Does trade lead to economic growth in the most basic models of trade?

It depends on how you define "growth". Trade models have multiple goods in them, so there's no unique way to define "growth" - you are just moving along a fixed production possibility frontier. You can calculate nominal GDP, but then you have to come up with a measure of inflation so that you can deflate the nominal figure into something real.

One of the results here is that if your economy which produces $ n $ goods has a production possibility frontier given by the zero set of a $ C^2 $ function $ Q : \mathbb R^n \to \mathbb R $ (with the usual properties, i.e increasing in each coordinate and with positive definite Hessian at every point), then a simple application of the chain rule gives (here the $ dx $ notation denotes derivatives with respect to time, so we're assuming that the economy shifts continuously along the ppf instead of a unit step jump from one point to another - this assumption is important, otherwise we can't lose the higher order terms in the Taylor expansion of $ Q $...)

$$ 0 = dQ = \sum_{k=1}^{n} \frac{\partial Q}{\partial x_k} dx_k $$

if you're moving along the ppf, so that the value of $ Q $ is conserved. However, we also know from the first order conditions of producers in a competitive market that

$$ \frac{\partial Q/\partial x_i}{\partial Q / \partial x_j} = \frac{p_i}{p_j} $$

where $ p_i, p_j $ are the nominal prices of good $ i $ and good $ j $ respectively. Substituting this into the above identity gives

$$ 0 = \sum_{k=1}^n p_k dx_k $$

This is an important result: it says that if you calculate growth by looking at changes in production and hold the prices fixed, then you will detect no growth if the economy moves along a fixed production possibility frontier. Furthermore, this identity combined with the identity

$$ NGDP = \sum_{k=1}^{n} p_k x_k $$

gives, using the product rule, that

$$ d(\log(NGDP)) = \frac{1}{NGDP} \sum_{k=1}^n x_k d p_k = \textrm{GDP deflator} $$

In other words, if we deflate GDP by using the GDP deflator which is defined as the percent change in the price of a basket of goods weighted by the share of total production they represented in a given time period, we will measure zero RGDP growth so long as the economy remains along the same production possibility frontier $ Q(x_1, x_2, \ldots, x_n) = 0 $. The results hold true in autarky as well as in an environment of free trade, so in this sense free trade does not lead to "more growth", at least not due to the effects present in Heckscher-Ohlin.

We can get away from this result if we choose to deflate nominal GDP by an alternative measure, but I think this should be enough illustration that you should be thinking about the welfare effects of trade and not about the "growth" effects. If you do come up with an appropriate notion of CPI inflation in a model, it will most likely be a "cost of living index" of the kind you find in Dixit-Stiglitz type models, so you will still indirectly be measuring welfare. The basic proof of the first welfare theorem carries over to the case of international free trade vs autarky, so under the usual assumptions (locally nonsatiated preferences and such) free trade outcomes are Pareto optimal, whereas autarky outcomes are in general not Pareto optimal.


Can I Upload my Preprint on Arxiv Before Submitting it to JPE

Most papers I have read in the JPE circulated in form of preprint before (not on arxiv, but this should not be an important point). See for instance https://ideas.repec.org if you want to find examples of working papers later published in JPE:

But it is probably better to read the journal's policy before submission.


What's the advantage of repurchase agreements over collateral backed loans?

Repo and Securities Lending
https://www.newyorkfed.org/medialibrary/media/research/staff_reports/sr529.pdf
Nominal page 2:

A repurchase agreement is the sale of securities coupled with an agreement to repurchase the securities, at a specified price, at a later date (see Duffie (1996) and Garbade (2006)). Securities lending agreements are economically similar to repo agreements.4 Both agreements resemble a collateralized loan, but their treatment under the U.S. bankruptcy law is more beneficial to cash lenders:    In the event of bankruptcy, cash lenders can typically sell their collateral, rather than be subject to an automatic stay as would be the case for a collateralized loan.

Two advantages are

Repo are short term self-liquidating positions which automatically run off the balance sheet under the terms of the deal; and

Collateral under Repo deals is available for sale by the lender if the counterparty goes bankrupt.


Could the upper quintile pay the annual federal personal income tax for all other quintiles from their annual income?

Here is the latest report of the Congressional Budget Office on federal taxes by households. The main result you are looking for is this one (page 31):

The "Federal Taxes" row shows that the upper quintile tax bill is higher than the tax bill of the other four quintiles combined. (the data are averages but each quintile has same number of households so those numbers are enough to conclude this).

The figure below show this more clearly, in percentage terms:

Check here for older reports.

PS: amazingly, the top quintile gets more government transfers than the poorest quintile, which get the lowest. An entirely regressive system!


Why are future expenses considered less costly than expenses in the present?

Let's simplify this with an example. If I gave you $100 right now, at the very worst, you could collect interest rate on it (let's assume 5%).

Let's compare two cases:

A) I gave you $100 a year ago.

B) I give you $100 now.

In the first case, you'd have collected interest and now be at $105.

In the second case you'd only have $100.

Note this same logic applies to expenses. You can pay me $100 now.

Or you can pay me $100 next year (and keep the interest for yourself).


Do banks really have people on the boards of corporations, and why?

What you describe is neither unusual nor pernicious, and occurs more commonly at smaller companies when the company has borrowed significantly from a single bank. Larger companies have direct access to capital markets (in addition to stock, they can issue bonds and commercial paper, which are both forms of debt), and are thus less reliant on their relationship with a single bank.

When companies are reliant on a relationship with a bank for access to debt, the bank will often seek a position on the board as a way of monitoring the activity of the company. From the bank's perspective, this is totally reasonable, but to understand why, you first have to understand a tiny bit of industrial organization theory.

In a nutshell, firms have what we refer to as "capital structures," which are a way of allocating risk and return.

At the top of capital structures is senior debt, which is first in line to be paid, but which receives the lowest (though a fixed and predictable) rate of return. If you hold senior debt, you want the company that issued the debt to take little risk, as you'll be paid the same amount so long as the company remains solvent. If the firm goes into bankruptcy, you'll still be paid first, but your money may be tied up for a time in bankruptcy proceedings, and you face having to reinvest the cash you get back, potentially at a lower rate of return. As a result, you want to avoid unnecessary risk.

At the bottom of capital structures is equity, which is high-risk and high-return. The equity portion gets paid the upside if a firm does well, and it bears the first loss if a firm performs poorly. If a company goes bankrupt, the equity is worthless. Equity investors have a lot of incentive to take on risk, because for each dollar invested, the downside is a maximum of a dollar (this is known as "limited liability"), but the upside is limitless.

In between these two parts of the capital structure, there are all kinds of hybrids of debt and equity, but we can set them aside for simplicity. Most important is that there is a fundamental conflict between debt holders, who want limited risk and who only have control of a firm once it is bankrupt, and equity holders, who favor risk and control the firm in the normal state of the world.

Once you understand this conflict, it's easy to see why a bank would ask for a seat on the board of a company when it holds that company's debt: by having a member on the board, it can ensure ongoing, regular access to the information it needs to be comfortable that the firm isn't taking on too much risk. If it sees that the firm is moving in a direction that it isn't comfortable with, it'll decline to roll over its debt, getting its money back and forcing the firm to find financing elsewhere.

So: is this unusual? No. Directors who don't directly "represent" equity holders aren't at all uncommon. In fact, the majority of directors at large firms in the US are "independent" directors who have no direct relationship to the firm, other than serving on the board. Many non-independent directors are "executive" directors, who are company management— a group that, like creditors, may have incentives that differ from those of shareholders. Other groups, like labor unions, are also sometimes represented on boards. So if in addition to equity holders, outsiders, management, and unions are all represented on boards at times, what is the specific objection to creditors being represented?

It's additionally worth noting that the only case where a creditor's vote matters is solely when the other directors are evenly split on an issue. If a particular matter clearly benefits creditors at the expense of equity holders, a significant fraction of equity holders would have to vote against their own interests for this to be the case.

Finally, to your question as to why regulators and corporations allow it. Simply put, regulators have no reason (from a market conduct, safety and soundness, consumer protection, or financial stability perspective— the general scope of regulators' mandates) to object to this sort of arrangement. For a regulator to intervene in this way would interfere with the company's freedom to contract however it chooses. Creditor board members are not a thing that banks do to corporations, who passively allow it; rather, the boards of directors have to vote to add the board member representing the creditor. So if one views positively the judgment of a board composed of equity holders, why should one question the judgment of that board when they vote to add a creditor as a member?

So in summary: banks as board members is a way of monitoring/limiting risk-taking by firms, firms "allow" this because they want to borrow money from the banks, whether it is socially optimal depends on whether firms should be taking more or less risk, and it's more about correcting information asymmetries than about asserting day-to-day control— boards of directors don't tend to have lots of measures passing or failing by a single vote.


Effect of lower Government spending on Loanable Funds Market

If the opposite were true and the government proposes spending $1 trillion on infrastructure spending we will see a shift outward of the demand curve and rates will tick upward to a new equilibrium point. The crowding out occurs when we have movement along the demand curve (from the old interest rate --> new demand curve intersection up to the new intersection of the S and D curve).

With a decrease in government spending your demand curve for the loan-able funds market will shift inward and push the interest rate lower. When a fall in the interest rate leads to higher investment spending, the resulting increase in real GDP generates exactly enough additional savings to match the rise in investment spending.

It is somewhat agreed upon that contractionary fiscal policy leads to a surplus and the government can act as a creditor rather than a debtor. It will also leave more money for private investments for people such as small borrowers.

(There are some economists that believe that under the right circumstances that expansionary government policies may produce crowing in instead of crowding out. Keynesian economists may suggest that if the policy is strong enough to change the economic climate then businesses may find it profitable to add capacity -induced investment- which works to counteract and possibly out weigh the crowding out effect.)


Utility maximization question setting up.

Consider a consumer whose preferences can be represented by the following utility function: $$u(x_1,x_2)=\dfrac{x_2}{(1+x_1)^2}.$$

    Assume the agent's income is $y=5$. The price of one unit of good $1$ is $p_1=1$. For each unit of good $1$ the agent buys, he qualifies to buy up to one unit of good $2$ at an additional price of $p_2=1$. In other words, to buy one unit of good $2$ the agent has to first buy one unit of good $1$. The agent must consume everything he buys. Using this information, sketch the feasible set. Is it convex? Derive the utility maximizing bundle.
    How does your answers to question 4. change if the agent does not have to consume everything he buys ("free disposal")?

For Q 4 :

Utility maximization problem of the consumer is :

\begin{eqnarray*} \max_{x_1, x_2} & \ \ \frac{x_2}{(1+x_1)^2} \\ \text{s.t.} & \ \ x_1+x_2 \leq 5 \\ \text{and} &  \ \ 0 \leq x_2 \leq x_1  \end{eqnarray*}

Here is the constraint set of the consumer, along with a few indifference curves:

Observe that the constraint set is convex and the consumer does not spend all his income in optimum. His optimal consumption bundle is $(x_1, x_2) = (1,1)$.

For Q 5 :

Utility maximization problem (with free disposal) of the consumer is :

\begin{eqnarray*} \max_{x_1, x_2, b_1, b_2} & \ \ \frac{x_2}{(1+x_1)^2} \\ \text{s.t.} & \ \ b_1+b_2 \leq 5 \\  &  \ \ 0 \leq b_2 \leq b_1  \\ \text{and} &  \ \ 0 \leq x_1 \leq b_1, 0 \leq x_2 \leq b_2\end{eqnarray*}

Here $b_1$, $b_2$ denotes the amount of the two commodities bought by the consumer, and $x_1$, $x_2$ denotes the amount consumed. In this case, the consumer will try and maximize his consumption of commodity 2 $(x_2)$ by buying as much amount of commodity 2 $(b_2)$ as he can. Clearly, the solution to this utility maximization problem is
$b_1 = b_2 = x_2 = 2.5, x_1 = 0$.

For Q 4, here is a way to solve the optimization problem using Lagrangian method :

Given the utility maximization problem of the consumer :

\begin{eqnarray*} \max_{x_1, x_2} & \ \ \frac{x_2}{(1+x_1)^2} \\ \text{s.t.} & \ \ x_1+x_2 \leq 5 \\ \text{and} &  \ \ 0 \leq x_2 \leq x_1  \end{eqnarray*}

We set up the Lagrangian as follows:

$\mathcal{L}(x_1, x_2) = \dfrac{x_2}{(1+x_1)^2} - \lambda(x_1+x_2-5)  +\mu_1(x_1-x_2)+ \mu_2x_2 $

Necessary conditions for optimality are as follows :

$\dfrac{\partial \mathcal{L}}{\partial x_1} = \dfrac{-2x_2}{(1+x_1)^3} - \lambda + \mu_1 = 0$

$\dfrac{\partial \mathcal{L}}{\partial x_2} = \dfrac{1}{(1+x_1)^2} -\lambda - \mu_1 + \mu_2 = 0$

$x_1+x_2 \leq 5$, $\lambda \geq 0$ and $\lambda(x_1+x_2-5) = 0$

$x_1 \geq x_2$, $\mu_1 \geq 0$ and $\mu_1(x_1-x_2) = 0$

$x_2 \geq 0$, $\mu_2 \geq 0$ and $\mu_2x_2 = 0$

Solving the above system, we get

$x_1 = 1$, $x_2 = 1$, $\mu_1 = \frac{1}{4}$, $\mu_2=0$, $\lambda = 0$

Alternatively for Q 4, the problem can also be converted to a single variable optimization problem. This is because the individual will always consume equal amounts of $x_1$ and $x_2$ in equilibrium. Substituting $x_2=x_1$ yields :

\begin{eqnarray*} \max_{x_2} & \ \ \frac{x_2}{(1+x_2)^2} \\ \text{s.t.}  &  \ \ 0 \leq x_2 \leq 2.5  \end{eqnarray*}

Differentiating $\dfrac{x_2}{(1+x_2)^2}$ with respect to $x_2$ gives the following necessary condition

$\dfrac{(1+x_2)^2 - 2(1+x_2)x_2}{(1+x_2)^4} = \dfrac{1-x_2^2}{(1+x_2)^4} = 0$

which yields $x_2 = 1$, and the corresponding value of $x_1 = 1$.


First, you should note that there is a Revenue Equivalence Theorem that says under a set of conditions, the seller's revenue from using different forms of auctions will be the same. This same result holds also in the multi-object case.

Thus, if you have $n$ identical objects, you can sell them using a ($n+1$)th price sealed bid auction, where

each bidder submits a bid,
the highest $n$ bidders each gets $1$ object, and
they all pay a price equal to the ($n+1$)th highest bid.

In such an auction, it is a weakly dominant strategy for every bidder to bid their true valuation of the object. The reasoning is similar to the second price auction.


In dealing with problems like this, it's helpful to do it in stages. You know that the two dealers take the wholesale price $r$ as a given (they can't change that), and car dealer $i$ choose retail price $q_i$ to try to maximize their profits $q_i \cdot (P - r)$.

You also know that these car dealers are exactly the same... they have the same cost, set some price, face the same demand (we say the two firms are "symmetric"). So in the end you know that whatever is the best quantity $q_i$ for the first car dealer would be the best choice for the second dealer as well, so we know $q_1 = q_2 = Q/2$.

With this, the dealers are choosing $Q$ to maximize $\pi(Q, r) = \frac Q 2 \cdot (P - r)$, with $P$ given by the demand function (solve the inverse total demand for $P$). Let's call this profit-maximizing quantity $Q^* = 2q_1^* = 2q_2^*$ and the resulting price $P^*$.

Now that we're done with the car dealer, let's step in Toyota's shoes. You want to make as much money as you could, so the higher $r$ is the better off you are. But you also know that if you set the retail price too high, the dealers will lose money with each car they sell and will likely stop ordering from you (maybe switch to selling Hondas instead). What's the maximum you could charge while still keep the dealers not losing money?


How does profit correlate with the duration of options?

You may profit more off of short-term options in terms of returns. The reason is simple: with less time to expiry, the time value of an out-of-the-money option is near 0. However, this is only true if the options expire in-the-money -- a huge "if."
There are a few caveats that are crucial:

Options impound the entire future expected distribution of returns, so you are unlikely to make easy money.
Any post on Reddit or other social media on money made trading or trading opportunities is likely to be worthless and could be trying to lure you into a pump-and-dump or other manipulation. I would stop reading such posts on Reddit.
You mention options maturing in 7 and 21 days as well as LEAPS. Weekly expirations tend to be very illiquid as do LEAPs. Quarterlies are your most liquid expiries; I'd stick to those.
The bid-ask spread can be quite wide for weeklies, LEAPS, and out-of-the-money options near expiry. You could easily pay a bid-ask spread that is nearly the quoted price of the option -- so a 100% bid-ask spread.

Glad to hear you aren't planning to trade this on your own. I've traded options for years professionally and even I will not trade options in my personal account.


What is the difference between microeconomics money and macroeconomics money?

There is no distinction between money in microeconomics or macroeconomics. In both fields money is  medium of exchange, unit of account and store of value.

The misconception you have probably arises from the role money plays in microeconomics and macroeconomics.

In microeconomics money is almost always neutral - that is it has no impact on the real variables like equilibrium output, labor supply, demand etc. Many micro models even assume money away and prices are simply given relative to some numeraire good. There are exceptions of course.

In macroeconomics money is also assumed to be neutral in the long run but not in the short run. Hence in macroeconomics money is analyzed explicitly by including price levels, inflation etc. For example, as you mentioned one of the special effects of money is that it can provide much needed liquidity that can boost aggregate demand.

If you are looking into name for the distinction between cases where money matter and when not, that is when it is not neutral and when it is (both in micro and macro) the name is classical dichotomy although the concept is again mainly used in macro since in micro money is almost always neutral


Utility maximization in a 2-good scenario with an option to buy a combo of the two

Your answer to Q1 is correct.

For Q2, a little graphical intuition would help.

Suppose you spend all your income on the combo, you'd be able to purchase $80$ movies and $20$ books. This is the point $(20,80)$ on the $(x,y)$-plane.

If you buy $1$ less combo, you'll have $76$ movies, $19$ books, and $\$50$ cash. Spending the entire $\$50$ on movies would move you to $(19,81)$; spending all the cash on books would move you to $(21,76)$. (Of course you can split the cash between the two goods, but it's easy see on a graph that such a split is not efficient.)

Therefore, trading one combo for movies implies a relative price of 1 book for 1 movie, and trading one combo for books implies a relative price of 4 movies for 1 book. Using the information, we can plot the budget constraint as below:

The green segment, given by $y=100-x$ ($x\in[0,20]$), represents trade-off between the two goods implied by switching from combos to movies. The red segment, given by $y=160-4x$ ($x\in[20,40]$), represents the trade-off implied by switching from combos to books. It is clear from the graph that no other ways of using the income is more efficient. (As a reference, the budget line without the combo is plotted using the black dashed line.)

Given this budget constraint, it should be straightforward to solve for the optimal consumption bundle.


How long does it take for shocks to propagate through the price system / market?

This is likely only part of the story, but I found an interesting lecture by Milton Friedman that shows a graph of the quantity of money per unit of output (likely GDP) graphed alongside a normalized consumer price index (normalized so that the index intersects with the other line in 1970).

The graphs all clearly shows changes in the consumer price index lagging about 1-3 years after discontinuities in the money supply.

https://www.youtube.com/watch?v=THAaIZmxfNA&t=11m2s

In this other graph, you can see the lag is even longer than that, at anywhere from 4 years between 2008 and 2001 to 6 years from 1985 to 1992:

3-4 years is probably a reasonable average estimate for how long it takes for money created by the Fed to "trickle down" to the average consumer in the form of inflation.


Adverse Selection: Positive Selection of Worker Types (Mas-Collel)

In PO, you want all types with opportunity cost $r(\theta)\leq \theta$ to trade, because the firm gets more productivity than the worker has to give up on home productivity ("opportunity cost").
By your assumptions $r(\theta_H) < \theta_H$ and $r(\theta_L)>\theta_L$ such that in PO, L shall stay at home and H shall work. However, you need to offer H a wage of at least $w\geq r(\theta_H)$.
The highest wage a firm would offer is the expected $\theta$ of a worker who is willing to accept this wage. Suppose that $\mathbb E[\theta]=\lambda \theta_H + (1-\lambda)\theta_L >r(\theta_H)$ and, by definition,  $\theta_L<\mathbb E[\theta]<\theta_H$.
Now, it could be that $w=E[\theta] > r(\theta_L)$ such that L would also want to accept this wage, but this is inefficient as
$$w > r(\theta_L) > \theta_L.$$
Low types should not work, but because there are so few of them ($\lambda$ is large), the firms would still profit if they also accepted $w$.
An uniformed third party could Pareto improve the allocation by changing the wage to $w' \in (r(\theta_H),r(\theta_L))$ such that L types would reject $w'$.


Does salary contribute to GDP under the expenditure approach?

No under expenditure approach neither salary or wages are directly counted. An expenditure approach to GDP calculates GDP as follows:
$$GDP=C+I+G+NX$$
Where $C$ is consumer spending on final goods and services at market prices, $I$ is the investment spending, $G$ is government spending and $NX$ are net exports.
All wages and salaries are of course indirectly included in as prices of final goods and services indirectly may include them but under spending approach they are not counted directly. Even if you would hire lets say language tutor from some company where company pays almost everything directly to that tutor it would be the cost of service that would appear on national accounts not the wage of the tutor under spending approach.
If you would want to count wages & salary directly you could use income approach to GDP where the GDP would be calculated by summing all returns to labor (wages & salary), profits, rental income and interest.


What are the causes of increase in productivity of labour?

In one word: Technology.

It's a bit abstract of course, but it summarises a lot of things very well. It's usually the A parameter in the models we have in growth theory (we augment e.g. Labour by Technology).
In empirical studies it has been tried to asses productivity by coming up with a measure called TFP (Total Factor Productivity).

A lot of specific things could potentially be subsumed under this broad term. Most importantantly it is the stock of knowledge (e.g. books, blueprints,...) that we improve and build upon. The growth of technology usually depends on the actual stock of knowledge in a certain economy. It can be achieved by Research & Development (in case the economy is on the technology frontier) or simply by technology adoption (e.g. in the case of catch-up economies).

Example: Knowing how to build a tractor, we only need one unit of labour (the guy who drives the tractor) to harvest, say, 100 kilos of grain per day instead of 50 workers that have to cut the same amount by hand. This would be a tremendous increase in productivity because the other 49 people can now do another job in which they potentially add to the economy's output. For building a better tractor (that drives with less fuel, can harvest more, ...) the economy needs the blueprint of the first tractor to build upon, and so on and so forth...


If a fluctuating cryptocurrency becomes the new world coin, would it stop fluctuating?

Your question is very similar to "if we were back on the gold standard, what mechanisms exist to regulate inflation/deflation".

The question is a difficult one because it brings into scope the role of banks and credit and how they would fit into the world. Crypto-systems (in the sense of cryptocurrencies) don't address credit today.

You'd likely see the world separated into two categories of 'user':

Those who manage their own wallets
Those who do not

For the second category banks would continue to exist. Cold wallet storage, data centers and the like. Those banks would still be licensed to issue notes on the basis of the 'digital gold' they hold, so we'd be back to square one with banks being literally issuers of notes with cryptocoin as the new gold.

The value of the trustless gold, and its trust based proxy, would be regulated by the same mechanisms that existed under the old gold framework: trade imbalances result in cross border settlements of gold influx/egress that adjust bank exposure to circulating 'paper'. This results in credit interest rate adjustments, which accelerate or brake the economy depending.

So yes in short international trade would act as a mechanism for adjusting things and tending towards some regulation of the currency value. Similarly central banks would adopt policies to manage the availability of coins. I don't know what policies, but we should never underestimate the power of states (eg now under C19) to simple dictate that coins be donated/given/taxed etc to a central authority.


The only use of money (other than by humans— who, it should be noted, lived for most of human history without it) in the animal kingdom that I'm aware of has been when researchers have taught other primates (Capuchin monkeys) how to use it. While the introduction of currency was apparently successful (there was Capuchin prostitution), the research did not go so far as to show that equilibrium prices would arise endogenously through exchange.


perfect bayesian nash equilibrium is simply nash equilibrium

The quick answer is Yes.

A Bayesian Nash equilibrium (BNE) is Nash equilibrium (NE) generalized to situations where there is uncertainty over players' payoffs; that is, games with incomplete information. For example, in the following zero-sum game, $t$ is a "state variable" that may take values $\epsilon>0$ or $-\epsilon$, each with probability $1/2$, say. Then the equilibrium you solve for would be a BNE.

On the other hand, if $t$ is not a random variable (or a degenerate random variable, as the jargons would have it), e.g. $t=0$ with probability $1$, then the equilibrium would simply be a NE.

In other words, a NE (of a complete information game) is a special case of a BNE (of a game of incomplete information, where there is only one payoff-relevant state, so to speak).

$$
\begin{array}{|c|c|c|}\hline
  & H & T \\\hline
H & 1+t,-1+t & -1+t, 1+t \\\hline
T & -1,1 & 1,-1\\\hline
\end{array}
$$

Adding perfection to BNE puts more restrictions on the players' off equilibrium beliefs about their opponents' strategies. In the case of matching pennies (the game above with $t=0$), which I assume you have in mind when saying "zero-sum games", this consideration wouldn't matter.


What is the equation $\mathbb{E}[mR]=1$?

This is an important result in financial economics (asset pricing) but not trivial to explain intuitively. I do my best to give you the big picture and get you started on your research.

R is the return on an asset or portfolio.  Any asset or portfolio.

m is the stochastic discount factor or pricing kernel.

The expectation here is over possible states of the world given information at time t (m and R are as of t+1).

These are physical, not risk-neutral, probabilities. In fact, a (positive) stochastic discount factor is what is used to translate physical to risk-neutral probabilities.  In a discrete case, the risk-neutral probability is the risk free rate times the physical probability times the value of the stochastic discount factor in that state.

The stochastic discount factor takes on a different value for each possible state of the world (that's the sense in which it is stochastic).  But it doesn't change for each different asset.  If it exists, a stochastic discount factor can be applied to every asset in the economy and price them all correctly.

Facts about the SDF:

The existence of such a factor is equivalent to the law of one price holding.
If it is always positive, there are no arbitrage opportunities in financial markets.
If and only if markets are complete, then the stochastic discount factor is unique.
The expectation of the SDF is the risk-free rate

There are a lot of theoretical uses of this relation and important related results---many fundamental results in finance can be interpreted in an SDF context, though many of those results came before development of SDF theory.  For many common utility functions we can derive the form of the SDF for a representative agent theoretically.

I'm sorry if my answer isn't super accessible.  The equation is simple, but stochastic discount factor theory can be a little opaque.

Resources to consider reading:

Cochrane and Culp's chapter in Modern Risk Management: A History

Asset Pricing and Portfolio Choice Theory, by Kerry Back


Lump sum taxation in the overlapping generations model

If I understand correctly the set up, the government does not destroy real goods, it just collects fiat money and destroys it to reduce the available quantity of money (since money is in the hands of the individuals). The fact that the government calculates the amount of fiat money to collect by using a real-goods base, $\tau$, does not alter this fact. But then, all endowment is consumed and the government's actions just affect the price level.

Some assumptions:
a) population is constant at $N$ per generation
b) all individuals are identical, so we can carry the analysis by setting $N=1$
c) There are no altruistic motives, so no inheritance
d) Individuals do not carry over periods real goods: they consume some of their endowment, and what they do not consume, they sell in exchange for money
e) the subscript $1$ denotes consumption when Young, and the subscript $2$ consumption when Old.

Assume that we are in economy's period $t+1$. The Young come to the Goods market as suppliers, bringing with them $y-c_{1,t+1}$ goods. The Old come to the Goods market as buyers, bringing with them fiat money.

Market clearing implies in real terms that all real goods will change hands, and then that the Old that bought them will consume them. So we obtain

$$c_{2,t+1} = y- c_{1,t+1} \implies c_{2,t+1} + c_{1,t+1} = y  \tag{1}$$
In nominal terms equilibrium requires that Total Revenue of Suppliers equals Total Expenditure of Buyers. Total Revenue is

$$TR_{t+1} = p_{t+1}(y-c_{1,t+1})  \tag{2}$$

The Old will want to spend all their available money (since money is not consumed per se, and they don't leave any inheritance). The money they got in the previous period (prior to being taxed) was Total Revenue of Suppliers in period $t$. The tax is paid on $t+1$, has a real base $\tau$ and so its nominal value is $T_{t+1} = p_{t+1}\tau$. So Total Expenditure is

$$TE_{t+1} = p_t(y-c_{1,t}) - p_{t+1}\tau  \tag {3}$$

Equating $(2)$ and $(3)$ we get

$$p_{t+1}(y-c_{1,t+1}) = p_t(y-c_{1,t}) - p_{t+1}\tau \implies p_{t+1}(y-c_{1,t+1} +\tau) = p_t(y-c_{1,t}) $$

and using $(1)$ we obtain an expression that involves the components of life-time consumption of the same individual

$$p_{t+1}(c_{2,t+1}+ \tau) = p_t(y-c_{1,t})\tag{4}$$

We are moreover told that what fiat money the government collects as taxes, it destroys, targeting a fixed rate of reduction. So we have

$$m_{t+1} = m_t - T_{t+1} = m_t - p_{t+1}\tau = zm_t$$

$$\implies m_t = \frac {p_{t+1}\tau}{1-z} ,\;\; 0<z<1 \tag {5}$$

We know that $m_t = p_t(y-c_{1,t})$, so we obtain

$$p_t(y-c_{1,t}) = \frac {p_{t+1}\tau}{1-z} \tag {6}$$

Combining $(4)$ and $(6)$ we see that consumption of Old is determined exogenously

$$ (4), (6) \implies p_{t+1}(c_{2,t+1}+ \tau) = \frac {p_{t+1}\tau}{1-z}$$

$$\implies c_{2,t+1} = \frac {z}{1-z}\tau \tag{7}$$

Since the endowment $y$ is also exogenous, consumption when Young is also determined exogenously (from $(1)$)

$$ c_{1,t+1} = y-\frac {z}{1-z}\tau \tag{8}$$

But if $z$ and $\tau$ are constant over time, then the time subscripts become irrelevant. Each generation consumes exactly the same when Young and when Old as the previous and the next ones:

$$c_1 = y- \frac {z}{1-z}\tau ,\;\; c_2 = \frac {z}{1-z}\tau \tag{9}$$

Using $(9)$ together with $(4)$ we can determine the evolution of the price level

$$(4), (9) \implies p_{t+1}\left(\frac {z}{1-z}\tau+ \tau\right) = p_t\frac {z}{1-z}\tau$$

$$\implies \frac {p_{t+1}}{p_t} = z <1 \tag{10}$$

as should be expected. Constant amount of goods with shrinking amount of fiat money $\implies$ deflation.

As regards the issue of whether the equilibrium here is welfare maximizing.

Eq. $(4)$ combined with $(10)$ gives us the life-time budget constraint. Assuming logarithmic utility, the individual would solve

$$\max V = \ln(c_1)+  \frac 1{1+\rho}\ln(c_2)\\
s.t. \;\; y - zc_{2} - z\tau  = c_{1} \tag {11}$$

The first order condition with respect to $c_2$ is

$$-z\frac 1 {c_1} + \frac 1{1+\rho}\frac 1 {c_2} = 0 \implies c_1 = (1+\rho)zc_2 \tag{12}$$

Combining with the life-time budget constraint as well as $(9)$ we end up with a quadratic in $z$ in order for the equilibrium to be utility-maximizing.

$$(11), (12) \implies (1+\rho)zc_2 = y - zc_{2} - z\tau \implies (2+\rho)zc_2^* = y  - z\tau \tag{13}$$

$$(9), (13):\;\; c_2^{eq.} = c_2^* \implies \frac {(2+\rho)\tau z^2}{1-z} = y  - z\tau$$

$$ \implies (1+\rho)\tau z^2 +(y+\tau)z - y = 0$$

This has a single positive real solution

$$z^* = -\frac {y+\tau}{2(1+\rho)\tau} + \left[\left(\frac {y+\tau}{2(1+\rho)\tau}\right)^2+\frac {y}{(1+\rho)\tau}\right]^{1/2} $$

Now, for this value of $z$ to be feasible, not only do we want it smaller than unity, but we must also consider the non-negativity constraint on the level of consumption of the first period.

We want
$$c_1 > 0 \implies y- \frac {z}{1-z}\tau > 0 \implies z < \frac {y}{y+\tau} <1 $$

So one should check whether
$$-\frac {y+\tau}{2(1+\rho)\tau} + \left[\left(\frac {y+\tau}{2(1+\rho)\tau}\right)^2+\frac {y}{(1+\rho)\tau}\right]^{1/2} < \;? \; \frac {y}{y+\tau}$$

which holds. So $z^*$ is feasible, and for this single value the imposed allocation of consumption will equal the utility-maximizing solution (at least for logarithmic utility).


Your statement "Portfolio Equity is about stocks/shares, whereas Portfolio Debt is about bonds" is essentially correct, with the associated income being dividends and coupons respectively

Direct investment (FDI) has the associated income of profits, while other debts may have the associated income of interest (if any)

Chapter 6 and other chapters of the IMF Balance of Payments Manual discuss these issues


If we talk about a run on one bank, I think it's the other way around. Back then, people need to stand in line and wait for their queue to take their money out of the bank (and deposit it to another bank). Now they can transfer their money from anywhere and anytime. So I think the risk of a run is greatly increased.


Unemployment Claims vs Monthly Jobs Report

On the last inference of the first graph, the unemployment rate is at 6.7%. This means that out of the 164.6 million Americans that the labor force is comprised, 11.2 million are unemployed as of November 2020 (data: Bureau of Labor Statistics2).
This graph and the graph to its right, both indicate that people are beginning to get employed at a dropping rate.
On the following (red) graph you may notice that as of August, fillings for unemployment have remained relatively constant (data: Department of Labor3).

So the answer I can quickly come up with goes like this; the US is not actually losing 2,000,000 jobs monthly, as the 2nd chart has shown. There might be other incentives at play that encourage people to file for unemployment - perhaps they need to do so in order to receive government handouts even if they didn't work and didn't need to file for unemployment prior to the crisis; meaning they weren't part of the active labor force.


Calculating Consumer Surplus Given Table

The key to this is in the hint. It lets you know supply and demand are both linear, so you need to figure out what the functions of those two curves are. In this case, you have
$$ Q_D = 18 - 1.5P $$
$$ Q_S = 3P $$
Plot these on a supply/demand graph (P on the vertical axis, Q on the horizontal), and the consumer surplus is the shaded area (note, it stops at Q=6 because only 6 units were traded in the question):

Using the formula for the area of a trapezoid, we have:
$$ CS = \frac{1}{2} [(12-4)+(8-4)]*6 = 36 $$


How does a Pareto efficient equilibrium scale for n individuals?

If you are doing pairwise comparisons, then with $n$ individuals there will be $$\dfrac{n(n-1)}{2} $$ such comparisons.  You can write this as $\binom{n}{2}$ or as $\frac12n^2-\frac12n$ if you prefer.

$1$ for two people, $3$ for three, $6$ for four and so on, known as triangular numbers


From reading a selection of writings by Marx, I have come to understand the following three as the core elements of Marxian economics.

The labour theory of value. It asserts that the exchange value of a commodity $x$ (what quantities of $x$ that can be exchanged for another commodity) is determined by that labour time which is socially necessary to produce $x$.

Surplus value theory of profit. I.e., capitalists derives profit from their exploitation of workers. Suppose a capitalist purchases labour power from a worker for one day. To Marx, then, the value of one day's usage of this labour power is the value of the commodities which is necessary for the worker to live through one day. Say that it takes one fourth of a day to produce this amount of commodities, the value of which will be equivalent to the wage paid to the worker by the capitalist. This one fourth of a day is called necessary labour. Any work above this level is surplus labour from which the capitalist derives profit.

The theory of alienation. In the following I quote a passage from Marx's Economic and Philosophical Manuscripts of 1844. Relating this passage to the above core elements, I note that Marx somehow seems to think that the institution of capitalism structures parts of the social world.

The worker becomes poorer the more wealth he produces, the more his
production increases in power and extent. The worker becomes an ever
cheaper commodity the more commodities he produces. The devaluation of
the human world grows in direct proportion to the increase in value of
the world of things. Labour not only produces commodities; it also
produces itself and the workers as a commodity and it does so in the
same proportion in which it produces commodities in general.
This fact simply means that the object that labour produces, its
product, stands opposed to it as something alien, as a power
independent of the producer. The product of labour is labour embodied
and made material in an object, it is the objectification of labour.
The realization of labour is its objectification. In the sphere of
political economy, this realization of labour appears as a loss of
reality for the worker, objectification as loss of and bondage to the
object, and appropriation as estrangement, as alienation.
So much does the realization of labour appear as loss of reality that
the worker loses his reality to the point of dying of starvation. So
much does objectification appear as loss of the object that the worker
is robbed of the objects he needs most not only for life but also for
work. Work itself becomes an object which he can only obtain through
an enormous effort and with spasmodic interruptions. So much does the
appropriation of the object appear as estrangement that the more
objects the worker produces the fewer can he possess and the more he
falls under the domination of his product, of capital.


Why does a firm in a perfectly competitive market not need to cover fixed costs in the short run?

Your question is somewhat wrong in the sense that a firm needn't cover its unavoidable costs (sunk costs). If there are fixed costs that are somewhat avoidable or recoverable (for instance selling an old machine for a salvage value), we need to cover them.

Now coming back to the main question. We can express the profit of a firm using the following equation
$$\pi = R-VC-FC$$

Assume that a firm has already incurred an unavoidable FC of \$5000. The VC that the firm is incurring is \$2000 and the revenue (R) of the firm is \$2500. If the firm continues to operate, its net loss from the business will be \$4500, however, if the firm ceases its operation, its loss would amount to \$5000. Thus as long as a firm has a positive contribution ($R-VC$), the firm must continue its operations.

All in all, Microeconomics says that there is no use crying over a spilt milk. If you have already incurred a sunk cost, it is better to get whatever you can.


Existence of maximum utility with two bads

I assume, by a strongly monotonically decreasing preference $\succsim$, you mean: for all $x,y\in\mathbb R^2_+$,
\begin{equation}
x\succ y \text{ only if $x\le y$ and $x\ne y$}.
\end{equation}
Then it is clear that $(0,0)$ is the most preferred bundle in $\mathbb R^2_+$, i.e. $(0,0)\succ x$ for all $x\in\mathbb R^2$ such that $x\ne (0,0)$.

Therefore, any utility function representing $\succsim$ must also satisfy $u(0,0)\ge u(x)$ for all $x\in\mathbb R^2_+$. Hence,
\begin{equation}
(0,0)=\arg\max_{x} u(x)
\end{equation}


What is the reason behind the demand function of a perfect complement good?

Suppose you have $M=20$ to spend on shoes. Left shoes cost $p_L=5$ and right shoes cost $p_R=5$. A bundle, $(x_L,x_R)$ consists of $x_L$ pairs of left shoes and $x_R$ pairs of right shoes.

How would you figure out how many of each type of shoe to buy? Probably the reasoning would go like this:

You only ever want to buy shoes in pairs (consisting of a left and a right shoe). So instead of asking how many left shoes and right shoes to buy, you can define a new good called "pairs of shoes" and ask how many units of this new good to buy.
The price of a pair is $p_P=p_L+p_R=10$. Thus, you can afford to buy up to $\frac{M}{p_P}=\frac{M}{p_L+p_R}$ pairs (two pairs in our example).
Since you need one left shoe to make each pair, the number of left shoes you buy is simply the same as the number of pairs calculated in step 2: $x_L=\frac{M}{p_L+p_R}$, or two left shoes. Likewise, you need to buy $x_R=\frac{M}{p_L+p_R}$ right shoes.


fall in price arising from the labor savings

The price of a product is usually determined in part by its cost. (This holds in almost all theoretical models without strict capacity constraints, even in the extreme case of monopoly.)
If automation can replace some of the labor with cheaper machines, cost will decrease, hence the product will be sold at a lower price.


Is there an economic analysis of how piracy allows a product to take a hegemonic position?

The model is a variant on penetration pricing (effectively $0 for pirates) which is used to gain market share.

The problem is while monopoly may be desirable, the path to monopoly is generally not, as the cost to acquire market share rises relative to the value of each additional unit of market share.

While there is no specific study of Adobe that I know of, penetration pricing and market share are generally well understood concepts in economics.

That being said while the linked MIT paper implies an "uncanny valley" of sub-optimal over-penetration before a monopolistic (~100%) market share/maximal profit scenario, to my knowledge the empirical data on ologopolistic firms and cartels is scant, outside of the energy industry. Historically, however, OPEC and the Seven Sisters, have never actually combined to a monopoly, nor engaged in penetration pricing by giving away oil (again, to my knowledge).

One thing that is agreed on is that a monopoly is only as valuable as the barrier to enter the market (broadly speaking).

Combining those facts one could surmise that for Adobe to maximize profits they would need to choose a strategy that allowed for either:

a) optimal penetration

$$or$$

b) monopolistic dominance of an industry that would need to produce profit per year less than roughly the barrier to entry divided by the discount rate they erected through a combination of R&D costs and $0 penetration pricing strategies.

$$P_{Total} <  \frac{B}{r}$$

Monopoly of a more valuable market would entice a rational actor to break it and leave both in the sub-optimal valley of oligopoly (unless they colluded), hence the prevalence in niche technical markets you mentioned.

Not to say graphic design is a small niche, but of the population at large they represent rouhgly 0.08% (at least in America).


Finding Mixed Nash Equilibria in a $3\times 3$ Game

Assume the pure strategy $s_i$ is strictly dominated by $s_i'$. This means that $s_i$ is not a best response to any of the opponent's pure strategies. For whatever strategy the other player is playing, the strategy $s_i'$ would yield a larger payoff. Even if $s_i'$ is a mixed strategy, for any strategy the opponent is playing you can select a pure component of $s_i'$ with a larger payoff than $s_i$ as the payoff of $s_i'$ is the average of the payoffs of its pure components.

A reversal of this is that if a pure strategy is a best response to any pure strategy of the opponent, it cannot be strictly dominated.

From the "underline the best response" method you will see that all the strategies of all the players are best response to at least one strategy of the opponent, hence none of them are strictly dominated.

Unfortunately, this still does not tell you much about the mixed equilibria. With different supports it is possible to have more than one.

An example:

In the game
$$
\begin{array}
\\&L&C&R\\
T&1,1&0,0&0,0\\
M&0,0&1,1&0,0\\
B&0,0&0,0&1,1
\end{array}
$$
the strategy profile where all strategies are mixed with probability $1/3$ is a mixed equilibrium, but the strategy profile where the Row player mixes $T$, $M$ while the Column player mixes $L$ and $C$ with probability $1/2$ is also a mixed equilibrium. (There are two other mixed equilibria and three pure ones.)

So there is no trivial method to find all the mixed equilibria, one usually has to use a trick or work through the different cases for all the supports.

Wilson's oddness theorem tells you that there are an odd number of equilibria, except in special "degenerate" cases.


When do workers versus capital owners share of income increase?

From the shares equation, we obtain:
$$
1 + \frac{rK}{wL} = \frac{1}{s_L}
\to tk = \frac{1 - s_L}{s_L}
$$
where I defined $t = \frac{r}{w}$ and $k = \frac{K}{L}$.
then taking logs gives:
$$
\ln(k) = -\ln(t) + \ln(1-s_L) - \ln(s_L)
$$
Then take the derivative of this expression with respect to $\ln(t)$:
$$
\begin{align*}
-\sigma &= -1 - \dfrac{\dfrac{\partial s_L}{\partial \ln(t)}}{1 - s_L} - \gamma,\\
&=-1 - \gamma - \frac{s_L}{1 - s_L} \gamma = -\left(\frac{\gamma}{1 - s_L} +1\right)
\end{align*}
$$
Where we defined $\gamma$ to be the elasticity of the labour revenue share with respect to the $r/w$ ratio.
Solving for $\gamma$ gives:
$$
\gamma = \left(\sigma-1\right)(1 - s_L)
$$
which means that $\gamma > 0$ if $\sigma < 1$. In other words, if the elasticity of subsititution is smaller than one, then the labour share increases when $r/w$ increases.


Why isn't there an "ideal value" for a given currency?

Because money is really just arbitrary numbers, and those numbers only really have any meaning in relation to the other numbers in the set. For example:

Average salary is \$50,000 and a entry-level car costs \$10,000. If the price drops by \$5,000, it's well known that sales will go up, ceteris paribus. But what we increased all prices and all wages by a factor of 100,000? Would that change people's behavior in an otherwise identical system? E.G. now the average salary is \$5,000,000,000 and the average car costs \$1,000,000,000 and the price drops 500,000,000, then what happens?

Well there is a fancy economic name for the theory that covers this (which I forget at the moment), but the answer is No, nothing would change. It's quite obvious that humans are very insensitive to absolute price and very, very sensitive to relative price. (There have been several economy-wide currency revaluations in other countries that serve as examples of this.) Adding or taking away a couple zeros at the end of the numbers doesn't change the really make a difference (provided you apply the change uniformly to all numbers everywhere).

The two major problems that hyperinflation a la Weimar Republic cause are 1.) governments are slow moving beasts and don't react quickly enough to start printing 1 billion dollar notes, etc. so you end up having to use a wheelbarrow full of cash for purchases (which in today's economy isn't as big of a concern since almost all money is just electrons in a computer somewhere,) and 2.) changing prices and adjusting to changing prices and adjusting your forward looking predictions about prices are all not costless actions. It takes time and energy to do any one of those three things. Because of that fact, it imposes a very real dead weight loss on the economy as a whole when prices increase or decrease very rapidly. The faster things move, the more costly it is to adjust and the more often you have to do so, so the DWL becomes exponentially greater than a more slowly-paced situation.


The question should be: Does there appear to be enough heteroskedasticity so that not taking it into account would lower the quality of inference? And this is because "taking heteroskedasticity into account" (even if only for robust standard errors) is not without costs -with small sample sizes it may worsen the reliability of results. And it becomes even riskier if you want to implement the more traditional approach where you will specify a functional relationship between the error variances and the regressors.

From the graph, my initial conjecture would be "no, not enough". But peform an "influential observations" analysis, since you obvioulsy have a few large residuals.
Also, with today's software, it is cheap to estimate a heteroskedasiticity robust variance covariance matrix, and see whether standard errors differ visibly or not, whether they reverse statistical sigificance tests or not, etc.


Where can I find data related to foreign remittances?

In the World Bank website you can find data on remittances such as

Annual and Monthly Remittance Flows to Selected Countries,
Bilateral Remittances Matrices,

and

Household survey data on migration and remittances.


Non-cooperative Nash Equilibrium in political game

You could add up the equations
\begin{align}
\delta(1-\underline{S})S_j=e_i(S_i+S_j)^2 \\
\delta(1-\underline{S})S_i=e_j(S_i+S_j)^2
\end{align}
to get
\begin{align}
\delta(1-\underline{S})(S_i+S_j)=(e_i+e_j)(S_i+S_j)^2
\end{align}
which simplifies to
\begin{align}
\delta(1-\underline{S})=(e_i+e_j)(S_i+S_j).
\end{align}
This gives you $(S_i+S_j)$. From the initial equations
\begin{align}
\delta(1-\underline{S})S_j=e_i(S_i+S_j)^2 \\
\delta(1-\underline{S})S_i=e_j(S_i+S_j)^2
\end{align}
you can also get
$$
\frac{S_j}{e_i} = \frac{(S_i+S_j)^2}{\delta(1-\underline{S})} = \frac{S_i}{e_j},
$$
which gives you
$$
\frac{S_j}{S_i} = \frac{e_i}{e_j}.
$$
Now you have both the sum and ratio of the variables $S_j,S_i$. Calculating their individual values should be a straightforward matter.


Term for fraction of economic value earned?

You seem to be looking for the phrase 'wage share' or 'share of labour compensation'.

Wage share:

  The wage share (or labor share) is the ratio between compensation of employees (according to the system of National accounts) and one of the following variables:

  gross domestic product at market prices
  gross domestic product at factor cost
  net domestic product at factor cost (domestic income at factor cost)

Here is some US data on share of labour compensation.


Is risk aversion bounded in CRRA utility?

A practical or mathematical bound? For a practical bound, people have argued that very large values of $\rho$ are impossible to reconcile with non-financial risk taking. Indeed, some values of $\rho$ may not be compatible with crossing the street.

Perhaps that's a bit glib, but consider Neilson and Winter (2001) which find that CRRA utility and observed wage premia for dangerous jobs are compatible with values of $\rho$ such that  $0<\rho<<1$. They then point out that if $\rho$ were very very large and the model properly specified then the dangerous job wage premium would also be large or alternatively (and implausibly) that people place a very, very low value on their own lives (like $3,000).

Generally speaking, data and models implying very high values of $\rho$ like a basic equity premium setup and the relative volatility of equity returns and consumption) are often taken as signs of model miss-specification instead of alternative measures of risk preferences.

While we can't talk about globally infinite risk aversion, we can consider what happens as If you take the limit $\rho\rightarrow\infty$ around a particular value of wealth $W_0$. As it goes to infinity you get a step-function utility function with $U(W) = -\infty, W < W_0$ and $U(W) = C, W \ge W_0$.


Set of consumption over all period is convex in $\mathbb{R}^T_+$?

Assuming the following:

$ S > 0 $, $T > 0 $,
$ c(S) = \left\{c\in\Bbb{R}^T_{+} : \sum_{t=1}^T{c_t} = S \right\} $, and
$ U(\cdot) $ continuous.

Let's start with convexity. To demonstrate convexity, we need to show that for any two points $c^1$ and $c^2$ in $c(S)$, any linear combination of the two is also an element of $c(S)$.
Let $\lambda \in [0,1]$, $c^1 \in c(S)$, $c^2 \in c(S)$. Then, we have
$$ \sum_{t=1}^T{\lambda c^1_t + (1-\lambda) c^2_t} $$
$$ = \sum_{t=1}^T{\lambda c^1_t} + \sum_{t=1}^T{(1-\lambda) c^2_t} $$
$$ = \lambda \sum_{t=1}^T{c^1_t} + (1-\lambda) \sum_{t=1}^T{c^2_t} $$
$$ = \lambda S + (1-\lambda)S $$
$$ = S $$
$$ \Rightarrow \lambda c^1 + (1-\lambda) c^2 \in c(S) $$
Since any linear combination of any two points in c(S) is itself in c(S), we have c(S) convex.
For non-emptiness, we just need to show that a single point exists in c(S):
$$ \sum_{t=1}^T{\frac{S}{T}} = S $$
$$ \Rightarrow \left\{ \frac{S}{T} \right\}^T \in c(S) $$
If $U(\cdot)$ is continuous on $c(S)$, then continuity of $W(c)$ follows from the triangle inequality (see e.g., here or here). If we don't know more about $U(\cdot)$, then there's not much we can say about $W(c)$. Berge's Theorem of the Maximum has to do with the continuity of an optimized function, not the sum of functions.


The Public Choice III by D. C. Mueller has whole chapter with mathematical exposition of the Coase theorem. His exposition includes both the general case and practical examples. Also I think Mueller strikes very good balance between rigor and simplicity.
You will also find some treatment of the theorem from Game Theory perspective in Game Theory: An Introduction by Steven Tadelis or the Tirol's Game Theory but only relatively briefly when compared to Public Choice III. Also Tirol's The Theory of Industrial Organization has some treatment but again relatively short. Nonetheless despite that these other texts only deal with the theorem briefly expositions of the theorem in different situations which itself has wide application may help you.
If you would prefer some paper then for example you can see application and exposition of Coase Theorem here, but papers are rarely written with didactic motive in mind so if you are self-studying I would recommend the textbook sources above.


Inverse/negative relationship between investments and interest rate

The investment I earns income in the future. This income can be compared to the risk free rate of interest in the market to determine whether the income earned on I exceeds the opportunity cost of investing in I.

When interest rates rise, it is more worthwhile for an investor to purchase risk free bonds rather than invest in I. This decreases the demand for I, hence its price decreases.


Skepticism about the claims of instrument variable validity/exclusion through a statistical test—the Arellano-Bond Test

If yes, then how does this square with the general point that
causality/exclusion cannot generally be established with statistical
tests...

It seems to me that "[exogeneity of IV] cannot generally be established with statistical tests" does not imply that it cannot be tested in specific cases. In this (very specific) context, the exogeneity claim rests on absence of serial correlation, which in principle can be tested as a null.

Or, is it merely a diagnostic that may be used as a secondary
argument along with primarily qualitative arguments for exclusion?

I would agree with you there. A non-rejection of the no-serial correlation null is by itself insufficient to establish exogeneity.
(If, hypothetically, the null hypothesis is presence of serial correlation, then yes, but such tests seems statistically unfeasible.)


Value of money (incomes) question - what is the "r" here?

This question implies to me that you want to convert historical incomes to “2020 dollars”. The standard way to do that is to use the consumer price index (CPI). Note that this is the level, not the rate of change (“inflation rate”).

The formula for a year T is straightforward:

(Income in “2020 dollars”) = (income in year T) *(CPI index in 2020)/(CPI in year T).

For example, if the income was \$100 in a year where the CPI was 60, and the 2020 CPI is 110, the income in 2020 dollars is \$100*(110/60).

If you wanted to convert future incomes to 2020 dollars, you would need to create an assumed inflation rate. You would either use the PV formula, or use the above formula with an extrapolated CPI. How the future inflation rate is determined is your decision.


How is the "value added" of firms distributed

Interesting question.

Indeed, it's somewhat difficult to model value added because in many models firms don't make profits all the time.  But therein lies your answer I think. Model VA as $Z$ in a distribution $g$ with two dimensions $(Z,Y)$, where Y is the size of the firm, Pareto distributed,  and then for any given $Y$, $Z$ is distributed as $Z \sim Y \cdot (1+ X)$ with $X$ a normal variable or something like that, so that $Z$ can be negative, but it is still on average proportional to $Y$. You can maybe then solve for $f_Z=\int g(Z,Y)dY$ analytically or numerically otherwise.


Time Series Analysis - What lag level is appropriate?

In your course they were likely talking about some specific case. Generally you often see more than one lag across applied work.
For example, one of standard time series models is the Autoregressive Distributed Lag (ARDL) model. In such model you often need to include several lags. The general ARDL(p,q) model looks like:
$$ y_t = \beta_0 + \sum_{i=1}^p \beta_i y_{t-i}+ \sum_{j=0}^q \gamma_jx_{t-j} +e_t$$
where you would typically want to choose the lag order to remove autocorrelation from the model. This is just an example, in many time series models you will want to include multiple lags for example also in VAR, AR, ARIMA and many other models. It is impossible to provide exhaustive overview of these models but you can have look at Verbeek (2008). A Guide to Modern Econometrics. 4th ed chapters 8-9 for an overview.
One way of how to select optimum number or lags is with information criteria such as Akaike or Schwartz Bayesian Information criterion (again just examples there are more of these - see the above mentioned source).
In R there are multiple ways how to implement these. One example is "tsDyn" package - this is mainly used to select optimum number of lags for VAR. For example you could run:
install.packages("tsDyn")
require("tsDyn")
data(barry)
test1 <- lags.select(barry, lag.max = 12)
test1
Best AIC:  lag= 12
Best BIC:  lag= 2
Best HQ :  lag= 2

So in the data above based on Akaike (AIC) we should include 12 lags in VAR model run on these data, according to Schwartz (BIC) and Hannan-Quinn (HQ) only 2 lags. As you can see this criterions do not always agree together so you also have to exercise some discretion but it is beyond scope of this SE answer to fully explain in all cases how to apply these. You can again have a look at the textbook I recommended.


Why is the Cobb-Douglas production function so popular?

As you hint at in your question, the real underlying reason (in my opinion) behind the C-D production function's popularity is mathematical convenience. The fact that the sum $\alpha + \beta$ is a nice, intuitive representation of "returns to scale" is very convenient.

I think of its use as similar to the use of exponential or power utilities in mathematical finance. Unrealistic? Perhaps, but oh so friendlier to work with.


Proving that constant absolute risk aversion and relative risk aversion implies independence of initial wealth

The foundation for my exposition comes from Mas-Colell's examples in Ch. 6.

The maximization problem for your specific question can be generalized pretty easily. Consider the case with one risky asset and one riskless asset. Let $\beta$ be the wealth invested in the safe asset, normalized to 1 dollar per dollar invested. Let $\alpha$ be the wealth invested in the risky asset, which has some random payout $z$ such that:

$$\int z \ \text{d}F(z) > 1$$

So that the mean return is greater than the riskless asset. So we express our maximization problem as:

$$\max \ \int u(\alpha z + \beta) \ \text{d}F(z) \\
\text{s.t.} \quad\alpha + \beta = w$$

You can take advantage of the fact that $w - \alpha = \beta$ $\implies \alpha z + \beta = w + \alpha(z - 1)$ and find first order conditions. If $u$ is concave (risk averse) then Kuhn-Tucker FOCs which combine to make:

$$\int u'(w + \alpha(z - 1))\cdot(z - 1) \ \text{d}F(z) = 0 \quad \text{iff} \quad \alpha \in (0, w) $$

So for the generic case, you can do the same setup with $N$ risky assets and one riskless asset that is better than whatever other riskless assets are out there. Let's normalize it again to payout of 1.

So the maximization is now:

$$\max \int u(\alpha_1 z_1 + \cdots + \alpha_N z_N + \beta) \ \text{d}F(z_1, \cdots z_N) \\
\text{s.t.} \quad \alpha_1 + \dots + \alpha_N + \beta = w$$

Notes:

If you have already done the easy case, this general case should not be so bad, just some more work. For other readers, I'll state the definitions of constant absolute and relative risk aversion below, respectively.

$$r_A(x) = -\frac{u''(x)}{u'(x)} = n \quad \forall x$$
$$r_R(x) = -\frac{x \cdot u''(x)}{u'(x)} = n \quad \forall x$$


Migration wage differential, Borjas (2013) textbook

Yes, that is an approximation one does. You have stumbled upon the log-approximation, which is very commonly used in Economics.
It states that $log(1+x)≈x$, for very small $x$. Hence, typically $x$ is a fraction. It is also used in macro to approximate growth rates.
There are lots of resources out there that explain it, e.g. here.
The way I remember it is that $log(1) =0$ so $log(1+x)≈x$ for small $x$. This is mathematically totally wrong, but I find it to be a useful memory device personally.


Interpreting how graphs of Cobb-Douglas utility functions. How does MRS vary as we vary $\alpha$?

If $\alpha$ rises, the utility puts more weights to the $x$. Then you must give up more $x$ for one $y$ (for the same utility). Your MRS, for a given $y$, increases in absolute value. Graphically:

If you set $y$ and $x$, the slope is lower for higher $\alpha$ (be careful to only change one thing at a time).

Concerning your "loop" problem, look at the function $1/x$, the first derivative is also a function of $x$, like the 2nd and so on. What matter is the value at one particular x.


Monopoly/Monopolistic Competition Productively Efficient or Inefficient?

No contradiction. All points in the AC curve indeed reflect the production of the corresponding quantity at minimum cost. This is conditional efficiency, conditional on arbitrarily specifying an output level.

Then we ask: what is the output level for which this product is produced at an average cost that it is lower than the average cost for all other output levels, the minimum minimorun, the least of all minima?

And we get the minimum of the Average Cost curve. At this output level we cannot do better by varying the quantity (either increase it or decrease it). So it is this quantity that achieves "universal" efficiency.


What is meant by the abbreviation 'MSV solution', used in the context of DSGE modeling?

Minimum state variable (MSV) solution is a special technique used to find an unique equilibrium with desirable properties in DSGE models. Often DSGE models can have multiple paths that will satisfy the conditions given by the system you are modelling.

Hence to provide some meaningful results you have to somehow choose between the all possible paths/solutions possible. There are multiple ways how to do that. MSV solution is a solution that helps to avoid sunspot equilibria and bubbles (see McCallum (1999)). In addition huge advantage of this approach as pointed out by McCallum is that  "the MSV criterion is designed to yield a single bubble-free solution by construction." It can be also shown that this kind of method is quite efficient and guaranteed to yield unique solution for wide ranges of models (albeit not all).

As you can see in Bullard and Mitra (2002), they talk about MSV solution when they want to show that there is unique solution that can determine their model coefficients $a,b,c$, so when they talk about MVS solution they are just saying that they applied this approach to select an unique solution this way as opposed to some other one.


Is Investopedia wrong about a Collar's maximum loss and profit?

The Investopedia article is correct.

Isn't your maximum profit when P≤77? Then you can exercise your 77P, but the call holder can't exercise his 97C. Your profit =77−P− options premiums.

You haven't accounted for the loss on the stock (Investopedia did).  It is:

stock price + strike price - collar cost (-80 +77 -1.50)

Isn't your maximum loss when P≥97? Then you can't exercise your 77P, but the call holder can exercise his 97C. Your loss =P−97+ options premiums.

You haven't accounted for the gain on the stock (Investopedia did).   It is:

stock price + strike price - collar cost (-80 +97 -1.50)


Replacing Value-added Tax with more Income Tax

In the long run there would be no difference as income will always eventually equal consumption so in the long run, flat 30% VAT tax would be equal to flat 30% Income tax. This is so because income can be only consumed or saved but saving is just deferred consumption. Assuming away some dynamic changes in taxation in future it does not make any difference from pure economics perspective.

However, in practice it is much better to have multiple sources of taxation instead of one. Having two sources of taxation helps combat tax avoidance (some people might easily avoid income tax like for example baby sitters or tutors but they can’t just as easily evade VAT). Also, in cases you care about redistribution it’s worth noting that it is very difficult to redistribute income with VAT if you apply it to individual products and not to total spending (which is virtually impossible in real life). So in cases where you want to have progressive non-linear tax to redistribute income, income tax is far superior instrument to VAT.

There is also some recent empirical evidence that shows that they might not be equivalent and which suggest labor taxes have higher welfare loss (you can see one example of such evidence here)). However, I would take the paper with a grain of salt. As the paper itself states it’s considered well known result in public economics that the two taxes are equivalent and laboratory experiments can have they own problems, and it’s hard to say what would be a convincing reason why they should not be equivalent. Still it’s worth knowing that the result is being challenged.


What's the minimum number of datapoints in order to run a diff-in-diff?

Diff-in-Diff is a parametric model. A rule of thumb for parametric models is that you should have at least 25-30 observations (different authors might disagree but it is around 30) per independent regressor used in your model (see discussion in Verbeek A Guide to Modern Econometrics pp 36). However, note this is a rule of thumb and as Verbeek points out (my comments are in []):

Unfortunately, there is no unambiguous definition of what is ‘sufficiently large’ [talking about sample sizes to justify asymptotic properties]. In
simple circumstances a sample size of 30 may be sufficient, whereas in more complicated or extreme cases a sample of 1000 may still be insufficient for the asymptotic
approximation to be reasonably accurate.

This being said getting precise number for every individual case would be too difficult (it would involve Monte Carlo simulations) so often people  follow the rules of thumb unless they have suspicion something is awry.
Consequently, if you would just follow the rule of thumb in a diff-in-diff model with 5 independent regressors you should have at least 150 observations, in model with 7 independent regressors 210, in model with 10 independent regressors 300 and so on.


What does 'establishment' mean in real estate industry?

Establishments are business locations (source). The real estate industry does not include construction (under either NAICS or SIC). It’s not really possible to tell what exactly would be included without more detail on the specific industry group you are looking at; it could be anything from realtors’ offices to apartment rental buildings, or both+other types of establishments. A summary of the NAICS groups can be found here.


Beginner level books on market economics

For introductory learning nearly any introductory economics textbook would do. For example, Mankiw's 7th edition has a section on "How Markets Work" and "Saving, Investment, and the Financial System". The latter discusses stock and bonds.


In the real world, how does a firm know their marginal costs and revenue? How do they mathematically calculate MC=MR?

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

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

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


Why did the stock of consumer loans held by commercial banks jump in March 2010?

Because of a change in regulation about how different accounts are to be recorded. That change was stated in the Financial Accounting Statements No. 166.

If you look into Notes on Data relating the "Assets and Liabilities of Commercial Banks in the United States - H.8", you find the following statement from April 9, 2010:

  As of the week ending March 31, 2010, domestically chartered banks and foreign-related institutions had consolidated onto their balance sheets the following assets and liabilities of off-balance-sheet vehicles, owing to the adoption of FASB's Financial Accounting Statements No. 166 (FAS 166), "Accounting for Transfers of Financial Assets," and No. 167 (FAS 167), "Amendments to FASB Interpretation No. 46(R)." Domestically chartered commercial banks consolidated \$363.4 billion in assets and liabilities. The major asset items affected were: other securities, mortgage-backed securities, -\$6.4; other securities, non-MBS, -\$23.5; commercial and industrial loans, \$19.1; real estate loans, revolving home equity loans, \$5.5; real estate loans, closed-end residential loans, \$20.9; real estate loans, commercial real estate loans, \$1.2; consumer loans, credit cards and other revolving plans, \$321.9; consumer loans, other consumer loans, \$24.6; other loans and leases, all other loans and leases, \$46.1; allowance for loan and lease losses, \$36.4; cash assets, \$4.1; trading assets, other trading assets, -\$1.7; and other asset items, -\$11.9. The major liability items affected were: borrowings, borrowings from banks in the U.S., \$2.6; borrowings, borrowings from others, \$379.8; and other liability items, -\$1.6. The residual (assets less liabilities) decreased \$17.4. The major memoranda items affected were: securitized consumer loans, securitized credit cards and other revolving plans, -\$322.7; securitized consumer loans, other securitized consumer loans, -\$24.5; and securitized real estate loans, -\$23.9. Foreign-related institutions consolidated \$20.7 billion in assets and liabilities. The major asset items affected were: other securities, non-MBS, \$1.1; commercial and industrial loans, \$0.7; and other loans and leases, \$18.9. The major liability items affected were: deposits, other deposits, \$0.5; borrowings, borrowings from others, \$21.0; net due to related foreign offices, -\$12.0; and other liabilities, \$11.2.

I highlighted the one corresponding to the graph you show. The magnitude of the jump precisely matches the amount the statement above mentions.


Who first formulated the Political Business Cycle Theory?

Quick Wikipedia found :

https://en.wikipedia.org/wiki/Business_cycle

Look at the section of Politically based business cycle

  The political business cycle theory is strongly linked to the name of
  Michał Kalecki who discussed "the reluctance of the 'captains of
  industry' to accept government intervention in the matter of
  employment."

https://en.wikipedia.org/wiki/Micha%C5%82_Kalecki


Why does Venezuela have such a high Big Mac Index?

It is worth noting that Venezuela has had currency controls in place for quite some time now, so the locals cannot get dollars at the world exchange rate.

  On January 2010, the government of Venezuela created a two-tiered official exchange rate system. Imports designated as "non-essential" receive a rate of 4.3 bolivares per dollar, and "essential" goods are exchanged at a rate of 2.6. There is also a third and unofficial exchange rate in the black market valued at around 6.8 bolivares per dollar (March 2010).
  The Venezuelan government decides what "essential" goods qualify for the 2.6 rate. They include imports for sectors related to food, health, education, equipment, and technology; remittances to relatives settled abroad; students' academic expenses abroad; expenses related to health, sport, culture, and scientific investigations; payments to retired and resident pensioners abroad; and currency conversions related to diplomatic activities.

The dollar value of a Big Mac at the official Venezuelan exchange rate is less than \$7.15.
(If my deeming it non-essential is correct it would be around \$2.50.)

Truth be told I am surprised they still have McDonald's restaurants in Venezuela, because long time president Chavez chastized US companies strongly and frequently and I think everyone associates McDonald's with the US.


What are the consequences of deflation in Japan?

Here is some evidence from third-party colleagues who have been to Japan in the 90s and recently visited again last year. He informed me that prices were the same as in the 90s or lower.

This is definitely the result of deflation working in Japan.

Another example of deflation in action is a report you can read here:
https://www.ozy.com/fast-forward/back-to-the-future-japanese-youth-flee-cities-for-the-countryside/87547

While here in the United States, most people don't get why someone like myself who works in the tech/economics field would want to own a small farming business, the youth of Japan do get it and they are moving back to rural villages and renting large traditional farm houses and adjoining land for $200/month, which is a fraction of what they were paying in cramped up studios in the city. This is an example of deflation in action where young Japanese citizens are abandoning costly housing, transportation, etcetera and adopting lifestyles that generate far less income and far lower expenses, both are deflationary.

So what are the consequences for these young people in Japan? Less stress, better health, higher quality of life overall? You would have to read the article and make your own conclusion.

You were not specific on consequences for whom.

The Bank of Japan has tripled its asset purchases:

This has had little apparent effect on conventional measures of inflation. The purpose is to offset deflationary declines but it has not worked and its starting to burden the State. So those are consequences the State of Japan will have to deal with. They may try to pass that off to its citizens the way our government does here, but as you have seen above, more and more Japanese are fleeing to a simpler life in the countryside.

All this new spending on the part of the BOJ may lead to mal-investment and systemic asymmetries that will eventually destabilize the entire financial system.


Yes, GDP can indeed be inflated by meaningless circular transactions. Furthermore, GDP growth completely ignores underlying changes to a country's asset base. Corporate accounts are considerably less naive. Any company that only ever reported growth in turnover wouldn't be trusted: turnover is vanity, profit is sanity.

Real increases in GDP can be a useful proxy for economic growth, because it measures the total transaction value within the economy, and historically that's been closely correlated to general perceptions of economic growth. So much so, that it's now pretty much used as the definition of economic growth.


How to estimate the labor time required to produce all necessities in a given economy (assuming all goods/services are produced domestically)?

Open question: how would you best make such an estimate?

Normally, if you would not care about closing the economy, such estimate would be simply done by calculation of real wages per hour and then based on prevailing prices of the necessities you could calculate how many hours would average worker need to work to afford them.
However, the fact that you want to do this for closed economy complicates everything because:
A) Real wage depends on whether economy is open or not.
B) Prices depend on whether economy is open or not.
There are trade models that can help you estimate what prices and wages would be in autarky. For example, you could create some Heckscher-Ohlin or Ricardian model which would be parametrized based on some auxiliary estimates of productivity of various factors at home and abroad (which will have to be done separately) and then simulate what would happen to prices and output based on parameters of the model  (and by extension factor returns (e.g. wages)).
For review of literature on how to estimate factor productivities see (Van Beveren, 2012). Alternatively, you could just do a literature review on countries you are interested in and make separate simulations for range of possible parameters that the model could have (there will always be some theoretical and also empirical estimates that can quite narrow the range of what parameters in the model can be).  For tutorial on how to do numerical simulations of trade model that can then help you determine autarky outcomes you can see this excellent tutorial for GAMS (program specifically built for trade simulations). Even if the tutorial is written for GAMS you can easily use most of what is written there in Python or R.
I can't recommend a specific way of doing the numerical simulation that would be the best because that is very case dependent and this is simply too broad question to go and cover advantages and disadvantages of all possible models you could use here.

Is my idea a good way to make the estimate? If no: why not?

No the idea is not good because of what was mentioned in answer to previous question. Trade generally allows country to consume outside its production possibility frontier (PPF) in autarky and it changes wages and prices.
Generally speaking, countries trade goods that can be produced more efficiently abroad (see Krugman et al. International Economics: Theory and Policy, 11th Edition). This in turn makes prices of those goods lower than they would be absent trade, and in addition it allows country to specialize in what it can do best leading to higher output and consequently real incomes. Of course, there is more nuance to it and trade can also depress wages in some sectors, but the point is that ignoring this would give you wrong and inaccurate results either way.
In your example you are trying to calculate what would happen in autarky using data on prices and wages that exist for open economy, and assuming they would still hold in closed one. This is very unrealistic and will give you wrong results - especially if the economy is small open economy such as let's say Netherlands or South Korea. For very closed economy with minimum amount of international trade such as North Korea (which only has some limited trade with China) it would not matter but I cannot think of any other economy that would not be significantly engaged in international trade. It would be less of an issue for big open economies such as US but only slightly less so.
Given the above the questions in 3 and 4 are moot questions.


What is Tinbergen's "all other income when paid out" measuring?

Very obscure indeed. If you see his paper in English (published in 1959), you can see the following (page 43):

By checking other variables, you can confirm that the variable letters are equivalent in this and your reference. Then, $E$ stands for "all non-wage income, taken at the moment received, instead of take when earned".

This is a standard differentiation between the time an income has been accrued ($Z$) and when it has actually been received ($E$).

You can see more about these definitions later on in the document. In page 49, you find an explanation on how these are calculated (19 is $Z$, 20 is $E$; see table in page 47-48):

It seems the author is assuming that income is received one further year after is is accrued. However, the text seem to me rather obscure. He mentions a "pamphlet regarding this working method which will be published later." (p.42). You might want to trace that one down.


Why are complementary goods defined as when a good's demand is inversely correlated with the price of another?

We know that (in this model) the price of patties influences the demand of buns. This is because a cheaper patty makes the whole burger cheaper, which makes me demand more burger and therefore more bun .

  Why would the demand of hamburger buns increase when the demand of
  hamburger patties decreased?

Other things being equal, the price of patties influences the price of buns. The theory says nothing about how the price change came about. You implicitly assume that something changing patty demand would change bun demand, also. In the model world, there are only explicit assumptions.


why is real wage divided by marginal product of labor often called real marginal cost?

This relation can be obtained if we realize that Total Cost can be written as a function of output (which in turn is a function of input factors), but also, that Total Cost equals total payments to factors of production. In real terms,

$$TC=TC(Q) = (w/p)N + (r/p)K$$
with the payment rates exogenous. But $Q = Q(N,K)$. Differentiate both sides by $N$:

$$\frac {\partial TC(Q(N,K))}{\partial N} = w/p $$

$$\Rightarrow \frac {\partial TC(Q(N,K))}{\partial Q}\cdot \frac {\partial Q(N,K)}{\partial N} = w/p$$

$$\Rightarrow MC \cdot MPN = w/p \Rightarrow MC = \frac {w/p}{MPN}$$

One can perform the same exercise to obtain the analogous relation with respect to capital:

$$MC = \frac {r/p}{MPK}$$


The total number of matched applications is

$$\sum_{x=1}^\infty g(x)xv = E[x]v$$

The number of applications matched with vacancies with $x$-many applications is

$$g(x)xv$$

Hence

$$Prob(\text{matching with x} | \text{matched} = \frac{g(x)x}{E[x]}$$

Using this in the equation in the question gives

$$ \tilde f = a \sum_{x=1}^\infty \frac{g(x)x}{E[x]}\frac{1}{x}\\
= E[x]a \sum_{x=1}^\infty g(x) \\
= \frac{v}{u} (1-g(0)) = f $$


if all income is spent on consumption how will it change or affect the economy

No, at least not if you mean "final consumption", that is spent on goods and services solely for our own enjoyment.

We produce goods and services with capital and labour (and natural resources and technology and human capital, but for simplicity let's put those under capital as well). The thing is that capital detoriates over time. My computer at my work place will have to be replaced at some point. The plough of the farmer will break down eventually. The natural gas that I use to heat my house will run out eventually, although potentially not during my life time.

To make up for these losses we need to invest some money in replacing the lost capital and finding replacements for non-natural resources. That is we need to invest money. If we'd only spend all our money on final consumption goods eventually we'd run out of capital and wouldn't be able to produce all those goods.


Which single letter abbreviations are typically used for the production factors Land, Labour and Capital?

In economic models, K is commonly used to represent "capital." This is presumably due to the fact that German for capital is "kapital," and also to the fact the C is more commonly used to represent consumption.

There is no difference in meaning between labor and labour. Labor is the preferred spelling in American English, and labour is preferred throughout the rest of the English-speaking world.

I believe it really depends what country the economic authors or book writers are from.


Convert DSGE paper to structural econometrics, anything to be noted？

There's no correct method to do this.  DSGE guys will estimate the model by doing something like this: plugging in priors for their parameters and then running an optimization that minimizes the distance between model and data moments while maximizing the log likelihood from the priors.  The data moments for them will be well known values of variances and covariance of the deviations of c, y, i, p from trend.

Structural estimation guys will do any number of things: a)  take a model predicted relationship,  like a investment to q relationship,  $i=b\cdot f(q)$ where b is a parameter,  estimated the relationship and infer the value of b; b)  general method of moments or simulated method of moments or indirect inference: use a characterization of the data into moments or auxiliary statistics,  and then find parameter values such that the model (analytically or with simulations)  replicates the values of these moments or auxiliary statistics.

Which moments / statistics to use?  Who knows!  Follow tradition!

Which macro variances or covariances?  Who knows!  Follow tradition! (plus the values of these maybe change dramatically depending on how and when you calculate them....)


Is MRS supposed to have a minus sign or not? What's there to say about convexity of preferences?

The Marginal Rate of Substitution is not just the "ratio of the partial derivatives": it represents the slope of an indifference curve. In order to obtain it, you must guarantee that you remain on the same indifference curve. How do we do that? One way is by taking the total differential of the utility function and requiring this total differential to be equal to zero:

$$dU(x_1,x_2) = 0 \implies \frac {\partial U(x_1, x_2)}{\partial x_1} \cdot dx_1 + \frac {\partial U(x_1, x_2)}{\partial x_2} \cdot dx_2 =0$$

and rearranging (and putting in this case $x_2$ on the vertical axis)

$$\implies \frac {dx_2}{dx_1} = -\frac {\partial U(x_1, x_2)/\partial x_1}{\partial U(x_1, x_2)/\partial x_2} = MRS_{x_1,x_2}$$

If we look at two goods and (not say, one good and one bad), and we have non-satiation, then the partial derivatives are strictly positive. So the minus sign is the one that tells you that the slope of the indifference curve should be algebraically negative, so a "downward slopping" curve. And this is  an argument in favor of keeping it. But indeed sometimes, people omit the minus considering the negativeness of the slope to be "understood". But this can create confusion.

But hey, the symbol $dx_2/dx_1$ can also be read as "the derivative of $x_2$ w.r.t. $x_1$", which is the first derivative of the indifference curve equation. Imagine now a convex to the origin indifference curve. As you increase the value of the variable measured on the horizontal axis ($x_1$ in our case), a convex indifference curve becomes flatter. This means that in absolute terms the slope becomes smaller. But in algebraic terms it becomes larger, because its value moves from a negative number, closer to zero.

I prefer to keep signs visible at all times (many short-lived scientific discoveries have sent their discoverer into depression five minutes later, when (s)he realized that it was just a sign mistake).

So, if I want to determine whether the indifference curve is convex, I want to see the MRS (sign included) increase in value, i.e. become less negative, and so also decrease in absolute value. So we see that "MRS decreases as we move down and to the right in a convex indifference curve" uses "decrease" in the absolute value sense.

And the confusion may get worse.

In any case, one should calculate

$$\frac {d^2x_2}{dx_1^2}$$

But with, or without the minus sign?

Assume that, like me (and unlike many others) you want to keep the minus sign in front of the $MRS$. In that case, you want to see the MRS increase. For this to happen, the derivative of the MRS must be positive. This in my eyes is the 2nd argument in favor of including the minus sign, because we know that positive 2nd derivative means convex function (referring here to the indifference curve equation).

So with the minus sign in front when calculating the derivative, positive $MRS$ derivative $\implies$ $MRS$ increase in algebraic value $\implies$ the slope of the $MRS$ gets larger in algebraic value, $\implies$ it gets smaller in absolute value, $\implies$ the curve gets flatter, and so it is convex indeed.
I leave the chain for the case of ignoring the minus sign as a for anyone interested. You will certainly get "negative 2nd derivative $\implies$ convex indifference curve", which will tend to mess with pre-existing knowledge in your brain.

And check this post.


Australia and Saudi Arabia - Structural Similarities

It's an analogy. The Saudi economy as an analogy for the Australian economy. Like any other analogy, you can stretch it to a certain point, but then it breaks.

The article sets out the ways in which the analogy holds. Stretching it further will probably break it. Let's look at how the article says the analogy holds:

both economies have a significant dependency on exports of a very small number of commodities: for Saudi, it's oil and gas; for Australia, iron and coal.
both boomed during the good times of the commodity super-cycle, both fuelled in part by huge and rapidly-rising demand from China; this flattered their domestic economies.
both now see a large drop in export revenues from falling demand and falling global prices in their key commodities; this exposes weaknesses elsewhere in each of their economies.
In Australia's case, this has caused a rapid increase in its net foreign debt; for Saudi, they're eating into their foreign reserves. So in each case, they're taking a big hit to the balance sheet, and for both countries, net foreign reserves have decreased a lot.