer to their knowledge of states of the world or other player's decisons. Now, you ask whether an equilibrium can be reached. As stated before (and of course this is just to provide you with a basic set up) under certain assumptions the existence of an equilibrium is shown. Now, figuring out that equilibrium, as well as the as the conditions under which each agent would have the incentive to invest as much as possible depends entirely on the complexity of the strategic interaction you model, and on the assumptions you make (among admittedly many others). You can make it as easy (e.g. take to agents with some payoff functions, impose known priors, make it a one shot game, and so on and you will easily find some equilibria, as uninteresting as this may be), or as difficult as you'd like it to be. In general, game theory is concerned with agents that interact, so it seems to me that there is a lot of use for you in this area given what you want to model. As you can see the notion of utility maximization is there, so this is not an issue.

Now, as far as the paper you mentioned and the general area I am not able to provide any help. But there are many papers out there on investment decisions and competition from that use game theoretical settings (see for example the work of Smit and Ankum, and i'm sure some key search such as game theory investment competition will bring you many more).

I am aware this is not a satisfying nor complete answer, but I am sure that if you look around at literature even outside your topic you will get a good idea on how you can develop a model (in my own experience this requires a lot of creativity and you often require "inspiration" from the work of others even if it does not necessarily relate to your topic so I would recommend you look at papers in general to gain some insight into the modelling component)

This other papers may provide an interesting perspective on competition (and investment) and game theory models:

https://link.springer.com/chapter/10.1007/978-3-319-05951-8_37

https://www.researchgate.net/profile/Lee_Cooper/publication/227442687_Modeling_Asymmetric_Competition/links/09e415140c23842b7f000000.pdf

https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1745-6622.2001.tb00332.x

https://ieeexplore.ieee.org/abstract/document/977512

https://ieeexplore.ieee.org/abstract/document/4438796

I hope this is of some help


Does random sampling cause zero conditional mean?

The $E(u_i|X_i) = 0$ can hold even without having simple random sample or random assignment. However, random assignment guarantees this will hold (in expectations). A violation of $E(u_i|X_i) \neq 0$ is typically consequence of omitted variable bias. For example, in regression of education on wages reason why $E(u_i|X_i) \neq 0$ can be that experience also affects wages and if people with different education have different experience this will result in $E(u_i|X_i) \neq 0$.
Random assignment of $X$ solves this issue. For example, lets consider $X_i$ to be treatment status $X_i = \{0,1\}$ following Angrist and Pischke Mostly Harmless Econometrics. Now let $Y_{i}$ be potential outcome based on $X_i$ so that we have:
$$ Y_i \begin{cases}
Y_{1i} \text{ if } X_i=1 \\[2ex]
Y_{0i} \text{ if } X_i=0
\end{cases} $$
As a consequence observed outcomes will be given by:
$$Y_{i} = \underbrace{Y_{0i}}_{\beta_0 } + \underbrace{(Y_{1i}-Y_{0i})}_{\beta_1}X_i\tag{1} $$
Now the above in expectation notation implies that:
$$\underbrace{E[Y_i | X_i = 1] - E[Y_i | X_i = 0]}_{\text{observed difference}} = \underbrace{E[Y_{1i} | X_{i} = 1] - E[Y_{0i} | X_i = 0]}_{\text{Average treatment Effect on Treated}} + \underbrace{E[Y_{0i} | X_{i} = 1] - E[Y_{0i} | X_i = 0]}_{\text{Selection Bias}} \tag{2}$$
Presence of selection bias would cause $E(u_i|X_i) \neq 0$.
However, note if we use random assignment of $X_i$ it eliminates the problem as it makes  $X_i$ independent of potential outcomes. If they are independent then $E[Y_{0i} | X_i = 1] = E[Y_{0i} | X_i = 1]$. Substituting this back to the 2 we get:
$$E[Y_i | X_i = 1] - E[Y_i | X_i = 1]= E[Y_{1i} | X_{i} = 1] - E[Y_{0i} | X_i = 1] \\
= E[Y_{1i}- Y_{0i}  | X_{i} = 1] \\ = E[Y_{1i}- Y_{0i}] .$$
This eliminates selection bias and ensures $E(u_i|X_i) = 0$ (although note random assignment is no silver bullet,  there could be various different issues that are still).


Doesnt convexity prevent thick indifference curves aswell?

My approach would be to define thick indifference curves in terms of a stronger form of local non-satiation:

  Definition (thick indifference curves) Preferences are said to have thick indifference curves if there exists at least one bundle $A\in\mathbb{R}^l$ and an open ball
  $\mathscr{B}(A)$ around $A$ such that $A'\sim A$ for every
  $A'\in\mathscr{B}(A)$.

Let's set up a definition for strict convexity to be sure we are on the same page:

  Definition (strict convexity) Preferences obey strict convexity if

  $$A\sim B\implies \lambda A+(1-\lambda) B\succ A\sim B$$

  for $A,B\in\mathbb{R}^l$ and any $\lambda\in(0,1)$.

Now we can state the desired result:

  Proposition If $\succsim$ is strictly convex then $\succsim$ does not have thick indifference curves.

Proof Suppose that $\succsim$ has thick indifference curves. Then there exists an $A$ such that $A'\sim A$ for all $A'\in\mathscr{B}(A)$. Thus, fix two bundles $A$ and $B$ such that $A\sim B$ and $B\in\mathscr{B}(A)$. We know that
$$\lambda A+(1-\lambda) B \in \mathscr{B}(A).$$
Thus, if the indifference curve is thick around point$A$ we must have $\lambda A+(1-\lambda) B\sim A\sim B$. But this contradicts the strict convexity hypothesis. QED

That should do the job for strict convexity. As frage_man already pointed out, the proposition fails to hold generally under weak convexity. Indifference everywhere is a very elegant example why.


Can PPP adjusted values be compared over time?

There are two questions here:

Whether it is meaningful to compare PPP-adjusted values intertemporaly

Whether the "Big Mac" index is a good index for PPP-adjustment.

I will occupy myself with the first question.
Why are we doing PPP-adjustment? We do it to make inter-country comparisons against a benchmark country (usually the US). So this country's magnitude remains unadjusted. Note that what we adjust is nominal magnitudes not "real" ones (that are magnitudes adjusted of in-country inflation).
So at time $t$, and using country $A$ as the benchmark, we can obtain the PPP-wage of country $B$, say $\bar w_{B|A}(t)$. We could think of that as "I live in country B. My living standard at time $t$ is as if I was living in the benchmark country $A$ and my wage was $\bar w_{B|A}(t)$". And comparing that to the actual (nominal) average wage in country $A$, $w_A(t)$, we get a measure of the living standards in country $B$ compared to those in the benchmark country $A$.
Consider now the PPP-adjusted wage from the same country at time $t+k$. It says " I am living in country B in period $t+k$. My living standard at time $k+t$ is as if I I was living in the benchmark country $A$ and my wage was $\bar w_{B|A}(t+k)$".
It is evident that in order to meaningfully compare $\bar w_{B|A}(t)$ and $\bar w_{B|A}(t+k)$, we must deflate the latter by the price deflator of the benchmark country A, covering the period $(t,\, t+k)$. Doing that, makes the two comparable.
But the main purpose of PPP-adjustment is inter-country coparisons, not same-country intertemporal comparisons. By considering the ratio in two different time periods,
$$\frac{\bar w_{B|A}(t)}{ w_A(t)},\qquad \frac{\bar w_{B|A}(t+k)}{w_A(t+k)}, $$
we can see whether living standards have converged or diverged.
A discussion on recent PPP issues can be downloaded here.
A visual tool can be accessed here.


Search for 'Milton Friedman' on youtube, he was a nobel prize winning economist, probably the best advocate of capitalism, especially his Q&A's with college students are very entertaining and also very accessible.


The Euler Equation typically refers to the interior optimal choice between consumption today and tomorrow (or some similar intertemporal choice).  That is, it equalizes the marginal utility of consuming a unit today vs saving that unit in order to consume tomorrow.  This yields the first equation you listed (where $\beta=\frac{1}{1+\rho}$):

$$
u'(C_{t}) = \frac{1+r}{1+\rho}u'(C_{t+1})
$$

I have always referred to the second equation you describe as the "New Keynesian IS Curve" as in Gali's textbook and several NK model papers.  But it does come from  the same intertemporal consumption choice Euler Equation.

Based on the equation you've provided, my guess is the model assumes a few things:

CRRA utility of the form:  $ u(C) = \frac{C^{1-\sigma}}{(1-\sigma)}$
No capital investment such that $ Y_{t} = C_{t} $

Using these assumptions, the Fisher Equation ($ r_{t} = i_{t} - E_{t}[\pi_{t+1}] $), and the Euler Equation you gave above, you would arrive at what you're calling the NK Model's Euler Equation. (Note that your second equation is also probably in log terms)

At the end of the day, the two are the same thing: The first order condition that is making consumption equivalent to saving in terms of utility.  The latter version just tacks on additional features of the NK model to arrive at an equation that's useful for describing aggregate dynamics (along with the Phillips Curve and a monetary rule).


How to linearize the following difference equation?

We have the recurrence relation

$$x_{k+1} = \frac{x_{k+8}}{x_{k+1}}$$

If the denominator is nonzero, this recurrence relation can be rewritten as follows

$$x_{k+7} = x_k^2$$

Assuming positivity and taking the logarithm of both sides, we obtain a linear recurrence relation

$$\ln (x_{k+7}) = 2 \ln (x_k)$$

Shifting,

$$\ln (x_{k+1}) = 2 \ln (x_{k-6})$$

Let

$$\eta_k := \left( \ln (x_k), \ln (x_{k-1}), \dots, \ln (x_{k-6}) \right)$$

be a $7$-dimensional state vector. In matrix form,

$$\eta_{k+1} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 2\\ 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0\end{bmatrix} \eta_k$$

Hence, we need $7$ initial conditions. We recover $x_k$ via $x_k = \exp( \mathrm e_1^{\top}\eta_k )$.


Data Set for Mankiw, Romer, and Weil 1992

Have you seen the GitHub Project Replicating Mankiw, Romer and Weil 1992? It seems to have both the data and a replication of the original results.

For the curious, the paper is A Contribution to the Empirics of Economic Growth.
Abstract:

  This paper examines whether the Solow growth model is consistent with
  the international variation in the standard of living. It shows that
  an augmented Solow model that includes accumulation of human as well
  as physical capital provides an excellent description of the
  cross-country data. The paper also examines the implications of the
  Solow model for convergence in standards of living, that is, for
  whether poor countries tend to grow faster than rich countries. The
  evidence indicates that, holding population growth and capital
  accumulation constant, countries converge at about the rate the
  augmented Solow model predicts.


First Order Stochastic Domination and lottery preferences

Suppose $L \succ_{FOSD} L'$ and let $L$ have CDF $F$ and $L'$ have CDF $G$.
That is, let
$$F(x) \leq G(x) \quad \forall x \quad \mbox{or} \quad 1-F(x) \geq 1-G(x) \quad \forall x. $$
Now consider two lotteries, one is $L'$ and the other one is $L'$ but instead of paying out $x$ it pays out
$$y(x) = F^{-1}(G(x)) \geq x,$$
where the inequality follows by FOSD. That is, the lottery always pays out more. So everyone with an increasing utility function (more is better) prefers the second lottery. How are the payoffs from this lottery distributed?
$$Pr(y(X) \leq y) = Pr ( F^{-1}(G(X)) \leq y ) = Pr ( X \leq G^{-1}(F(y)))=
G(G^{-1}(F(y)))= F(y).$$
Thus,
$$\mathbb E[u(L)]= \int u (y(x)) d F(y(x)) = \int u (y(x)) d G(x)
\geq \int u (x) d G(x) = \mathbb E[u(L')]$$
for any increasing $u$ by the $y(x)\geq x$.

The other direction would be that $\mathbb E[u(L)]\geq \mathbb E[u(L')]$ for all increasing $u$ implies
$L \succ_{FOSD} L'$. Let's show the equivalent statement that if $L$ does not FOSD $L'$, then there is at least one increasing $\widetilde u$ such that  $\mathbb E[\widetilde u(L)]< \mathbb E[\widetilde u(L')]$.
If there exists some $z$ such that $1-F(z) < 1-G(z)$, then there is the increasing utility function
$$ \widetilde u (x) =\begin{cases}
1 \mbox{ if } x> z\\
0 \mbox{ if } x \leq z \end{cases}$$
such that
$$\mathbb E[\widetilde u(L)]= 1-F(z)  < 1-G(z) = \mathbb E[\widetilde u(L')].$$
That's it.


How do takeover deals create arbitrage opportunities that involve necessarily selling short the acquirer’s stock?

In simple terms, an acquisition usually happens when a successful (read: low-risk) company purchases an unsuccessful (read: high-risk) company. It can be viewed as a transference of risk from the acquiree to the acquirer, and all things being equal, the market demands higher returns from riskier assets. In the short-term this is effected through the price, and one channel for this is that risk-averse holders of the acquiring company will move to safer harbors. The acquirer's stock thus falls, implying a higher return in line with the new level of risk.  The market also demands lower returns from lower-risk assets, and so more investment will flow into the acquiree, driving its price up.

Obviously, there's more going on and this is not investment advice.  It's just a rule of thumb.


This seems like a basic issue about probability calculus/theory.

The intuition behind the uniform distribution over $[a,b]$ is that all outcomes between and $a$ and $b$ are equally likely. Because of this, one can get an intuitive grasp about the probability that the outcome $x$ falls in a subinterval $[a_1,b_1] \subset [a,b]$. As the ratio of their length is
$$
\frac{b_1 - a_1}{b - a},
$$
the subinterval $[a_1,b_1]$ contains $(b_1 - a_1)/(b-a)$ share of all the points in $[a,b]$. Each point is an equally likely outcome, therefore
$$
P\left(x \in [a_1,b_1]\right) = \frac{b_1 - a_1}{b - a}.
$$
A problem arises if the subinterval is in fact a single point, $a_1$. What is
$P\left(x \in \left\{a_1\right\}\right)$?
 By the above intuitive reasoning
$$
P\left(x \in [a_1,a_1]\right) = \frac{a_1 - a_1}{b - a} = 0.
$$
A more rigorous approach:
 Let us denote this probability $P\left(x \in \left\{a_1\right\}\right)$ by $\alpha$. As the distribution is uniform, all points are equally likely outcomes and hence for any $a_i \in [a,b]$ we have
$$
P\left(x \in \left\{a_i\right\}\right) = \alpha.
$$
There are infinitely many (disjoint) points in $[a,b]$. Let $A$ be an arbitrary set of infinitely many such points, that is
$$
A = \bigcup_i^{\infty} a_i.
$$
Then as probability is sigma additive over disjoint sets
$$
P\left(x \in A \right) = P\left(x \in \bigcup_i^{\infty} a_i \right)
= \sum_i^{\infty} P\left(x \in \left\{a_i\right\}\right)
= \sum_i^{\infty} \alpha.
$$
However as $A \subseteq [a,b]$, $P\left(x \in A \right) \leq 1$. But for any $\alpha > 0$ the sum $\sum_i^{\infty} \alpha$ would be infinite. Therefore $\alpha$ cannot be positive, it is zero.


Wouldn't reducing Fed Balance Sheet lead to Inverted Yield Curve?

As a preface to my answers, I want to point out that bond yields are set in the market. We can think of bond yields as being (roughly) the sum of the expected path of short rates over the life of the bond, plus a term premium.

QE may or may not have lowered term premia (which would lower yields), but that is not enough for the Fed to completely set interest rates independent of what bond market participants expect the path of short rates to be.

The Fed may want to keep bond yields low when raising rates, but bond market participants will not want to own bonds that they expect to lose money on. Therefore, it would be extremely difficult for the Fed to keep bond yields down if they are in the process of raising short rates. As a result, it is difficult for the curve (overnight - bond maturity) to invert when rate hikes are expected.
Yes, inverted yield curves have been historically associated with recessions in the United States. This is an indication that the bond market is pricing future rate cuts. (The curve will have difficulties inverting if short rates are zero, as has been the case in Japan.)
The raising of bond yields is done by people who expect rate hikes, not by a Fed decision. Since the private sector does not exclusively borrow on a short-term basis (for example mortgages), this presumably has an effect on the economy. As a result, it is part of the "transmission mechanism" from the policy rate to the economy, and so it is not really better or worse than raising the short-term rate.


How can we know how PPF of our country looks like?

You can estimate the PPF just by technically estimating what’s the production function for all goods and what’s the resource constraint.

For example, let’s for the sake of simplicity assume that the only factor of production is labor so the problem does not get too complicated. Also, just for a sake of simplicity let’s assume there only two goods apples and oranges.

An example of simple production function for apples would be $A=L^{1/2}$ and for oranges $O=2L^{1/2}$ and let’s assume that the maximum amount of labor hours country has available is 100 to make math simpler. This would give you PPF with corners at 10A and 20O and then some curve in between them. In real life the same calculation would be performed but production function would use more factors of production and would be estimated for all industries. That would be difficult to visualize because the graph would have dimensions equal to the number of industries and anything above 3D is very hard to plot but it would work the same way.

In this calculation you take production function and resource constraint as it is. There is always possibility that a country discovers new technology that increases efficiency or new resource, but you have to define PPF at some point of time otherwise it does not have any meaning - in long run the whole universe will became just cold lifeless void because of the entropy so for most of the time universe exists the PPF of any life form will be 0. But at present or different points of time it will be non-zero.

For example a new technology in above example for apples could change in time $t+1$ the apples production function to $A=4L^{1/2}$. Now the PPF would have corners at 40A and 20O at the time $t+1$ even though before it was smaller. However, at both periods of the time people could choose to produce not at the boundary of PPF but that would not be due to the technology or resources but due to preferences or perhaps some bad policies that prevent them to do that. Culture could affect preferences but also resource constraint - if you for example really have to observe shabbat and can’t work no matter what, but again at a point of time that’s fixed and implicitly included in PPF.

PPF is by its definition constructed by already taking technology (via production function) and amount of resources available (via the resource or budget constraint) at some time into the account so it’s not like you somehow have to add that to PPF. Of course, in real life people could argue that different econometric techniques might not capture these factors properly and might dispute your statistical methodology, but that’s whole different discussion


You have to be careful when comparing different utility functions.
For example the function
$$
v(C_A) = - \frac{1}{C_A},
$$
and the function:
$$
u(C_A) + u(C_B) = - \frac{1}{C_A} - \frac{1}{C_B},
$$
cannot simply be compared. The first represents preferences over one good, while the second gives preferences over two goods.
So if you say:
$$
u(C_A) + u(C_B) < v(C_A),
$$
you are in fact comparing two different utility functions.
If you want to know what happens if you only consume one of the goods, you need to compare, for example $u(C_A) + u(C_B)$ with $u(C_A) + u(0)$.
Also if you increase $k$ in the function:
$$
-\frac{1}{C_A} - \frac{k}{C_B},
$$
you are in fact making $C_B$ more valuable. To see this let's compute the marginal utility of $C_B$ relative to the marginal utility of $C_A$:
$$
\frac{MU_B}{MU_A} = \frac{\dfrac{k}{C_B^2}}{\dfrac{1}{C_A^2}} = k \frac{C_A^2}{C_B^2},
$$
which does increase in $k$. So the higher $k$ the higher the value of an additional unit of $B$ compared to an additional unit of $A$.


Well going through the derivatives, simply put. View $U$ as a function. A.K.A $U=U(q_1,q_2)$ and let $U_1=U(q_1)$ and $U_2=U(q_2)$. So to find $U_1$ and $U_2$ respectively, just take the partial derivatives in respect to the  given variables

$$U_1=\frac{\partial U(q_1,q_2)}{\partial q_1 }=(1-\beta)q_1^{-\beta}+ \frac{\partial}{\partial q_1} \gamma q_2=(1-\beta)q_1^{-\beta}$$

Since $\frac{\partial}{\partial q_1}\gamma q_2=0$

Similarly,
$$U_2=\frac{\partial U(q_1,q_2)}{\partial q_2}= \frac{\partial}{\partial q_2}q_1^{1-\beta}+\gamma=\gamma$$

since $\frac{\partial}{\partial q_2} q_1^{1-\beta}=0$

Hope this helps


Let me start from the bottom up.

"[...] is there a bigger picture which I am missing that all this global recession can be a planned activity by major economies every few years/decades to achieve something bigger?"

No, global recessions usually hurt all economies and though in theory one can come up with an example where someone might benefit from a global recession, arguing that such actors actively caused a global recession for personal gains, is simply a false conspiracy theory. I don't think there is a single example of history where this has happened.

4."How does the world come out of global recession? If it has happened before many times in history, why haven't the major economies taken lessons from it and only do things which helped them to come out of recession last time?"

Global economies are complex systems, it is not like you pull a trigger and you're out. People have won Nobel prices for coming up with ideas to come out of a recessions. However, each recession is different and might need different actions to come out of it. However, we have learned a LOT from previous recessions, nowadays,  global recessions are less frequent, and a lot less severe than they were in the past.

1-3.

A global recession is a complicated phenomena, it might be caused because there are problems in the financial sector, or firms are investing less, or consumers are worried about the future and consuming less.  It could be because there is a war in an oil producing country which will increase the production costs for a lot of countries, etc, etc.
It is absolutely false that a country's loss is always some other country's profit. Though there are situations where it might be true. For example Mexico losses economic growth because Americans are now substituting Mexican goods for Chinese (this is a fictitious example), but there are far more situations where this is false. Usually the growth of one country fosters the growth of its trading partner countries, and vice versa. In the crisis of 2008, to put a concrete example, most economies were worse off. Higher unemployment everywhere, less goods and services where produced and consumed, lower wages, etc.

If you are interested in the last financial crisis of 2008, I recommend you to watch or read "The big short". It is a well-balanced narrative that explains how the causes of a crisis can be complicated. And how a crisis usually occurs due to multiple mistakes in human actions, but not because someone wanted to cause a crisis. The way out of that crisis turned out to be quite complicated since the crisis looked nothing like previous ones.


What are the economic perspectives regarding the game of salary negotiations?

I will provide a simple game-theoretic modelling of the situation.
A new year starts and a company wants to make a wage-increase offer to an existing employee. Let $e$ be the employee's current efficiency and the corresponding wage $h(e)$ (which, represents an increase over previous wage). Let $v$ be the premium observed in the market for new hires (so if the employee goes to another employer he will earn $h(e) + v$). Let $c$ be churn costs (recruitment plus loss of efficiency etc) to the current employer, if the employee leaves and needs to be replaced.

This is a sequential game so we have to use the extensive form.

A) Firm offers a wage

First, the case where the firm ($F$) offers a wage and the employee ($E$) decides what to do:

The first outcome refers to the firm's cost, the second to the employee's wage. We have assumed that if the employee is offered the new hire premium it stays with the firm.
Let $p_l$ be the probability that the employee will leave if he is offered $h(e)$ only. The firm faces the following expected costs:

$$EC [h(e)] \equiv EC_{A1}= (1-p_l)h(e) + p_l[h(e)+v+c] = h(e)+p_l[v+c]$$

$$EC [h(e)+v] \equiv EC_{A2} = h(e)+v$$

Then in order for the firm to nevertheless offer $h(e)$ it must be the case that

$$EC_{A1} < EC_{A2} \implies h(e)+p_l[v+c] < h(e)+v$$

$$\implies p_l < \frac {v}{v+c}$$

and it should offer $h(e) + v$ if the inequality points to the other direction.

Let's move now to the OP idea, to tell the employee to ask for a wage. Here we have

B) Employee asks for a wage

Here too the first outcome is the firm's cost. We have allowed for the possibility that the employee asks only the efficiency wage. This is crucial.

Since we are in the "decision about the process structure" phase, we assign some probability $p_e$ that the employee may actually ask for just the efficiency wage. This is important.
It is also important to note that, assuming that the employee has asked for $h(e) + v$, while the criterion for whether the firm should accept or counter-offer only $h(e)$ has exactly the same expression as before, we are looking at a different probability. Here the firm must decide whether the employee is "bluffing" (he does not have an offer from another firm), or not. This is a different probability than the previous one. Here, the firm has additional information (for better or for worse), and so it has to make a difference assessment. Call the probability of not bluffing $p_c$.

We have that

$$p_l < p_c$$

because the former is a conditional probability of the same event ("the employee leaves"), while the latter is the unconditional probability. Keep this inequality for later.

Assume that the firm has estimated somehow ("if the employee asks for $h(e)+v$ there is $p_c$ probability that he will leave if I counter offer $h(e)$ only).
If this is estimated, then the firm already knows what it will do if the employee asks for the new hire premium -and it will depend on the specific values of the various quantities here.

So given an estimate for $p_c$ we are looking at two possible expected costs for the firm
B.1. The firm will counteroffer $h(e)$
here the expected cost is

$$EC_{B1} = p_eh(e) + (1-p_e)\cdot [(1-p_c)h(e) + p_c(h(e)+v+c)]$$

$$=  h(e) + (1-p_e)p_c(v+c)$$

B.2. The firm will accept $h(e)+v$

$$EC_{B2} = p_eh(e) + (1-p_e)(h(e) +v) = h(e) + (1-p_e)v$$

WHAT STRUCTURE TO CHOOSE?

Now we want somehow to compare the two structures and select the one that is more profitable for the firm. This requires to examine various cases characterized by the relation between the various probabilities.

CASE 1 : $p_e = 0 , p_l < p_c < v/(v+c)$

Here the firm will offer $h(e)$ in structure $A$ (so expected cost $EC_{A1}$), and will counter-offer $h(e)$ in structure $B$ (so expected cost $EC_{B1}$).
Given the assumed values of the probabilities we have that

$$EC_{A1} = h(e)+p_l[+v+c] < h(e) + p_c(v+c) = EC_{B1}$$

an so we should stick with the traditional structure $A$ where the firm offers first a wage.

CASE 2 : $p_e = 0 , p_l  < v/(v+c) < p_c$

Here the firm will offer $h(e)$ in structure $A$ (so expected cost $EC_{A1}$), but will accept $h(e)+v$ in structure $B$ (so expected cost $EC_{B2}$).
Given the assumed values of the probabilities we have that

$$EC_{A1} = h(e)+p_l[v+c] <  h(e) + v =EC_{B2}$$

and again we should stick with structure $A$.

CASE 3 : $p_e = 0 ,  v/(v+c) < p_l  < p_c$
Here we compare $EC_{A2}$ with  $EC_{B2}$
$$EC_{A2} = h(e)+v  = EC_{B2}$$

No winner here, but overall we see that
the incentive to adopt structure $B$ hinges on whether the employee may after all ask only for $h(e)$. ($p_e>0$ is necessary but not sufficient condition to adopt structure $B$).

CASE 4 : $p_e > 0 , p_l < p_c < v/(v+c)$

Here too we compare $EC_{A1}$,with $EC_{B1}$ but with $p_e>0$ so

$$EC_{A1} = h(e)+p_l[v+c] < >h(e) + (1-p_e)p_c(v+c) = EC_{B1}$$

We stick with structure $A$ if $p_e < (p_c -p_l)/p_c$, and we adopt structure $B$ if the inequality runs the other way.

CASE 5 : $p_e > 0 , p_l  < v/(v+c) < p_c$

Here we compare  $EC_{A1}$, with $EC_{B2}$ but with $p_e > 0$

$$EC_{A1} = h(e)+p_l[v+c] <  h(e) + (1-p_e)v = EC_{B2}$$

as one can verify. So here we stick with structure $A$.

Finally

CASE 6 : $p_e > 0 ,  v/(v+c) < p_l  < p_c$

Here we compare $EC_{A2}$ with  $EC_{B2}$
$$EC_{A2} = h(e)+v > h(e) + (1-p_e)v = EC_{B2}$$

and we should go with structure $B$.

VERBAL SUMMARY

1) If we expect that employees will always ask for the new hire premium if they get to ask first, then we should stick with the structure where the firm offers first a wage.  (Cases (1,2,3)

2) If there exists a positive probability that the employees may just ask for $h(e)$ then :
2a) If the firm will stand by $h(e)$ in any case and structure, we should keep the structure where the firm offers first a wage if $p_e < (p_c -p_l)/p_c$ (Case 4)
2b) If the firm will go for $h(e) + v$ in any case and structure, we should choose the structure where employees ask first (Case 6).
2c) If the firm will play differently in the two situations, we should keep the structure where the firm offers first a wage. (Case 5).

As is usually the case reality is more complex than that: negotiations may have more rounds, and the firm and the employee may not even agree on $h(e)$ although such disagreement is less common than "accepted wisdom" would have it.

But the general feeling I get from all the above analysis is that the main reason I would consider implementing a structure where employees "ask first" is if I thought that there exists a high enough probability that they won't ask for the new hire premium -and still, if at the same time I think that they won't try to bluff (i.e. I expect $p_c$ to be close to unity), again it would be likely preferable to stick with the traditional model.


How can the bond market be overvalued when the value of a bond is a function of interest rates?

How can the bond market be overvalued?

To value a bond, you need, as you mention, to discount future cash flows. The cash flows (or coupons) are fixed, but the discount rate is not, and every investor may have a different view on interest rates and hence price bonds differently.

Therefore, someone thinking that interest rates will remain at rock-bottom levels will find different prices for bonds from someone who thinks that interest rates will rise sharply in the near future (the first investor will find higher prices).

So, in your case, someone thinking that the bond market is overvalued simply has a different view on interest rates than the consensus on the market. To be more precise, they expect interest rates to increase faster than the market consensus.

Bond Prices and interest rates

Taking into account what we have said above, you can view the bond market as follows: every investor forms a view on discount rates, for a given bond. Doing so, they actually compute the yield, that is the discount rate that makes the NPV of the cash flows of the bond equal to 0.

However, investors disagree on where the yield should be. They will thus trade the bond between them until they all agree.

Let's take a numerical example :

Investor A owns a Bond, which has a face value of 100, a coupon of 1 and will mature in one year.
Investor A thinks that the yield of this bond should be 2%
Investor B thinks that the yield of this bond should be 0.5%

The value of this Bond to Investor A is : $(100 + 1) / 1.02 = 99.02$
The value of this Bond to Investor B is : $(100 + 1) / 1.005 = 100.50$

Therefore investor A and Investor B will both improve their current state by trading the bond anywhere between \$99 and \$100.5, ie. Investor A should sell the Bond to Investor B.

If the market price of the bond is \$100, then Investor A thinks that the bond is overvalued, and Investor B thinks that the bond is undervalued.


Non-negative deviations from Nash Equilibrium

It depends on whether the other players' strategies in the initial equilibrium are still best responses to the deviator's new strategy. For example, $(T,L)$ is the unique NE in the game below. Player 1 (the row player) can deviate from $T$ to $B$ without hurting his own payoff, but $(B,L)$ is not a NE, since $L$ is not a best response to $B$ in this game.

\begin{array}{|c|c|c|}\hline
&L&R\\\hline
T&1,1&2,0\\\hline
B&1,0&0,1\\\hline
\end{array}

Of course you can easily construct a game where such a unilateral deviation still leads to a NE. As the game below shows, player 1's deviation from $T$ to $B$ is still without loss of profit, but it changes the NE from $(T,L)$ to $(B,L)$. There are also infinitely many other possible deviations for player 1 if you consider deviations in mixed strategies.

\begin{array}{|c|c|c|}\hline
&L&R\\\hline
T&1,1&2,0\\\hline
B&1,2&0,1\\\hline
\end{array}


IV：Quasi-random variance V.S. 'Placebo' check

There is no contradiction in two approaches.
Baum-Snow (2007) looks at the effect of the Interstate Highway construction on those MSAs which happened to lie on its way. The randomness comes from the fact that when the plan for the Interstate Highway project was considered, the aim was to connect distant areas. Through which MSAs the highway would go was more or less random (at least this is what the author claims). Thus, some MSAs received treatment by lying on the way whereas other didn't.

Donaldson (2012) uses planned railway routes that were NOT implemented for a placebo test. The concern which Donaldson addresses is that there could exist some unobservable economic factors (different economic development, agricultural income etc) affecting the decision about where to construct the railway. To overcome this, the key assumption he uses is that, on the stage of planning, the several proposed routes were equally desirable from the strategic point of few, but some of them were more costly to build, hence these routes were dropped at some stages of construction.

Now, if we saw an effect on income in areas where railroads were planned but never built compared to areas where railroads were not even planned, this would suggest that railways are probably not the reason to explain main findings in the paper, but rather the selection of more developed areas into treatment (receiving or at least planning to receive railroads) was in place. However, such placebo test shows that areas that were initially selected into construction of railway, but due to technical reasons didn't receive the railways - have no significant income effect. Hence, the selection is argued to be not an issue in Donaldson's identification.


A case where the solution to Expenditure Minimisation Problem is not a solution to Utility Maximistion Problem

If the utility is continuous and locally insatiable then the Hicksian demand equals the Walrasian demand.

So you'll need to look for a utility function which violates atleast one of these.

Lets try a simple violation of non-satiation $$
u(x,y) = \left\{
        \begin{array}{ll}
            x+y & \quad x+y \leq 1 \\
            1 & \quad \text{ otherwise}
        \end{array}
    \right.
$$

With $p=(1,1)$ and $w=2$ Then $x(p,w)=\{(x,y): 1\leq x+y\leq 2\}$. This is becuse the consumer's problem is $$\max_{x,y} u(x,y)$$ $$\text{st }x+y\leq 2$$

$u(x(p,w))=1$ so apply the EMP to this utility $$\min x+y$$ $$\text{st } x+y\geq 1$$
This gives $h(p,1)=\{(x,y): x+y= 1\}$. Hence, $h(p,v(p,w))\neq x(p,w)$.

I think a violation of non-satiation is much easier to construct a counter example with. If we have a 'thick' indifference curve then the UMP will choose the entire curve but the EMP will choose at most the lower envelope, as why spend more to get the same utility?

Conversely, we can use continuity to find an example of where $h(p,v)$ is not a subset of $x(p,e(p,v))$. Take $p$ and $w$ as above and

$$
u(x,y) = \left\{
        \begin{array}{ll}
            x+y+1 & \quad x+y \geq 2,\ x\geq y \\
            x+y & \quad \text{ otherwise}
        \end{array}
    \right.
$$

The EMP with $v=2$ is

$$\min x+y$$
$$\text{st }u(x,y)\geq 2$$

$h(p,2)=\{(x,y): x+y=2\}$. Now look at the UMP at $p,w$

$$\max u(x,y)$$

$$x+y\leq 2$$

$x(p,w)=\{(x,y):\ x+y=2, x\geq y\}$. Hence, $x(p,e(p,v))$ is a proper subset of $h(p,v)$.

I found it alot easier to come up with a counterexample without continuity (our $u$ is locally non-satiated). Perhaps with continuity you get $h\subset x$ and local non-satiation you get the reverse? Ill leave that to you.


Is Artificial Intelligence a completely new (and underestimated) production factor?

My guess is that it is not a new factor of production, but simply a type of total factor productivity. This is because:

factors of production are a stock, which produce services. What is the stock of AI? The stock of labour is easy to calculate. The stock of capital is more complex, but at least all capital goods (even robots) have a market price. As such, the heterogenous capital in an economy can be converted into a unique variable, via prices. Does different AI components have prices? I don't think so.
factors of production are rival goods. A machine cannot be used simultaneously by the same firm. Same for workers. Instead, AI are knowledge, codes, algorithms, which are non-rival.
AI is produced by labour. As such, from a endogenous growth perspective (a lá Romer), resources need to be invested in order to augment the level of AI in the economy.
factors of production get paid. How do you pay an algorithm? Even more, why would you pay an algorithm at all?

Therefore, if I were to model AI, I would do so as part of TFP, and not as a new factor of production. In my view, the report is highlighting is that AI is a different and new component of TFP, one which could revitalise growth.


Why don't governments ban businesses from passing tax costs onto their customers?

I don't see how the government taking X% of a business's profits is stopping the business from increasing prices (or is this just the penalty in the enforcement clause?), so I am going ignore that part.

What you are describing is essentially setting prices, determining what prices a business can and cannot set. (If you merely prohibited passing on costs, the business would claim that they are not doing this, they merely happen to be raising prices because of other reasons.)
This usually means a planned economy rather then a market based one, as the most important function of the market is to set the prices. Allowing the market to do this results in all sorts of nice things: it balances supply and demand, provides incentives for innovation, etc.
To some extent even market based economies engage in price setting. Sometime governments set price ceilings on some products. The usual reasons behind this are

Fairness: at the lower price demand will outstrip supply (in fact the low price will likely diminish supply), but everyone will have a chance of buying some of this good, not just the wealthy who could afford the higher equilibrium prices. (See rent control.) The dynamic effects are assumed to be negative: the lower price will disincentivize future investment.
Non-competitive markets: if the market is not functioning properly, e.g., someone has a monopoly on the good, price setting may actually increase the quantity sold and have a positive effect on net welfare.
Anti-price gouging: if there is a short term demand spike and supply is strongly inelastic, prices may skyrocket in a given market. (Such was the case in some good markets at the start of the Covid-19 pandemic.) In this case, a price ceiling may transfer welfare from producers to consumers without significant welfare loss (if supply is really strongly inelastic). The term "anti-price gouging" is somewhat unfortunate as it is emotional with sensationalistic overtones.
There are also non-economic reasons: such a measure may be politically popular, even when there are downsides, as these are not immediately visible.

Recommended further reading:
REVIEW OF ECONOMIC THEORIES OF REGULATION


I’ll offer brief answers to the sub-questions.

“Who owns the debt?” You might be able to find information in Canada’s financial accounts data, but all provinces are likely aggregated. Finance Quebec may voluntarily post some information on holders, but I am unaware of any sources in the public domain.

“What happens when a province falls in the red?” Canadian provincial finance is very different than American state finance. If “fall in the red” means “runs a fiscal deficit,” the province will just issue bonds.

“Is it possible for a province to no longer borrow?” Unlike American states, there are no balanced budget laws in Canadian provinces. The “limit” is the inability to float bonds in the bond markets. The Social Credit government in Alberta was forced to default in the Great Depression (1930s), and I believe that was the only post-Confederation (1867) provincial default. (It’s hard to find an authoritative reference for that statement, sorry.)

“Is there a finite anount it can borrow?” The only limit is what it can get from auctioning bonds to investors. There is no iron law determining an upper limit for that, but realistically debt levels will be compared to past history and the situation of other provinces.

A single province is relatively small when compared to aggregate borrowing/lending in the Canadian dollar financial system. Therefore, a province is unlikely to hit some aggregate “limit,” rather it is a question of whether investors are willing to allocate their portfolios towards the province. For example, if investors fear default, they will not be happy to lend, no matter how much balance sheet capacity they have to buy bonds. Discussing the aggregate lending capacity is controversial, and I am deliberately ducking that question. That should be a separate question.


Elasticity of Durable Goods vs Non-Durable Goods

For the theory, House (JME several years ago), Barsky, House and Kimball (AER, circa 2006) Barsky, Boehm, House, and Kimball (working paper on SED website).  Johannes Wieland has a new paper on this, maybe not yet available.

The key theoretical point (put somewhat loosely) is that purchases of an "idealized durable", one with a very low depreciation rate, should have a near-infinite intertemporal elasticity of substitution.


Will periodical increasing and decreasing cryptocurrency supply peg cryptocurrency to dollar effectivelly?

No, it's by no means a guarantee of a peg.

The cryptocurrency's actual exchange rate will be determined by people's collective expectations of the present and future value of the cryptocurrency relative to the dollar.

Most importantly, the exchange rate will be driven by how much people trust that they will actually be able to get a real dollar in exchange for a unit of the cryptocurrency.


If both $xRy$ and $yRx$ $\forall x,y \in X$, then the indifference relation $I$ is symmetric by definition. Not all symmetric relations are equivalence relations. However, you are given that $R$ is also reflexive and transitive. So we have that $I$ is symmetric, reflexive, and transitive. Therefore $I$, or $\sim$, is an equivalence relation.

You can verify that $\sim$ is an equivalence relation by reading off the definitions of the three properties. $R$, or $\succsim$, is a (weak) preference relation, also known as a total preorder. Total preorders are transitive, complete, and therefore reflexive.

Keep in mind that reflexivity and transitivity does not necessarily imply symmetry: $R = \{(x,x), (y,y), (x,y)\}$ given $X = \{x,y\}$ is clearly reflexive and transitive but not symmetric because $xRy$ but not $yRx$.


How to model best combination of resources

It sounds like you could simply use the Cost Minimization Problem:
$$\underset{z_1,...z_N}{min}\sum_{i=1}^N q_iz_i$$
$$s.t.\quad f(z_1,...,z_N)\geq \bar{y}$$
$$z_1,...z_n\geq0$$
Where $z_i$ and $q_i$ are the quantity and price of input $i$, respectively, $\bar{y}$ is some predetermined level of output, and $f(\frac{}{})$ is the production function.

A production function relates physical output of a production process to factors of production. Of course, it seems to me that it may be a challenge to characterize a production function for your case. However, the production function can be any function which satisfies the following:

1) Strict Monotonicity: If $ z'>z$ then $f(z')>f(z)$

2) Quasi-concavity: $V(y)=\{ z:f(x)\geq y\}$ is a convex set

3) $V(y)$ is closed and non-empty

4) $f(z)$ is finite, nonnegative, real valued, and single valued $\forall z\geq 0$

5) $f(z)$ is a $C^2$ function

To be more specific to your case (data is the input and output) the problem reduces to:
$$\underset{d_i}{min}\; d_iq+wl+rk$$
$$s.t.\quad f(d_i)\geq \bar{d_o}$$
$$d_i,l,k\geq 0$$

Where $d_i$ is the data used as input, $d_o$ is the data output, $q$ is the price of input data, $w$ is wage, $l$ is labor hours, $r$ is the rental price of capital and $k$ is quantity of capital.

Again, this type of production process is foreign to me, so I cannot make an informed suggestion regarding specification of the production function, but maybe someone who is more apt on this site may offer a suggestion. I hope this helps!


What causes the US media obsession with tariffs; The EU and others have charged tariffs on US stuff forever?

Nations frequently have tariffs between each other. However, the trend since WW2 has been fewer tariffs and other trade barriers. For example, even recently the EU has significantly reduced tariffs.

Likewise, the US has also both practiced and preached free trade, reducing tariff rates.

Economists have strong consensus that tariffs are bad. This isn't dependent on political leanings, it has wide agreement. "The economics profession continues to show a consensus in favor of unfettered international trade, as 83 percent agree and only 10 percent disagree that the United States should eliminate remaining tariffs and other barriers."

The reason the new tariffs are reported by the US media are because they are a new policy change which oppose the existing trends and expert opinion, although they do have public suport


Does anybody knows where I should look for a proxy for consumption to estimate a two factor C-CAPM?

I would recommend you read Kroencke (2017). That paper dismantles the problem in a lot of prior (flawed) consumption-based asset pricing research as well as explaining some findings for what were seemingly better proxies. It is a fantastic bit of work and likely not yet in some of the texts you might encounter.


In Diamond's OLG model, why is capital equal to saving by the young in previous period?

Property rights and assuming away bequest motives (the Old do no care about the Young).

Capital of the previous period belongs to the Old, and since they will die at the and of old age, the finite-horizon transversality condition for utility maximization requires that either all assets have been consumed at the time of death, or that they have no utility-enhancing potential. Since the utility function reflects the "more is better" property, assets do have value as consumption down to the last bolt, and since there is no bequest motive, no utility from "leaving something for your children", it follows that the Old will consume all their capital, being utility maximizers.


Interpreting coefficients with Delta ln as independant and dependent variables

Your equation can be re-written (through the properties of logs) as:

$$
\ln(y_{it} / y_{it-1}) = \beta\ln(x_{it} / x_{it-1})
$$

So it becomes a log-log regression equation.  In this form, it means that if we change the ratio of $x_{it}$ to $x_{it-1}$ by one percent, the ratio of $y_{it}$ to $y_{it-1}$ will change by $\beta$ percent, on average.  In other words, if $x_{i}$ grows by one percent between two periods, $y_{i}$ is expected to grow by $\beta$ percent in that same time span.


In the original problem, for $L=2$, the consumption set was $(-\infty,\infty) \times \mathbb{R}_{+}$. Now, the consumer is restricted to $[0,\infty)\times \mathbb{R}_{+} = \mathbb{R}_{+}^2$. Fix a price $p = (1,p_2)$, and now see what happens to Walrasian demand (i.e. $x(p,w) = (x_1,x_2))$ as we vary $w$.
You saw from part (a) that a consumer with these preferences consume in the following order.

(i) Spend all money on good 2 until we get to $x_2(p,0)$ (since the amount of good 2 demanded is independent of $w$).

(ii) If you have money leftover, spend it on good 1 (i.e. $x_1(p,w) \geq 0$). Otherwise, in order to balance the budget, consume negative amounts of good 1 (i.e. $x_1(p,w) < 0$).

Let us now fix $p = (1,p_2)$. Now, we are unable to do case (ii) when you have overspent on good 2. You can only overspend when:
$$p_2 \times x_2(p,0) > w$$

Using the usual marginal rate of substitution equal price ratio, we have:
$$
\frac{\eta'(x_2)}{1} = \frac{p_2}{1}
$$
Hence,
$$x_2(p,0) = \eta'^{-1}(p_2)$$

Going back to the first equation, $p_2\times \eta'^{-1}(p_2) > w$.

Since she spends all her money on good 2, the budget constraint
$$x_1(p,w) + p_2 x_2(p,0) = w$$
becomes
$$p_2 x_2(p,0) = w$$
or
$$x_2(p,0) = \frac{w}{p_2}$$

Hence, her Walrasian demand (up to $p_2\times \eta'^{-1}(p_2) > w$) is
$$x(p,w) = (0,\frac{w}{p_2})$$
And once $w$ is large enough such that $p_2\times \eta'^{-1}(p_2) \leq w$,
$$x(p,w) = (w - x_2(p,0),x_2(p,0))$$

EDIT: Lets try a different way to get the answer using the calculus approach.

Our new maximisation problem is:
\begin{align*}
\max_{x_1,x_2} \, &x_1 + \eta(x_2) \\
\text{s.t. } &x_1 + p_2 x_2 = w &\text{ (Budget constraint)}\\
&x_1 \geq 0 &\text{ (Non-negativity constraint) }
\end{align*}
I put equality on the budget constraint because I assume monotonicity.
The associated Lagrangian is
$$
\mathcal{L} = x_1 + \eta(x_2) + \lambda(w - x_1 - p_2 x_2) + \mu(x_1)
$$
Taking first order conditions:
$$
1 - \lambda + \mu = 0 \\
\eta'(x_2) - p_2\lambda = 0
$$
Now, suppose the non-negativity constraint does not bind (i.e. $x_1 > 0$). Then, by complimentary slackness, $\mu = 0$, which implies $\lambda = 1$ and $x^*_2 = \eta'^{-1}(p_2)$.
We can clearly see that the optimal choice of $x_2$ does not depend on $w$ at all when the constraint is non-binding.

Now, suppose the non-negativity constraint does bind (i.e. $x_1 = 0$). Using the budget constraint, $0 + p_2x_2 = w$, thus $x_2 = \frac{w}{p_2}$.

Finally, when does the non-negativity constraint matter? It matters only when we would like to set $x_1 < 0$. Again, using the budget constraint,
$$
0 > x_1 = w - p_2x_2^* = w - p_2 \eta'^{-1}(p_2) \Leftrightarrow w < p_2 \eta'^{-1}(p_2)
$$

Hence, when $w < p_2 \eta'^{-1}(p_2)$, the constraint binds thus $x_1 = 0$ and $x_2 = \frac{w}{p_2}$.

When $w \geq p_2 \eta'^{-1}(p_2)$, the constraint does not bind thus $x_2 = \eta'^{-1}(p_2)$ and $x_1 = w - \eta'^{-1}(p_2) \geq 0$.


What happens to consumer surplus and profits as firms get more information in Hotelling's duopoly model?

The article Quality of Information and Oligopolistic Price
Discrimination by Liu and Serfes covers this topic in great detail. It also has a rather nice literature review.


Do property values capture producer choice in agriculture?

Suppose that, initially, crop A is the most profitable crop that can be grown on a particular farm, yielding an average annual profit of \$X, profit being measured after deduction of all costs (including a normal return to the farmer) other than the cost of the property (ie the land).  In a competitive property market, the capital value of the property might be expected to approximate to the discounted present value of a stream of \$X per year.

Now suppose that, due to climate change, the relative profitability of different crops changes, and that crop B becomes the most profitable crop, yielding an average annual profit of \$Y. This could represent either an increase or a decrease in profitability. The former is possible for a farm in a cool climate, where perhaps crop B was previously too unreliable, but with a higher average temperature becomes more profitable than crop A.  The latter is possible for a farm in the tropics, where perhaps both crops used to be reliable but now crop A suffers from heat stress.  In either case the capital value of the property might be expected to change to the discounted present value of a stream of \$Y per year.

So in principle your colleague's suggestion is correct.  However, there are many reasons why it might be hard to identify such an effect in practice:

Some changes in crop in response to climate change might result in the new profitability being similar to the old.
Average annual profitability may be hard to estimate given random fluctuations in weather conditions.
Profitability depends on market prices for crops and for inputs such as seeds, labour, fuel and fertilisers which may change.
The relevant discount rate for converting annual profit to a capital sum may also vary with market conditions.