Why can anyone buy insurance of an, without owning the, underlying asset?

In a free market, two parties should be free to contract/trade whatever they like, under whatever conditions they like, unless the state has good reasons -- be it economic, moral, religious, or otherwise -- for intervening.

So I would instead rephrase your question as the following two questions:

Question 1. Why is it that in life insurance, the state intervenes by imposing the requirement known as insurable interest?

Question 2. Are there good reasons for the state to likewise impose a similar requirement on credit default swaps (CDS)?

Question 1. Actually, during the early (modern) history of insurance (which involved primarily life and marine insurance), you could buy insurance even without having an underlying stake in the insured object or what we now call an insurable interest.

As you can imagine, this led to a number of abuses, including murders and ships getting lost or destroyed. The term moral hazard actually originated in this context (and has since been appropriated by economists to mean something slightly different).

One result of such abuses was that in 1746, the Marine Insurance Act came into effect in Great Britain and introduced the doctrine of insurable interest:

  WHEREAS it hath been found by Experience, that the making Assurances, Interest or no Interest, or without further Proof of Interest than the Policy, hath been productive of many pernicious Practices, whereby great Numbers of Ships, with their Cargoes, have either been fraudulently lost and destroyed ... no Assurance or Assurances shall be made ... without further Proof or Interest than the Policy, or by way of Gaming or Wagering, or without Benefit of Salvage to the Assurer ...

Question 2. With CDS, the arguments in favor of imposing the insurable interest doctrine are weaker. For example, the risk of moral hazard is lower, because it is harder to cause a firm or sovereign state to default than to murder someone or burn a ship.

So by and large, CDS are treated by most governments as being just like any speculative asset that anyone is free to trade in at one's own risk.

Nonetheless, Germany (2010) and the EU (2012) have imposed this doctrine on CDS, though only for debt issued by European governments and not by firms  -- see e.g. Aldasoro and Ehlers (2018).

Some of the blame for the Global Financial Crisis has been rightly ascribed to CDS. However, I think the two big problems with CDS were the lack of transparency and the systemic risk (or "domino effect"), rather than the lack of an insurable interest requirement.

Imposing the insurable interest requirement would not have much mitigated these two problems, except perhaps by increasing the transaction costs associated with CDS and thus decreasing their use.


Intuition behind a cournot duopoly nash equilibrium producing a higher output than a monopoly?

Producing more will decrease the price and therefore the profit per unit sold. For a monopolist, all units are their own units. For a duopolist, many units will be the competitor's units.

Viewed differently, producing more produces an externality between producers via prices in a duopoly. A monopolist internalizes these externalities completely.


What are the 'Public Account Liabilities' of the Government of India?

As per Factly:

Public Account liabilities
include liabilities on account of National Small Saving Fund (NSSF),
Provident Funds, Reserve Funds and Deposits, and Other Accounts.
Liabilities under ‘Other Accounts’ include, among others, special
bonds issued to oil marketing companies, fertilizer companies and
FCI.


Why is sugar cane still harvested manually in Brazil?

Although it seems that it necessarily boils down to the idea that labor is very, very cheap for some reason or another, it is often the case in these situations that workers unions, somewhat perversely, will block the firm´s replacement of labor with capital, in order to keep the workers employed.


Terminology: _______ financing (meaning financed through the private sector)

“Private sector” financing or “non-government” financing will certainly be most clear. “Market” financing is succinct but it can be confused with “capital market” financing, that is, borrowing by issuing tradable debt instruments. “Alternative” financing is generally used to refer to non-bank, non-capital-market financing, such as borrowing from private equity or hedge funds.


As you say the first step is to take log of both sides after that you are just applying the rules for logarithms and rearrange.

For example:
$$\ln (XZ)=\ln X + \ln Z$$
$$\ln X/Z= \ln X - \ln Z$$
$$\ln X^a = a \ln X$$
$$\ln 1 = 0$$

Also an important approximations that hold close to zero are applied here as well these are:

$\ln(1+x) \approx x $ for $x$ close to zero (which for interest rates and inflation which are usually just couple of percents applies).

Also Taylor approximation is actually a different way how to linearize relationship so although it’s an example of linearization it’s not necessary log-linearization. In fact the result $\ln(1+x)$ is based on Taylor approximation but it’s not log linearization because just applying logs there won’t produce loglinear expression.

Using these rules you can prove all the above solutions. I will leave the first equation for you as an exercise, for the other equations you can see that:

Log linearizing  $1=\beta(1+r)$ gives: $ \ln 1= \ln (\beta(1+r))$ which after simplification gives us $0= \ln \beta + \ln (1+r)$ or $\ln \beta = -r $

From the second equation  $\beta=(1+\rho)^{-1}$ log linearizing gives us $\ln \beta =-\ln(1+\rho) \implies \ln \beta = -\rho$. Hence you get the equality that $-r=\ln \beta = -\rho$ then you can multiply all sides by -1 to move the minus in the middle of the equality.

The $1=\beta (1+r)$ comes from the fact that rational person would want the marginal utility of consumption today and in future to be equal so actually the equation properly reads:

$$u_t^{\prime} = \beta (1+r) u_{t+1}^{\prime}$$

Which can be rewritten as: $u_t^{\prime} / u_{t+1}^{\prime} = \beta (1+r) $ and if $u_{t}^{\prime}= u_{t+1}^{\prime}$ you get the result that $1=\beta (1+r)$. Again this is because in steady state you want marginal utility of consumption to be equal in each and every period.


Doubt on the meaning of real money balances

The level of a price index (like the GDP deflator) is essentially arbitrary. The usual choice is to set the reference date to 100, but it could just as easily be set to 1. So the level of the real money balance is essentially an arbitrary number. If you were using it in an economic model, you would need to adjust the scaling in the functions to account for this.

For an interpretation, pick a base time $t$. You then divide the monetary aggregate $M(t)$ by annualised nominal GDP at time $t$. This converts the money holding into a fraction of annual GDP. Since GDP is a flow, it has units of \$/(year), and the money number has units of \$. The ratio of the division has units of years. The interpretation is that the ratio is the number of years worth of the goods and services that go into the “GDP basket” that the monetary aggregate could buy. (For example, if the monetary aggregate was 0.5 of annual nominal GDP, it can buy 6 months worth of whatever goes into GDP.)

Let us fix the base period $t$. If we rebase the deflator $P(t)$ so that it equals annualised nominal GDP at time $t$, the “real money balance” using that deflator gives us that same ratio discussed above. (Since the level of the deflator is arbitrary, we can multiply the the entire time series by a constant, and nothing much changes. So in this case, set the scaling constant to be equal to the level of nominal GDP at tine $t$ divided by the value of the original deflator at time $t$. The result is that the new deflator is equal to nominal GDP at time $t$.) So we have an interpretation for the real money balance for period $t$.

If we go to the next period ($t+1$), the interpretation gets more complicated. The problem is that the mix of goods and services in GDP changes. So it may be that I am over-simplifying things here. (This does not happen for bread, which is presumed to be uniform over time.)

My interpretation of the real money balance is that it is volume of real GDP (in years) that can be purchased by the money balance $M(t+1)$ using the prices prevailing at time $(t+1)$, but using the time $t$ real GDP basket. So if moves to 0.75, that means that the money balance at time $t+1$ could buy 9 months worth of the goods and services that were produced at time $t$. However, this will probably not correspond to 9 months worth of the annualised production at time $t+1$, since real GDP may itself have grown (or shrunk, if there was a recession).

One can argue that this interpretation is complicated, but it reflects the difficulties with the definition.


What is the reason for the U3 rate being the more popularised and widely emphasised measure of unemployment in the United States and also for other developed economies?

Historically, data for U-3 have been collected for longer than U-6.

For example, in the US, the former have been collected since 1948, while the latter only since 1994. So, it is "traditional" to report the U-3 data.

In many other countries, the equivalent of U-6 data may not even be collected.

So, when making comparisons across time and countries, U-3 data are the best we've got.

  This broad measure is deemed by numerous economists to be the "true" unemployment rate.

This is incorrect. Economists do not consider any single measure to be the "true" unemployment rate. Instead, to arrive at any conclusion, economists try to look at a broad variety of measures and data, where available.

(But again, for reasons already given above about data availability, economists will often use only the U-3 data. This though should not be misconstrued as them thinking that this is the "true" unemployment rate. In fact, the deficiencies of the U-3 data is exactly why economists encourage collecting also other measures of unemployment.)


How does market work when there are few buyers and few sellers at the same market?

Hendricks and McAfee (2007) offer a theory of bilateral oligopoly. They consider the example of the wholesale gasoline market on the west coast of the United States, which is composed of a small number of large sellers and large buyers who compete against each other in the downstream retail market. They are specifically interested in the effect of a merger of vertically integrated firms on the whole sale and retail markets. They discuss price and cost formation. They also offer a review of the literature.

The extreme case is bilateral monopoly with one buyer and one seller. Tirole's Industrial Organization  textbook (pp. 21-25) offers a nice discussion of how this market works and in particular price behavior under bargaining and contracting.


Based on the first and second number it seems that your investment has more costs than benefits, so if all those flows were to be brought to present value, the project will lose 200 Million. This is consistent with the fact that it has a negative return of $60\%$ on average. However, the pay-back period makes no sense. I think that you probably just applied a formula and since on average you are losing money every period ($60\%$), being in this situation one less year is considered by the formula as making that money (instead of loosing it), hence "the time that will take you to pay-back the investment" is -4 years. You probably don't want to present that number, since this is not its natural interpretation, and instead say that the project is never paid back.


Because as your covariance-variance matrix shows the covariance is 0 between them.

The question e asks you to use the “true” covariance matrix from d presumably.

It’s not that you exclude it, that’s sort of misinterpretation, it is just zero ($cov(R_1,R_2)=0 \implies 2w_1 w_2 Cov(R_1R_2) =0$) and per convention zero is usually dropped from equations.


Does vNM rationality depend on the good chosen?

In the Von Neumann-Morgenstern theory, arguments of the utility function are final consumption goods. The utility function is extended over lotteries by linearity but it does not really make sense to assume that the individual "consumes" the lotteries L1 and L2 in your scenario with the raffle tickets: what he ultimately consumes is the 100 dollars monetary prize. Therefore there is no reason to expect something like diminishing marginal utility over lottery tickets - but only over final monetary prizes.

This property comes from the independence axiom: intuitively, the decision-maker views randomizations over lotteries and randomizations over final consumption goods differently. This behavioral pattern (that you find paradoxical) is usually called Reduction of Compound Lotteries. In your scenario, it says that the agent views the lottery L2 and the lottery that is computed by compounding the second-order probabilities as equivalent objects. There are some reasons to question the relevance of this axiom if you want to dig deeper (for instance, it has been found that Ellsberg-like behavior is associated with an aversion to compound lotteries).

I hope it helps.


The BLS details the construction of the CPI Index in detail here
Moreover, you can find some notes on methodology changes here
You may find the package IndexNumR in R useful for index number calculation, you can find the documentation here


First, note that the growth rate of $\mu$ is defined as $\dot\mu = \frac{ d\mu }{ \mu }$. Therefore, you will have to

take the total differential of the equation; and
divide by $\mu$.

For the first step, as in your previous question, simply use the rules of total differentials, specifically the product rule that states that for variables $X$ and $Y$, the total differential of $XY$ is given by

$$d(XY) = Y \, dX + X \, dY$$

Assuming that $s_\pi$, $s_W$ and $\zeta$ are (constant) parameters (so that by the constant rule, $ds_\pi = ds_W = d\zeta = 0$), we can apply this to your equation:

$$\begin{align}
d\mu & = d \left[ s_\pi - v (s_\pi - s_W) + \zeta \right] \\
& = ds_\pi - d \left[ v (s_\pi - s_W) \right] + d\zeta \\
& = -\left[ (s_\pi - s_W) \, dv + v \, d(s_\pi - s_W) \right] \\
& = -(s_\pi - s_W) \, dv
\end{align}$$

Now divide both sides by $\mu$, noting that since $\dot v = \frac{ dv }v$, we also have $dv = v \dot v$:

$$\begin{align}
\dot\mu = \frac{ d\mu }\mu & = -\frac{ dv }\mu (s_\pi - s_W) \\
& = -\frac v\mu (s_\pi - s_W) \dot v
\end{align}$$

which is the desired result. As before, see  chapter 8 of Chiang and Wainwright, Fundamental Methods of Mathematical Economics, 4th ed., McGraw-Hill 2005, for more on total differentials. Alternatively, see this Q&A post of mine explaining the rules of total differentials and showing another example of how to apply them to an equation.


Why FDI stock in the same year is different in the same reports?

In note 1 of the World Investment Report 2017 (page 39), it is mentioned that

  FDI data may differ from one WIR issue to another as data are
  continually revised, updated and corrected by the responsible
  authorities, such as central banks and statistical offices, that
  provide FDI data to UNCTAD.

The last figures are the most accurate ones.


Formula for Effective Discount Rate of Discount Rate Series

$$\displaystyle 1-\prod_i(1-d_i)$$ should give you the answer, with the discounts treated as fractions, so for example $10\%$ being $0.1$


Is foreign-financed Basic Income Guarantee inflationary?

Assume that a big country decides that, from here on, it will distribute for free 1 Kgr of bread to every citizen of a neighbor small country. The bread will be made, transported and distributed with big country resources (raw materials, labor, equipment, wheels, etc). Moreover, big country's representatives will sit around while local citizens eat it.

Will this create inflation? It doesn't seem so.

Now assume that instead, the big country gives for free to the local government electronic funds which the local government transfers to its citizens which in turn transfer it to the bakeries of the big country in order to buy bread and have it delivered (and being watched at, while they eat it). Will this create inflation? It doesn't seem so.

But assume now that the funds are given, always for free and always from the big country, and the citizens of the small country are free to do what they want with them. Will everybody buy bread from abroad? Hmmm, not likely. They may want to go cut their hair with that money. Lots of them. So they rush to all the barber shops and beauty parlors in the country, whose owners start to live this dreadful paradise of every supplier, to be drowning in demand and not have the capacity to fulfill it. And the barbers may think, "hey these people are dying to have their hair cut now. I have two hands and 16 hours a day -They're not enough for all these customers... what if I ask for a little more money per haircut in exchange, say, for a reservation?"

The moral here is that injecting resources into a country may result in excessive demand in local sectors where the productive capacity is not enough to cover the demand, leading to inflation. This is why the ever-sung mantra is "productive investments" rather than "consumption transfers".

If one extends the horizon of course, excessive demand will signal "hey! you, productive resources over there! Come over here! There is more demand than anyone can cover!" -but relocating productive resources takes time, more time than it takes for prices to rise.

(I apologize for the fairy-tale approach, it just came out and for once, I thought "let MathJax lie in peace". But it is perfectly valid scientifically.)


Is there a universal type space involving infinite players?

Countably infinite is easy to extend to, no obstacles. You can use results by Moss and Viglizzo (2006). I will also shamelessly promote my own work Universal type structures with unawareness that mathematically treats each of (possibly countably infinitely many) awareness levels as a player.


How often do banks borrow from the Federal Reserve?

When banks borrow from the Federal Reserve they can do so through the discount windows:

  The discount window helps to relieve liquidity strains for individual
  depository institutions and for the banking system as a whole by
  providing a reliable backup source of funding. Much of the statutory
  framework that governs lending to depository institutions is contained
  in section 10B of the Federal Reserve Act. The general policies that
  govern discount window lending are set forth in the Federal Reserve's
  Regulation A. As described in more detail below, depository
  institutions have access to three types of discount window
  credit--primary credit, secondary credit, and seasonal credit. All
  discount window loans must be collateralized to the satisfaction of
  the lending Reserve Bank.
  Discount Window Lending

The FRB has published discount window lending activity since 2010. Though this data is published with a two year lag, it may be of use. You can download that data at from their website at the link above.

There is price discrimination at the window:

  In the United States, there are actually several different rates
  charged to institutions borrowing at the Discount Window. In 2006,
  these were: the primary credit rate (the most common), the secondary
  credit rate (for banks that are less financially sound), and the
  seasonal credit rate. The Federal Reserve does not publish information
  regarding institutions' eligibility for primary or secondary credit
  Wikipedia: Discount window

It is somewhat rare to use the window. There are about 8,000 banks in the United States and about four to five hundred banks use the window in a typical quarter (according to that data).

The amounts borrowed are very small. The largest amount borrowed was about 47 million in the latest report. It was borrowed by First National Bank and Trust of Rochelle, a bank with 270 million in assets, so those window loans amount to 18 percent of assets.

But that is not representative of most banks. The second heaviest borrowing bank on the list is Xenith Bank which borrowed less than two percent of assets. More generally, there is stigma for discount window lending. The overall stigma is that since only those frozen out of capital markets use the window, users of the window must be in serious trouble. As shown above, 90 percent or more of banks don't use it in a given quarter.


Whats the rationale for using taylor series in economics?

You will gain in flexibility and generality by adding more terms and avoid to impose restrictions on substitution between inputs and returns to scale, that are usually rejected (when tested).


Is economic growth measured in real GDP or in real GDP per capita?

Usually GDP growth rates are publicly talked about in real terms but not per capita, see for example World Bank where, above the relevant table we read

  Annual percentage growth rate of GDP at market prices based on constant local currency. Aggregates are based on constant 2005 U.S.
  dollars. GDP is the sum of gross value added by all resident producers
  in the economy plus any product taxes and minus any subsidies not
  included in the value of the products. It is calculated without making
  deductions for depreciation of fabricated assets or for depletion and
  degradation of natural resources.

Real GDP levels are also used in empirical research papers and books -for example, in Blanchard & Fischer, the first figure in the book is real GNP (not per capita).

A technical issue regarding per capita magnitudes is that population is not measured quarterly, or yearly (in fact it is usually measured per decade). So any attempt to provide quarterly or yearly "GDP per capita" figures would require extrapolating the population from sparser data. While this is not necessarily catastrophic as regards accuracy, it requires more resources to be devoted to its calculation.

Per capita concepts certainly appear more often in theoretical research papers. The main reason is that theoretical models look also for possible "steady-states" (as attractors of the economy's evolution). Since GDP in levels exhibits a long-run upward trend, the next step is to consider "GDP per capita" (to "take out" the effect of population growth), and then the next step is to consider efficiency-indexed growth, like "GDP per labor efficiency unit", in order to look for the model's steady-state.


Compute evolution of a distribution over time

Here's my best guess. I haven't checked to thoroughly if this is right, but maybe it will help.

Evolution of population density

I understand the model as follows. $f(a,t)$
is the density of people of age $a$ at time $t$. Suppose at time $t=0$, the density of the population is $f_0(a)$. To model the aging process as well as the mortality rate, the density $f$ must evolve over time so that it satisfies the condition that you derived. Thus, $m$ must satisfy the following partial differential equation
$$
\frac{\partial f}{\partial t} + \frac{\partial f}{\partial a} + d(a, t) f(a,t) = 0
$$
and the initial condition
$$
f(a,0) = f_0(a).
$$
I believe that PDEs of this form can have relatively straightforward solutions depending on the functional form chosen for $d$. Here are some cases.

Case 1: Constant mortality rate, $d(a,t) = d_0$.

Suppose a constant mortality rate, $d(a,t) = d_0$. Then (using Mathematica),

DSolve[
 {D[f[a, t], t] + D[f[a, t], a] + d0 f[a, t] == 0,
  f[a, 0] ==  f0[a]},
 f[a, t], {a, t}]

results in

{{f[a, t] -> E^(-a d0 + d0 (a - t)) f0[a - t]}}

So, as we can see, this gives a simple solution
$$
f(a,t) = \exp\{-a d_0 + d_0 (a-t)\} \cdot f_0(a-t)
$$
Case 2: Log Mortality, $d(a,t) = \log(a+1)$.

Here we have

DSolve[
 {D[f[a, t], t] + D[f[a, t], a] + Log[a+1] f[a, t] == 0,
  f[a, 0] ==  f0[a]},
 f[a, t], {a, t}]

which results in

{{f[a, t] -> (1 + a)^(-1 - a) E^t (1 + a - t)^(1 + a - t) f0[a - t]}}

This also gives a simple solution
$$
f(a,t) = (1+a)^{-a-1} e^t (1+a-t)^{1+a-t} \cdot f_0(a-t)
$$

Case 3: The general case

For the case where we do not yet specify $d(a,t)$,

DSolve[
 {D[f[a, t], t] + D[f[a, t], a] + d[a, t] f[a, t] == 0,
  f[a, 0] ==  f0[a]},
 f[a, t], {a, t}]

gives us (in TeXForm for clarity)

$$
\begin{align*}
\left\{\left\{f(a,t)\to \\
   \text{f0}(a-t) \cdot \\
   \exp \left(\int_1^a
   -d(K[1],K[1]-a+t) \, dK[1]-\int_1^{a-t} -d(K[1],K[1]-a+t) \,
   dK[1]\right)\right\}\right\}
\end{align*}
$$


Would the military be more efficiant if they were unionized?

European countries offer a variety of different unionisation models, so provide reasonable evidence on the usual consequences of unionisation.  Dutch military personnel are represented by a union, for instance.

The European Association of Military Organisations appears sanguine about the idea, although many European military unions don't have the right to strike (emphasis added):

  The President concluded by emphasizing that EUROMIL strongly believes
  that military trade unionism does not harm military efficiency. On the
  contrary, being represented in trade unions and professional
  associations can enhance the [morale] and loyalty of the troops.

Less information is available about the consequences of unionisation on working hours; in general, it seems that unions appear to understand and accept the reasons for long working hours under certain circumstances.


You did not apply your formula correctly.

Your top formula is correct. But in your bottom formula, the numerator on the left side of the equation is a percentage. It should, instead, be a monetary unit of £ in this case.

The annual coupon payment should be denominated in £. Not a percentage. Otherwise, the units don't balance each other and don't match the units of the original equation.

Edit:

I noticed in your comment you include some information that was left out of the original question: that the par value is £100.

That's why I asked for a link to the source of the Bloomberg report. These numbers would make more since if the par value were £1,000


Let's put the succinct answer by @TheAlmightyBob into an abstract model:

We want to model the labor market.

Markets' structure assumptions: goods market and labor markets are perfectly competitive. All participants are "too small" economically, and they cannot affect equilibrium price through their quantities demanded/supplied - they are "price takers". Markets "clear" - i.e. prices adjust so that quantity actually supplied equals quantity actually bought.

Agents assumption: There are $n$ identical workers, and $m$ identical firms, that participate in the market. Both populations are fixed.

Other assumptions: a) deterministic environment, b) one perishable good produced, c) model in "real terms" (real wage etc, scaled by the price of the good produced).

The typical firm produces according to the technology
$$Y_j = F_j(K_j,L_j;\mathbf q)  \tag{1}$$

where $\mathbf q$ is a vector of parameters. Perfect competition in the goods market, and a perishable good imply that all output produced is sold.The goal of the firm is maximization of capital returns over the choice of labor.

$$\max_{L_j} \pi_j = F_j(K_j,L_j;\mathbf q) - wL_j$$

We are modelling the labor market, so we are interested in the first-order condition

$$\frac {\partial \pi_j}{\partial L_j} = 0 \tag{2}$$
and the corresponding input demand schedule

$$L_j^* = L_j^*\left(K_j, \mathbf q, w\right) \tag{3}$$

Total Labor demand is $L_d = m\cdot L_j^*$.
The labor market equilibrium assumption implies

$$ L_d = L_s \Rightarrow m\cdot L_j^*\left(K_j, \mathbf q, w\right) = L_s \tag{4}$$

which implicitly expresses the equilibrium wage as a function of technology constants, of per-firm capital, and of labor supplied. In order to fully characterize the labor market, we need to derive also the optimal labor supply.

Each identical worker derives utility from consumption and leisure, subject to a biological limit of available time, $T$, and the budget constraint that consumption equals wage income:

$$\max_{L_i} U(C_i, T-L_i;\mathbf \gamma),\;\; \text{s.t.} \;C_i= wL_i$$

where $\gamma$ is a vector of preference parameters, indicating the relative weight between utility from consumption, and from leisure.
This will give us individual labor supply as

$$L_i^* = L_i^*(T,w, \mathbf \gamma)  \tag{5}$$

and total labor supply is $L_s = n\cdot L_i^*$. Plugging this into $(4)$  we obtain

$$mL_j^*\left(K_j, \mathbf q, w\right) =n L_i^*(T,w, \mathbf \gamma) \tag{6}$$

If we stop here, we have a partial equilibrium model that examines the labor market. We have fully described the market, and the goals and the constraints of the participants in it (firms and workers), related to the specific market. We can perform comparative statics in order to see how the various components of $(6)$ affect the equilibrium wage. Among them, there is the capital-per-firm term, whose effects on wage we can also consider based on $(6)$, by treating it as varying arbitrarily.

In order to turn this model into a general equilibrium model:
a) We need to specify things about capital: who owns it/controls it/makes decisions on it. What is the objective functions of these decision makers. This will lead us to an optimal $K_j^*$ as a function of the structure we will impose here. Then, comparative statics with respect to $K_j$ will turn into comparative statics with respect to the factors that affect the determination of $K_j^*$, which may very well prove to involve also $\mathbf q, w$ and even the other parameters in $(6)$, changing in this way the comparative statics results obtained in a partial equilibrium setting.

b) We also need to take into account any macroeconomic identities that characterize this economy, something along the lines of $mY_j \equiv ...$ where the right hand side will be determined by the assumptions we make related to capital, but also, for example, by whether we will assume that the economy is closed or open, or partially open to the outside economic system.

So, apart from being more complicated as a model, it may also lead us to different conclusions than partial equilibrium analysis.


Productive Allocative Efficiency (Competitive Equilibrium)

Equilibrium conditions will require, among other things,:

1.) $MRS_{a,b}^i=MRS_{a,b}^j \forall [i,j] \in N$ , $N$ the set of agents

2.) $MRTS_{x,y}^i=MRTS_{x,y}^j \forall [i,j] \in J$ , $J$ the set of firms

3.) $MRT=MRS$ (Assuming that 1,2 hold)

And I note that 1,2 are efficiency conditions. 3 is efficiency in the output market.

Why? Suppose 1,2 hold at:

$MRS^i_{y,x}=3$
$MRT_{y,x}=2$

Obviously, condition 3 does not hold.

Can you see why this cannot be an equilibrium outcome? Think about what should happen in the economy and the forces that will drive this economy toward a state s.t. condition 3 holds.

So:

$MRT_{x,y}=\frac{MC_x}{MC_y} \equiv \frac{P_x}{P_y} = MRS$

where I assume 1 holds so MRS is same for all agents.


The short answer is that the bank could do that, but it prefers to hold the $30,000 as collatoral on the LOC since presumably this clothing store is a relatively high-risk client.


Is increasing the money supply possible with a gold standard? How about controlling interest rates?

Yes,  in principle,  under a gold standard,  the central bank can buy more gold, or build a mine and mine it, which increases money supply. This lowers  the short run interest rates. But the issue is that it typically can't fight the market price of gold: when the price of gold increases beyond the parity, people rush to exchange their paper currency for gold at the bank. This happens until money becomes so scarce that ir regains its value relative to gold.

This is very different from what happens with fiat currency.  With fiat currency,  the central bank can buy bonds, or gold,  but it can even just print money and give it away. ( The act of giving it for free to individuals is called a "helicopter drop". Dropping it from a helicopter is one way to actually do it...)

Even though in a gold standard,  the central bank can increase money supply,  this is different than buying bonds. This is because since money is backed by gold,  it can't loose value relative to gold.  In this case,  it's really hard for the central bank to lower the value of money if it wants, for example, to stop a deflation. Instead, because bonds are also denominated in the country's currency, making money lose value also makes bonds lose value and vice versa.


Quantity restriction in model with fixed factor of production

Summary The equilibrium wage will decrease. The reasoning is that, due to the output constraint, total labour demand will go down. Then because of the fixed labour supply (and the fixed price of capital), this makes it that the equilibrium wage falls.
Derivation
We have the first order condition for the (aggregate) labor demand.
$$
l = (1-\alpha)^\alpha W^{-\alpha} R^\alpha \alpha^{-\alpha} y \tag{1}
$$
Take first the case, where there is perfect competition and no output restrictions. Then equalising (inverse) demand and (inverse) supply for the output market gives:
$$
\begin{align*}
&\frac{\phi}{y} = p = R^\alpha \alpha^{-\alpha} W^{1- \alpha} (1-\alpha)^{1- \alpha}. \\
\to &y = \phi R^{-\alpha} \alpha^\alpha W^{\alpha-1} (1-\alpha)^{\alpha-1} \tag{2}
\end{align*}
$$
Subsituting $(2)$ in $(1)$ gives:
$$
l = (1-\alpha)^{2 \alpha-1} W^{-1} \phi
$$
This is the (aggregate) demand for $l$. Labour supply equals $1$ so equilibrium on the labour market requires that:
$$
\begin{align*}
&1 = (1-\alpha)^{2 \alpha - 1} W^{-1} \phi,\\
\to &W = (1-\alpha)^{2 \alpha - 1} \phi. \tag{3}
\end{align*}
$$
This is the equilibrium wage.
Plugging $(3)$ back into $(2)$ gives the equilibrium output.
$$
\begin{align*}
y &= \phi R^{-\alpha} \alpha^\alpha (1-\alpha)^{\alpha-1} (1-\alpha)^{(2 \alpha-1)(\alpha-1)}\phi^{\alpha-1},\\
&= \phi^\alpha R^{-\alpha} \alpha^\alpha (1-\alpha)^{2\alpha(\alpha-1)}
\end{align*}
$$
Notice that we do not need to consider ourselves with the equilibrium on the capital market as $R$ is fixed.
Now the alternative with a fixed supply (use subscripts $f$ for the equilibrium prices). If $\overline{y} < y$ then the price on the output market will be given by:
$$
p_f = \frac{\phi}{\overline{y}}.
$$
The price will no longer equal marginal costs but will be higher. Then aggregate labour demand will be equal to:
$$
l = (1- \alpha)^\alpha W_f^{-\alpha} R^\alpha \alpha^{-\alpha} \overline{y}.
$$
Setting this equal to aggregate supply (which equals 1) gives the equilibrium wage if output is restricted:
$$
W_f = (1-\alpha) R \alpha^{-1} \overline{y}^{1/\alpha}.
$$
Then:
$$
W_f < W,\\
\iff (1-\alpha) R \alpha^{-1} \overline{y}^{1/\alpha} < (1-\alpha)^{2 \alpha-1} \phi,\\
\iff \overline{y} < \phi^\alpha (1-\alpha)^{2\alpha(\alpha-1)}R^{-\alpha} \alpha^\alpha = y
$$
So the wage on the restricted market is lower as we assumed that $\overline{y} < y$.


Errata for Wooldridge graduate econometrics book

I know I've referenced that text before so I ran a quick search. Here's the instructor's solution manual.

Wooldridge Instructor Edition - Cross Section & Panel


If the bank sells me the bond they bought from a municipality, I am now simply entitled to the bond's remaining cash flows from the municipality. The price I pay for the bond from the bank is equivalent to the discounted present value of all remaining cash flows.


Best way to measure inflation in real time?

The highest frequency real time data that I am aware of the Billion Prices Project, which is daily. Link to website. These are prices on the internet, which can be mapped to the US CPI relatively well.

Otherwise, consumer prices in bricks and mortar stores are Generally not set “minute by minute,” nor is there a way to sample them at that frequency.

The closest you can get is countries in hyperinflation, where prices end up being indexed to a foreign currency. That is, the price paid is a fixed price in a foreign currency, multiplied by the exchange rate. This mechanism results in rapid changes, even during the time to drink a coffee (anecdote from the Weimar hyperinflation). Foreign exchange rates are the usual way of estimating a hyperinflation, since there is no good way to survey prices.


Can the Fed control the shape of the entire yield curve?

Yes, and they don't need any "unconventional" policy tools to do this, all they need to do is precommit to a future path of short-term nominal interest rates which they will maintain no matter what. The reason that in practice the Fed is unable to control the long end of the yield curve is because the Fed is not an institution which precommits to these kinds of policies - their policy decisions are always contingent on the information available at the date they take the decision, which means bond investors will try to forecast Fed funds rate movements not by looking at what the Fed says they will do in the future, but by forecasting the relevant macroeconomic variables that enter the Fed's decision process and thinking about how the Fed will react to what they expect will happen. Long-term bond yields reflect investors' future short-term nominal interest rate expectations, and by Fisher you may split that into inflation and real interest rates, so inflation expectations are a part of it, but they are not the whole story. Could the Fed set the entire yield curve to zero? Sure, if it could convince investors that it would maintain a zero interest rate policy for 30 years.

If government bonds bear no liquidity premium, then buying up lots of long-term debt and replacing it with short-term reserves will have no effect on long-term bond yields given the short-term nominal interest rate path the Fed intends to follow in the future. If long-term debt is important for some other purpose, for instance to meet some regulatory safety requirements, as collateral in secured lending, etc. then the Fed buying up long-term government bonds may create a shortage of long-term debt, increase the liquidity premium and force long-term bond yields down, but this policy is not optimal. When the Fed tries to reduce long rates, they intend this to have a stimulative effect on the economy today; reducing yields through this kind of mechanism would instead lead to inefficiency a la Friedman's rule.

In practice, a lot of the correlation between QE announcements and bond yield movements comes from what Bernanke called the forward guidance effect: investors interpret QE as a sign of the central bank's commitment to maintain lower policy rates in the future. According to Bernanke, this effect was the primary reason for the "taper tantrum" in 2013, and when the Fed clarified that "tapering" would not mean that the Fed would start raising policy rates sooner, bond yields came back down again.


Correct way to measure Defence budget of a country like India

So the budget is simply measured by adding up all the expenses.

The comparison to GDP that you reference is that we can measure how 'intensive' that sector is in the economy by what size of the economic pie is spent on that sector. So a country could spend a seemingly large amount of money, but if their overall economy is large enough, the military spending to GDP ratio would be small, indicating that the country isn't too devoted to that sector.

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

As we can see in the link, India spends 2.3% and the US spend 3.3%, while the actual number is different by an order of magnitude. So the ratio gives us a way to compare across countries as the absolute numbers ignore a country's size.


Elasticity Cobb-Douglas production function

For elasticity calculation why don't you try this:
\begin{align}
\frac{x_2^*}{x_1^*}&=\frac{\beta w_1}{\alpha w_2}  \\
\ln\bigg(\frac{x_2^*}{x_1^*}\bigg) &= c - \ln\bigg(\frac{w_2}{w_1}\bigg) \tag{$c=\ln(\beta/\alpha)$}
\end{align}
It's easy to see from above that elasticity is $-1$
For second part, cost of production:
\begin{align}
C&=w_1x_1^* + w_2x_2^* \\
&=w_1x_1^*(1+\beta/\alpha) \\
&=\frac{w_1x_1^*}{\alpha} \tag{$\beta=1-\alpha$}
\end{align}
So the production cost per unit of input 1 is just the function of $w_1$ (given $\alpha$).
EDIT: As requested for clarification in comments:
Note that elasticity of $y$ w.r.t $x$ is defined as:
\begin{align}
\epsilon &= \frac{\partial y/y}{\partial x/x} \\
&= \frac{\partial \ln y}{\partial \ln x}
\end{align}
So elasticity is simply the slope on the log-scale for $y$ and $x$.
Now substitute $y=x_2^*/x_1^*$ and $x=w_2/w_1$ in your case.


Potential solutions of the $n$-player Prisoner's Dilemma

All the insights to your question about the n-player game carry over from the 2-player game.

The validity in the real world is justified to the extent and in the same way as for the 2-player game, with all its caveats. An n-player game, however is even more realistic, because it is more general and many economic issues involve more than 2 players. Consider for example global geopolitical issues that require coordination, such as climate change or nuclear bombs.

The potential solutions are the same as for the 2-player game as well. Notably, in repeated games over time, reputational effects and retaliation can matter. If those who play the non-cooperative strategy (which is the "bad" action that leads to the prisoner's dilemma outcome that is worse for everyone) are punished over time, then the problem you describe can be overcome. Denesp in his comment to this answer gives some examples.

The reason a static prisoner's dilemma always has a non-pareto-optimal equilibrium (i.e. why it poses a societal problem) is because it is played once or is a one-shot game without the possibility to punish deviators later on. If players interact frequently with each other, (the threat of) retaliation can help overcome the dilemma. There are a number of such strategies. One of the more famous ones to look up would be "tit for tat".


The relation between the Black-Scholes model and quantum mechanics

I have more often heard of how the Black-Scholes equation is just the heat equation. You can find that information here. I haven't heard of a relationship between Black-Scholes and quantum mechanics before, but this post on the Physics stack exchange seems to have the details you're looking for. (Maybe you've seen this post before.)


Marginal rate of technical substitution for multi-dimension inputs?

\begin{equation}
MRTS_{x_{i};x_{j}}=\frac{\partial f(X)}{\partial x_{i}}/\frac{\partial f(X)}{\partial x_{j}} \qquad \forall i\neq j
\end{equation}


What would cause the velocity of money (V) to increase / accelerate (or at least stop falling)

[...] what event(s) or scenerio(s) can you think of that would cause the velocity of money (V) to start increasing / accelerating again [...]

That commercial banks stop using injected liquidities to "consolidate" their balance sheet instead of credit-stimulating demands. Extract from one of my previous answer, illustrated on Europe:

What I mean by "consolidate" is that the subprime crisis and its consequences have contaminated the Euro-zone's commercial banks through securitization, interrogating the default risk associated to their assets and liabilities. Injected liquidities have thus often been used by commercial banks to anticipate these potential defaults on the one hand, and to elevate their fractional-reserve ratios on the other hand so as to comply with the Basel III framework.

Indeed, there is a strong relation between fractional reserve banking (FRB) and the velocity of money, see, e.g. this.

Would V be expected to increase as a result of a mass exiting of the stock market (frigtened stock market investors liquidating their retirement holdings into cash and then spending it)?

I do not think so. Simply because I do not see stock market investors redirecting their capital toward commercial banks, i.e. toward FRB-to-velocity engines. That being said, why not (!). It all depends on their psychology.

For instance, the stock market starts falling [...] ?

The mechanism that you are so involving can only very indirectly change the velocity of money. According to me, this is not to be considered as key driver.


How exactly could you protect your debt, by enclosing it in larger debts with better security?

(Please note that is guesswork based on the linked text.)
From the Inky Fool page:

Then, in the early seventeenth century John Donne promised a chap called Sir Henry Goodere that he would send him a letter in verse. He didn't get round to it, and to make matters worse Goodere wrote him one, which meant that Donne really owed him. Or, as Donne put it in an apologetic letter:

I owed you a Letter in verse before by mine own promise, and now that you think that you have hedged in that debt by a greater by your Letter in verse, I think it now most seasonable and fashionable for me to break.

John Donne already owed him a letter, then that obligation for a letter got rolled into a larger obligation.
This is obviously a non-financial obligation. To what extent we can extend this to finance, an existing debt is rolled into a larger new loan, with the new loan having better collateral than the old. In a modern context, it would be something like rolling a credit card debt into a home equity line.
(The only other interpretation I can think of - which does not fit this example - would be to reduce risk in a book of loans by diversification. I.e., replace riskier loans with ones secured by stronger collateral. This would not be considered “hedging” by modern portfolio managers, but it might match a loose interpretation of “hedge your bets.”)


The paper "The impact of scale on energy intensity in freight transportation" by Gucwa and Schaefer has some of the information you need. Figure 5 from that paper is presented below:

The site ShipMap.Org has a data visualisation of shipping routes, where you can filter by type of freight.

(Disclosure: these are all colleagues of mine except Gucwa)


Derivative of definite integrals - how did MWG arrived at this result? Microeconomics

The minimization problem
$$\min_{w(\cdot)}  \int^{\pi_{max}}_{\pi_{min}}w(\pi)f(\pi|e)d\pi$$
s.t
$$\int^{\pi_{max}}_{\pi_{min}}v(w(\pi))f(\pi|e)d\pi-g(e) = \bar{u},$$
is evidently an infinite dimensional optimization problem. The FOC, a la Lagrange, comes from standard consideration for such problems.
To make this more explicit, define the objective and constrain functionals $\Phi$ and $G$ (on an appropriate function space, say the Banach space $C[\pi_{min}, \pi_{max}]$) by
$$
\Phi(w(\cdot)) =  \int^{\pi_{max}}_{\pi_{min}}w(\pi)f(\pi|e)d\pi
$$
and
$$
G(w(\cdot)) = \bar{u} - \int^{\pi_{max}}_{\pi_{min}}v(w(\pi))f(\pi|e)d\pi + g(e).
$$
Then the problem is simply
$$
\min_{w(\cdot)} \Phi(w(\cdot)) \;\; s.t. \;\; G(w(\cdot)) = 0,
$$
which is a standard sort of optimization problem over an infinite dimensional space.
The necessary Lagrangian FOC (as in the finite dimensional setting) would be
$$
D_{w} \Phi + \lambda D_{w} G = 0,
$$
where $\lambda$ is the Lagrange multiplier.
Here, the appropriate notion of the derivative $D_{w}$ is the Frechet derivative.
In this specific case, $\Phi$ is a linear functional. Just as in the finite dimensional setting, the derivative of a linear functional is itself, i.e.
$$
D_{w} \Phi = f,
$$
and, an integration by parts calculation would tell us
$$
D_{w} G = v'(w(\pi))f(\pi|e).
$$
So one arrives at the FOC
$$
-f(\pi|e) + \gamma v'(w(\pi))f(\pi|e)=0.
$$

...this situation ...happens often in microeconomics.

By "this", you mean "solving" such problems by differentiating under the integral sign then setting the integrand equal to identically zero.
Yes, this somewhat loose approach---with various accompanying hand-wavy justifications---often suffices in economic contexts.


How to demonstrate that a game always have a subgame-perfect equilibrium in pure strategies?

You could represent the game in simultaneous form (matrix) to see all possible equilibria. If there is only one, then that's your subgame-perfect Nash equilibrium (since there are no other equilibria). If there are multiple equilibrias in the simultaneous representation, the extensive form allows you to see if some paths would never be reached given the structure of the game.

Essentially, the extensive form "proves" which paths would never be reached through ruling them out. The one path that is reached is your SPNE. An SPNE has a more strict definition. There is really no shortcut around the arithmetic in the extensive form since the definition of SPNE forces you to check for the one viable path.


Effect of prediction of later catastrophe

This is a very open ended question, that could be researched. I can start to give some direction. For a realistic answer, one of the most important assumptions to your questions is how strongly everyone believes a meteor will hit 100 years from now, and second, how strongly everyone believes it is preventable/management.

Consider Two Scenarios:

A. The meteor will hit in 100 years. In the first 2 years, we developed a solution to it all, and everyone is convinced it will work. Will markets budge much? Probably not much afterward. Depending on the cost of the solution (\$1 million vs. \$10 trillion) markets may have moved.

B. The meteor will hit in 100 years. Everyone has fully accepted there is nothing we can do about it, and we all will be annihilated. Market will budge significantly, especially as the date gets closer. People's investment for future will change, how we think about raising kids, etc. How markets will change would be pretty hard to describe, but there may be outside literature that postulates ideas.

Overall, the biggest factor is humanities belief in their future. You can start by modeling the distribution of humanity that believes we will exist for $t$ more years. This now gets into probability theory.


The Fed Lowered Interest Rates, Are they going to buy Bonds?

There have been no such announcements as far as I know. The press release from the Federal Reserve mentions the following implementation details:

Decrease in interest rate on required/excess reserve balances to 1.80 percent
Repo operations for up to an aggregate amount of $75 billion
Decrease in the primary credit rate to 2.50 percent

My guess is that if these efforts do not have enough of an effect, then quantitative easing might resume soon as hinted by Powell:

  "It is certainly possible we will need to resume the organic growth of the balance sheet sooner than we thought," Powell told reporters during a press conference. - CNN


Intuition Behind the Definition of Prudence?

From the Wikipedia article:

  Economists describe a consumer as "prudent" if he or she saves more when faced with riskier future income. This additional saving is called precautionary saving.

For risk averse consumers with thrice differentiable utility  $u^{\prime\prime\prime}>0$ is a necessary and sufficient condition for consumers to save more for tomorrow when they are really unsure what is going to happen in the future.


Variable cost as the sum of marginal cost

You can obtain variable cost as the sum of a series in this case because the given formula for marginal cost is linear. But there is a much simpler way.

First the series approach.  The first term of your series was $10+1/4$ which is the marginal cost at $q=1$. This ignores the fact that, according to the formula, marginal cost is less than that whenever $0 \leq q < 1$. The first term should instead be the average marginal cost over the range $[0,1]$. Since marginal cost at $q=0$ is $10$, this is given by:

$$(1/2)[10 + (10 + 1/4)] = 10 + 1/8$$

Now the idea of a marginal cost for a quantity of less than one unit may seem nonsense if one thinks of marginal cost as the additional cost attributable to the marginal unit. But it does make sense for a good that can be produced in quantities that are not exact numbers of units of measurement (eg liquids, powders, measured in units of capacity or volume).  In such a case, and given linearity, marginal cost can be defined as $\Delta C/\Delta q$, where the $\Delta$'s indicate changes in cost $C$ and $q$ respectively, and $\Delta q$ may be less than one unit.

By a similar logic, the second term in the series should be the average marginal cost over the range $[1,2]$ which is:

$$(1/2)[(10 + 1/4) + (10 + 2/4)] = 10 + 3/8$$

Continuing in this manner up to the final term which is the average marginal cost over the range $[199,200]$ we have:

$$VC = (10 + 1/8) + (10 + 3/8) \dots + (10 + 399/8) = 2000 + (100)(400)/8 = 7000$$

(The formula $(100)(400)$ comes from summing $100$ pairs: $1+399$, $3 + 397$, and so on.)

Now the simpler way. Given linearity, there is no need to consider intervals of one unit.  The averaging formula can just be applied to the whole range $[0,200]$ as below:

$$\text{Variable cost = No. of units x Average marginal cost}$$

$$VC = (200)(1/2)[10 + (10 + 200/4)] = (200)(35) = 7000$$

Note however that neither of these methods would work if the formula for marginal cost were non-linear.  Given a non-linear formula you would need to calculate an integral as illustrated in Herr K's answer.


How can you tell if any given preference structure is continuous or not?

By trying to prove whether the following holds.

Note: Image taken from http://www.econ.ucla.edu/iobara/lecturepreferenceandutility201a.pdf


Contradictory FOC and maximizing solution

As alluded to by Bertrand in his +1 comments this is because FOCs do not tell you where maximum or minimum occurs. This is common misconception among some students but it simply does not hold.
FOCs give you location of stationary points. They give you points where $df(x,y)/dx = df(x,y)/dy= 0$.
A maximum or minimum will occur at points where $df(x,y)/dx = df(x,y)/dy= 0$, but so will saddle points. FOCs cannot distinguish between whether point is maximum, minimum or saddle point, it can just tell you that you found points at which slope is zero.
What even more a function can have multiple local maxima/minima and saddle points. In optimization you will generally want want to find the global maximum not just a local maximum. You can see this visualized at a figure below taken from EMEA by Sydsæter, Hammond & StrØm.

If you want to know whether the point is maximum or minimum you need to examine second order conditions (SOCs) which tells you whether the function is concave or convex (or some people like to call it concave down) then based on the concavity of the function you can infer if it is a local maximum, minimum or saddle point. For non-constrained (e.g. Lagrangian SOCs are different) multivariable optimization SOCs are given by:
A function will be concave (implying local maximum) if the conditions for $C^2$ function in $\mathbb{R}^2$ are satisfied:
$$f_{11}''(x, y) \leq 0, f_{22}''(x, y) \leq 0, \text{ and } f_{11}''(x, y)f_{22}''(x, y) −
(f_{12}''(x, y))^2
\geq 0$$
A function will be convex (implying local maximum) if the conditions for $C^2$ function in $\mathbb{R}^2$ are satisfied:
$$f_{11}''(x, y) \geq 0, f_{22}''(x, y) \geq 0, \text{ and } f_{11}''(x, y)f_{22}''(x, y) −
(f_{12}''(x, y))^2
\geq 0$$
However, the conditions above only tell you if you found local minimum or maximum. In order to find global maximum/minimum, you have to examine all maximums or minimums as well as the boundaries of your function (e.g. maximum could occur at some endpoints for example $y=x+10$ defined on interval $x\in[0,10]$ will have maximum at the upper boundary $x=10$ even if FOC does not find any stationary point at $X=10$) and check which of their output value is highest or lowest (although most introductory problems will typically be set in a way there is only one max or min - this is likely why some students confuse FOCs as conditions for max or min rather than just conditions for stationary points).


Approximately Optimal Mechanism Design vs. Revelation Principle

A key part of the paper is that they are interested in

  computationally efficient truthful mechanisms

The VCG mechanism is a truthful and efficient direct mechanism, but it is not computationally efficient. In many combinatorial auctions, it is impossible to compute the VCG transfers in a reasonable amount of time using current technology.

In the past, economists have largely ignored the problem of computational efficiency in mechanism design. This problem has been mainly addressed by computer scientists. Read for instance the introduction to Milgrom and Segal (2020) Clock Auctions and Radio Spectrum Reallocation. JPE


The fact that "trade in the United States seems to have grown substantially faster than production" could be explained by reduction in trade costs. For instance, a decrease in importing partners' tariffs tends to favor US exports. So, for the same level of production or GDP, US sales will be more export oriented. It works similarly for US imports following a reduction of US tariffs.

A first explanation to the reduction in trade costs is the reduction of tariffs and non-tariff barriers (such as administrative fees or quotas), that is changes in trade policy. Trade liberalization policies could be unilateral, multilateral, regional or bilateral. Regional trade agreements for instance have become increasingly prevalent since the early 1990s, according to the WTO.

A second explanation to the reduction in trade costs is related to technological progress such as faster communications.

Note that it is common to compute a country's trade openness index as the ratio of a country's trade (exports plus imports) to its GDP. Controlling for GDP allows economists to evaluate the outward or inward orientation of a given country's economy. Moreover, this ratio could be computed over time to measure the extent of globalization. For instance, according to the World Bank data Chinese trade openness ratio raised from 2.6% in 1970 to 39% in 2006. The US ratio increased from 9.7% in 2001 to 13.5% in 2011, which is consistent with the facts you mentioned.


How was the Cobb Douglas function derived?

What is the proof of this formula?

There is actually no proof for what the production function should be. There are infinite many possible production functions and to discover which one is the most appropriate we need to make some empirical observations. In different cases different production functions are appropriate. Cobb-Douglas is popular production function but I seen many others as well.
What you provide below in your update is not so much of a proof that production has to be Cobb-Douglas rather its a proof that if we make some specific assumptions about production (which although quite general might not always necessary hold actually) we get function that is Cobb-Douglass.

$ln(P)=α∗ln(cL)$
For example here, from where it comes the constant "c"?,

the $c$ is an integration constant. Whenever, you have indefinite integrals you have to add some constant $c$ to the solution because constants get eliminated during differentiation we can never know if there was or wasn't some constant previously so after integrating we always add $c$.
In this case when you integrate this separable differential equation the solution would actually look like:
$$\ln(P)= \alpha (\ln(L) + C) \implies\ln(P)= \alpha \ln(cL) | C= \ln(c) $$
(actually the variables should even be in absolute values - but since the function is only defined for non-negative values of $P,L$ and $K$ we can omit them).


How do you find out how much money is in circulation?

The video gives the impression that the Fed "sets" the money supply. However this is not true at all. Surprisingly the money supply is mostly determined by private banks and the amount of lending that they do. The Fed then has some things it can do to influence the money supply. It's a system called fractional reserve banking and its very hard to get your head round. For a less technical explanation, watch this.

You will learn that there is more than one possible definition of "the money supply" (M1, M2 etc). The amount of each can be found at the St Lois Fed website - try doing a search for "money".


What are volatility swaps?

Before the introduction of what is now volatility swaps, investors gained exposure to the market's volatility (yes, they already wanted to) through call and put options, products that depend on volatility, but also heavily on the price level of the underlying asset.

A volatility swap is a forward contract on future realized price volatility. Similarly, a variance swap is a forward contract on future realized price variance, variance being the square of volatility. It is basically an agreement between two parties on exchanging $N_{var} (\sigma_{realized}^2 - \sigma_{strike}^2)$ after a given period.

  How is the fair value calculated?

A good way to catch the meaning of that fair value is to suppose both actors had to make a consensual prediction on volatility. For example, they expect a realized variance of $\sigma^2$ and they agree on $B$ giving $A$ $N_{var} (\sigma_{realized}^2 - \sigma_{strike}^2)$ at expiry. Then the fair value would be $N_{var} (\sigma^2 - \sigma_{strike}^2)$.

The strike is usually chosen such that the fair value of the swap is zero, i.e. $\sigma_{strike} =\sigma$. This is not always the case, though. This was for the second, simpler question.

  How can volatility swaps be used to reduce risk of volatility in the
  markets?

Let's take the example of a life insurance company. The company offers many products with guaranteed benefits and these expose them to short volatility positions, that is, they are better off if volatility is low than if it is high. To balance this risk (they are insurance companies after all), they will buy long positions, that is, volatility swaps that gives them green notes if the volatility is higher than they expected. This does not modify their expected outcome, however it contributes to risk reduction. As they are risk-adverse, they may be willing to lower their expected value in order to bring risk down, which mean they are prepared to pay more than the fair price for a volatility swaps, which means other players will be interested in buying them whatever the composition of their portfolios, hence giving birth to the volatility market, which now-a-days has a daily trading volume of about $30-35 million.

Reference: https://www0.gsb.columbia.edu/mygsb/faculty/research/pubfiles/3967/pricing_hedging.pdf


why is Fed creation of fiat money valid?

If it is good for society to have stable prices then the measure of "valid" money is price stability. The debate over, for example, free banking laws versus central banking customs is one where both teams argue that stable prices are good and each team does not agree on the political-economic customs which would cause the good. So we see that economics involving finance relations (systems of money, credit, and debt) cannot be independent of ethical arguments in the sphere of public debate.
Regarding central bank balance sheet start with the balance sheet identity:

Assets = Liabilities + Equity

The Federal Reserve monitors the sources and uses of Reserve Balances which are liabilities on the Fed balance sheet:
https://www.federalreserve.gov/monetarypolicy/bst_fedsbalancesheet.htm
Reserve Balances and Currency are components of the base money or monetary base which are "valid" means of payment because banks use Reserve Balances to clear interbank payments and society uses Currency to make withdrawals or deposits into banks and for hand-to-hand payment.
The simplified Federal Reserve balance sheet with assets on left hand side of equation:

Treasury Securities + Discount Loans = Reserve Balances + Currency + OLE

where Reserve Balances and Currency are liabilities and where OLE means Other Liabilities and Equity. Fed uses Treasury holdings and Discount Loans as control factors to (1) service the drain of Currency; and (2) provide Reserve Balances to the aggregate bank.
Service the currency drain prior to late 2008 financial crisis:

Treasury Securities = Currency

where if Fed makes its holdings of Treasury Securities approximately equal to the Currency it has provided to banks and nonbanks over the history of Fed operations then this holds Reserve Balances constant to a first case approximation. Before late 2008 there were few excess reserves in the system so Fed had to use gross and fine control over the level of Reserve Balances to enforce control over the federal funds rate.
Fine control over Reserve Balances prior to late 2008 would be accomplished by three mechanisms: (1) Discount window loans; (2) securities repurchase agreements; and (2) reverse repurchase agreements. Since the dual mandate of the Fed is to maintain price stability and employment levels it would not take loans or open market operations (security dealing) down to zero unless political unrest disrupts the nation and puts the Fed out of business as the central bank or monetary authority.

[S]ome people suggest that the government prints money to pay its bills and that the increased amount of money in circulation lessens the value of $1. i hope to get an explanation of why both assertions are false[.]

When people criticize a government for "printing money" this often means the central government engages in deficit spending and the central bank fails to act to curb inflation as a matter of political policy. The inflation is not "caused" by the central bank printing too much money it is "caused" by the failure of the central bank to tighten credit when the bank and nonbank sector drive inflation via too much credit expansion that uses up and makes scarce the slack or excess capacity in the economy.


Have you tried solving the maximization problem? Normally you do this taking into account some market clearing condition or budget constraint and you should be able to solve for equilibrium prices to get future consumption as a function of present consumption. Consumption smoothing will simply show the path of consumption.

Getting this equation you can show how consumption varies for different levels of $ \gamma $ (in your case = 0 and = 1)


Business cycle stylized facts are not country specific?

There are several aspects to this question:
General aspects:

What determines a stylized fact?
Are stylized facts country-specific?
Why do economic studies often use the United States as a benchmark?

Specific aspects:

What do we know about the macroeconomic quantity $C_t / Y_t$?

It will benefit our understanding to consider each of these questions separately.
1.
The term "stylized fact" is not very precisely defined in economics, and it is understandable that it confuses you. In my personal experience, most economists have an intuitive understanding of what constitutes a stylized fact, but would struggle to provide a precise definition when prompted.
Intuitively, stylized facts are the closest economics, and social science at large, can come to empirical predictions arising from scientific laws. The study of what precisely determines a scientific law constitutes a whole branch of philosophy of science by itself. Let me, for the sake of discussion, simplistically assume that scientific laws are statements about causation.
Since most social systems, such as the economy, are extremely complex, it is practically impossible to observe phenomena under controlled conditions. Any causal force that can be observed is usually counteracted by myriad other causal forces, and scientific experimentation aimed to isolate a single causal effect is usually ethically and practically out of bounds. As a result, it is exceedingly difficult to falsify economic "laws" (insofar as these exist) by way of precise empirical predictions, as is for example common practice in physics (see, e.g., the long history of attempts to falsify Einstein's general relativity theory). Stylized facts constitute a less precise measuring stick for the falsification of economic laws: they are robust empirical regularities observed in a certain social system, which any proposed causal mechanisms (in the context of a scientific law or model) characterizing that system have to generate as a baseline. In your case, if I write down a model of the macroeconomy, I typically want it to generate $C_t / Y_t < 1$, unless I have a good reason otherwise -- that is, unless I can posit a causal mechanism that explains my deviation from that stylized fact.
The linked philosophy of science article provides a more formal discussion of the use of the term stylized fact in the social sciences. The author distinguishes four features of stylized facts, which they say involve (p.607-608):

Claims about the kinds of things that exist in the world: the categories that divide the world into an analytical ontology.
A way to operationalize those categories for analysis, usually in relation to a particular dataset or statistical classification (e.g. $C_t / Y_t$ only has meaning in the context of national accounting).
A claim of non-robust dependence ("Y usually depends on X", "$C_t / Y_t$ is usually smaller than 1").
A normative claim that the "particular regularities identified are the ones most important to study and are preferable to other potential characterizations of the evidence" (p.608). (A silly example: $C_t / Y_t$, the consumption share of output, tells us something important about an economy; the share of the value of cat food in output less so.)

2
Stylized facts are not supposed to be country-specific. Rather, they are meant to capture some general feature of an arbitrary economic system. However, due to the fact that economics often focuses on select sets of countries, and that stylized facts typically hold generally but not universally, it may appear differently. This brings us to...
3
It is true economics has traditionally focused excessively on advanced economies, and in particular the US. This has shifted dramatically in recent years, though. Still, a lot of economics teaching has not caught up with this shift in research focus, and it may certainly seem strange to students in countries / economies that are very different from the US that it is so often used as a benchmark.
This recent article in The Economist discusses this question in detail, and provides several reasons for economics's focus on particular countries:

The size of a country's economy accounts for 80% of variation in research attention.
The quality and availability of data.
A country's use of the English language.
Professional incentives: papers focusing on the United States have been published more frequently in top journals.

4
Now, to get to your question about $C_t / Y_t$. Though your other questions were interesting and important in their own right, they didn't really need to be answered: simple logical thinking could have already gotten you a long way.
$Y_t$ denotes income, $C_t$ consumption. Typically, it is not possible for a country to consume more than it earns, except through borrowing or transfers. Hence, as a baseline, $C_t / Y_t < 1$. This is actually a very unsurprising empirical regularity, and not that interesting of its own right (as a matter of fact, the deviations from the baseline are much more interesting). See the data here and here (depending on your definition of C_t) to discover which countries constitute the exception to the rule.


Imports/Exports in the equation of exchange

Mine is not an expert opinion, but just adding my thought since the question went unanswered.
MV=PQ has got nothing to do with trade deficit. Nor the equation will be impacted by varying import/export.
This is because PY is the nominal GDP. Which means, PY is inclusive of real gdp(C+I+G+(X-M)), where as (X-M) is the export-import.
Putting it plainly will be like, it doest matter if the exports or import varies or if it is impacted. This variation will be adjusted in the RHS of the equation. So when we calculate Velocity it should not matter theoretically.


How much is someone willing to pay to own a monopoly?

I'm not super clear about the policy here regarding self study questions. I'll give a hint for now.

You can translate the question into: finding the present value of a perpetuity, which is the present value of a constant and infinite sequence: (450*365, 450*365, ...)


"Stated Capital" is the nominal value (or "par" value) of all the outstanding shares of a company.

When a company issues shares each has a nominal price, say EUR $1.00$. But of course they may be sold for more (as far as I know, it is forbidden to issue a share and receive an amount less than its nominal value. Afterwards of course, the share of a company may be re-sold for less than its nominal value -but not when it is issued originally).

If it is sold for more, say EUR $1.50$, then the EUR $1.00$ is designated and presented as "Shareholder's Capital" in a specific line in the financial statements (usually the first line of Equity), while the "above par" value, the remaining EUR $0.50$, goes into another Equity line.


Interest rate determination in the real world

There is one more element to take into account into this demand and supply schedule: how does the central bank conduct it's (traditional) monetary policy? There are two options:

decide on base money supplied and let the money market interest rate be determined endogenously as the money market equilibrium. In this case, for a given money multiplier, the supply of money is vertical and the interest rate adjusts to clear the market. In this case: an increase in money demand does affect money market rates;
decide on the appropriate interest rate and let the amount of base money supplied to the market be endogenously determined by the money market equilibrium. In this case, supply of money is horizontal and, at the central bank will supply all the base money that is demanded by the financial sector at the chosen interest rate. An increase in money demand is now completely accommodated by increased money supply.

The role of nominal incomes in shifting money demand seems, at least to me, mostly related to periods of volatile inflation rates: if prices suddenly increase by 5% you'll need 5% more money to finance all transactions. So, households, firms, banks, ... will want to keep a slightly higher cash balance.
So, two reasons why rates may not respond to nominal income:

nominal income affects money demand, but increases in money demand may be, depending on the policy instrument, completely accommodated by increased supply;
fluctuations in nominal income may not be a dominant driver of changes in money demand. In this case there may still be a causal effect, but it's just of crazy small importance compared to other factors that shift money demand.


Why are log returns used in finance?

It really is about compactness when devising models. The mathematical property of logarithms $$log(S_{t+n}/S_t)=log(S_{t+n})-log(S_t)$$ makes log returns more convenient for modeling than the traditional $$\frac{ S_{t+1}-S_t}{S_t}$$ because the former allow the modeler to think in terms of subtraction.

The only "con" for using log returns is that logarithms are more difficult to compute or approximate "by hand" than a subtraction and division. However, with our widespread access to computers, that is not really a drawback for practical purposes.


Consider the utility function U(x,y) = y√x

Hint: This suggests $y=\dfrac{U}{\sqrt{x}}$

so draw the three curves in the usual way

$y=\dfrac{10}{\sqrt{x}}$; three of the points are $(1,10)$ and $(4,5)$ and $(9,3.333)$
$y=\dfrac{15}{\sqrt{x}}$; three of the points are $(1,15)$ and $(4,7.5)$ and $(9,5)$
$y=\dfrac{20}{\sqrt{x}}$; three of the points are $(1,20)$ and $(4,10)$ and $(9,6.667)$


Converting price for service in the foreign market to local price

Converting the amount from USD directly to the domestic currency using market exchange rates is an easy solution, but does not take into consideration the PPP (purchasing power parity) between the countries.

One possibility is to use instead the PPP rates, which are rates of currency conversion that equalize the purchasing power of different currencies by eliminating the differences in price levels between countries.

The date can be obtained from: https://data.oecd.org/conversion/purchasing-power-parities-ppp.htm#indicator-chart, from which you can calculate the price of the service, instead of using a currency converted based on market prices. On the same link you can check for the differences between the PPP rates and the market exchange rate, which in some cases are substantial.

One limitation should be pointed. The PPP rate is not calculated for a specific kind of service, but for a basket of products and services. However, this is a limitation that also affects the market exchange rates.

Therefore, at least in my opinion, this method allows to better capture the differences in purchasing power between the countries, and allow you to apply a more reasonable and justifiable price.


What is the use of knowing elasticity of average product?

If you know Elasticity of average product you can identify how your output per worker or machine will change as $L$ or $K$ increases.

Recall the definition of elasticity is:

$$\mathcal{E}=\frac{\Delta y/y}{\Delta x/x}$$

average product of labor and capital:

$$AP_L=\frac{Q}{L}$$
$$AP_K=\frac{Q}{K}$$

therefore the elasticity of average product of labor and capital is:

$$\Gamma_L=\frac{\Delta AP/AP}{\Delta L/L}$$
$$\Gamma_K=\frac{\Delta AP/AP}{\Delta K/K}$$

This is useful for evaluating how existing worker or capital productivity will be impacted as a result of an increase in capital or labor.


Can an appreciating currency lead to inflation?

You are wrong/vague here:

the rupee (INR), is appreciating against, say, the USD, it possibly
means that there is a high demand for the former.

If the rupee becomes more expensive w.r.t. the dollar, demand for the rupee usually decreases, as Indian exports are now more expensive to pay for when measured in dollars, and hence fewer people seek to convert their dollars to rupees to pay for these exports.


Which part of the market value a thief can expect to get for the stolen item?

I would even say, you might be able to make more profit on the black market, and sell it for more than the market price. the fact that you do not pay any taxes on the black market gives you the opportunity to make profit in the range of what the tax would be.

Surely there are cases where that might differ.
Depending on supply and demand. Still i think it is not far fetched to assume, you can sometimes make a lot more money by selling something on the black market. Whether it is stolen or not.

  Most likely it should be much less than $1000, because why someone would buy a used item from the questionable sources for the same price.

That would then depend on the "kind" of black market. Surely there are black markets considered "safe" und "trustworthy" in the eyes of some dealers and thieves.

  Also, if a thief would use some black market handler, that handler also needs to earn.

Here you could split the profit made due to a lack of taxes and national regulations between all the people involved in that trading process.


Subsidies Incidence on Producers and Consumers

According to standard textbooks on public finance (e.g., Tresch, Public Finance, 3rd ed., p. 272) the tax incidence includes the deadweight loss. This is because we are interested in the burden of the tax on consumers and producers and the deadweight loss can be a large part of this. (However, see https://en.wikipedia.org/wiki/Tax_incidence which does not include the deadweight loss.)

There are some difficulties, however. First, the deadweight loss is only a meaningful concept in partial equilibrium settings. The incidence as you described it cannot be used in general equilibrium settings as supply and demand curves shift with the introduction of the tax. Thus, the more common approach in general equilibrium is to determine the welfare loss of different individuals.


How could the economic cost of the world not speaking the same language be estimated?

To measure the costs of different people speaking different languages, researchers use a "linguistic distance" metric, see for example this paper. However, measuring the cost of linguistic diversity appears to be challenging. Some effects are shown along the following lines.

Impact of linguistic distance on international trade

In this paper, they construct new series for common native language and common spoken language for 195 countries, which they use together with series for common official language and linguistic proximity in order to draw inferences about

the aggregate impact of all linguistic factors on bilateral trade,
the separate role of ease of communication as distinct from
ethnicity and trust, and
the contribution of translation and interpreters to ease of communication.

Results show that the impact of linguistic factors, all together, is at least twice as great as the usual dummy variable for common language, resting on official language, would say.

Impact of linguistic distance on international migration

In this paper they examine the importance of language in international migration from multiple angles by studying the role of linguistic proximity, widely spoken languages, linguistic enclaves and language-based immigration policy requirements. To this aim they collect a unique data set on immigration flows and stocks in 30 OECD destinations from all world countries over the period 1980–2010 and construct a set of linguistic proximity measures.

Results: Migration rates increase with linguistic proximity and with English at destination. Softer linguistic requirements for naturalisation and larger linguistic communities at destination encourage more migrants to move. Linguistic proximity matters less when local linguistic networks are larger.

This paper on the Costs of Babylon—Linguistic Distance in Applied Economics
may also be relevant.

Impact of linguistic distance on literary translations

Interestingly, one of the latest Freakonomics Radio episode is called "Why Don't We All Speak the Same Language? (Earth 2.0 Series)". There are 7,000 languages spoken on Earth. What are the costs — and benefits? They quote (with references)

  There's also the roughly \$40 billion a year we spend on global
  language services — primarily translation and interpretation. And
  another \$50-plus billion a year spent learning other languages.

Useful references

Victor Ginsburgh and Shlomo Weber, Culture, languages, and
economics, in Victor Ginsburgh and David Throsby, eds.,  Handbook
of the Economics of Art and Culture, vol. 2, Amsterdam: Elsevier,
2014.
Victor Ginsburgh and Shlomo Weber, How Many Languages Do We Need? The Economics of Linguistic Diversity, Princeton, NJ: Princeton University Press, 2011


Reference request: Gender wage gap / Minority wage gap

There are a lot of possible places to start when looking at the literature on gender or racial wage discrimination/differentials. Preferably, it'd be good to look at the literature on labor supply as a whole (e.g. Mincer equation, work-leisure tradeoffs etc.), but we could be here all day discussing papers related to modelling wages in general (I am happy to provide some if you would like.) I will do my best to summarize the theoretical and empirical literature I give reference to below. Let us look at differentials.

Becker (1973), A Theory of Marriage: Part I was some of the earlier work on family economics, where men and women would tend to pick each other for marriage based on their relative strengths. To put it roughly, though people tend to like marrying others who are the same height, race, education level, etc., the theoretical model set up by Becker predicted that because families care about work (employment), leisure, and household production, there'd end up being some sort of division of labor so that all these needs were met.

Mincer and Polachek (1974), Family Investments in Human Capital: Earnings of Women and others would build off of this earlier body of work when looking at the effects of education and work training on wages. This paper in particular argued that the expectations of market activity, particularly regarding raising children, would affect how much women would pursue education or work training, and would effect how employers would want to pay women over time. If women are more likely than men to leave the workforce to spend time raising a child, then their wage-age profile will change. Intuitively, even before a child is born, women might get paid less because future expectations of productivity affect wages and individual investment into skills.

Becker (1971), The economics of discrimination is a gigantic book written as a lot of the basis for modelling economic discrimination. There's a lot to sift through and I'm not super well-read in that area, but the gist of the model set up here is you can distinguish between prejudice which is where employers may have a preference for certain type of worker that shows up in the utility function, and discrimination, which is the explicit unequal treatment of equals; that is, unequal outcomes. Although there may be prejudice from employers, depending on the search model used, if there are enough jobs by non-prejudiced employers, we may not see any discrimination in outcomes.

Even after controlling for confounding variables that could affect the wage gap between women and men, or between races, there is some discrimination in outcomes on the basis of race and gender.

Dougherty (2004) Why is the rate of return to schooling higher for women than for men? uses the Oaxaca decompositions in E. Sommer's answer to look at why the rate of return to education seems to be higher for women than for men. In this paper, one theory looked at was that better educated women are more willing/able to compete with men to push back against gender discrimination, hence why it seems the rate of return is higher. The paper has some support for this finding.

Lang and Lehmann (2011) Racial Discrimination in the Labor Market: Theory and Empirics notes that while taste/preference frameworks combined with search models are able to explain a lot of differences between wages and unemployment duration, these models are not so good at showing differences in unemployment rates themselves. Statistical models (employers have some sort of imperfect information that leads to discriminatory outcomes through, for example, self-confirming stereotypes) can explain magnitudes of wage differences well, but not much about employment. This paper tried to combine elements of the two approaches.

This above article is a particularly good one to look at (and should not be too much of a problem to sift through if you have a good math background). I don't think I can do justice trying to summarize the breadth of their results here in a reasonable amount of time, so check it out on your own if you like.

There are a lot of good articles in labor economics that look at gender and racial discrimination but those are a few I've picked out to give some brief summaries on. Some of the ones I've not mentioned, but which are important to the literature are listed below. Just because I chose not to summarize them does not mean the below articles are particularly less important. I just have my own time constraints. My presentation of the existing literature is also not comprehensive, but hopefully a good starting point for others.

Hope some of these resources prove useful to you and future users.

Further Reading

Oaxaca-Blinder Decomposition:

Blinder, Alan S. "Wage discrimination: reduced form and structural estimates." Journal of Human resources (1973): 436-455.
Oaxaca, Ronald. "Male-female wage differentials in urban labor markets." International economic review (1973): 693-709.

Labor Supply/Compensating Differentials:

Oaxaca, Ronald L., and Michael R. Ransom. "On discrimination and the decomposition of wage differentials." Journal of econometrics 61, no. 1 (1994): 5-21.
Rosenzweig, Mark R., and T. Paul Schultz. "The demand for and supply of births: Fertility and its life cycle consequences." The American Economic Review 75, no. 5 (1985): 992-1015.
Henderson, Daniel J., Solomon W. Polachek, and Le Wang. "Heterogeneity in schooling rates of return." Economics of Education Review 30, no. 6 (2011): 1202-1214.
Goldin, Claudia, Katz, Lawrence F, and Kuziemko, Ilyana, “The Homecoming of American                         College Women: The Reversal of the College Gender Gap”, Journal of Economic                             Perspectives 20(4), 2006, 133-156
Blau, Francine D., and Lawrence M. Kahn. "The US gender pay gap in the 1990s: Slowing convergence." ILR Review 60, no. 1 (2006): 45-66.
Mobius, Markus M., and Tanya S. Rosenblat. "Why beauty matters." American Economic Review 96, no. 1 (2006): 222-235.
Lang, Kevin, Michael Manove, and William T. Dickens. "Racial discrimination in labor markets with posted wage offers." American Economic Review 95, no. 4 (2005): 1327-1340.


Does the growth rate of a neoclassical production function converge as all input factors grow with constant, but different growth rates?

I am not sure that elasticities converge for arbitrary growth rates.

Here is a treatment of the issue. Subscripts in $F$ denote partial derivatives. I will omit the time subscript. By definition, the elasticity of the $i$-th input is the ratio of marginal product ($F_i$) over Average Product $(F/x_i)$. So for an elasticity to stabilize intertemporally, it must be the case that the growth rate of the marginal product  is equal to (or equalizes a the limit with) the growth rate of the average product.

For input $x_{1}$ we have

$$\frac {d}{dt} F_1 = F_{11}\dot x_{1} +F_{12}\dot x_{2} +...+F_{1n}\dot x_{n}$$

All inputs are strictly positive so we can multiply and divide, and also divide by $F_1$ to obtain the growth rate (denoted by a tilde)

$$\tilde {F_1} = \frac {dF_1/dt}{F_1}  = \frac{F_{11} x_1}{F_1}\frac{\dot x_{1}}{x_1} + ... +\frac{F_{1n} x_n}{F_1}\frac{\dot x_{n}}{x_n}$$

The first ratio in each term is the elasticity of the $F_1$ with respect to each input, say $\eta_{1k}$. So,

$$\tilde {F_1} = \sum_{k=1}^n\eta_{1k}g_k $$

and analogously for the other inputs.

Turning to the average product of the first input, we have

$$\frac {d}{dt} (F/x_1 ) = \frac {F_1x_1 - F}{x_1^2}\dot x_1 + \frac {F_2}{x_1}\dot x_2+...+\frac {F_n}{x_1}\dot x_n$$

Manipulating,

$$\frac {d}{dt} (F/x_1 ) = \left[F_1-\frac {F}{x_1} \right]g_1 + \frac{F}{x_1}\frac{F_2x_2}{F} \frac {\dot x_2}{x_2}+...+\frac{F}{x_1}\frac{F_nx_n}{F} \frac {\dot x_n}{x_n}$$

$$= \frac {F}{x_1}\cdot \Big( (e_1-1)g_1 + e_2g_2 + ...+ e_ng_n\Big)$$

$$\implies \frac {d(F/x_1)/dt}{(F/x)} = \tilde F - g_1 = \sum_{k=1}^ne_kg_k - g_1$$

So for all elasticities to stabilize (and so for $\tilde F$ to stabilize) we want, at least eventually, that

$$ \forall i=1,...,n\;\;\;  \sum_{k=1}^n\eta_{ik}g_k = \sum_{k=1}^ne_kg_k - g_i$$

Denoting $\mathbf H$ the $n\times n$ the matrix of the elasticities of the marginal products, $\mathbf I_n$ the identity matrix, $\mathbf e$  an $n \times 1$ vector containing the output elasticities $e_i$, $\mathbf i$ a column vector of ones, and $\mathbf g$ the $n \times 1$ column vector of growth rates of inputs, we can write the conditions for all elasticities to be stable compactly , as

$$\big(\mathbf H + \mathbf I_n - (\mathbf i \oplus \mathbf e')\big)\mathbf g =0$$

I note that the sum-of-matrices term does not depend at all on the growth rates vector. So the above orthogonality condition cannot hold for arbitrary $\mathbf g$ (except if I am neglecting some property stemming from the homogeneity of $F$).


It is just an opaque way to denote that the non-definite integration limit will take the value of the variable itself. Namely,

$$\int_a g(x) dx \equiv \int_a^x g(s)ds$$

... where $s$ is just the dummy variable of integration. Since sometimes we use the notation $\int _D f(x) dx$ to denote fully definite integration "over the domain $D$ of $x$", I would avoid the left-hand side notation and I would stick with the more clear right-hand-side one.

I do hope that the authors where you find this notation mean what I wrote above, because this is what mathematicians mean by it.


Aren't all cost functions step functions?

Analyzed over (0,5) these would coincide, and an analyst might not
  have any expectation that increasing by a small factor would
  dramatically increase cost, and while the (simple) derivative at any
  point on the line is the same for either line (inside the step
  interval), this is very misleading.

Good question, but mainly of theoretical validity. The "analyst" in question would most probably have some indications of the cost for most if not all production scales of interest. There are few cases when a production plan is about producing on a scale no-one has seen before (not only the specific company, but any company). So data to compare and get an idea will be around, and also, as a last resort, "engineering" data can, and do, assist also (and engineering is much more of an exact science than Economics is).

Firms' practitioners have a rather clear view of "Fixed Costs" that are not really Fixed but step cost functions, trust my corporate experience on this one. They "sense" when the step is about to be taken -and they sit down and calculate it. The whole Cost-Accounting business is essentially about learning how to detect, separate and handle Fixed and Variable Costs, as well as Variable and Marginal Costs (in the real world, the distinction between Variable and Marginal Cost is very existing, and very important).

In the case of aggregate/average cost functions, the individuality of each firm alongside the common structural base that exists in all production activities, helps in smoothing the relationship.


Do stock markets price in existential risk (i.e. global nuclear war)?

I don't have anything to say on the empirical side.

On the theoretical side, I think that is an interesting question, even in a standard framework with "rational" agents. The possibility of the end of the world reduces the expected stream of income from assets (e.g. a share of a firm) because maybe the firm will not be there any more to distribute dividends.

Maybe you will not be there (that's the existential risk). About this the prediction of the standard microeconomic theory is the same as in the science fiction movies : your preference for present should rise, the 'natural' interest rate and the actualization rate with it. (If you fear to be dead tomorrow, you will not lend your money today... unless for a very high interest rate.) That means the value of future income falls and you want to consume everything now.

So both effects go in the same direction and "standard" prediction would be that the stock prices should fall.


Individuals vs. groups as decision makers

Many fields of economics focus on collective decision making. For example public choice theory is almost solely focused on collective decision making as opposed to individual one.

Many areas of economics focus on individual decision making because that’s where actually decisions are made in real life. When you go to the store buy shirt you go there as an individual not as some kind of collective or hive mind. This holds for many economic interactions as well.

Furthermore, collective decisions are not more socially optimal then individual ones. For example, as Arrow impossibility theorem suggests, collective decisions where there are more then 2 options will never be optimal from social welfare perspective (at least in a democracy where there is no benevolent dictator).

However, the Arrow impossibility theorem is derived in idealized conditions. In real life there are areas and cases where collective action is preferable, but as a general rule it’s not possible to say that collective action is more likely to yield socially optimal outcomes.


Perfectly competitive constant MC Questions

Theoretically, if all firms are identical and have the same cost schedule, then the supply curve for the market should be perfectly elastic, as you stated. The side-by-side market and firm graphs would appear as follows:

This is an odd situation. Typically, when there exists a surplus of a good, firms competitively lower their prices until enough firms go out of business for the surplus to no longer exist; and when there exists a shortage, firms enter the industry until the supply increases enough for the shortage to vanish.

However, in this case, since firms are not distinguished by the price that they can afford to charge, the natural tendency towards equilibrium cannot occur this way. Instead, when there is a surplus, firms cannot compete by price lowering and will either survive or perish based on the random purchase choices of  buyers (if prices are the same everywhere, consumers will probably choose where they buy their goods based on other factors, like geographic proximity, aesthetic preferences, etc, but “rational” buyers in the simplified model will be indifferent). For our purposes, which firms survive and which close down will be essentially random - whichever firms happen not to sell enough of their product for whatever reason will close.

During a shortage, it gets even weirder. Whether or not firms enter the industry depends on the location of the ATC curve (which is why I did not draw it in the above graphs). If $ATC < P_F$ at $Q_F$, firms would enter since profits are possible and people will buy the good they produce. If $ATC = P_F$ at $Q_F$, there is no telling whether firms will enter, since they will all be indifferent.

And if $ATC > P_F$ at $Q_F$, whether there is a surplus or shortage, firms will drop dead like flies until not a one is left. If the good being produced is of any real import, the government would probably offer subsidies to incentivise the production of the good.

So, to answer your question - it would make sense for you (a firm) to enter if $Q<Q_M$ and $ATC < P_F$ at $Q_F$. You could also enter when $Q\ge Q_M$, but then your success or failure would be left up to chance or other factors as consumers decide whether to buy your product or your competitor’s identically-priced product.


A survey on the economics of loyalty programs

A relevant literature is that on the economics of switching costs. There are many papers in this literature and a number of surveys, but I would recommend this paper by Klemperer for a concise combination of survey and theory. A more comprehensive survey can be found in this handbook chapter by Farrell and Klemperer.

The basic idea is that loyalty programmes (among other things) create a cost for consumers who contemplate switching to another supplier. This literature studies how such costs influence strategic behaviour and competition between firms.