Show that an equilibrium in strictly dominant strategies is a unique Nash equilibrium

It will be useful to spell out the relevant definitions.

Let $A_i$ be the set of possible actions for player $i$, $A_{-i}$ be the set of possible joint actions of all players except for player $i$, $A$ be the set of possible joint actions of all the players. For $a=(a_i,a_{-i})\in A$, let $u_i(a)$ denote the payoff to player $i$ from action $a_i$, given that the other players play $a_{-i}$.

Definition 1: An action $a_i\in A_i$ is weakly dominant for player $i$ if for every $a_{-i}\in A_{-i}$, $u_i(a_i,a_{-i})\ge u_i(\overline{a_i},a_{-i})$ for every $\overline{a_i}\in A_i$, i.e. no matter what the other players do, action $a_i$ yields a payoff at least as high as any other action available to player $i$. [Strict dominance given by $>$ instead of $\ge$.]

Definition 2: An action $a_i\in A_i$ is a best response to action $a_{-i}\in A_{-i}$ if $u_i(a_i,a_{-i})\ge u_i(\overline{a_i},a_{-i})$ for every $\overline{a_i}\in A_i$, i.e. If I fix other players' actions to some particular joint action, then $a_i$ yields a payoff to player $i$ at least as high as any other action available to player $i$.

Claim 1: If an action $a_i\in A_i$ is strictly or weakly dominant, then it is a best response for any joint action $a_{-i}\in A_{-i}$. [This follows from Definitions 1 and 2 immediately.]

Definition 3: A joint action $a\in A$ is a Nash equilibrium if for each player $i$, action $a_i$ is a best response to $a_{-i}$.

Claim 2 (your question): A strategy profile (i.e. joint action) $s=(s_1,...,s_n)\in A$ in which each $s_i$ is a (strictly or weakly) dominant strategy is a Nash equilibrium.

Note that Claim 2 is a consequence of Claim 1.


Do machines reduce labor... or so they redistribute, alter, and globally increase labor?

The only "perpetual motion machine" is life itself.

No it is not. We also convert energy to heat as we go through life. We are certainly not immune to the second law of thermodynamics, we increase entropy all the time.

  Machines cannot run or reproduce themselves.

Why not? There is already factory robots making other robots. But of course, they run on fuel. But again, humans also run on fuel, we just call it food.

  [...] someone must obviously make, fuel, and maintain the machines. Someone else must [...] mine the additional metals, feed the miners who mine the metals, [...]

Yes, there is a kind of dependency graph, and for now there are many spots for humans, there are many things the machines just cant do. But there are no fundamental reason why a machine can't fix another machine, and there is no reason why there can't be a "universal repair machine" which also repairs copies of itself.


The supply and demand of Virtual Products

The book Information Rules by Google Chief Economist Hal Varian (with Carl Shapiro) deals with many issues raised by the particular features of the digital economy. In general, he finds you don't need new models, and that things are well approximated by high fixed and low or zero marginal cost of production.


Alternative way of deriving OLS coefficients

The $\mathbf M = \mathbf I-\mathbf X(\mathbf X'\mathbf X)^{-1}\mathbf X'$ matrix is the "annihilator" or "residual maker" matrix associated with matrix $\mathbf X$. It is called "annihilator" because $\mathbf M\mathbf X =0$ (for its own $X$ matrix of course). Is is called "residual maker" because $\mathbf M \mathbf y =\mathbf {\hat e}$, in the regression $\mathbf y = \mathbf X \beta + \mathbf e$.

It is a symmetric and idempotent matrix. It is used in the proof of the Gauss-Markov theorem.

Also, it is used in the Frisch–Waugh–Lovell theorem, from which one can obtain results for the "partitioned regression", that says that in the model (in matrix form)

$$\mathbf y = \mathbf X_1\beta_1 + \mathbf X_2\beta_2 + \mathbf u$$

we have that

$$\hat \beta_1 = (\mathbf X_1'\mathbf M_2\mathbf X_1)^{-1}(\mathbf X_1'\mathbf M_2)\mathbf y $$

Since $\mathbf M_2$ is idempotent we can re-write the above by

$$\hat \beta_1 = (\mathbf X_1'\mathbf M_2\mathbf M_2\mathbf X_1)^{-1}(\mathbf X_1'\mathbf M_2\mathbf M_2)\mathbf y$$

and since $M_2$ is also symmetric we have

$$\hat \beta_1 = ([\mathbf M_2\mathbf X_1]'[\mathbf M_2\mathbf X_1])^{-1}([\mathbf M_2\mathbf X_1]'[\mathbf M_2\mathbf y]$$

But this is the least-squares estimator from the model

$$[\mathbf M_2\mathbf y] = [\mathbf M_2\mathbf X_1]\beta_1 + \mathbf M_2\mathbf u$$

and also $\mathbf M_2\mathbf y$ are the residuals from regressing $\mathbf y$ on the matrix $\mathbf X_2$ only.

In other words:
1) If we regress $\mathbf y$ on the matrix $\mathbf X_2$ only, and then regress the residuals from this estimation on the matrix $\mathbf M_2\mathbf X_1$ only, the $\hat \beta_1$ estimates we will obtain will be mathematically equal to the estimates we will obtain if we regress $\mathbf y$ on both $\mathbf X_1$ and $\mathbf X_2$ together at the same time, as a usual multiple regression.

Now, assume that $\mathbf X_1$ is not a matrix but just one regressor, say $\mathbf x_1$. Then $\mathbf M_2 \mathbf x_1$ is the residuals from regressing the variable $X_1$ on the regressor matrix $\mathbf X_2$. And this provides the intuition here: $\hat \beta_1$ gives us the effect that "the part of $X_1$ that is unexplained by $\mathbf X_2$" has on "the part of $Y$ that is left unexplained by $\mathbf X_2$".

This is an emblematic part of classic Least-Squares Algebra.


Is choosing an investment by a bank only due to its regulatory capital charge a form of moral hazard?

Wikipedia:

  In economics, moral hazard occurs when someone increases their exposure to risk when insured.

Investopedia:

  Moral hazard is idea that a party protected in some way from risk will act differently than if they didn't have that protection. We encounter moral hazard every day—tenured professors becoming indifferent lecturers, people with theft insurance being less vigilant about where they park, salaried salespeople taking long breaks, and so on.

Economics Help:

  Moral Hazard is the concept that individuals have incentives to alter their behaviour when their risk or bad-decision making is borne by others.

A moral hazard occurs when acting causes the behavior against which you are acting to increase.

For example, bailing out a bank due to risky loans that the bank took encourages future banks to make riskier loans.  If the loans pan out, they make a lot of money.  If the loans fail, they get a bailout.  A bailout is less harmful than a bankruptcy, so they view it as less of a risk.  The bailout avoids a bankruptcy today, but it puts more companies at risk of bankruptcy tomorrow.

In your example, you have two equally risky companies (or types of companies).  But the regulations regard one as riskier than the other.  So they encourage investment in one investment over the other.  That's the intended goal of the policy.  It's not a moral hazard if it is encouraging the intended behavior.

Now, it may be that the policy is misguided and distorts the market in ways that are not intended.  But that is not a moral hazard.  It would only be a moral hazard if the rule were intended to encourage increased investment in A but due to the way it changed the incentives, it actually increased investment in B.


Need help reading early 20th century stock market list

I am unfamiliar with this style of notation, but in my opinion it seems likely that the double fractions denote buy and sell prices. I.e. on Nov 29th 1911 the sell price was 104 3/8 and the buy price was 104 7/8. This gets you an average or mid price of 104 1/2 = 104.5.

It is also possible that the two numbers denote the daily opening and closing prices, but the variation seems small for this.


MLE estimation with serially correlated errors

This is model is known an GARCH(general autoregressive conditional heteroskedasticity) where the AR component of the observable series $\{a_t\}$ is equal to unity.

The likelihood is determined by your model assumptions on the innovation terms $\varepsilon_t$ and $e_t$.

If you are looking for a canned function to estimate an GARCH see https://www.kevinsheppard.com/MFE_Toolbox. If you are looking for some more theory on how to form the likelihood see http://www.math.elte.hu/probability/markus/AlkmatTS1/ARCH_GARCH_Eload.html.


Technically, a stock broker brokers stock purchases: they execute transactions for clients. That is, they buy on the behalf of clients. (This is not true for bonds, where the practice is that they sell clients bonds from their inventories.)

Direct stock purchase plans are aimed at individuals, to the best of my knowledge. Brokers might help set up such purchases, but historically they existed as a way to buy equities without using a broker.

A broker will transact in two forms.

Primary market: buying shares directly from the company, such as in an initial public offering (IPO) or secondary offering.
Secondary market: buying shares from other entities on stock exchanges or other markets (“dark pools”, etc.).


Why wasn't the wage of the entrepreneur included in explicit costs?

Entrepreneur's wages are his return from his investment (his labour, leisure time, etc.) and his current wages are implicit costs. You need to be able to differentiate between accounting and economic profits to understand why this is so. This can be explained with an example.

Assume that you currently work for a dentistry. You are now considering operning your own practice. You may earn 100,000 per year as an entrepreneur. The running business costs are, say, 40,000.

Step 1. Calculate Explicit costs.
Explicit costs: rental, utility, admin etc. costs = 40,000.

Step 2. Accounting profit. Accounting profit = Revenues−Explicit costs = 100,000 - 40,000 = 60,000. Looks good!

But these calculations consider only the explicit costs. Since running your practice means leaving your current job where you are earning $40,000, your current earnings will be opportunity costs or things that you sacrifice to get the next best alternative. Thus, your current salary is an implicit cost to your business. You will compare implicit costs and potential profit from the new business (your net earnings from the new business).

Step 3. You need to subtract both the explicit and implicit costs to determine the true economic profit: 60,000 - 40,000 = 20,000.

Note that, here, your wages are not explicit costs; as the owner of the business you are receiving return on your investment in the form of wages.


If 1% of the solar energy of the Sahara can power the entire world, why isn't it happening?

It's lots of things, which added together means it's not going to happen.

Firstly, the initial calculation of area required was just to illustrate how small the world's energy use is, compared to the amount of energy we receive in sunlight, even after compensating for the 20% efficiency of PV modules. So it wasn't actually intended as an engineering solution.

Secondly, there is little political interest globally in becoming completely dependent on a handful of North African countries for electricity supply: most countries made that mistake with oil and OPEC, creating decades of geopolitical mess.

Thirdly, it's not a great renewables scenario, because the amounts of energy generated in any two spots of that 1.2% of the Sahara are very closely correlated over all timescales from hours to decades, which means the whole output of the theoretical farm would not show a lot of smoothness. It doesn't span that many timezones, so there'd be several hours every day with no output, for the entire world. Now, that can be addressed with back-up power - just use excess PV to generate hydrogen which is stored, then burnt when its dark to generate electricity then - but that's when it starts getting expensive.

It's cheaper to diversify both the types of energy harvested (wind, tidal, geothermal, hydro, biomass), and the geographic area over which it's harvested (both latitude and longtitude), which significantly reduces the storage requirements needed, and reduces the risks associated with intra-annual and inter-annual variability of supply. This also hugely reduces the geo-political implications, and means that all countries get a decent chunk of internal investment from the project.


Do trade policies have to change after Brexit?

The UK do it the "Norway way" this would mean they are part of the
European Economic Area, they need to pay into EU budget (roughly 80% of current pay) but
have no Seat in Brussels)
The Switzerland way - bilateral deals and roughly 50% of the
Norway way costs.
Lone Member of WTO.

In any way, Britain would gain the freedom to make its own trade deals (how about TTIP?!).


GDP Calculation and treatment of Personal Savings

The GDP, as you implied, is represented as follows

  Y = C + G + I + nX

Where Y is the GDP, C and G are private and government consumption, and I is the invest in fixed assets

This formula represents the demand for GDP

In a closed market I = S since all savings, for a certain amount of real interest r are allocated to investments in projects that would return r or higher

S then is a supply factor for the demand for I

In an open market, local savings can be invested abroad and foreign savings can be invested locally, via financial assets

nX are net exports of goods and services, these can be financial or other. tourism is, in a way, an export / import service


What are examples for the phenomenon that more (or better) information makes everybody worse off?

Traffic reports are another type of information that potentially makes everyone worse off.

A recent working paper (Wiseman and Wiseman 2019, https://drive.google.com/open?id=10B6VudYJOQB5w2UVBrYEjYc5wwzGZc8Z) shows this in a simple model of traffic.

I quote from the paper, "...public information about road conditions tends to increase average travel time when...3) the scale of the traffic network is large." This corresponds to their Theorem 3, where "...in a scaled-up version of any environment with excess capacity, traffic reports increase expected travel time."

Yet another broad way of thinking about your question is in terms of monitoring structures of repeated games. A recent paper (Sugaya and Wolitzky 2018 JPE, http://economics.mit.edu/files/11455) look at imperfect monitoring in cartels and show that less transparency can facilitate collusion, making all participants better off.


How to find interest rates with fractional reserve banking?

This question is sufficiently confused about basic concepts that I'm reluctant to go through the effort of clarifying everything enough to provide a meaningful answer, but here is a quick stab...

First, you need to understand that interest rates are not set by the supply of cash provided by banks and the demand from borrowers. They're set in money markets, which is the other side of the credit intermediation that banks perform. Banks lend to borrowers and borrow in money markets (if they didn't borrow, then it wouldn't be fractional reserve banking) to fund a significant portion of their lending.

Second, reserve requirements are far from the only constraint on bank money creation, and I frankly cannot think of a single bank for which they're the binding constraint. Banks are more likely to be constrained by risk-weighted capital requirements, the liquidity coverage ratio, or stress test scenarios than by reserve requirements.

So to answer your question as to how interest rates are determined under fractional reserve banking: bank demand for funds is determined by their set of profitable lending opportunities under a set of constraints including capital requirements (and they face some competition for these funds from firms that have direct access to capital markets and which issue short-term liabilities), while the supply of funds is determined by the amount of uncommitted cash held by firms (and to a much lesser extent, households). These intersect at some point, which determines the interest rate. The Fed affects the risk-free rate by engaging in transactions which swap safe, usually interest-bearing collateral for cash, affecting the relative supply of each.


What will be the effect on velocity of money when demand for real money increases?

(a) The effect of this policy change sounds theoretically ambiguous to me. It ultimately depends on the consumers' preferences.

An increase in interest payments lowers the opportunity cost of having cash, which might make some consumers better off by saving more and some might see this is a relief and hold more cash. We need an utility function in order to know how consumers would react to the policy change.

(b) Again, ambiguous. It depends on the liquid effect of the policy change on money demand. Let's first remember the Equation of Exchange:

$$MV=PQ$$

If consumers deposit more money on checking accounts, money supply goes up. Given the price level and output are fixed (we're in the short run), an increase in $M$ makes the money velocity $V$ decrease. On the other hand, if consumers deposit less on checking accounts, money supply goes down and velocity of money increases.

(c) The correct answer - at least in my understanding - is that it's ambiguous. If the money supply increases, the LM shifts to the right and if it decreases, to the left.

(d) If the LM shifts to the right and the price level is fixed, output goes up and interest rates go down, if it shifts to the left, output goes down and interest rates go up.


It works like any other exchange rate:

Divide ₹50,000 by 70 to get about US$714 / month at market exchange rates
Divide ₹50,000 by 18 to get about PPP$2778 / month at PPP rates (so not a poor farmer)
Multiply USD$30 by 70 to get ₹2100 at market exchange rates
Multiply PPP$30 by 18 to get ₹540 at PPP rates


Interaction between productivity growth and interest rates

Starting at the end, the simple answer is that intreset rates are decided as a counter measure for slow growth, which may derive from productivity slump. Meaning interest policy is a result, not a cause

On a wider level, the interntion of Central bankers is that there is a productivity rate, internally set in the economy set it's pace of growth. Imagine the added value of internet applications, as opposed to the world before that, as an example.

Their goal, is to set interest rate to match the proper appetite for risk, to encourage that productivity growth: take loans to pay for internet services, as per the example

The main parameter lacking in your equation, is that monitary policy cannot be the main economic policy to encourage productivity. It's the government in general: investinh in education, lowering barries for good healthy markets, provide a supporting legal system for business disputes etc.

Monitary policy can only be effective to a certain extent, in a way, like a wide (regressive) tax deduction


Why has gold dropped during the same week as Deutches Bank faces great difficulties

As Brian Romanchuk commented, there are a number of factors that can affect the daily price of an asset.  Gold is one of the most traded futures contracts on the CME so the performance of one bank (assuming you mean Deutsche Bank) might not be enough to offset those other factors.

But there is a strong correlation between gold and fiat currencies.  This is because gold is used to hedge against inflation.

Regarding your question about economic theory:

In a fractional reserve banking system, the supply of money is controlled by a central bank such as the Federal Reserve.  According to the Quantity Theory of Money, the supply and demand of money determines inflation.  However, John Maynard Keynes rejected this theory because it doesn't account for interest rates.

Keynes suggested that governments could offset recessionary / inflationary periods by directly decreasing / increasing interest rates.  According to Keynes, lowering interest rates in a recessionary period will create an incentive to borrow (cheaper capital) and short-term demand for money increases.  Prices change slowly in response to the change in interest rates as lenders increase the money supply to meet demand.  This slow change allows money supply and changes in interest rates to encourage or discourage borrowing and lending.

I'll attempt to unite economic theory and the futures market:

Generally speaking, a commercial bank generates most of its revenue through loans and consumer lending.  You could think of it as essentially selling / producing money.  They are inherently long money just like a gold miner is inherently long gold.  So their "product" becomes cheaper when interest rates are cut.  Since the price of gold behaves similarly to currencies, I suppose a bank could short gold futures in order to hedge their position.

The Fed's chairman Jerome Powell spoke recently. Many interpreted his economic outlook as a dovish signal which could imply lower interest rates.  If that happens, banks' revenue would decrease and the value of the dollar would be expected to decrease and the value of gold would be expected to decrease as well.  But nothing is guaranteed :)

Regarding irrational markets

A former coworker would say to me, "the market can remain irrational longer than you can remain solvent".  If you are able to predict irrationality then it would no longer be irrational. I recommend you read the book Fooled By Randomness by Nassim Taleb.


Suppose the marginal cost is constant and equal to $c$, that fixed costs are $K>0$, and that revenue is $R(q)$. You seem to understand that MR=MC must be true for profits to be maximized: $R'(q)=c$. We also know that average costs are given by $AC=(K+qc)/q$. But note that $AC=(K+qc)/q>c=MC$. Thus, when profits are maximized we have $AC>MC=R'(q)$. So it is not true that $AC=MC$ when profits are maximized in this example. We can conclude that $AC=MC=R'(q)$ holds only in certain special cases.


Is corporate tax an argument in equal income tax (from labour and capital) debate and does the return on capital depend on rate of the corporate tax?

There are too many questions in your post, and I advise to edit out most of them, especially those that are not answerable (like "is the system just" etc).

For just the first question the answer is No, you don't have the same tax rate on income from labor and from capital.

Let $\pi$ be profits before tax, $\tau_{\pi}$ the tax rate on profits, $\tau_d$ that tax rate on dividends. Then capital owners will receive (as a percentage of profits)

$$(1-\tau_d)(1-\tau_{\pi}) =  1-\tau_d-\tau_{\pi}+\tau_d\tau_{\pi}$$

since the dividend tax is applied to profits after tax. So the total tax rate on capital is

$$\tau_c = \tau_d+\tau_{\pi}-\tau_d\tau_{\pi}$$

In your case $\tau_d\tau_{\pi} = 0.10\times 0.15 = 0.015$

So $\tau_c = 0.235$ while tax on labor income is $0.25$. This, in general, is not considered a negligible difference.


How should I define most of the import or export when doing analysis of the import and export of a country?

If I understand you correctly, you are asking how to find out which countries' exports have most significantly contributed to a given country's (say, France's) imports.

I have recently completed some research on gravity models of trade, and in the process I ended up having to compile various datasets on precisely the topic you are raising.

The United Nations Comtrade International Trade Statistics Database has an easy-to-use interface where you can download bilateral trade data between a "reporter" country (France) and "partner" countries from Rest Of World.

Here: https://comtrade.un.org/data

Set the relevant year, set "Reporter" $\to$ "France", set "Partner" $\to$ "All", and set "Trade flows" $\to$ "Imports". Then you can download a cleaned .csv file on which you can run basic summary statistics and do some data analysis (such as finding the top-10 country exporters to France, etc.).

Broadly labeling where "most of France's imports come from" is an ambiguous task. Are you referring to a measure that uses a single country or perhaps a continent? It really depends on what your research question/methodology is. Can you provide some more details?


Why does a higher risk aversion leads to a lower intertemporal elasticity of substitution?

Mathematically, a higher risk aversion leads to a lower intertemporal elasticity of substitution (there is an inverse relationship). But why?

The (relative) measure of risk aversion is measured as:
$$
r(x) = -x\frac{u''(x)}{u'(x)}
$$
Consider an intertemporal utility function $u(x_1) + \beta u(x_2)$. Maximising this with respect to an intertemporal budget constraint gives the following first order condition:
$$
\frac{u'(x_2)}{u'(x_1)} = \frac{1+r}{\beta}.
$$
where $r$ is the interest rate. This states that the marginal rate of substitution should equal the interest rate.
Now, for a given utility level, the indifference curve $x_1(x_2)$ can be implicitly defined as:
$$
u(x_1(x_2)) + u(x_2) = \bar u.
$$
Using the implicit function theorem, we can compute its slope:
$$
\frac{\partial x_1}{\partial x_2} = - \frac{u'(x_2)}{u'(x_1)}.
$$
The slope of the indifference curve is therefore equal to the negative of the marginal rate of substitution. Now, differentiate this again with respect to $x_2$ :
$$
\begin{align*}
\frac{\partial^2 x_1}{\partial x_2 \partial x_2} &= - \frac{u''(x_2) u'(x_1) - u'(x_2) u''(x_1) \dfrac{\partial x_1}{\partial x_2}}{(u'(x_1))^2},\\
&= - \frac{u''(x_2)}{u'(x_1)} - \frac{u'(x_2)}{u'(x_1)} \frac{u''(x_1)}{u'(x_1)}\dfrac{ u'(x_2)}{u'(x_1)},\\
&= -\frac{u''(x_2)}{u'(x_2)}\frac{u'(x_2)}{u'(x_1)} - \left(\frac{u'(x_2)}{u'(x_1)}\right)^2 \frac{u''(x_1)}{u'(x_1)},\\
&= \frac{r(x_2)}{x_2} \frac{(1+r)}{\beta} + \left(\frac{(1+r)}{\beta}\right)^2 \frac{r(x_1)}{x_1}.
\end{align*}
$$
This shows that the curvature of the indifference curve at the optimum is increasing in the measure of risk aversion.
Concerning it's relationship to the elasticity of intertemporal substitution, we intuitively see that the higher the relative risk aversion, the higher the curvature of the indifference curve, so the more the slope at the indifference curve, $\frac{u'(x_2)}{u'(x_1)}$,  will change as a response to a change in the ratio $\frac{x_2}{x_1}$. Vice versa, the higher the relative risk aversion, the less the ratio $\frac{x_2}{x_1}$ will change in response to a change in the slope $\frac{u'(x_2)}{u'(x_1)}$ (e.g. due to a change in interest rates).
To see this mathematically, consider he intertemporal elasticity of subsitution:
$$
\varepsilon_s = -\frac{\partial \ln(x_2/x_1)}{\partial \ln(u'(x_2)/u'(x_1))}
$$
This measures the percentage change in $x_2/x_1$ due to a one percentage point change in the marginal rate of substitution.
We can write everything in terms of $x_2$ (as we did above), then take derivatives of the numerator and denominator with respect to $x_2$ and finally evaluate it at the optimum:
$$
\begin{align*}
\varepsilon_s &= -\frac{\dfrac{x_1}{x_2}\left(\dfrac{x_1 - x_2 \dfrac{\partial x_1}{\partial x_2}}{(x_1)^2}\right)}{-\dfrac{1}{\dfrac{u'(x_2)}{u'(x_1)}} \left(\dfrac{r(x_2)}{x_2} \dfrac{1+r)}{\beta} + \left(\frac{1+r}{\beta}\right)^2 \dfrac{r(x_1)}{x_1}\right)},\\
&= -\frac{\dfrac{1}{x_2} - \dfrac{\dfrac{\partial x_1}{\partial x_2}}{x_1} }{\dfrac{1}{\dfrac{\partial x_1}{\partial x_2}} \left(\dfrac{r(x_2)}{x_2} \dfrac{1+r}{\beta} + \left(\frac{1+r}{\beta}\right)^2 \dfrac{r(x_1)}{x_1}\right)},\\
&= -\frac{\dfrac{1}{x_2} - \dfrac{- \dfrac{1+r}{\beta}}{x_1} }{ -\dfrac{\beta}{1+r} \left(\dfrac{r(x_2)}{x_2} \dfrac{1+r}{\beta} + \left(\frac{1+r}{\beta}\right)^2 \dfrac{r(x_1)}{\beta}\right)},\\
&= \frac{\dfrac{x_1}{x_2} + \dfrac{1+r}{\beta}}{ \left(\dfrac{x_1}{x_2}r(x_2) + \dfrac{1+r}{\beta} r(x_1)\right)},\\
\end{align*}
$$
The last expression gives the elasticity as a function of the relative slope $x_2/x_1$ and the relative risk aversions. Now, if the relative risk aversion is constant, say $r(x_2) = r(x_1) = \sigma$ this simplifies to $\varepsilon_s = 1/\sigma$.
Nevertheless, we see that if the risk aversion increases, i.e. $r(x_2)$ and $r(x_1)$ increase, then $\varepsilon_s$ decreases.

If I become more risk-averse, I would like to smooth my consumption. So I will increase my participation in the financial market to avoid or reduce the risk. But how does that lower my intertemporal elasticity of substitution? Like why would I become less responsive to changes in the interest rate if I become more risk-averse? What's the intuition here?

There is a difference between consumption and saving. The higher your risk aversion, the bigger the curvature of the indifference curve, i.e. the lower $\varepsilon_s$. And so the smaller the change of $x_2/x_1$ due to a change in, for example the interest rate $r$.
You can think of the curvature of the indifference curve as the degree of complementarity between $x_1$ and $x_2$. The higher the risk aversion the more $x_1$ and $x_2$ become complements. So I will try to consume $x_1$ and $x_2$ together which is equal to increasing consumption smoothing.
In order to smooth my consumption, I will need to lend and borrow on the financial market. So if my income over time changes a lot, I will have to save and borrow a lot. On the other hand, if my income is reasonable smooth over time, I will not need to borrow or save a lot in order to smooth out out my consumption.

Or because I just wanna smooth my consumption, I don't really care about changes in the interest rate?

This is a mis-understanding, risk averse consumers actually do care about changes in the interest rate.

First of all, a change in interest rates affects their saving decision. It is true, however, that changes in the interest rate will not change $x_2/x_1$ a lot. So although my relative consumption over time will not be affected a lot by changes in the interest rate, the amount that I save or borrow can change substantially.

A change in the interest rates also changes my utility. Assume for simplicity that I am infinitely risk aversion, so I will always always choose to set $x_1 = x_2$. In this case, my intertemporal budget constraint becomes:

$$
\begin{align*}
&x_1 + \frac{x_2}{(1+r)} = y_1  + \frac{y_2}{(1+r)},\\
\to &x_1\frac{2+r}{1+r} = y_1 + \frac{y_2}{1+r},\\
\to &x_1 = \frac{(1+r)y_1 + y_2}{2+r}
\end{align*}
$$
Then:
$$
\frac{\partial x_1}{\partial r} = \frac{y_1(2+r) - [(1+r)y_2 + y_2]}{(2+r)^2} = \frac{y_1-x}{2+r}.
$$
So if my consumption $x_1$ is smaller than my income in period 1 $y_1$ (i.e. I save in period 1), I will be able to consume more if interest rates increase. On the other hand, if $x_1 > y_1$ (So I borrow in period 1), then I will be worse of if interest rates increase. So changing interest rates will change my utility depending on whether I am a net saver or borrower.

Risk-lovers, on the other hand, although also participate in the financial market, do not care about smoothing their consumption? Like if the interest rate increases, they can drastically reduce their current consumption for more consumption in the future, but risk-aversers do not care about the higher interest rate?

Again risk loving (or neutral) consumers also care about interest rates. Consider for simplicity a risk neutral consumer with utility function $u(x_1, x_2) = x_1 + \beta x_2$. Then it is clear to see that if:

$\beta < \frac{1}{1+r}$, I will consume everything in period 1, so $x_1 = y_1 + \frac{y_2}{1+r}$ and $x_2 = 0$. Utility will be equal to $y_1 + \frac{y_2}{1+r}$,  which is decreasing in $r$. Higher interest rates lower my discounted future income, so I will be worse off.
$\beta > \frac{1}{1+r}$, then I will consume everything in period 2, so $x_1 = 0$ and $x_2 = (1+r) y_1 + y_2$. Then utility is equal to $\beta (1+r)y_1 + \beta y_2$ which is increasing in $r$. Higher interest rates increase my consumption in period 2.

As a conclusion, the consumer will gain by an interest rate increase if she is a net saver and will loose if she is a net borrower.
It is true that normally a risk neutral consumer will not respond her saving decisions by a lot if interest rates change. However if, for example due to the interest rate increase we shift from a situation where $\beta < 1/(1+r)$ to a situation where $\beta > 1/(1+r)$ then suddenly she will move all consumption from period 1 to period 2, which will mean a big shift from borrowing to saving.


What are directional derivatives used for in economics?

Recall, that the directional derivative of a function is defined as the the rate at which the function $f(x,y,z)$ at a given point $(x_0, y_0, z_0)$ changes in  direction of the vector $v$, where $||v||$ is the norm of $v$1.
What comes to mind
Applications of the directional derivative can be used in determining the rate of switching inputs in production functions, which can be very helpful in determining/forecasting switching costs for a given bundle of inputs. Though it does not tell you where the optimum points are it does tell you the rate of change (i.e. the switching costs) of  moving to a different bundle.
Hope this is helpful
1 http://mathworld.wolfram.com/DirectionalDerivative.html


What would be the economic impact of the "fair tax" plan?

Well, one advantage to a sales tax is that it affects everyone (even those who don't pay income taxes). However, some people may see that same fact as a problem. This problem lies in analysis of the equity of the tax. When we consider every single person being taxed 30% on everything they buy, we need to think about how that affects different income groups. Lower income groups spend a much larger portion of their income on consumption, which means that they will be taxed relatively much more heavily than high income groups. This fact alone is enough to discourage this sort of action because the government tries to encourage saving in the lower income groups (so they can move up to higher income levels). This is one large reason why a tax like this may not be as fair as the name makes it seem.

Another piece of information that must be considered is the incidence of the tax. As we know, the incidence of a sales tax depends upon the relative own-price elasticities of supply and demand. This means that in a market with very inelastic demand (such as cigarettes or oil), the buyer is paying most of the tax whereas in a market with elastic demand the sellers would pay most of the tax. This fact would make it so that different industries would end up paying a different amount of the sales tax compared to one-another. For example, an oil company would end up paying much less of the percentage of the tax than a company making a selfie stick. This, again, seems to be unfair.


If you look at generic games, that is games where players are not indifferent between any choices such a classification is easy to produce. Let the strategy sets be
$$
S_1 = \left\{a,b\right\}, S_2 = \left\{c,d\right\}
$$
and let the utility/payoff functions be denoted by $U_1$ and $U_2$.
The characterization will be based on the dynamics of the game, that is given a strategy profile $(s_1,s_2)$ will any player wish to alther his strategy. This will also give you all the equilibria. Without loss of generality you can assume
$$
U_1(a,c) > U_1(b,c).
$$
(Should this not hold, switch the labels of the strategies $a$ and $b$.)
 We still need to know if player 1 prefers $(a,d)$ to $(b,d)$ or vice versa, and also if player 2 prefers $(a,c)$ to $(a,d)$ and if he prefers $(b,c)$ to $(b,d)$. This gives eight possible combinations which will characterize the game classes. (But due to symmetry actually there will only be four classes.)

Category 1.
$$
U_1(a,d) > U_1(b,d) \mbox{ and } U_2(a,c) > U_2(a,d) \mbox{ and } U_2(b,c) > U_2(b,d).
$$
In this category $a$ and $c$ are strictly dominant strategies. A game like this will play out like the prisoners dilemma in that it has a single Nash-equilibrium which is also a dominant equilibrium.

Category 2.
$$
U_1(a,d) < U_1(b,d) \mbox{ and } U_2(a,c) < U_2(a,d) \mbox{ and } U_2(b,c) > U_2(b,d).
$$
This game will play out the matching pennies game, in that there are no equilibria.

Category 3.
$$
U_1(a,d) < U_1(b,d) \mbox{ and } U_2(a,c) > U_2(a,d) \mbox{ and } U_2(b,c) < U_2(b,d).
$$
This game will play out like a coordination game, e.g. Battle of the Sexes or Chicken. There are two equilibria, $(a,c)$ and $(b,d)$.

Category 4.
$$
U_1(a,d) > U_1(b,d) \mbox{ and } U_2(a,c) > U_2(a,d) \mbox{ and } U_2(b,c) < U_2(b,d).
$$
In this category $a$ is again strictly dominant. $c$ is not, but a rational player 1 will always play $a$, and $c$ is best response to that. I don't know if this game has a well accepted name, it has one Nash-equilibrium $(a,c)$.

Due to symmetry, all other combinations belong to one of these categories.

If you wish to admit indifference between several outcomes, it is still possible to use this kind of classification, but the number of categories will be much larger.


Does consumer surplus over-weigh the interests of high-income consumers?

Consumer surplus in Microeconomics 101 is indeed often used in the way you describe and to illustrate a point about the allocation efficiency of a free market.

However, it is important to keep in mind the assumptions and limitations of the framework you describe.

First, the case you describe is a partial equilibrium case. This framework does not allow for such general statements as you are making about all goods always going to richer people. In partial equilibrium we look at one small market and forget about everything else. To use such a framework we must accept that it is reasonable for our question at hand to analyze one purchase in isolation. That assumption is clearly not reasonable for the question you have on all goods and as such the framework you describe cannot be used as it will lead to false conclusions.

With partial equilibrium we can look at this purchase in isolation from other markets. If income effects and the purchases in question are important then we cannot use the consumer surplus idea as done in Micro 101. Certainly this is an issue for the point you raise about all goods always going to a wealthier consumer. All goods cannot be analyzed with partial equilibrium.

The good allocation properties that economist often talk and that you mention about are based on general equilibrium and the proofs do not really involve the concept of consumer surplus.

To fully ensure avoidance of income issues, we often work with quasilinear utility functions when talking about consumer surplus and allocation, as you are doing. It is important, however, to keep in mind that these utility functions are special cases. They imply that income effects do not matter so they are widely used in Micro 101 for illustrative purposes.

Consumer surplus has a nice interpretation for quasilinear utility functions: it’s equal to the gain in utility. This makes quasilinear utility functions especially instructive. If everyone’s utility function has this form then the Pareto efficient allocations are exactly the allocations that maximize total consumer surplus (in the absence of production).

So when using consumer surplus to talk about efficient allocation you need to accept the assumptions that go along with quasilinear utility. However, such utility function are only realistic for certain goods and certainly not for all goods.

Luckily, what you learned about free markets achieving allocation efficiency is still true even when considering more complex cases. However, these results are then true for different reasons and under different assumptions than what is taught through consumer surplus in introductory lessons.


Situations where revelation principle may not hold

You might want to be a little more precise about what you mean by "Revelation Principle" as there are many formulations of the "Revelation Principle" out there, some of which are stronger than others. Each of these formulations makes a different claim and relies on a particular set of assumptions. Of course, the claim will often fail to be true if some of the assumptions are false.

(The following is from notes I got from a microeconomics class.)

Consider for instance the following version of the revelation principle, which is from Repullo (1985), Review of Economic Studies:

  Repullo's Revelation Principle : Let $g$ be a dominant strategy mechanism for the game $\Gamma \equiv ( g, U_1, \dots, U_n)$, where $g$ is some game form. For each equilibrium selection function $s : \Theta \rightarrow S$, there exists an equivalent direct dominant strategy mechanism $h$ to $g$ (where $\Theta$ is the set of types). If in addition the equilibrium selection function $s^* : \Theta \rightarrow S$ is surjective, then the dominant equilibrium outcome under $h$ are a subset of the dominant equilibrium outcome under $g$ for all $\theta \in \Theta$.

The bold part is important. If it is not satisfied, there may still be a non-truthtelling  equilibrium in the equivalent direct mechanism. An example is provided in Repullo (1985), Review of Economic Studies pp 223-229.

$$A \equiv \{a,b,c,d\}$$
$$ \Theta_1 \equiv \{ \theta_1', \theta_1''\}$$
$$ \Theta_2 \equiv \{ \theta_2', \theta_2''\}$$

$$ \begin{array}{c |c c c c}
 & a & b & c & d \\ \hline
u_1(\cdot, \theta_1') & 2 & 4 & 2 & 4\\
u_1(\cdot, \theta_1'') & 1 & 0 & 2 & 4\\
u_2(\cdot, \theta_2') & 2 & 2 & 4 & 4\\
u_2(\cdot, \theta_2'') & 1 & 2 & 0 & 4\\
\end{array} $$

$$S_1 \equiv \{s_1',s_1'',s_1'''\}$$
$$S_2 \equiv \{s_2',s_2'',s_2'''\}$$

The game form is

$$ \begin{array}{c |c c  c}
 & s_2' & s_2'' & s_2''' \\ \hline
s_1' & a & b & b \\
s_1'' & c & d & c \\
s_1''' & c & b & a \\
\end{array} $$

On can check than the following is an equivalent direct mechanism

$$ \begin{array}{c |c c  }
 & \theta_2' & \theta_2''  \\ \hline
\theta_1 & a & b  \\
\theta_1 & c & d \\
\end{array} $$

Yet, when the types are $(\theta_1',\theta_2')$, although telling the truth is a dominant strategy, any other report of preferences is also a dominant strategy. This may be quite bothersome as it means that for some configuration of the types, saying the truth is only one equilibrium among others. As a consequence, we have no real guarantee that "telling the truth"  will be played (it may even be that truth-telling is Pareto dominated by another equilibrium. Using a focal point argument, this may further undermine the relevance of the truth-telling equilibrium).

The above issue is due to the fact that in the original game, some strategies are never played, which is ruled out if $s^*$ is surjective. So Repullo's version of the Revelation Principle (requiering that every dominant strategy equilibrium outcome of the equivalent game be among the equilibrium outcome of the initial game for every possible configuration of the types) only holds if the equilibrium selection function be surjective, and fails otherwise.


Why did the EUR/USD go down immediately when the Fed cut interest rate?

One possibility is that although market participants had generally expected a 25-basis-point cut, they had also assigned a small probability to a bigger 50-basis-point cut.

And so, when it was confirmed that the cut would be 25 rather than 50 points, market participants then reacted accordingly to this (mild) surprise by strengthening the USD.

Edit: Actually, a bigger factor may have been the Fed's failure to promise more stimulus/rate cuts in the coming months (as some/many may have hoped).

For example, Wall Street Journal:

  Stock markets sold off and the dollar strengthened Wednesday afternoon after Fed Chairman Jerome Powell disappointed investors in his post-decision news conference when he didn’t more explicitly ratify expectations of additional stimulus in the months ahead.


What does a supply+demand curve look like for fare evasion?

The way to get the demand curve is by deriving the utility function.

First, the utility function needs to be defined.

$E(u)=(u(0)(r)+u(200)(1-r))$

In this case, the expected utility is the utility of spending \$0 on the fare vs the utility if you get a \$200 fine. in this case, r is the probability of getting away with it.

If this is greater than the utility of paying the fare, the fare will be evaded.

$E(u)=(u(0)r+u(200)(1-r))\geq {u(5)}$

The exact utility function can take many forms. Popular choices are Decreasing Absolute Risk Aversion (DARA), Constant Absolute Risk Aversion (CARA) or others. Wikipedia is OK on this.

The next steps in this analysis are to decide on a utility function.
You will need inputs on risk of getting caught and fines.

After defining the utility function you will be able to derive the demand curve.

You will also need to define the supply curve. If the fare is something like a train ride, you may be able to set up the marginal cost=0. That allows you to only worry about the fixed cost.

A useful resource will be the wiki article mentioned earlier. This Mas Collel book is a very popular graduate text. I think it treats risk aversion relatively well, although I did find it not incredibly accessible.


Should I construct balanced data with only necessary variables or unbalanced data with many potentially usable variables?

I would construct and maintain only the full (unbalanced) data set with many variables with many missing values. Why? We can always extract balanced panel data sets from the full one; we can't go the other way around. When a balanced set is needed (in order to reduce the file size or send it to others), I would use Stata or R and never do it manually. To err is human.


Does a private company that is 100% financed by a bank loan have a WACC equal to the interest rate?

Let's address your question in a canonical manner. Say that:

the required rate of return of (private) investors is denoted by $r$.
the interest rate or the cost of debt is denoted by $r_D$ (naturally $r_D < r$)
the share of debt in the capital structure of the company is denoted by $X\%$
the corporate tax rate is denoted by $T_c$

It follows that the average cost of capital (average over equity and debt), $r'$, is such that

$r' = X\% r_D (1-T_c) + (1-X\%) r$

In your case, $X\% = 100\% = 1$, then $r' = r_D(1-T_c)$.

  [...] will the cost of debt be simply equal to the interest rate? If not why?

No. Because of $T_c$, the corporate tax rate being potentially non-zero.


How to integrate a piece-wise function and what are semicolons in function notation?

CDF = integral of PDF from $-\infty$ to $x$, or:

$F\left(x\right) = \int_{-\infty}^{x} f\left(u\right) du$.

Functions like $f\left(x;\beta\right)$ take $\beta$ as the parameters, the $\beta$ index different probability density functions.  You don't need to integrate over them to get the CDF.

$F\left(x;\beta\right) = \int_{-\infty}^{x} f\left(u;\beta\right)du$.


Why would option chains' IV differ, for the same stock and expiration?

With illiquid stocks, the B/A spreads for the options are Holland tunnel wide.  For some it's maybe 50 cents, for others, over $4.  With such a wide range of price, where is the actual market?  What's fair value on which to base an IV calculation?
Most likely, the broker is basing IV on the midpoint.  However, the midpoint isn't an accurate representation of price.  For example, the $8 put is 4.00/7.80 with a midpoint of 5.90.  Suppose I offer to sell at 6.80?  That changes the midpoint to 5.40 and therefore lowers the IV.  It's all meaningless.
As for arbitraging the difference, how do you arbitrage when there's a spread of anywhere from 50 cents to over 4 dollars?  It's impossible.


Difference between social choice function and mechanism outcome function

A social choice function presumes the individuals' preference parameters $\theta_i$'s are observable, whereas in a mechanism, such knowledge is not presupposed. Therefore, in a mechanism, the arguments of the outcome function are strategies of the players, which are observable, not their preference parameters, which, although indirectly determine the players' strategies, are not observable.

As an example, suppose $3$ individuals are either rich or pool ($\Theta_i=\{R,P\}$ for $i=1,2,3$) and there are two policies, $X=\{A,B\}$. Assume rich people prefer $A$ and the poor prefer $B$. A social choice function would take a profile of the three people's wealth status and produce a policy outcome, for instance,
\begin{equation}
f(R,R,P)=A,\quad\text{or}\quad f(P,P,R)=B.
\end{equation}
However, if individual wealth status is private information which cannot be directly observed, and the policy must be chosen by a simple majority voting mechanism, then each individual's strategy space would be $S_i=\{A,B\}$ and their strategy $s_i:\Theta_i\to X$ would map from their own type to a policy choice. The mechanism outcome function would turn a profile of the strategies into a policy outcome, e.g.
\begin{equation}
g(A,B,A)=A,\quad\text{or}\quad g(A,B,B)=B.
\end{equation}

There is a special mechanism: when each player's strategy space is also their type space, i.e. $S_i=\Theta_i$, and the mechanism's outcome function is the same as the social choice function $g(\theta)=f(\theta)$, we have what's known as a direct mechanism.


Why is there consumption sharing in Arrow-debreu time-zero complete markets trading model?

I guess it just follows by definition.

Assume that in the high state (for household $i$, aggregate endowment is 10, and household $i$ consumes $1/5 = 2$ of this, while having an endowment of $5$. Now consider what happens if we were in the low state (for household $i$), where household $i$ only gets $1$ of whatever endowment. However, if this is a high state for some other household(s), such that the aggregate endowment is still $10$, since the fraction is constant household $i$ still consumes $2$, even though they have a smaller endowment.

Similarly for across time consumption sharing (although a bit weird since the model has no storage...). I guess it works that, if aggregate endowments are the same across two time periods, a household will consume the same amount regardless of whether their endowment was large or small (basically, they contribute to others when they have a lot (they lend), and they take contributions when they have little (they borrow). Thus, consumption smoothing.


Why exchange rates of national currency are usually stable, where as cryptocurrency like bitcoin and ethereum is highly volatile?

The value of currencies (including most popular cryptocurrencies) is set by a market. If more people want to sell the currency (i.e., exchange it for a different currency) than to buy it then some of the sellers will be unable to find a buyer. Their only option will be to lower the price they demand until someone is willing to buy.

Likewise, if more people want to buy, say, euros than are willing to sell them then they will have to be willing to pay a bit more for the euros in order to persuade someone to sell.

In this way, price rises or falls until the supply and demand of the currency are equalised. So when the price fluctuates a lot, it is because either the supply or demand (or both) are changing a lot.

At the moment, cryptocurrencies have very little practical use. It's quite difficult to use them for many kinds of transaction. Thus, most of the trading of those currencies is driven by speculation: people buying and selling them in the hope of making money through changes in the price. This means the price can fall very rapidly when people think there is about the be a crash because everyone wants to sell to protect the value of their money. This is called a speculative bubble.

While euros (and dollars, and pounds,...) are sometimes used for speculation too, the main reason most people want euros is to buy stuff from/in Europe—whether that's German cars, or Bordeaux wine, or just stock in an Italian company. Likewise, the main reason people want to sell Euros is because they are Europeans who are trying to buy stuff made in America or China or elsewhere. They sell their Euros for dollars so that they can buy Ford cars or Coca Cola. In normal times the value of euros tends to be quite stable because the amount of stuff people buy from Europe and the amount of foreign-made stuff that Europeans buy doesn't change very quickly.


What happens when a country's central bank significantly reduces the available currency?

According to standard monetary equation:

$$MV=PY$$

Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ real output. And where the‘value’ of money is inversely proportional to the price level $P$, we can rearrange the above for $P$ to see how it changes when parameters change and from that we can infer what happens to the value of money:

$$P=\frac{MV}{Y}$$

Hence if the money supply goes down and velocity and output stays the same the $P$ will decrease and money become worth more. Important caveat is that both velocity and output might not stay constant. If real output decreases faster than the decrease in money supply then the price level would still increase and value of money decrease. Also, in more complex models expectations play big role. If the change in money supply is expected to be reversed in near future it might not affect price levels and the relationship might not hold at 0 lower bound (see the influential paper from Krugman on this here).


Are Trade Blocs regionalism rather than Globalisation

This is an important question in the trade literature. The regionalization of trade is of serious concern to many international economists who view multilateralism as far superior to regionalism for improving welfare.

In a nice paper, Caroline Freund and Richard Baldwin survey the available theoretical and empirical evidence on the relationship between regionalism and multilateralism, with the aim of discerning whether the spread of regionalism is likely to be a threat to, or an opportunity for, broader trade liberalization. I summarize here their main insights:

Two main related concerns raised by the "multilateralists":

The first is trade diversion: preferential trade agreements, by diverting trade away from the most efficient global producers in favor of regional partners, may prove welfare reducing.
Regionalism may hinder multilateralism, leading to a bad equilibrium in which several regional trade blocs maintain high external trade barriers. Regionalism can also undermine multilateralism simply by diverting limited government resources from multilateral negotiations.

However the "regionalists" argue that

it is precisely because the trade diversion is costly to bloc members that there is an incentive to reduce external tariffs. As tariffs fall, trade diversion disappears, and regionalism becomes a force for general liberalization.

Although the verdict is not yet in, the evidence indicates that regionalism is broadly liberalizing. So regionalism does not go against globalization.


For what demand function is a monopoly most harmful?

An arbitrarily large ratio should occur with demand curve

$P=\begin{cases}
\frac{1}{Q} & \text{if } Q>1 \\
2-Q & \text{if } Q\leq 1 \\
\end{cases}$.

The monopolist prices at $P=1$, but the consumers' surplus if $P=0$ is infinite, because the area under the demand curve contains $\int_1^\infty \frac{1}{Q}dQ=\infty$.


Have there been any attempts to turn free-trade agreements into Pareto improvements?

The US has a  "Trade Adjustment Assistance" for (almost) this exact purpose.

  Trade Adjustment Assistance (TAA) is a federal program of the United
  States government to act as a way to reduce the damaging impact of
  imports felt by certain sectors of the U.S. economy. The current
  structure features four components of Trade Adjustment Assistance: for
  workers, firms, farmers, and communities. ...it was proposed by President John F. Kennedy as part of the total package to open up free trade. President Kennedy said: "When considerations of national policy make it desirable to avoid higher tariffs, those injured by that competition should not be required to bear the full brunt of the impact. Rather, the burden of economic adjustment should be borne in part by the Federal Government."...As a 2012 Report by the Joint Economic Committee states: "TAA needs to remain an integral part of trade policy because it compensates those harmed by import competition without sacrificing the larger demonstrable benefits of trade

Trade Adjustment Assistance at Wikipedia

Canada had a program like this into the 1980's. The comparable EU program is called The European Globalisation Adjustment Fund.


The total cost was 291 m.u. for 1 unit given $MC(q) \equiv \frac{dC(q)}{q} = 3 q^2 - 40 q + 220$.

First, the area under the Marginal Cost gives the Total Variable Cost, $TVC(q)$. To find this area, we will integrate the Marginal Cost:

$$
\begin{align}
TVC(q) &= \int_0^q\frac{dTVC(q)}{dq} dq = \int_0^q (3q^2 - 40 q + 220) dq\\
&= \left. \left(q^3 -20q^2 +220q  \right) \right|_q - \left. \left(q^3 -20q^2 +220q\right)  \right|_0\\
&= {q^3 -20q^2 +220q}
\end{align}
$$
Now, you have to use the additional information of 291 m. u. to find the Fixed Cost. The Total Cost equals the Variable Cost plus the Fixed Cost (which does not depend on q):
$$\begin{align}
TC(1) = TVC(1) + FC &=  291\\
1^3 -20 \cdot 1^2 + 220 \cdot 1 + FC &= 291\\
FC &= 90
\end{align}
$$

Hence, for the 8 units:
$$
\begin{align}
TC(8) = TVC(8) + FC &= 8^3-20 \cdot 8^2 + 220 \cdot 8 + 90\\
&= 1082
\end{align}
$$


Connection between negative interest rates and negative bond yields

I cannot see the mechanism as to why if central banks set rates in positive territory when long term bonds yields are negative how this will cause the yield curve to invert.

No need to seek a mechanism because the inverted yield curve occurs by definition in the scenario you describe. This Investopedia video defines inverted yield curve as follows:

  An inverted yield curve occurs when short term interest rates are higher than long term interest rates.

If short term rates are positive and long term rates are negative then short term rates must be higher than long term rates. Therefore, the yield curve will be considered inverted by definition.

Remember, central banks set the interbank overnight lending rate as a matter of policy. In the U.S., this is called the Federal Reserve Funds Rate. This is the shortest possible maturity. So this is the leftmost point on the yield curve.

By contrast, supply-demand equilibrium (i.e., the market) sets long-term bond prices. And, as we know, bond prices are inversely correlated to their yields. So when the market demand increases for long term bonds, their prices rise and their yields fall.


Is there a cost to running the NewYork Stock Exchange? Does it affect the price of stocks sold on the exchange?

Yes, there's a cost to running the New York Stock Exchange. There's the premises, utility bills, staff, systems, and so on.

Yes, this affects the price of stocks on the exchange. The exchange recoups its fees, ultimately, from the companies listed on it, pushing up their costs.

However, for pretty much all companies listed on it, the stock exchange fees are a tiny part of their overall costs. Which means that the impact on the stock prices is pretty much negligible.


Is there an alternative metric to GDP that measures a country's assets rather than flows?

You are correct in pointing out the flaws of GDP. But honestly, it's the best we have came up with; measuring the stock of wealth is just impossible.

You say you can estimate the value of your own assets, and of course it seems that way. We all feel that way, until we actually try. How much is your cellphone worth? Easy to know, go to ebay and look for similar phones in similar conditions and you get a price for that phone, right? Why don't we do that for all the country's cellphones? Well, because if we put all the cellphones in the market prices would drop down dramatically. Now, we're not obviously sending all the cellphones to the market, we just want to know how much its value is, but the point is that in order to have an estimation, we have to make assumptions regarding the market. The value of one cellphone is easy to calculate because it doesn't really affect the market.

Additionally, who says that you have to sell it in your local market? Why don't you offer it in Australia or Japan or Algeria or wherever the price is higher? And why not sell it another day when the price is up? The same goes for your car, house, computer, clothes, books, etc.

Think now of your financial assets. You can know that, right? All one needs to do is see what the bank says, and it says that my investment portfolio is worth \$100.... at least today... and exactly at the time I checked. But remember: the value of any financial portfolio is going up and down with high volatility even in the same day. Now try to figure out the value of the financial assets of all the country. Well, you can do that. But again: just today, and just at the time you checked.

Even though it doesn't seem like it, the same goes for goods. Take oil. Say that there is an oil reserve with x millions of barrels just waiting to be extracted and sent out to the market. How much is the value of that reserve? It depends on when you ask, whom you ask to, and who else is asking the same question (other sellers). What about gold? silver? copper? coal? wood? You can estimate a value in a given time, but that can be very different in another time.

What about land? Well, if I sow corn it will give me a given amount, but if I sow rice it will yield a different amount. Besides, would it be the same if my neighbor sows corn and I sow rice than if we both sow the same (competition)? What if I want to use it to generate electricity instead of growing food? Or what if I use it for livestock? What if I sell it? What if we all decide to do the same thing at the same time? Or what if we decide to do completely different things? Indeed to value farm land, you have to assume what use will it have.

Infrastructure is even more complicated. How much is a railroad valued? And what does value even mean in this context? There is no secondary market for railroads is there? There are opportunity costs, though. And we have the construction value too. But the we would have to make assumptions on its depreciation, and say how much of the revenue of the country is due to this specific segment of the railroad. To estimate this, one has to say how much the country would loss was the railroad not there (which is a fictitious value). What about parks?

Don't even get me started on natural resources (rivers, lakes, coast, jungles, biodiversity, etc). But even then, those are the easy ones. We still have left education, experience, skills and the like. But this post is long already, the point has been made and now I have a headache for thinking all this.

Indeed, there are many serious and very clever attempts to make estimations of these. But they are used to answer very specific questions. GDP, on the other hand is easy: just sum all the sells of goods and services. We (economists) don't like it very much either, but that's all we have. Just try to accept it, and don't take it to seriously.


What determines rent prices of houses? Does it depend only on interest rates?

What you're referring to is called the "rental yield", i.e. income generated from rentals as a portion of the amount you have invested in the house.

The interest rate is definitely one of the factors determining rental prices. Most other factors would be included in the house price (e.g. how big and nice the house is, how many bedrooms, are there many workplaces around, crime rate, etc.)

Other things that might be left out apart from the interest rate and house price is how competitive the house rental market is. You could imagine a neighborhood where there's only a few houses that are for rent. If there's demand, there's no reason the rental yield should be lower than another neighborhood with more competitive rental market.

If a person buys a house as an investment, then he/she would have taken into account the return he/she would get if this money was invested elsewhere. Say the average stock market return is about 7%, then it makes no sense for a person buying a house as an investment to seek for rental yield lower than that.

Having said that, of course, you need to take into account property price increase as well. You might be ok getting lower rental yield than your next best option if you foresee that the house price in your neighborhood would rise in the future.

Lastly, if you have a house and need to move away for a few years knowing that you'll eventually move back, it might be a good choice to rent it out in the meantime. For this, your primary reason of owning that house is not for investment, so you might be ok with getting lower rental yield (perhaps in exchange for being able to select suitable renters).


What is the result of the Bellman Equation

...the result of applying sup operator is a NUMBER...

Read it carefully. The equation is

$$
v(x_0) = \sup_{ \{x_t \}_{t \geq 1}} \cdots \quad (1)
$$
This defines a function $v$, called the value function, which is a type of indirect utility function. For given $x = x_0$, the value of $v(x)$ is defined to be the sup on the RHS, taken over feasible sequences $\{ x_t \}$.

  ...there is a functional equation, the result is a function...

Yes, the function $v(\cdot)$ defined by $(1)$ satisfies the function (Bellman) equation. Equality is in the sense of functions. The LHS is $v(\cdot)$ defined by $(1)$. The RHS is another function
$$
\nu(x) = \sup_{y\in\Gamma(x)}[F(x,y)+\beta v(y)].
$$
The claim---the dynamic programming principle---is that they are equal, $v(\cdot) = \nu(\cdot)$.


Is whatever they calculated a valid "demand elasticity"?

The way how they estimated elasticity is definitely not an accurate and unbiased way.

You are correct that when you estimate elasticity it is important to control for income, but not only for that you should also have controls for prices of competing products and  other relevant controls.

Supply-Demand is an endogenous system and every observed price and quantity is result of an interaction of the two. In order to estimate elasticity some method that can model them simultaneously or allows for endogeneity is required. The author does not specify methodology used in the blog post but there is no indication that this was acknowledged.

In order to make valid inferences from data when using parametric models you need at least 25-30 observations per independent regressor (see Newbold et al statistics for business and economics for example). The blog post tries to estimate the elasticities from 3-6 data points (depending on the model). That is woefully inadequate.

So in conclusion even though the blog you refer to does not share it's  methodology, it seems to be acknowledging neither of the issues above so their estimates of elasticities are likely biased.


You can use the PISA study from OECD. It evaluates the skills and knowledge in reading, mathematics and sciene of 15-year-old students. PISA does not combine these results, but commentators do so sometimes.

The most recent results are from 2012, you can find an overview of the rankings here. These rankings are composited from various tabels you can download using the links at page 297 from this document.
Results of 2009 are available in a xls-file. Scroll down this artcile to download.


Clearing price in a double auction market

In your example, the market clearing price is $25$ as there are two buying bids at that price or higher and two selling bids at that price or lower, and no other price achieves this match

So Buyer $1$ and Buyer $2$ both pay $25$ and Seller $1$ and Seller $2$ both receive $25$

If you want actual examples, you could look at the shares of Tottenham Hotspur Limited (the football club) on Asset Match and the Auction results (almost every month) in the right hand column.  The orders are shown cumulatively but it is easy enough to decipher the actual bids at each price.  For example in June 2019 they looked like this

  Cumulative  Actual           Actual   Cumulative
   wanted     wanted   Price   offered   offered
    11900         0     250p      875        875
    11900         0     260p      500       1375
    11900      7900     265p     9842      11217
     4000      2500     270p    16314      27531
     1500      1500     275p     1384      28915

The market clearing price was $265$ covering $11217$ shares as this is where the cumulative supply and demand curves cross.  The successful trades included all those shares offered at $250$ and $260$ and $265$, and all those wanted at $275$ and $270$ plus $7217$ of the shares wanted at $265$.  All successful transactions were settled at $265$, though the Assetmatch would have then charged a small fee for arranging the deals


When an estimator is consistent, the sampling distribution of the estimator converges to the true parameter value being estimated as the sample size increases.


factors of production (and their relative contribution)

People aren't ignoring things like raw material, energy, and land, they're abstracting them. They're saying, "We don't want to keep track of each of these separately, so we'll say that for our purposes, as an input to production, they're all really the same thing: capital." Whether that level of abstraction is appropriate for a specific problem can be questioned, and it is, and people come up with other models that refine the abstraction. For an example of this, which goes to your second question, rather than shovel all "technology" into either the production function itself or an investment which is reflected in capital, you can claim there is "labor augmenting technology", and that appears as a coefficient of the labor input- instead of y = f(K, L), you have y = f(K, AL).

When you talk about "disentangling" the parts of a production function, you have to be careful not to put something back into the problem that wasn't there when the abstraction was made, otherwise, you've changed your problem. To take your Pentium example, if you've made a model that includes a labor augmenting coefficient, you might want to put it there, otherwise, you might want to put that in capital, as an investment the firm has made. Although the production function represents "technology", the worker is still using the same basic production process, so it doesn't seem to belong there. But what you can't do is put it directly in labor, if you haven't allowed labor to be directly affected by technology.

The point is, however you claim it affects output, you have to make sure the abstractions you've made support it as an input.


When countries industrialise, why do jobs from agriculture shift to manufacturing?

This is a common area of study in Development Economics. There is for example the Dual-sector model, first developed in 1954. It is very well explained in the link provided, but basically:

[the] agricultural sector is typically characterized by low wages, an abundance of labour, and low productivity through a labour-intensive production process. In contrast, the capitalist manufacturing sector is defined by higher wage rates as compared to the subsistence sector, higher marginal productivity, and a demand for more workers. Also, the capitalist sector is assumed to use a production process that is capital intensive, so investment and capital formation in the manufacturing sector are possible over time as capitalists' profits are reinvested in the capital stock. [...]
The primary relationship between the two sectors is that when the capitalist sector expands, it extracts or draws labour from the subsistence sector. This causes the output per head of labourers who move from the subsistence sector to the capitalist sector to increase. [...]
The agricultural sector has a limited amount of land to cultivate, the marginal product of an additional farmer is assumed to be zero as the law of diminishing marginal returns has run its course due to the fixed input, land. As a result, the agricultural sector has a quantity of farm workers that are not contributing to agricultural output since their marginal productivities are zero. This group of farmers that is not producing any output is termed surplus labour since this cohort could be moved to another sector with no effect on agricultural output. [...]
The end result of this transition process is that the agricultural wage equals the manufacturing wage, the agricultural marginal product of labour equals the manufacturing marginal product of labour, and no further manufacturing sector enlargement takes place as workers no longer have a monetary incentive to transition.

In other words,, low productivity in agriculture due to unlimited land and workers and low use of capital means low agricultural wages, whereas high productivity in new capital intensive industries means high wages, thereby leading to a migration process that continues until wages equalise.
It might be worth noticing that this model was also used by Simon Kuznets to explain why industrialised countries saw a non-monotonic evolution of wage inequality between 1870 and 1950 (i.e. an increase and then a decrease in inequality), pattern that came to be known as the Kuznets Curve. As the article above states:

The Kuznets curve implies that as a nation undergoes industrialization – and especially the mechanization of agriculture – the center of the nation’s economy will shift to the cities. As internal migration by farmers looking for better-paying jobs in urban hubs causes a significant rural-urban inequality gap (the owners of firms would be profiting, while laborers from those industries would see their incomes rise at a much slower rate and agricultural workers would possibly see their incomes decrease), rural populations decrease as urban populations increase. Inequality is then expected to decrease when a certain level of average income is reached and the processes of industrialization – democratization and the rise of the welfare state – allow for the trickle-down of the benefits from rapid growth, and increase the per-capita income.


I assume that you're interested in gross value added as a national accounting concept (which is the most frequent use of the term). In that case, it is formally defined by the UN's System of National Accounts as "the value of output less the value of intermediate consumption". Here is a table from the freely available old version of Understanding National Accounts showing this breakdown for the nonfinancial corporate sector of France:

This definition is made on the production side, but gross value added can also be broken down into income. On the income side, it includes worker salaries, as another account excerpted from Understanding National Accounts shows:

As we can see above, gross value added is decomposed into compensation of employees (of which "wages and salaries" is the most important part), other taxes less subsidies on production, and gross operating surplus. Gross operating surplus can further be decomposed into consumption of fixed capital (basically depreciation) and net operating surplus.

Combining Tables 3 and 4, we see how the initial "output" accruing to a firm can be progressively whittled down:

First, we subtract intermediate inputs to obtain gross value added.
Then, we subtract compensation of employees and taxes less subsidies on production to obtain gross operating surplus.
Finally, we subtract consumption of fixed capital to obtain net operating surplus.

Of all these concepts, the final one ("net operating surplus") is closest to our usual concept of profit, since it subtracts payments for intermediates, compensation of employees, production taxes, and depreciation. Nevertheless, there are a number of conceptual and technical differences between this and profit as it is usually defined: for instance, the net earnings of a corporation subtract interest payments, while "net operating surplus" does not. Instead, "net operating surplus" can be viewed as a broader measure of capital income. (Indeed, when Piketty and Zucman (QJE 2014) plot "capital shares" of national income in Table XII, they are actually plotting net operating surplus as a share of national income for each country, taken from the national accounts.)

Finally, note that all these definitions are quite generic and meant to apply to many levels of aggregation: firms, industries, areas, sectors, or entire economies. The tables with gross value added above show the "production" and "generation of income" accounts for the non-financial corporate sector, but all national accounts constructed using the same principles provide an analogous account for other sectors—including households, government, the total economy, etc.


Pareto efficiency and maximization of sum of utilities

Finding Pareto efficient allocations via social planner's problems is a special case of scalarization in convex optimization.
Suppose agents' utility functions are  $u_i:\mathbb{R}^n \rightarrow \mathbb{R}$, for $i = 1, 2, \cdots, m$, and the feasible allocations are given by $g(x) \geq 0$ for some $g:\mathbb{R}^n \rightarrow \mathbb{R}^p$.
The general result is as follows.
Sufficiency under general conditions
If an allocation $x$ solves the social planner's problem with strictly positive social weights, then it is Pareto. This is true with no assumptions on utility functions and feasibility constraint.
In other words, if
$$
x \in \arg\max_{g(x') \geq 0} \sum_{i = 1}^m \lambda_i u_i(x'), \mbox{ for some } \lambda \in \mathbb{R}^m_{++}, \quad (*)
$$
then it is Pareto optimal.
This is easy to see. If $y$ Pareto-improves $x$, then
$$
\sum_{i = 1}^m \lambda_i u_i(y) > \sum_{i = 1}^m \lambda_i u_i(x)
$$
for any  $\lambda \in \mathbb{R}^m_{++}$. It is also easy to see that zero entry in $\lambda$ cannot be allowed---it would make the statement false.
The converse is not true in general---this sufficient condition is not necessary. There are Pareto allocations that do not solve $(*)$. This is because the set of achievable utility values
$$
\mathcal{U} = \{ (u_1 (x), \cdots, u_m(x)); g(x) \geq 0 \}
$$
is in general not a convex subset of $\mathbb{R}^m$. The social planner's weights $\lambda$ corresponds to certain supporting hyperplanes of $\mathcal{U}$. If $\mathcal{U}$ is not convex, its Pareto frontier cannot be recovered by varying $\lambda$.
However, there is a partial converse.
Necessity under concavity
Assume $u_i$'s and $g$ are concave. If an allocation $x$ is Pareto, then it solves a social planner's problem with nonnegative social weights.
In other words, if $x$ is Pareto then
$$
x \in \arg\max_{g(x') \geq 0} \sum_{i = 1}^m \lambda_i u_i(x') \mbox{ for some } \lambda \in \mathbb{R}^m_{+}. \quad (**)
$$
This is true because of convex sets and separating hyperplanes.
This is a partial converse because it's necessary but not sufficient. There are allocations that solve $(**)$ but not Pareto.
Comment
Even under concavity, $(**)$ is a only a partial converse of $(*)$. Notice the gap between strictly positive and nonnegative social weights.
To recover the Pareto frontier, the general procedure is to first find solutions to $(*)$. Then check the solutions to $(**)$ case-by-case for Pareto optimality. Alternatively, one can also take the closure (the limit points) of solutions to $(*)$.


What data/factors/attributes do the professionals monitor in a top-down analysis on the macroeconomics?

I am experienced with fixed income macro analysis. The short answer is that what you look at depends on what you want to know.

In rates analysis, the key driver for bond yields (hence prices) is the expected path for the overnight rate. The central bank sets the overnight rate. So you need to be interested in what the central bank is looking at.

The Bank of Canada is a fairly typical developed central bank, and has very good primers on their decision-making process. They have some pages describing indicators they look at here: link to BoC page. You could draw analogies to indicators in other countries.

(With regards to the 2/10 slope, that is indication of the expected path of short rates. The usual story is that if the slope is negative - 10-year yield less than the 2-year - rate cuts are expected, which normally happens in a recession. There is a huge literature on the “yield curve” for forecasting; should be easy to find something with a web search.)


How to derive the reaction function from these provided variables?

The reaction function tells you how a player in a game reacts to the strategy of another player.

As noted by denesp in the commentary, the statements provided in the question are verbal descriptions of reaction functions. The mathematical description (in millions) is given as follows:

Firm 1:
$x_1= 2 + (1/3) x_2$

Firm 2:
$x_2= 7 + (1/8) x_1$

There is not really any procedure to it, besides rewriting the verbal description into a mathematical description.


Conventional Monetary Policy and Income Inequality

Well, certainly you'll see literature saying it is possible to make monetary redistributive. This paper by Brunnermeier and Sannikov (2012), Redistributive Monetary Policy, says that deflation can cause redistributive effects in an economy, and that monetary policy can be used to correct that, but "shouldn't" be used as a redistributive tool beyond that correction.

  In general, conventional monetary policy focuses primarily on the short end of the yield curve. Expectations about future policy indirectly affect the long end of the yield curve. Unconventional monetary policy directly targets the long end of the yield curve and prices of specific assets. All these measures can redistribute wealth across and within sectors.

Notice how they emphasize the unusual nature of monetary policy meant to create (anti-)redistribution.

This paper by Faust (1996), Whom can we trust to run the Fed? Theoretical support for the founders' views, shows that while generally any policy that affects inflation will create redistributive effects, it's better that the Federal Reserve control monetary policy to mitigate those effects rather than let policy be determined by voting, because of conflicts of interest that can arise from voting.

The final paper I'd like to present is by Romer and Romer (1998). Their main findings in their paper, Monetary Policy and the Well-Being of the Poor, is that expansionary monetary policy is better for the poor in the short run, but less so in the long run. Low inflation and stable aggregate demand growth is better for the long run, and that usually means tighter monetary policy.

So it seems that monetary policy can be redistributive very easily. Also consider that optimal monetary policy usually wishes for the nominal interest rate to be close to zero. What does that do for capital owners who want to borrow for investments? How does that change long run growth and affect income inequality? Characterizing that can be a bit nebulous.


Proxies for technological change in an economy?

I would say these are some proxies

Number of patents
Number of published scientific papers
Human capital


Bounded rationality with Quantal Response Equilibrium model for an Extensive Form game

I'm not sure what capacity allocation games you're applying QRE to. But here's a very stylized example where QRE is applied to an asymmetric game where the strategy spaces of the two players are (nominally) different:
\begin{array}{c|cc}
&L&R\\\hline
T&1,0&0,9\\
D&0,1&1,0
\end{array}
This game can be easily represented in the extensive form as well.

In the standard formulation of QRE, each player $i$ plays a mixed strategy $\sigma_i$, where the probability of pure strategy $s_i$ being played is determined by the following formula:
\begin{align}
\sigma_i(s_i)&=\frac{\exp(\lambda u_i(s_i,\sigma_{-i}))}{\sum_{s_i'\in S_i}\exp(\lambda u_i(s_i',\sigma_{-i}))},
\end{align}
where $\lambda\in[0,\infty)$ measures the precision of the response; the larger the $\lambda$ the more precise the response.

Applying to the game above, let $p$ be the probability that player 1 chooses $T$, and $q$ the probability that player 2 chooses $L$. Note that $p$ and $q$ parameterize the two players' respective mixed strategy. Then, player 1's quantal response to any given mixed strategy by player 2 (parameterized by $q$) is to play $T$ with probability $p$ and $D$ with $1-p$, where
\begin{equation}
p=\frac{\exp(\lambda\cdot (1q+0(1-q)))}{\exp(\lambda\cdot(1q+0(1-q)))+\exp(\lambda\cdot(0q+1(1-q)))}.\tag{1}
\end{equation}
Similarly, player 2's quantal response to any given mixed strategy by player 1 (parameterized by $p$) is to play $L$ with probability $q$ and $R$ with $1-q$, where
\begin{equation}
q=\frac{\exp(\lambda\cdot(0p+1(1-p)))}{\exp(\lambda\cdot(0p+1(1-p)))+\exp(\lambda\cdot(9p+0(1-p)))}.\tag{2}
\end{equation}

In a quantal response equilibrium, $(\sigma_1,\sigma_2)$, the two player's strategies must be quantal responses to each other; that is,
\begin{align}
\sigma_1(T)&=\frac{\exp(\lambda \cdot \sigma_2(L))}{\exp(\lambda\cdot\sigma_2(L))+\exp(\lambda(1-\sigma_2(L)))}&\sigma_1(D)&=1-\sigma_1(T)\\[12pt]
\sigma_2(L)&=\frac{\exp(\lambda\cdot(1-\sigma_1(T)))}{\exp(\lambda\cdot(1-\sigma_1(T)))+\exp(9\cdot\lambda\cdot\sigma_1(T))}&\sigma_2(R)&=1-\sigma_2(L)
\end{align}

In the following graphs, the dashed lines plot the best responses of each player; the solid curves represent the quantal responses under different levels of precision. That is, the solid curves plots equations $(1)$ and $(2)$ above. The intersection of the two solid curves is the quantal response equilibrium point, described by the last set of equations above.


What are the economics of bike sharing companies?

Interest is their profitability mechanism. Pretty much everything else is a cost.

Let's make an example:

I start a bike share business.
I buy 100 bikes. Costs me \$10K.
I make the fee of using the bikes so cheap, lots and lots of people have no problem using them as its so cheap.

Here is the catch: To cover my liability (lost bikes, stolen, wrecked, etc.) there is a  one time fee (before you can use any of my bikes) that is a 100% REFUNDABLE security deposit of \$50 (you get your \$50 back whenever you want, so it's still cheap for you).

Everyone's \$50 adds up to quite a lot of money just sitting earning no money. But I want to make money. So what do I do? I invest it in super safe short/medium term bonds or whatever I want at a low/decent interest rate and collect my profit.


Why do some perfectly competitive, loss-making firms shutdown and others don't?

$AVC<AR$ means, without considering fixed cost, the firm is making a profit of $AR-AVC>0$ per unit of output. Compare the two options: keep producing vs  shutdown:

Keep producing: $\text{Avg Profit}=\underbrace{AR-AVC}_{>0}-AFC$
Shutdown: $\text{Avg Profit}=0-AFC$

Since $AVC<AR$, staying in production is better since the revenues can be used to offset part of the fixed costs already incurred.

$AVC>AR$ means, without considering fixed costs, the firm is making a loss of $AR-AVC<0$ per unit of output. Compare the two options: keep producing vs shutdown:

Keep producing: $\text{Avg Profit}=\underbrace{AR-AVC}_{<0}-AFC$
Shutdown: $\text{Avg Profit}=0-AFC$

Since $AVC>AR$, shutting down leads to a smaller loss than staying in production.


Coefficients of Cubic Total Cost Function

There is no deeper meaning than what you already discuss.

The Total Cost function is always made up of fixed costs plus the variable costs (each of which may be zero). We have:

$TC(Q) = F \; + \, VC(Q)$

where $TC$ is the total cost function, $F$ are the fixed costs and $VC$ are the variable costs.

Fixed costs do not depend on the quantity, unlike variable costs. There is nothing more to it. In your case, $a$ is the fixed cost and the variable costs are $bQ + cQ^2 + dQ^3$. The relation between quantity and variable costs is somewhat complicated and non-linear in your example, but there's nothing wrong with that. The exact form of the cost function comes from the technical aspect of the production process, which may very well be complicated.

So $b, c, d \,$ are just exogenous parameters (imagine them as numbers), which give the shape of the cost function, but do not have any deeper meaning. Furthermore, they will not be solved for or calculated and are simply taken as given. This is always the case with parameters. In solving economic models, you take the parameters and exogenous variables as given and solve for the endogenous variables, which in your case probably involves the quantity $Q$.


The role of reinforcement learning in Economics

I would say applications of RL in economics are still in their infancy, though probably have applications in many state-based models, agent-based models, or fields with repeated interactions like game theory.
There has been some work done on what you might call the "microfoundations" part - which is studying "regret". Susan Athey, for example, does a lot of work in this space, and an example paper is "Policy Learning with Observational Data" found here: https://arxiv.org/abs/1702.02896
There are also some papers that shoes RL results converge to what one might obtain via rational expectations / equilibrium. For example, "Learning leads to bounded rationality and the evolution of cognitive bias in public goods games" by Leimar, O. & McNamara, J. M.


Conjecture Steady State from limit properties

Upgrading an exchange of comments, a critical point in the question is the expression
\begin{align}
\frac{\partial u(\tilde x)}{\partial \rho}=\frac{1}{(\rho + u'(\tilde x) + 1)^2}
\end{align}

which is mistaken , because from

\begin{align}
u(\tilde x) = \frac{\rho + u'(\tilde x)}{\rho + u'(\tilde x) + 1}.
\end{align}

we obtain

\begin{align}
\frac{\partial u(\tilde x)}{\partial \rho}=\frac{1+\partial u'(\tilde x)/\partial \rho}{(\rho + u'(\tilde x) + 1)^2}
\end{align}

The OP notes that the sign of $\partial u'(\tilde x)/\partial \rho$ is indeterinate, and so $\partial u(\tilde x)/\partial \rho$ cannot be said to be monotonically increasing, which was what drove the pinning-down of the steady state.


As a first point, in general, the dotted line indicates that the nodes are in the same information set. So in your example, there is a single information set for the Red player and as such she has 2 actions---since she cannot tell which of the 3 nodes she is in, she cannot condition her action on this information, and therefore must take the same action (either left ($l$) or right ($r$)) irrespective of her position.

If, instead, there was no dotted line, then the Red player would know which action the Blue player had taken: lets call them $L$, $M$, or $R$. Contrary to your understanding, in this situation, the Red player does indeed have 8 strategies. Specifically, she can choose to play $l$ or $r$ conditional on each of the three actions she might observes Blue play: letting $(x,y,z) \in \{l,r\}^{3}$ denote the strategy where Red plays $x \in \{l,r\}$ conditional on $L$, $y \in \{l,r\}$ conditional on $M$, and $z \in \{l,r\}$ conditional on $R$, the 8 strategies are

$$(l,l,l),(l,l,r),(l,r,r),(l,r,l),(r,l,l),(r,l,r),(r,r,r),(r,r,l)$$

To understand more generally what is going on, it might help to think less formally and more philosophically about what an information set is. An information set is a collection of game positions (nodes) that a player cannot distinguish between (you can think of the dotted line as representing this entanglement). Then, if a player cannot distinguish between positions, she certainly cannot choose different actions at different positions, for how would she practically make this happen. So the player makes one choice per info set. If, on the other hand, the player can tell the difference between positions, she has many info sets and thus must make correspondingly many more choices.

The combinatorial explosion in the strategy space comes from the fact that each of these choices can be made independently.1 So changing what the Red player will do after the choice of $L$ by the Blue player yields two strategies for
every fixed choice out of the other information sets. When there are more periods of play, this growth becomes very very fast (I remember in grad school calculating the number of strategies in tic-tac-toe to be much larger than the atoms in the observable universe). On the other hand, having many strategies does not make the game necessarily more complicated, as many of the strategies may be identified via symmetry or other heuristics.

1 There is a subtly here that some actions might preclude reaching other information sets: nonetheless, in general a strategy is taken as a choice of action at every information set---including the ones precluded by earlier actions in the same strategy


Does Direct Benefit Transfers make an economy poorer?

You seem to be having some misconception how these transfers work

  Printing money and giving to the poor causes inflation. Increases demand and hikes prices. This is basically certain.

This is not generally how government transfers are done. In fact it is very rare to see in practice transfers that are payed by directly printing money. However, if it would be done it would ceteris paribus increase price level and hence cause inflation. It might happen indirectly by central bank buying up government bonds, but governments issue debt for plethora of reasons so its not straightforward to attribute it to any particular transfer. Moreover, if the policy is just paid out of the budget without accompanying monetary expansion then in principle it should not lead to inflation per se. Even if it did, inflation in principle does not necessary decreases the economy's output making it poorer (especially if the inflation is very low, as its rate increases it can also negatively affect output but that usually requires high rates of inflation).

Next the article from Banerjee does not even discuses whether transfers make economy richer or poorer.  The article only shows that the labor supply responses of the recipients of cash transfers are not as bad as is often thought and that they are relatively efficient way of combating inequality and poverty compared to other transfers. Moreover, they also show they do not provide incentives for  some bad behaviors as much as was thought. This is not the same as saying transfers make society richer.

Now to your main question whether direct cash transfer makes economy poorer or richer depends on what is their effect on output. Unfortunately there is no straightforward answer as it depends. Transfers create efficiency loss because they distort incentives of people to produce goods and services in several ways. First, the transfers must come out of somewhere - so usually countries have to raise taxes. These in turn lead to dead-weight loss as these taxes distort peoples optimal choices when it comes to their consumption/labor supply and their economic activity in general (for example income tax creates substitution effect from working to leisure). At the same time they also distort the behavior of recipients, as people have less incentive to work when they receive transfers (this is not an exhaustive list of all effects just examples).

However, transfers also create several benefits for society. For example during economic crises some transfers (such as unemployment benefits) serve as automatic macroeconomic stabilizers and can help recessions to be milder. There is some tentative evidence that economic inequality is actually bad for economic growth although jury is still out on that (see Barro, R. J. (2000). Inequality and Growth in a Panel of Countries. Journal of economic growth, 5(1), 5-32. -  the research shows some evidence for inequality retarding growth in poor contries and boosting it in richer countries - but see also sources cited therein some authors claim that inequality retards growth even in rich countries). There might be other efficiency reasons why transfers, including cash transfers, might help to raise output.

Whether the benefits of transfer to the economy are larger than cost is complex question that is frankly poorly understood and not yet intensively studied as it depends on what are the effect on inequality on growth and productivity and the relationship between them is anything but straightforward. The answer will for sure depend on what is exactly the size of these transfers and also how exactly they are paid for. However, to the extent that our society wants to have transfers the research of Banerjee shows having direct cash transfers might be one of the more efficient ways how to do that, and thus even in case that the transfer would make economy poorer in general a direct cash transfer might have less negative effect than other type of transfers.


extensive vs intensive labor in US in recent decades

This should serve as a starting point, although it's too early to cover the crisis. Maybe you'll find something when looking in papers citing this one via Google Scholar.


What are some of the barriers to trading services?

If you look at the four different ways that international service trade is categorised by the GATS, that will help you think about what sort of constraints are faced.  The four GATS "modes" are:

Cross border (the most simple): a firm in country A supplies a service to customers in country B.
Consumption abroad: someone from country B visits A and uses a service there.
Commercial presence: a firm in country A sets up a subsidiary in B to supply B's customers.
Presence of natural persons: a firm in country A moves staff to B to provide services there.

Some example constraints, by mode:

Some countries don't allow their citizens to hold accounts with foreign banks.
The US historically not allowing its citizens to visit Cuba.  (Mode 2 constraints are the rarest.)
Rules specifying that service companies must be 51% owned by locals have been common in various countries.
All immigration law places restrictions on this.


The US BEA defines durable goods as those lasting three years or more:

  PCE is classified by   type of product as follows. Goods consist
  primarily   of tangible commodities that can be stored or inventoried,
  but they also include certain intangible products, such as software.
  Durable goods are goods that have an average useful life of at least 3
  years. Nondurable goods are goods that have an average useful life of
  less than 3 years. Services are commodities that cannot be stored or
  inventoried and that are usually   consumed at the place and time of
  purchase. ... Durable goods [include]: motor vehicles and parts, furnishings and
  durable household equipment, recreational goods and vehicles, and
  other durable goods.

CHAPTER 5: PERSONAL CONSUMPTION EXPENDITURES (Updated: November 2017)