Convergence in probability and convergence in distribution

I will attempt to explain the distinction using the simplest example: the sample mean. Suppose we have an iid sample of random variables $\{X_i\}_{i=1}^n$. Then define the sample mean as $\bar{X}_n$. As the sample size grows, our value of the sample mean changes, hence the subscript $n$ to emphasize that our sample mean depends on the sample size.

Noting that $\bar{X}_n$ itself is a random variable, we can define a sequence of random variables, where elements of the sequence are indexed by different samples (sample size is growing), i.e. $\{\bar{X}_n\}_{n=1}^{\infty}$. The weak law of large numbers (WLLN) tells us that so long as $E(X_1^2)<\infty$, that
$$plim\bar{X}_n = \mu,$$
or equivalently
$$\bar{X}_n \rightarrow_P \mu,$$

where $\mu=E(X_1)$. Formally, convergence in probability is defined as
$$\forall \epsilon>0, \lim_{n \rightarrow \infty} P(|\bar{X}_n - \mu| <\epsilon)=1. $$
In other words, the probability of our estimate being within $\epsilon$ from the true value tends to 1 as $n \rightarrow \infty$. Convergence in probability gives us confidence our estimators perform well with large samples.

Convergence in distribution tell us something very different and is primarily used for hypothesis testing. Under the same distributional assumptions described above, CLT gives us that
$$\sqrt{n}(\bar{X}_n-\mu) \rightarrow_D N(0,E(X_1^2)).$$
Convergence in distribution means that the cdf of the left-hand size converges at all continuity points to the cdf of the right-hand side, i.e.
$$\lim_{n \rightarrow \infty} F_n(x) = F(x),$$
where $F_n(x)$ is the cdf of $\sqrt{n}(\bar{X}_n-\mu)$ and $F(x)$ is the cdf for a $N(0,E(X_1^2))$ distribution. Knowing the limiting distribution allows us to test hypotheses about the sample mean (or whatever estimate we are generating).


Econ Intuition for Jacobian inverse in demand system

For the 2x2 case being considered, write
$$\mathbf{B}=\left[\begin{array}{cc}
b_{1,1} & b_{1,2}\\
b_{2,1} & b_{2,2}
\end{array}\right].\quad$$

It follows that the element (1,1) in $B^{-1}$ is given by $\frac{b_{2,2}}{b_{1,1}b_{2,2}-b_{1,2}b_{2,1}}$. Notice that
$$\frac{\partial q_1(p_1,p_2)}{\partial p_1}=(\frac{\partial p_1(q_1,q_2)}{\partial q_1 })^{-1}$$
implies
$$b_{1,1}=(\frac{b_{2,2}}{b_{1,1}b_{2,2}-b_{1,2}b_{2,1}})^{-1},$$
which can only occur if either $b_{1,2}=0$, $b_{2,1}=0,$ or both.

To better understand the intuition for this result, consider the demand function $q_1(p_1,p_2).$ Suppose we are interested in the demand curve for good 1 when $p_2=\tilde{p_2}$, which is a plot of the function $q_1(p_1,\tilde{p_2})$ with $p_1$ on the vertical axis and $q_1$ on the horizontal axis. The slope of this demand curve is given by $(\frac{\partial q_1(p_1,p_2)}{\partial p_1})^{-1}=-\frac{1}{b_{1,1}}$.

Now consider the inverse demand for good 1, which gives us maximum marginal willingness to pay for good 1 conditional on quantities of good 1 and good 2. If we want to plot this demand curve we must fix the quantity of good 2, say at $x_2=\tilde{x_2}.$ Suppose we want this demand curve to correspond to the demand curve we plotted for $q_1(p_1,\tilde{p_2})$, meaning $\tilde{x_2}$ must be the quantity demanded at $\tilde{p_2}$. However, to find the quantity demanded of good 2 at $\tilde{p_2}$,we must condition on the price of good 1. Likewise, to find the marginal willingness to pay of good 2 at quantity $\tilde{x_2}$, we must condition on the quantity of good 1. Both of the previous statements make no sense, as we are trying to plot the relationship between $x_1$ and $p_1$, so we cannot fix either of them.

The basic intuition is that when we invert $B$ to solve for inverse demands, we are accounting for the effects of cross-price effects in writing our marginal willingness to pay (inverse demand) functions. In economics terms, the following two expressions are not equal unless cross-price effects are zero:

Marginal effect of the quantity of good 1 on willingness-to-pay for good 1, conditional on the  quantity of good 2.
Inverse of the marginal effect of the price of good 1 on quantity demanded of good 1, conditional on the price of good 2.


Difference between Luxury/Premium/Value fashion segments? What does "value" mean here?

Consider the ratio
$$
\dfrac{\text{Benefit}}{\text{Price}}.
$$
This is not the same thing as your net benefit, $\text{Benefit}-\text{Price}$.  So a good that might give you a high net benefit might have a low benefit-to-price ratio.  For example, $100-50=50$ and $100/50=2$ versus $1000-900 = 100$ and $1000/900=1.1111...$.

Luxury goods have high gross benefit but low benefit-to-price ratio and possibly a low net benefit.  Affordable luxury is trying to improve on benefit-to-price ratio  and net benefit while still having a high gross benefit.

Value goods are trying to achieve a high benefit-to-price ratio, despite the products having a low gross benefit and a mediocre net benefit.  So you feel like you got a lot for what you paid, even if it's a $\$$20 acrylic sweater instead of a $\$$500 cashmere one.


Are there Nash Equilibria that aren't mixed strategies?

Nope, every pure strategy equilibrium can be characterized as a degenerate mixed strategy equilibrium.

That is, it is a mixed strategy in which a pure strategy is played with probability $1$.


Why GDP positive growth does not decrease inflation?

GDP growth would lead to deflation if the money supply remained unchanged. The government usually increases money supply as the economy grows to avoid deflation. Controlling money supply is not an easy task though because the economy responds with lag to every policy decision. But ideally, the government aims at keeping inflation low (1-4%) and avoiding deflation. Zero inflation is also undesirable because if you know your money will be worth just as much a year from now, you have less incentive to invest.


Optimization: Dynamic Programming vs Kuhn-Tucker

I would say that the main difference stems from the solution method, which results in your statement about all paths versus only the path at time t being true.

Dynamic programming (at least when done numerically) consists of backward induction. One tries to identify the optimal action for all possible values of the state variable in the final period, and then reasons backwards following the state equation. In this way one gets a solution not just for the path we're currently on, but also all other paths.

Similarly, if one uses the guess-and-verify method to solve the value function of the Bellman equation, the value function one guesses defines the optimal decision for all possible values of the state variable. Thus one gets a solution for all possible paths, including the current one.

Kuhn-Tucker basically works the other way around. One formulates the necessary and sufficient conditions, and solves the resulting difference equation(s) using the initial conditions as starting points.


Relationship between Real Interest Rate and Employment

Recall that central banks use interest rate to control inflation by inducing spending or saving  with decrease or increase in interest rate.

Also,note the theory proposed by the short-run Phillips curve.

Though this graph has  some controversy in terms of its empirical soundness. It is the basis of how we tie interest rate to unemployment.

abstractly you can think of it as a "causal chain" being:
$$\text{interest rate}\rightarrow\text{savings rate}\rightarrow\text{spending}\rightarrow\text{inflation}\rightarrow\text{unemployment}$$

Hope this helps.


There’s no formal meaning, but it would be the perceived safest government bonds. The quote indicates thar US Treasury bonds are part of that core. The 10-year yield fell, which meant that price went up.


What's the point of adding security features to new bill designs?

Bills don't last that long. The Federal Reserve estimates that $100 bills last about 15 years. Therefore, old bills are fairly rapidly rendered quite rare. Bills that look really odd (like the rare ones) are likely to warrant a second glance from the acceptor and later, if the counterfeiting is uncovered, much more likely to be traceable back to the forger. In The Great Paper Caper, Wells Tower tells the story of Frank Bourassa a counterfeiter who focused exclusively on modern \$20 bills because the enormous scrutiny on \$100 bills in general.


JOLTS actually goes back to 2000 (though years before 2005 are only available on a SIC basis). Prior to JOLTS, most people used data on the number of help-wanted ads in newspapers, such as this series from the 1950s-60s or this one that goes back to 1919.


Does it make sense to 'deflate' a sectoral price index in a regression analysis?

In that situation, how do you extract these fluctuations related to
  the general economy out of your price index being used as an
  instrument?

Regress the sectorial price index on the general price index, and use as instrument in the demand equation the "residuals plus estimated constant" from the previous regression.

These residuals will be by construction orthogonal to the general price index which indeed could be also a demand shifter, something that would invalidate the sectorial price index as an instrument.

As is the case with any time-series data, one should also deal with possible issues of non-stationarity, co-integration etc.


Why is such a high US national debt not resulting in another great depression?

You answered question (1) with your own chart. The United States had a “high” debt-to-GDP ratio in World War II - and there was no subsequent depression. (There was a sharp recession as a result of demobilisation, but that was short-lived.)

For a country that borrows in its own currency, there is no obvious reason how a high debt-to-GDP creates recession risk.

There is no easy answer to (2). We are currently headed into the greatest economic contraction since the Great Depression, and the cause was a pandemic, which would not show up in a standard mathematical economic model.


Is there a relationship between stock market hedge fund leveraging and the repo market?

"I can't use the stock I bought as collateral in the repo market right?"

In principle, you can.  Repos can and do happen between non-government actors, with any AAA-rated paper being preferred, but a slice of the repo market does deal in riskier instruments like corporate bonds and equities.   But that piece of the repo market likely isn't big enough to impact the money supply, and it isn't what's being alleged in the article you linked.

The article specifically addresses repo transactions between the Fed and financial institutions.  In these, the Fed buys Treasuries from banks under an agreement to resell them at a higher price, usually (but not necessarily) the next day.  The spread is the repo rate.  The intended effect is to increase short-term liquidity, by inducing banks to reduce prime rates and lend more freely.  This lending is supposed to support real economic activity that is otherwise not happening due to various frictions in the economy - it operates on the presumption that output is below potential, and that this is primarily because of credit constraints based in sentiment rather than fundamentals.

The issue is that if the market fundamentals have shifted so that the economy is in fact not producing below its potential (or not as far below as is believed), but is rather transitioning to a new and lower steady state, then there is no appetite for the liquidity being injected in this way.  So while banks are enticed to sell off their Treasury holdings because of aggressively low repo rates, because fundamentals in the economy are weak, they are reluctant to increase loans to economic actors.  Being bound to repurchase those treasuries in the short term, they pile the money into financial markets with the expectation of realizing their desired gains in this way instead.

Being collectively large enough to move share prices, when all of the banks are doing this together it produces a false signal: normally, markets rise because the underlying economic activity represented by stocks is rising.  Yet in this mechanism, markets are rising even as the underlying economic activity is stagnant or even declining.

This is what analysts mean when they talk of "decoupling".


hint:
sum equation (2) for all i and use the definition of $Q^c$, we get an expression for $Q^c$. then, substitute this expression into (2) and solve for $q^c$.


Differential equation for first-price auction

For notational simplicity, let me define the distribution $G(s) = F^{N-1}(s)$ with density $g(s)$. Let $\underline{s} = 0$ (for simplicity).
We have
$$
b'(s)G(s)+b(s)g(s)=s g(s)
$$
Integrating to $x$ gives us
$$
\int_0^x b'(s)G(s)+b(s)g(s) ds = \int_0^x s g(s)ds
$$
Notice that $\frac{\partial }{\partial s}(b(s)G(s)) =  b'(s)G(s)+b(s)g(s)$, so
$$
 b(x)G(x) = \int_0^x s g(s) ds
$$
where we use the boundary condition that $b(0) = 0$.
Now,
$$
 b(x) = \frac{1}{G(x)}\int_0^x s g(s) ds
$$
Finally, apply integration by parts to $\int_0^x s g(s)$ to get
$$
 b(x) = \frac{1}{G(x)}\left[ G(x) x - \int_0^xG(y) dy\right]= x - \int_0^x\frac{G(y)}{G(x)} dy
$$


The problem is that your supply-demand graph is wrong. You assume in your graph that the demand and supply are linear functions but in this case they are step functions.

For example, in your graph there would be 2.5 demand at price $\$0.55$ but the problem clearly states that the units are sold at discrete quantities of 1. So the drawing is incorrectly modeling the situation as there should be a steps in increases in demand not a simple linear function. The same holds for supply.

If you draw your supply and demand correctly you will see that in your case deadweight loss (DWL) is not a triangle but just two squares with areas 0.10 respectively which combine into a rectangle with area 0.2 and hence total DWL will be also $\$0.2$.

This is how your problem would look visualized (pardon the bad drawing):

Where the shaded area is the DWL. The two methods will always match but you have to use the same demand and supply functions in both problems if you assume different supply and demand of course results will change (in your original drawing also consumer surplus after tax would be wrong by the way as in your original drawing it would be area of two trapezoids both having areas of 0.15, and the pre tax consumer surplus in your picture would also not be 1.2).


What metrics would indicate a house bubble rather than genuine market values?

The first metric to look at is house-price-to-rent ratios. Rental prices capture the value of the housing (and housing-linked) services provided by a property, including things like how safe a neighborhood is, how good the schools are, et cetera. In contrast, house prices are the capitalized value of the future stream of housing services PLUS the price of an asset (mostly land; structure value beyond the housing services it provides is negligible). As a result, a higher ratio of house prices to rents means that, controlling for the value of housing services provided, we observe people paying more for housing assets (i.e., land).

This doesn't mean that the asset is necessarily overvalued, but it at least moves you in the right direction by separating out the asset price from the value of housing services. You can then start thinking about the supply/demand fundamentals that apply to housing assets, and what you'd have to assume to support a particular valuation of the asset (things like, as Jamzy noted, whether growth in price-to-rent ratios is linked with increased home purchase borrowing).


What is the difference between risk, uncertainty and ambiguity

Here is a decision-theoretic formalization of your definitions.

The usual framework to talk about objective risk is the situation where a decision-maker expresses preferences over objective lotteries. Formally, if $X$ is a prize space, objective lotteries are defined as elements of the space $\Delta(X)$ of probability distributions (usually with finite support) over $X$. For instance, the decision-maker might be asked to form preferences between the lottery that offers her/him an apple with probability 0.3 and an orange with probability 0.7, and the lottery that offers her/him an apple with probability 0.5 and an orange with probability 0.5. The standard result in that area (von Neumann-Morgenstern theorem) delivers a representation that identifies the agent's attitude towards objective risk (her/his utility function), while the probabilities are given as a primitive of the model.

The usual framework to talk about ambiguity is the situation where a decision-maker expresses preferences over uncertain acts. Formally, if $X$ is a prize space and $S$ is a state space, acts are mappings $f:S \rightarrow X$ from $S$ to $X$. For instance, the decision-maker might be asked to form preferences between the act that offers her/him an apple if Novak Djokovic wins the 2017 Australian Open and an orange otherwise, and the lottery that offers her/him an apple if Andy Murray wins the 2017 Australian Open and an orange otherwise. The standard result in that area (von Neumann-Morgenstern theorem) delivers a representation that identifies both the agent's probabilistic beliefs regarding the states and her/his attitude towards risk (her/his utility function).

There is a third widely used concept, usually called Anscombe-Aumann acts or horse races, which associates both objective lotteries and uncertain acts. Formally, given a prize space $X$, an Anscombe-Aumann act is a mapping $f:S \rightarrow \Delta(X)$ that associates an objective lottery to any state in $S$.

Notice that the definitions of objective risk and ambiguity are to some extent subjective. The fact that risk is called "objective" relies very much on the assumption that the decision-maker agrees with the underlying probability model. For instance, if you observe the outcome of a coin toss, you might believe that heads happen with an objective probability of 0.5. It is implicitly imposed in the theory that the decision-maker agrees with this statement.

Regarding ambiguity, you might yourself believe that the act "receive an apple if Novak Djokovic wins the 2017 Australian Open" is very ambiguous because you have no idea on how to compute a subjective probability for this event. That said, another decision-maker might very confidently believe that Djokovic has a 74% chance of winning the tournament, in which case she/he does not perceive this act as ambiguous at all. Ambiguity is a subjective notion, which is given by people's preferences and behavior and not by the choice situation itself.


How to intuit a necessary optimality conidtion involving growth rate of Lagrange multiplier in Ramsey growth model?

The multiplier is equal to marginal utility of consumption. In case of CRRA utility (i.e. $\lambda_t = c_t^{-\gamma}$), its rate of growth is then related to consumption growth:

$$
\frac{\dot \lambda}{\lambda} = \frac{\mathrm{d} \log \lambda}{\mathrm{d} t} = \frac{\mathrm{d} (-\gamma \log c)}{\mathrm{d} t} =  -\gamma \frac{\dot c}{c}
$$

The condition can be then rewritten as

$$
\frac{\dot c}{c} = \frac{1}{\gamma} \left[ (f'(k) - \delta - \xi) - \nu\right],
$$
so that the consumption growth rate is proportional to return on capital (less rate of time preference), which is pretty much the same as Euler equation in discrete time.


Regarding your first question, the space on which you can apply Lebesgue's theorem is $\mathbb{R}_{+}^{N}$. The relevant $\sigma$-algebra is the Borel $\sigma$-algebra and the integration corresponds to the Lebesgue measure. Formally, with your notation, the function $f_m$ is defined on vectors $x=(x_1,\cdots,x_N)$ by
\begin{equation*}
f_m(x_1,\cdots,x_N) = u(\sum_{n}{\alpha_n^m x_n})
\end{equation*}

Regarding your second question, it seems to me that constructing a counter-example would be difficult and tedious. Proofs of continuity are rarely done that way. You might ask your teacher what kind of argument he has in mind. A possibility that you might wish to explore is to go back to the definition of continuity. Basically, you want to show that:
\begin{equation*}
\forall \alpha, \forall \epsilon>0, \exists \delta>0 \text{ such that } \|\alpha-\beta\|<\delta \Rightarrow \|U(\alpha)-U(\beta)\|<\epsilon
\end{equation*}
You can try to replace $U(\alpha)$ and $U(\beta)$ by their expression and manipulate the resulting integral. But this essentially boils down to proving Lebesgue's convergence theorem again : this solution is therefore not going to be simpler, but it might help you to understand what is behind this result.


What are taxes for since law forbids printing of money?

"Why I don't hear nobody speaking about such idea?"

Because historical experience says it won't work.

By printing money instead of collecting taxes, what increases is the nominal disposable income. The "value of work" is certainly not increased. And the important question is, does this increased nominal income lead to higher consumption?

Consider a static society that produces food using labor $L_p$. This society wants also the government to offer some services, say law and order. So the government will also employ some of the available labor in the society, $L_g$ to offer these services. Now, policemen and judges must also eat to survive, so the food produced by the private sector must be enough also for them.

Say it is and part of it must be transferred to $L_g$ somehow. Initially, the government collects taxes for this purpose.
The private sector faces

$$P_c\cdot Q = P_c\cdot w_p\cdot L_p =P_c\cdot C_p + P_c\cdot T$$

where $P_c$ is the price level, $C_p$ is private-sector consumption in food units, $w_p$ is the wage per unit of labor in food units, and $T$ is taxes in food units The left-hand side is production the middle term is pre-tax nominal income of the private sector, the right-hand side is how this income is split between own consumption and taxes. The government looks at

$$P_c\cdot T=P_c\cdot w_g\cdot L_g $$

The government collects that tax, gives it to public servants as pay for their services and they in turn go to the private sector and purchase the food they need.

The economy uses money, and the current quantity of money printed is $M$, which is fully used to facilitate the transactions in a production-consumption cycle.

Then somebody comes up with the OP's idea: why collect taxes which nobody likes, and not print the money needed to pay the government employees?

So the government prints new money $M_n = P_c \cdot T$, stops collecting taxes and pays its employees with $M_n$.

Since taxes are not collected anymore, once production is finished and before consumption, the private sector has in its hands
a) Real quantity of food $C=C_p+T$
b) Money holdings $M = P_c \cdot C$

Government employees appear and show the new money quantity $M_n$ asking to buy the same quantity of food as before. The private sector complies. After the transaction, government employees have no money but they have food. On the other hand, the private sector has

a) Real food quantity $C - T=C_p$ as before
b) Increased money holdings equal to $M + M_n$

And the question is : how does the private sector experience its new situation? What does it mean for the private sector to have more money holdings than before? Well, the private sector accepted the new money brought by the government employees, so it must be the case that this new money is considered to bear the same value, the same credibility as the pre-existing money. This in turn means that all money bills are thought to have purchasing power. It follows that the private sector has in its hands more purchasing power than previously, $M+M_n$, but the same amount of output, $C - T$.

From here on, the standard story is that people will try to use $M+M_n$ to buy more food and consume more, but since available output is the same, $C-T$, this will increase prices (imagining a "bid war" to buy the output). This story is "standard", because it has been observed too many times around the world. Although governments have not eliminated taxes, they have on occasion printed additional money in an attempt to reduce the tax burden or increase government expenses without increasing taxes.

I will describe now two conceivable scenarios (alternatives or in sequel) that do not lead to increased prices:

Α) People do not attempt to increase their consumption with their newly increased purchasing power. They just stash away $M_n$.
Next year, the same cycle happens, the government prints an additional $M_n$ quantity of money to pay its employees, the private sector save this new money on top of its previous savings etc,: the price level stays the same, while the private sector has stashed $2M_n$ away. After $k$ years, the private sector will have saved $k\cdot M_n$ in paper money... To do what with it? Look at it and "feel rich" but never actually use it?

Well, historical experience tells us that people will tend to use this additional money, exactly because they accept that it carries purchasing power. It appears that people are not content to just look at stacks of money bills and rejoice. I mean, initially they do, but then they want to go out there and translate these stacks of money bills into increased consumption (the Scrooge McDuck type of person is, alas, a negligible minority).

B) People are not Scrooge McDucks and try to buy more than before with their increased money holdings. But sellers do not increase their prices, they even resist any attempts by the buyers to offer a higher amount of money,  and sell at the old prices until they run out of stock. Result: at the end of the day, people realize that their increased money holdings were not good enough to bring them higher consumption. But this will make them question and re-assess how valuable, after all, is this money. Which leads us to the exact same consequences with a price increase: all money will lose part of its carrying value in both cases.

People will enjoy the same consumption level as before, but they will feel happier now that they don't pay taxes. But since the government keeps printing new money year after year, prices will go to heaven. Again, historical experience says that people appear to dislike high inflation, probably because in the real world with millions of products the price system plays an important signalling role, which appears to be understood and be valued by the people. High inflation impairs this function, increasing the transactional costs of the people. So one should offset the gain in happiness from the feeling of not paying taxes, to the problems that will be created by high inflation... since recorded episodes of hyperinflation as well as high inflation were time and again ended by government action, it seems that people support such measures, which implies that even though they don't like paying taxes, they dislike more living in high inflation.

ADDENDUM
The OP seems particularly concerned about the tax collection costs, as he indicated in a comment to this answer, putting the matter "at the core" of its question. Here are estimates of the tax collection costs for many countries for three recent years, from OECD. OECD-30 mean value is at $1\%$ of revenue collected.

An excel file with the data series and the diagram can be downloaded here.

More info from OECD on the matter of tax collection efficiency, here.


Emerging Market dollar debt, FX swaps and exchange rate risks

However, the global bank presumably derives "utility" from dollar assets and hence will need to convert the EM currency proceeds that she receives from the FX swap into dollars using the future spot rate.

The bank already had the EM currency though, right? If they had invested it in the EM stock market or any other way they would have had to do switch the returns to dollars anyway, so this is not caused by FX swap.

If the bank did not have the EM currency before entering into the FX swap then it is buying the EM currency when the goal is actually future dollars that creates the exchange rate risk, not FX swap itself.


What can we know for sure among ambiguous impulse responses in RBC?

Yes and no.

Temporary Shock

It is correct, that a temporary increase in TFP has unclear effects on leisure:

The **income effect* will increase consumption and leisure
The **substitution effect* will decrease leisure.

On total, effects onto leisure are unclear. But, unless you have very specific preferences, consumption should increase. What are the cases in which it doesn't? Let $l$ denote leisure, $1-N$:

Violation of local nonsatiation: Households do not prefer more consumption at their current $(C,l)$ locus, so an income effect will not make them consume more (shut down income effect)
Strong and negative elasticity of substitution between $(C, N)$: People that work more, prefer more consumption. Hence, if substitution effect increases $N$, it also decreases $C$ (putting in $C$ into the substitution effect)

1. is generally not accepted, both empirically and theoretically (remember, we're here not talking about individual preferences, but under aggregation). 2. is more disputed, but elasticities of substitution found typically range between $0$ and small positive amounts, not strongly negative as required here.

So, to conclude, temporarily, $C$ responds positively to a positive TFP shock.

Permanent Shock

Yes, a temporary shock could end up however, given how we calibrate it. But here's the gist:

We calibrate models to data.

One standard calibration goal are the Kaldor facts, among which we will find:

Despite an strong rise in productivity during the 20th century, working hours per capita have remained constant.

This means that in calibrating an RBC model, we will chose specifications that give us exactly this result. A simple way of doing so is having an additive separate utility function of King-Rebelo-Plosser type (Note that Prescott in his original paper claimed that only log-log preferences would reproduce this independence, which is incorrect).

Temporary Shock revisited

To conclude, we calibrate the model such that an income and substitution effect on leisure cancel out exactly. The income effect of a permanent shock is bigger than the income effect of a temporary shock. Hence, if it holds that permanent increases in TFP do not affect working hours, it must be the case that temporary increases in TFP do positively affect working hours, since the substitution effect is the same, but the income effect is smaller.


$$y = \frac {a(x)}{b+a(x)} \implies \frac{\partial y}{\partial x} = \frac{\partial y}{\partial a}\cdot \frac{\partial a}{\partial x}$$

Since

$$\frac{\partial y}{\partial a} = \frac {b+a -a}{(b+a)^2} = \frac {b}{(b+a)^2} >0$$

because in your case "$b$"$>0$,

the sign of $\frac{\partial y}{\partial x}$ will be the same as the sign of $\frac{\partial a}{\partial x}$, which is what Romer is asserting.

$\theta >1$ would make $\frac{\partial a}{\partial x} <0$, so

$$\theta <1 \implies \frac{\partial a}{\partial x}>0\implies \frac{\partial y}{\partial x}>0$$


Properties on conditional demand correspondence from the textbook of Mas-Colell et al

Let $z_1$ and $z_2$ be $\geq 0$ and solution to
$$\min_z \{w^\top z\lvert f(z)\geq q\}$$
then clearly $f(z_1)\geq q$ and $f(z_2)\geq q$ and since $\{z\geq 0\lvert f(z)\geq z \}$ is convex it then follows that $z_3 := \lambda z_1 + (1-\lambda)z_2$ must satisfy the constraint $f(z_3)\geq q$.
Since $z_1$ and $z_2$ are both minimizers it cannot be the case that $w^\top z_1 \not = w^\top z_2$ rather it must be the case that $w^\top z_1 = w^\top z_2$. This implies that $w^\top z_3 = w^\top(\lambda z_1 + (1-\lambda)z_2) = \lambda w^\top z_1 + (1-\lambda) w^\top z_2 = w^\top z_1$ hence $z_3$ is a minimizer.
Since $z_3$ is minimizer and satisfy constraint it must be in $z(w,q)$ which therefore must be convex set.
If $f$ is quasi concave then by definition $\{z\geq 0 \lvert f(z) \geq q\}$ is convex and therefore using the above argument implies $z(w,q)$ is convex.


Macroeconomic formulas for global limited raw material economy exist?

I am not an expert on the domain. However, a popular extension in various domains of macroeconomics is to replace the standard Cobb-Douglas function

$$Y=AK^{\alpha}L^{1-\alpha}$$

by a production function that take a third resources (typically called land and noted $T$) which is supposed to be in a constant quantity. The production function hence becomes

$$Y=AK^{\alpha}L^{\beta}T^{1-\alpha-\beta}$$

With

$$\frac{\dot{A}}{A}=g $$
$$\frac{\dot{L}}{L}=n $$
$$ \dot{K}=sY-\delta K $$
$$\frac{\dot{T}}{T}=0 $$

It is -to my knowledge, typically used in growth theory, to account for malthusian theories, more here for instance, and some empirics Here, but i guess every textbook in growth has a note or more on it.

I also recall seeing it  in international trade, but less frequently.


Is there another name for marginal luxury?

The closest I can think of is Stone-Geary utility/preferences, where consumption is assumed to provide utility only above a specific level (the $\gamma_i$ parameter in the link), which is usually designated as the "subsistence level".

As for how much an economy is above this subsistence threshold, a way to roughly assess this is to calculate what would Total Income be if all people were at the poverty-line-income (each country determines its own poverty line). Then subtract this from the actual Total Income. What is left can be thought of as what the OP is looking for.

But calling it "luxury", implies that any consumption above the level guaranteeing mere survival is a luxury. Is it?


Testing if one regressor is a proxy for another

For the benefit of readers I note that, as the authors write at the bottom of p.2557

  "We define trust as the subjective probability individuals attribute to the possibility of being cheated."

So they actually measure "lack of trust" (the higher the value of the variable, the lower the trust), in order to see whether it co-varies positively with risk aversion, and so whether it can be considered a proxy for it.

But the correlation shown between "Trust" and Absolute Risk Aversion in the table you mention is $0.017$ which is not just "weak" -it is essentially non-existent, since with a finite number of (observational) data points, correlation even between random variables that are in reality uncorrelated, will not be exactly equal to zero (even with a sample size of $n=1,444$ as theirs).

This alone suffices for the argument that "Trust" cannot be considered a proxy for Risk Aversion, since in order for this to be a valid argument, we need the correlation between the two to be high.

The fact that Risk Aversion appears "statistically insignificant" means that Risk Aversion has no explanatory power on Diversification, but this cannot be attributed to the presence of Trust, since their correlation is so small.


Bertrand game with delivery risk and side payments

I am not going to give a solution to you question (as I am not able too). However, let me try to indicate the difficulty of obtaining one if you cannot find a closed form solution for $q_1$ and $q_2$ in the last stage of the game.
Let $\Pi(q_1, q_2, p_1, p_2, B)$ be the objective function that we want to maximize. Assuming we can totally differentiate this, we get:
$$
d\Pi = \Pi_{q_1} dq_1 + \Pi_{q_2} dq_2 + \Pi_{p_1} dp_1 + \Pi_{p_2} dp_2 + \Pi_B dB. \tag{1}
$$
Assume the first order conditions hold:
$$
\Pi_{q_1} = 0\\
\Pi_{q_2} = 0 \tag{foc-stage3}
$$
This then simplifies $(1)$ to:
$$
d\Pi = \Pi_{p_1} dp_1 + \Pi_{p_2} dp_2 + \Pi_B dB \tag{1.a}
$$
The problem here is that we can not drop the $dp_1$ and $dp_2$ terms as they are (in general) both going to be a function of $B$.
Assume that the first order conditions $\rm{(foc-stage3)}$ can be solved to give a solution $q_2(p_1, p_2, B)$ and $q_1(p_1, p_2, B)$. Then in stage 2 of the game, Firm 2 maximizes:
$$
p_2 q_2(p_1, p_2, B)
$$
Again, assume that the first order conditions hold:
$$
q_2 + p_2 \frac{\partial q_2}{\partial p_2} = 0 \tag{foc-stage2}
$$
We can totally differentiate this with respect to $p_1$ and $p_2$ and $B$ (notice that $q_2$ is a function of all three these variables). This gives:
$$
\left(2\frac{\partial q_2}{\partial p_2} + p_2 \frac{\partial^2 q_2}{\partial p_2\partial p_2}\right) dp_2 = -\left(\frac{\partial q_2}{\partial p_1} + p_2 \frac{\partial^2 q_2}{\partial p_2 \partial p_1}\right) dp_1 - \left(\frac{\partial q_2}{\partial B} + p_2 \frac{\partial^2 q_2}{\partial p_2 \partial B}\right) dB \tag{2}
$$
Let $p_2(p_1, B)$ be the optimal solution of firm 2 (by solving $({\rm foc-stage2})$. Then in stage 1, Firm 1 maximizes
$$
p_1 q_1(p_1, p_2(p_1, B), B)
$$
So:
$$
q_1 + p_1 \frac{\partial q_1}{\partial p_1} + p_1 \frac{\partial q_1}{\partial p_2} \frac{\partial p_2}{\partial p_1}= 0 \tag{foc-stage1}
$$
Assuming again that this can be totally differentiated with respect to $p_1$ and $B$, we obtain:
$$
\begin{align*}
\left(\frac{\partial q_1}{\partial p_1} + \frac{\partial q_1}{\partial p_1} + p_1 \frac{\partial^2 q_1}{\partial p_1 \partial p_1} + \frac{\partial^2 q_1}{\partial p_2\partial p_1}\frac{\partial p_2}{\partial p_1} + \frac{\partial q_1}{\partial p_2}\frac{\partial^2 p_2}{\partial p_1\partial p_1}\right) dp_1\\
=- \left(\frac{\partial q_1}{\partial B} + p_1 \frac{\partial^2 q}{\partial p_1 \partial B} + \frac{\partial^2 q_1}{\partial p_2 \partial B} \frac{\partial p_2}{\partial p_1} + \frac{\partial q_1}{\partial p_2} \frac{\partial^2 p_2}{\partial p_1 \partial B}\right) dB \tag{3}
\end{align*}
$$
Then we can use $(3)$ to substitute for $d p_1$ in $(2)$. Then $(2)$ and $(3)$ give $dp_1$ and $dp_2$ in terms of only  $dB$. This can be substituted into $(1.a)$ to give the total effect of $B$ on $\Pi$.


Liquidity trap and consumers' reaction to an increase in money supply

Good question. The answer depends on what exactly you mean by an increase in the money supply and how it is implemented.

Because standard monetary policy (ie. open market operations) is implemented through banks, it is functionally quite different from a pure infusion of cash to consumers, which is more the realm of fiscal policy, depending on how it is financed. Consider a central bank that buys bonds from commercial banks by crediting their accounts with reserves. Banks now have excess reserves which they are able to lend should they so choose. If interest rates are above zero, interest rates will fall enabling banks to offload the excess reserves. However, if interest rates are already at a nominal lower bound of zero, the price of loanable funds cannot fall to stimulate increased demand, and hence banks cannot offload the excess reserves.

It also depends on whether the increase in the money supply is temporary or permanent. A permanent increase in the money supply will permanently increase the price level, thereby inducing a one-period spike in inflation and a contraction in real interest rates ($i^R=i^N-\pi$). The quantity of loanable funds demanded will increase. If the increase in the money supply is temporary, though, this will not work. This is why economists view the implications of permanent versus temporary changes to the money supply very differently during zero bound episodes.

A few links that may be helpful:

http://www.voxeu.org/article/combatting-eurozone-deflation-qe-people
http://www.bruegel.org/nc/blog/detail/article/1527-permanent-qe-and-helicopter-money/


Getting rid of sanctions will boost the currency because it will allow Iran to export more oil, thus encouraging other countries to buy more rials. It also is a sign of stability, which further appreciates the currency.

However, there is no way for you as an individual investor to know when this will take effect or whether it has been already priced in. So I wouldn't buy rials.


Why does the Market Risk Premium increase in times of crisis?

Risk premium is the excess expected return of the market over that of the risk-free asset. (There is no "(expected) risk premium".)

It is a property of the expected returns of two assets. It is an indication of investor's willingness to bear risk. High risk premium means less risk-bearing capacity in the market, where market participants are in risk-off mode.

Risk premium is not a forecast of future returns of the risky asset.
In contrast, something like a P/E ratio (or Shiller's cyclically adjusted CA P/E) is designed to reflect future returns. High P/E means asset is over-priced and lower future returns.

Risk premium can go up when price of risky asset decreases and/or risk-free rate decreases. Both tend to occur when market perceives uncertainty and shifts from a risk-on to risk-off regime. For example, S&P500 price and US Treasury yield (risk-free rate) could go down as capital shift from risky equity to safe US government debt.

This is not restricted to equity markets. Credit spread in the corporate bond market is also a kind of risk premium, for credit risk instead of market risk. The higher perceived default risk of the issuer, the higher is the yield demanded by investors. Corporates whose debt has high yield (junk) rating are perceived to have higher credit risk than those who are low yield (investment grade).
During market uncertainty, credit spread widens.


Is the minimum of short run average cost equal to the minimum of long run average cost?

My textbook states that they are "tangential" to one another, but shouldn't they be equal

The cost curves are tangential (at least the ones the unnamed book seems to be talking about), and cost levels are equal given the optimal production level.


Calculating elasticity between terms in a regression equation

No that would not be correct definition of elasticity. First, mathematically in multivariate function elasticity is defined as follows:

$$ EL_x =\frac{ f_x '(x,y)}{f(x,y)}x$$ or in your case it would be: $$ \frac{ \partial \ln [w(age,Y,T,Mar)]}{\partial age} \frac{age}{\\ln [w(age,Y,T,Mar)]}$$

However, even if you would plug in the expressions in this formula you would get an elasticity of a log of wage with respect to age - I am not sure if thats what you actually want. You could exponentiate the original function but that would be messy as hell.

One way how to estimate the elasticity directly from the regression would be using log-log form for variables of interest. In your case (assuming there are no people with 0 age in your sample):

$$ln(w_i)=\beta_1+\beta_2 \ln (age) +\beta_4Y_i+\beta_5T_i+\beta_6Mar_i+\epsilon_i$$

In this specification $\beta_2$ will already give you an elasticity. This specification also already controls for possible non-linear effect of an age so no quadratic term is necessary. This being said logs are only rarely used for age, but you could estimate this specification as an additional robustness check and still keep the main specification with quadratic term.

Alternatively, already in your original equation which is in log-level form:

$$ln(w_i)=\beta_1+\beta_2age+\beta_3age_i^2+\beta_4Y_i+\beta_5T_i+\beta_6Mar_i+\epsilon_i$$

all the beta coefficients are already semi-elasticities (as also pointed out by @chan1142). So if you would only care about semi-elasticity you will already get your result from your original model as in your case semi elasticity of wages with respect to age is $\beta_2+2\beta_3age_i$.


Ambiguous definition of Substitution and Income effect?

First, you can make any definition you want as long as you use it consistently, but it's usually better to adopt the standard definition if you want to avoid confusion.   Similarly you could ask why we don't call a garage a dogpound and a dogpound a garage.  We perfectly well could, but it's better if we all use words the same way.

Second, assuming we agree to adopt the standard definition, your picture doesn't give enough information to determine a substitution and an income effect.  That depends on where you started and what changed, which you haven't told us.

IF you started at $A$ and the price dropped, then, according to the standard definition, the substitution effect is from $A$ to $B'$ and the income effect is from $B'$ to $B$.

If you started at $B$ and the price rose, then, according to the standard definition, the substitution effect is from $B$ to $A'$ and the income effect is from $A'$ to $A$.

Third, if you want to adopt your alternate definition, you will find that your definition and the standard definition agree arbitrarily closely for sufficiently small changes.  So if you imagine an infinitesimal price change, it doesn't matter which definition you adopt.  But for finite price changes, what matters is not that one definition is better than another, but that we find a definition we can all agree to use.


Need examples for confusion about rent-seeking. Who is the seeker and what is the rent?

Rent-seeking

Has no single agreed-upon definition (see the many definitions below).
Is a fancy synonym for "bad for society" (see especially Tullock, 1989 below).
Sometimes (but not always) refers specifically to lobbying (your example seems to be following this usage).

(In light of the above, my opinion/recommendation is that everyone should simply avoid using the term rent-seeking. If one simply means "bad for society", then one should just say that. If one is thinking of "lobbying", then one should just say that. However, I don't expect that my recommendation will be adopted because rent-seeking has the advantage of being an unfamiliar and technical-sounding term that helps writers sound more intelligent than they are.)

Krueger (1974) introduced the term rent-seeking:

In many market-oriented economies, government restrictions upon economic activity are pervasive facts of life. These restrictions give rise to rents of a variety of forms, and people often compete for the rents. Sometimes, such competition is perfectly legal. In other instances, rent seeking takes other forms, such as bribery, corruption, smuggling, and black markets.

Gordon Tullock credits himself with introducing the concept in 1967. Tullock himself gives various definitions of the term:
Tullock (1989):

My suggestion is that we use the term “rent seeking” (and I always have) solely for cases in which whatever is proposed has a negative social impact.

Contradicting himself, Tullock (1987):

‘Rent seeking’ refers to the investment of resources in efforts to create monopolies.

Definitions by some other writers:
Hyde & Borzutzky (2016):

political rent-seeking—the pursuit of unearned income streams by lobbying government for market privileges

Henderson (2008):

People are said to seek rents when they try to obtain benefits for themselves through the political arena. They typically do so by getting a subsidy for a good they produce or for being in a particular class of people, by getting a tariff on a good they produce, or by getting a special regulation that hampers their competitors. Elderly people, for example, often seek higher Social Security payments; steel producers often seek restrictions on imports of steel; and licensed electricians and doctors often lobby to keep regulations in place that restrict competition from unlicensed electricians or doctors.

Parson (2004):

the using up of real resources in an effort to secure the rights to economic rents that arise from government policies.

Bateman & McAdam (2003):

the act of trying to improve personal income at the expense of someone else, rather than by increased work or productivity

Khan (2000):

a person gets a rent if he or she earns an income higher than the minimum that person would have accepted, the minimum being usually defined as the income in his or her next-best opportunity ... Rents may take the form of higher rates of return in monopolies, the extra income from politically organized transfers such as subsidies, or the extra income which comes from owning scarce resources, whether natural resources or specialized knowledge. ...

Rent-seeking is the expenditure of resources and effort in creating, maintaining or transferring rents.

Black (1997):

Spending time and money not on the production of real goods and services, but rather on trying to get the government to change the rules so as to make one's business more profitable. This can take various forms, including seeking subsidies on the outputs or the inputs of a business. or persuading the government to change the rules so as to keep out competitors, tolerate or promote collusion between those already engaged in an activity, or make legally compulsory the use of professional services.

Rutherford (1995):

Monopolizing activity. This is much criticized as it produces a social waste rather than a social surplus.

Murphy, Schleifer, & Vishny (1993):

any redistributive activity that takes up resources

Pearce (1992):

The use of real resources in an attempt to appropriate a surplus in the form of a rent. An industry may use resources to lobby a government to impose a restriction such as a TARIFF on an imported good, so that the factors of productions in the industry earn payments in excess of their TRANSFER EARNINGS as a result of the higher market price for the good, which they produce. Consumers suffer two losses from rent seeking; the loss of CONSUMERS’ SURPLUS from the higher price and the loss of output from the resources devoted to rent-seeking. Resources may subsequently be devoted to rent protection. In international economics a related, though different, concept has been developed with the name, directly unproductive profit-seeking (DUP).

Bannock, Baxter, & Davis (1992):

Behaviour that improves the welfare of someone at the expense of the welfare of someone else. The most extreme example of rent-seeking behaviour is that of a protection racket, in which one group betters themselves without creating any welfare-enhancing output at all. Not all examples are criminal, however: the behaviour of labour or management when they put more effort into increasing their share of turnover, rather than into increasing the total volume of turnover, can be described as rent-seeking.


How does a budget deficit help grow a trade deficit?

It all boils down to the macroeconomic identity

$$ (G -T) \equiv (S – I) +(M-X), $$

where $X$ stands for exports, $M$ for imports, $G$ is government spending, $T$ are taxes, $S$ private savings and $I$ private investment.

Given $S-I$, the larger is the budget déficit, the larger the trade deficit is. $S-I$ can be thought of as the private sector excess financing ability. With this amount, it needs to finance both the budget deficit and pay for the difference between what it is able to sell abroad and what it buys abroad. If, as in the US, $S-I$ is not too large (due to low $S$), as soon as the government incurs in a budget deficit, a trade deficit will appear.

This does not tell us with which specific country the trade deficit will happen. However, it does tell us that those countries with high $S-I$ and low budget deficits (or even surpluses) need to be running trade surpluses, and this is the case of China (or Germany).


Are these preferences consistent with rationality?

You implicitly assume that the utility of $n$ units of $Y$ equals $n$ times the utility of 1 unit of $Y$, and there is no reason for that. For instance, if $Y$ is a fridge, the gain in utility from having 1 fridge compared to 0 is certainly larger than the gain in utility from having 2 fridges compared to 1. The formula $U("nY")=n*U("Y")$ that you use does not make sense: there are utility functions other than $U$ that represent the same preferences over $\{X,Y,Z\}$, and you will reach different conclusions if you make the same computation with these utilities.

The Von Neumann-Morgenstern theory does not make any assumption on how to value $n$ units of a good. A Von Neumann-Morgenstern representation attributes a utility level to all the consumption goods in the choice set. In your case, the representation would include objects such as $U("Y")$ (the utility of one unit of $Y$), $U("nY")$ (the utility of $n$ units of Y), $U("Z \text{ and  }(n-1)X")$ (the utility of $1$ unit of $Z$ and $n-1$ units of $X$), etc. There is nothing that allows you to relate these values with each other. The only behavioral restrictions of the Von Neumann-Morgenstern theory concern preferences over lotteries induced by these goods.

You need other tools to understand whether preferences over a finite set of commodities are rational. These tools are the Weak Axiom of Revealed Preferences and the Generalized Axiom of Revealed Preferences.

On a historical note, the reason why Von Neumann and Morgenstern developed a theory of preferences over lotteries is precisely because they wanted to identify a notion of "how much the good $Y$ is preferred to the good $X$". An obvious possibility was to observe preferences over pairs of goods such as: $n$ units of $Y$ versus $m$ units of $X$. But this is problematic for the reason stated above: nothing relates the utility of one unit of $Y$ to the utility of $n$ units of $Y$. Their trick was to consider preferences over lotteries that deliver $Y$ with a certain probability and $X$ with the complementary probability: the answer to the question "how much are you willing to pay to increase your chance of receiving $Y$ by 1%?" gives a good measure of "how much you prefer $Y$ to $X$". In contrast, the answer to the question "how many units of $X$ are you willing to sacrifice to receive one additional unit of $Y$?" does not make any meaningful sense.


Derive demand function $x(p,w)$ from utility function $u(x) = \min\{x_1, x_2\} + x_3$

Hint: Imagine that there are two coffee bars, $A$ and $B$. There is only one type of coffee in the world. My preferences are such that I always want 1 unit of sugar with 1 unit of coffee; if I consume units of coffee and sugar in the ratio $1:1$, additional units of only one of the two don't give me extra utility.

At coffee bar $A$ they sell coffee and sugar separately. That is: there is one counter for coffee, where coffee is sold for a price of $p_1$ per unit. There is another counter for sugar, where I can buy one unit of sugar for a price of $p_2$.

At coffee bar $B$ however, they only sell one unit of coffee with one unit of sugar in a bundle, the price of which is equal to $p_3$. They do not sell anything separately

At what prices do I prefer to buy my coffee with sugar at $A$, at what prices do I prefer to buy my coffee with sugar at $B$?


Is quantitative easing part of the ECB's and Fed's mandates?

It’s a means to an end, and to the mandate that these institutions imposed on themselves. We could also call it an “unofficial” mandate since politicians would likely put pressure on them in indirect ways to uphold their primary mandate. That primary mandate is a 2% inflation target. QE of course expands the money supply which can expand the velocity of money and keep prices relatively stable, as we already know. So QE is a means to that mandate end so that prices don’t fall sharply as we were beginning to see following the financial crisis. QE in it of itself is not a mandate.


Intermediate demand vs. Intermediate consumption

Intermediate consumption

total intermediate consumption is the difference between gross output (sales or receipts, which can include sales to final users in the economy or sales to other industries (intermediate inputs) and Value added GDP (total market value of all final goods and services produced).

This difference would be called intermediate inputs which are goods and services that are used in the production process of other goods and services and are not sold as final goods.

They are consumed or used up as inputs in production process by businesses, they are typically subcategorized as raw materials, energy, and purchased business services (which include various other operating expenses).

Intermediate demand

intermediate demand which is called Intrasector transactions or Intrasectoral transactions.  I will go with the name intrasectoral transactions (they all mean the same thing).  It is the value of production sold to other industries( i.e. NAICS 481, 482  ) within the same sector (NAICS 48).

To avoid double counting with the economic measurement within a sector intrasectoral transactions(the value of certain inputs produced within the same industry/sector) are removed.

The sectoral measures remain the only estimate that is useful at both the detailed industry level and the aggregate sector economy level.

When measuring the total economy one would use intermediate input in order to construct gross output  but when you are measuring sector level data and you are using the same intermediate inputs that are produced by industries within the same sector then those values need to be removed in order to prevent double counting.


What value do virtual traders of commodity add to the market?

There are various important social benefits of commodity trading even just on paper. The main ones are risk management, hedging, providing liquidity and secondary market, and price discovery.

For example, both producers and customers of commodities can use future market to hedge risks of adverse price movement. For example, in market for potatoes the potato store always faces risk that the price of potatoes will increase and conversely the potato farmer faces risk that the price will drop. This could be for example due to unpredictable weather patterns which can cause supply to vary from year to year.

One way how to avoid sudden price swings to make a future contract between farmer and seller of potatoes where the farmer agrees to deliver seller certain quantity of potatoes at per-specified price. However, pricing such future contract requires lot of skill and speculation that most farmers or potato store owners might not have so naturally through division of labor this part of work is outsourced to commodity trader. Esentially - your question is in its spirit the same question as what store adds to the market and why sellers exist and why commodity producers just dont sell directly to consumers and again answer would be division of labor as farmer might not necessary be good seller.

Moreover, most traders dont trade just in one commodity but in multiple so it is easier for them to manage risk associated with the trading the commodities, and it is always socially optimal that the party which has the least costs of managing risks takes the risk from parties which have the higher costs of managing risks.

Furthermore, the traders can also provide liquidity to the market. Again a bank or big financial company will certainly have a comparative advantage to the farmer or store when it comes to providing liquidity to the market and this function is connected to the above mentioned function of hedging and risk management so it makes sense that instead of farmers and stores keeping extra liquidity they outsource this task to the commodity trader.

Having a secondary market where you can sell your commodity is another good thing. For example imagine that potato farmer entered into a contract to sell a certain quantity of potatoes at a certain price. However, later one of the parties wants to back out. They can do so by selling the contract on the commodities exchange. Since the exchange is the counterparty to everybody, the farmer may not even know that the buyer has been changed and vice versa (also note that in some textbooks this is also mentioned as a part of market providing liquidity, but in others the provision of liquidity was used in more narrow sense of providing leverage to the market participants).

Price discovery is another important reason why especially spot markets exists. Especially for very highly perishable commodities it might be very hard for farmer or store to know exactly how to price the goods and hence even before advent of modern finance sent some representatives to market to haggle the price for them in one place so they dont need to leave their work to do that (That farmers do this is even mentioned in Plato's republic - so really spot markets existed even in ancient times albeit in less sophisticated form).


Quoting from the question:

  $F(aK,aL)$ : simultaneous and proportional inputs increase
  $aF(K,L)$ : output increase

But $F(aK,aL)$ is an output increase, that is why it starts with $F$. It is the output increase due to the simultaneous and proportional increase of inputs. At the same time, $aF(K,L)$ is a hypothetical output, the output were this if it was increased by the same proportion as the inputs. So
\begin{align*}
F(aK,aL) & > aF(K,L)
\end{align*}
shows that if you increase the inputs simultaneously and proportionally, the increase in output is larger than proportional.


What will happen to a poor country’s economy if it replaces its entire banking system with the Islamic banking system?

I see a lot of research on Islamic banking, but I don't see a lot of studies that speculate on the question framed here. You might be interested in this study where the author claims:

  It would be better if Islamic banks had the opportunity to work as a sole system in an economy. That would provide Islamic banking system to fully utilize its potentials. Studies show that Islamic banks can not operate with its full efficiency level if it operates under a conventional banking framework, their efficiency goes down in a number of dimensions. The deterioration is not because of Islamic bank’s own mechanical deficiencies.  Rather  it  is  the  efficiency-blunt  operations  of  the  conventional  banking  system  that  puts  obstructions to efficient operation of Islamic banks.

However, it doesn't look like these assertions have been proven. For example here is a study that suggests that Islamic finance works best on a small scale as larger-scale Islamic banks seem to have unique instability issues.


Can you take the moving average of quarterly data of an explanatory variable in a regression to smoothen noise and get more accurate coefficients?

Let's say $D_t$ is the stock of debt at time $t$.

The first difference is $D_t - D_{t-1}$.

The second difference is $(D_t - D_{t-1}) - (D_{t-1}-D_{t-2}) = D_t - 2D_{t-1} + D_{t-2}$.

The [4-period] moving average is then given by

\begin{align*}
&= \frac 1 4 (D_t-2D_{t-1}+D_{t-2}+D_{t-1}-2D_{t-2}+D_{t-3}\\
&\qquad+D_{t-2}-2D_{t-3}+D_{t-4}+D_{t-3}-2D_{t-4}+D_{t-5})\\
&= \frac 1 4 (D_t - D_{t-1} - D_{t-4} + D_{t-5})\\
&= \frac 1 4 ((D_t - D_{t-4}) - (D_{t-1} - D_{t-5}))
\end{align*}

Which would be the yearly increase in the stock of debt this period compared to the same thing a quarter ago. Only you could tell if this is what you want in your regression. Something to consider is only to use the first difference: $D_t - D_{t-4}$.


House allocation: Data notation correction with Top Trading Cycles (TTC) example

I think this is a typo in the paper. As far as I see, the houses $\{h_5, h_6, h_7\}$ are not occupied, so it should be that $H_V = \{h_5, h_6, h_7\}$.


Levelized cost of electricity, to calculate the absolute costs?

In general, an LCoE forecast for a year, is the unit cost of electricity for generators that come online in that particular year.

Whereas, for a given year's (e.g. 2030) forecast of the cost of electricity, the average unit cost of electricity is a volume-weighted arithmetic average of the levelised costs of all the generators that have been built previously, that are operating in the forecast year. Typically, those generators will have been built in the previous 20 years.

That's not the only challenge:

you will need to check the actual definition of the LCoE data that you have. It might be the unit cost for generators that get their Final Investment Decision in that year (and so will come online sometime later, probably in the range of 1-15 years)
usually (but not always), changes to the underlying cost will affect total demand, so you've got a two-way feedback mechanism between quantity and price.
that rate at which things are deployed, and the consistency of speed with which they are deployed, affects their LCoE. There are chokepoints in various supply chains, there's exogenous and endogenous cost reductions from learning by doing, and so on.
merely averaging all the individualised costs of energy, omits the costs that are borne by the system itself: the cost of transmission and distribution networks, the costs of storage, and the costs of mechanisms to keep demand and supply in perfect balance every 20 milliseconds (or so) of the year.

So you can take historic LCoE, and multiply it by the quantity of electricity demanded in a year, to calculate total cost of electricity consumed in that year, subject to the above caveats. Note that that will give you a different answer to the amount of capital investment needed.


Is it more costly to raise cattle for beef or raise fish in aquacultures?

Cost of production is somewhat difficult to pin down, since much of that information is private.  However, consumer costs are widely available: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2954450/
I have managed to find a comparison of beef, pork, dairy, poultry, and eggs to plants in terms of protein, and broadly beef is the most inefficient of those (See Figure 3): https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5899434/
This suggests (but does not prove) beef is probably the more inefficient of the two.


Axiomatic Bargaining: Kalai-Smorodinsky and Nash solutions coincide

Another sufficient condition for the two solutions to coincide, which is not necessarily the same as symmetry, is that the feasible set $S$ be rectangular. That is,
\begin{equation}
S=\text{convex hull}\{(d_1,d_2),(d_1,\overline x_2),(\overline x_1,d_2),(\overline x_1,\overline x_2)\},
\end{equation}
where $d_i$ is $i$'s reservation utility and $\overline x_i$ is $i$'s best possible utility from the bargaining. Both Nash and KS solutions are at $(\overline x_1,\overline x_2)$.

Using the property that the KS solution can be obtained at the intersection of a ray from the disagreement point $d$ and the (northeast) boundary of the feasible set $S$, we can derive the following necessary condition for KS and Nash solutions to match: let $(x_1^N,x_2^N)$ denote the Nash solution to bargaining problem $(S,d)$; the KS solution $(x_1^{KS},x_2^{KS})=(x_1^N,x_2^N)$ only if
\begin{equation}
\frac{x_2^N-d_2}{x_1^N-d_1}=\frac{\overline x_2-d_2}{\overline x_1-d_1}.
\end{equation}


What are examples of inversely (negatively) correlated stock prices in economics?

Have you tried Google? By copying a part of your question into Google I was able to find several examples. The best one may be this site which lists 20 positively and negatively correlated stocks for any stock you pick:
http://www.market-topology.com/correlation/


Why don't suppliers increase price after a positive demand shock?

This is because of competition. If you would keep the quantity supplied low someone else would produce as they can both earn profit and increase their market share by doing so.
This being said there are situations where producers actually keep the quantity limited so they can increase the price. We call this kind of situation collusion (or cartel).
However, collusion is forbidden by laws in virtually all countries around the world so it is not easy to achieve. Firms can still try to collude tacitly, that is they try to collude without making specific deal which would be unlawful. However, tacit collusion is very hard to sustain and becomes harder the more firms there are in the market. This is because tacit collusion requires a lot of unspoken coordination among the firms (and it has to be unspoken as in many countries even discussing collusion is illegal and heavily penalized. Furthermore, many laws incentivize firms to ‘snitch’ as usually firms that snitch are not punished yet their colluding competitors are). You can see some industrial organization textbook that showcases models in which that can happen - for example Industrial Organization: Markets and Strategies by Belleflamme and Peitz or the Theory of Industrial Organization by Tirole. This is also touched upon in most introductory undergraduate economics textbooks when they discuss duopolies, so you might just try to look for duopoly chapter in book/lecture notes you follow, but they often don’t do the subject enough justice.
If you are however, studying economics 101 competition will be for most part assumed to be perfect and under perfect competition it’s virtually impossible to sustain any collusion agreement that is not enforceable in law (and again all such agreements are illegal in almost all countries).


Price of Anarchy when the cost is to society and not to the selfish agents

If there is a free market for the product the agent who gets the resource becomes a monopolist. "Produce more efficiently"  is taken to mean "produce every quantity of output for lower cost". That is its usual meaning in economics. Thus if someone different from the most efficient producer by any means gets hold of the resource the best thing that someone could do for himself, is to sell this resource to the most efficient producer, who is going to make more profit than the initial owner could and hence is able to make the initial owner an offer he cannot reject. Hence in the end the product gets produced by the most efficient technology available and the prices for the product are the same in both worlds. The consumers are exactly as well off. There is only a shift of profits between the entrepreneurs.

If the resource is auctioned off to the highest bidder, the most efficient producer will win the auction, as he can outbid all others, because he is going to make the largest profits among all.


Which capital accumulation is right? $K_t = (1-\delta)K_{t-1} +I_t$ or $K_t = (1-\delta)K_{t-1}+I_{t-1}$?

Both are economically sound. The notation is just a question of convention. The reason behind the ambiguity is that capital is a stock and investment is a flow variable. You are looking at capital in two different instants. Investment happens during the time between the two instants and its index is either the starting or the ending instant.


The multiplication factors are there so that they can be charted and be seen on the same axis as the St. Louis Adjusted Monetary Base.

The factor of 47 means that the closeness of the match between that and the Wilshire 5000 in 2000-2015 is easiest to see, by overlaying the curves in that period.


Wouldn't abolition of cash give rise to a substitute currency?

"Cash" is an emergent phenomenon of human economic organization. It exists for lots of reasons, as a provider of economic anonymity, a low transaction cost solution to the double-coincidence of wants, a portable medium of exchange, and a tool of economic accessibility for all including those in the informal economy, foreigners, the unbanked, and those with poor credit, among others.

A country can make it difficult to use cash in the interest of thwarting anonymity and convenience, usually to prevent money laundering and tax evasion. They can do so by taxing the use of cash, failing to enforce contracts calling for cash, refusing to make more, even criminalizing the possession of cash.

But cash is unlikely to die easily. We know from the experience with drug and alcohol prohibition as well as the international ivory and rhino horn trades that while the regulation can can certainly influence prices and thereby quantities demanded, it is awfully difficult to destroy a market for a valuable product. For a more specific example, possession of money is forbidden to US convicts. Nevertheless, they have developed internal monetary economies based around canned fish, shelf-stable pastries, as well as cigarettes and stamps. Cigarettes also emerged as money in WWII P.O.W. camps, showing that even the Nazis couldn't keep a monetary economy from forming.

Sweden in particular has a significant additional obstacle to keeping currency out of their economy. They share boarders with Norway and Finland, each with their own currency. Go to practically any boarder region or tourist spot around the world and you'll see tellers happy to take dollars, euros, or yen in addition to the local currency. It is especially difficult to ditch cash when other sources of cash are available.


Original data Altman used for his z-score formula

The original paper (of Altman) is

Altman, E. I. (1968). Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. The journal of finance, 23(4), 589-609.

We read (p. 593)

  III. DEVELOPMENT OF THE MODEL

  "Sample Selection. The initial sample is composed of sixty-six
  corporations with thirty-three firms in each of the two groups. The
  bankrupt group (Group 1) are manufacturers that filed a bankruptcy
  petition under Chapter X of the National Bankruptcy Act during the
  period 1946-1965. (...) Recognizing that this group is not completely
  homogeneous, due to industry and size differences, a careful selection
  of non-bankrupt firms was attempted. Group 2 consisted of a paired
  sample of manufacturing firms chosen on a stratified random basis.
  (...) Firms in Group 2 were still in existence in 1966. Also, the data
  collected are from the same years as those compiled for the bankrupt
  firms. For the initial sample test, the data are derived from
  financial statements one reporting period prior to bankruptcy.
  (footnote 18: The data was derived from Moody's Industrial Manuals and
  selected Annual Reports. The average lead time of the financial
  statements was approximately seven and one-half months prior to
  bankruptcy.)

The paper does not contain, even in print form, the actual data sample, so BKay's suggestion to contact the authors is your best shot.


Profit maximization with Cobb-Douglas function

Should I solve for $L^∗$ by separating $K^∗$ from the equation and plugging into $pMP_{L}$

Yep, that's about it.

  Wouldn't this yield a very complicated solution?

Somewhat. The math is available at many places, like section 4 here. But you can surely do it yourself!


Difference between representation of LM function in different textbooks

The function $L(y,r)$ is a placeholder for any rule that says

  "give me a $y$ and and an $r$, and I will spit-out a value for $L$."

For example, we could have

$L(y,r)\equiv y+r$;
$L(y,r)\equiv y + \cos(y)+\sin(r)$;
$L(y,r)\equiv y \ln (r)$;
$L(y,r)\equiv yr$;
etc.

$L(r)+k(y)$ imposes an additional restriction. We con no longer pick any rule. Now we are only allowed to use rules where any terms involving $r$ are separated from those involving $y$ by a $+$. we call such function "additively separable".

Thus, in the Branson formulation encompasses rules such as 1 and 2 above, but not 3 and 4 (because the latter two are not additively separable).

This additional restriction means the Blanchard function is a more general formulation than that in Branson.

If we have a result that we know to be true for any function $f(x_1,\ldots,x_n)$ then we automatically know it must be true for additively separable functions of the form $f_1(x_1)+\ldots+f_n(x_n)$ (because the additively separable form is just a less general subcase of the more general form $f(\cdot)$).

But the same is not true in reverse. A result that we know holds for additively separable functions $f_1(x_1)+\ldots+f_n(x_n)$ need not hold for more general functions $f(x_1,\ldots,x_n)$.


Why is marginal cost not the cost of producing the last unit?

The full quote from the cited reference (p. 181) is

  Marginal cost is not the cost of producing the "last" unit of output. The cost of producing the last unit of output is the same as the cost of producing the first or any other unit of output and is, in fact, the average cost of output. Marginal cost (in the finite sense) is the increase (or decrease) in cost resulting from the production of an extra increment of output, which is not the same thing as the "cost of the last unit."

I think the boldfaced part is the important part that the Wikipedia editor left out.

What you have in mind is correct. Suppose we can produce 4 hats for the cost of \$8, and 5 hats for the cost of \$15, then the authors are saying that if you choose to produce 5 hats instead of 4, then it would cost you \$7 more. That's the marginal cost.

They argue that it's not right to say that, having chosen to produce 5 hats, that the cost of producing the 5th hat is \$7.

To me, this is a very minor distinction.


What is causing the difference between USA financial indexes and perceived economy?

Stock market prices depend on investors view on what the value of company is which in turn depends on what is its expected future profitability. For example, under a simple Gordon growth model the stocks would be valued using (See Miskin & Eakins Financial Markets and Instiutions pp 347):
$$ P_0 = \frac{D_1}{k-g}$$
where $D$ is the dividend, $k$ required rate of return and $g$ expected growth of dividend (and dividends directly depend on profitability). The model above is a simple model and there are more nuanced ones but the point from most models is that stock prices ultimately depend on firm's expected profitability.
Typically recession lower firm profitability but rarely shocks affect all firms the same way. For example, take Covid-19. Research shows that covid-19 primarily negatively affected industries like hospitality, transport, energy and so on (e.g. see Alonso et al (2020) or Shen (2020)). On the other hand many IT/technology firms benefited from the pandemic and lockdowns. More importantly, research shows that small and medium size businesses were affected mainly small and medium size businesses (e.g. see Dua et al (2020); Dua et al (2020);  Alekseev et al (2020))*
The NASDAQ100 and S&P500 are tracking equities of 100 largest and 500 largest companies listed on the stock exchange. Consequently, shocks like covid-19 that hurt profitability of predominantly small and medium size companies that are often not even listed on stock exchanges (not even mentioning getting all the way to status 100 or 500 biggest firms respectively) will not affect stock prices as much as shocks that directly large companies, such as let's say .com bubble which mainly affected large technological businesses and correspondingly the S&P500 fell much more than during covid-19 crisis. This does not mean that covid-19 crisis was less worse for the economy than .com crisis since most firms in almost every country are small and medium size businesses. For example, according to the Eurostat in EU in 2015:

In 2015, enterprises employing fewer than 250 persons represented 99 % of all enterprises in the EU.

In the US according to SBE Council working with US 2016 census data showed that:

there were 5.6 million employer firms in the United States in 2016. ... Firms with fewer than 100 workers accounted for 98.2 percent. Firms with fewer than 20 workers made up 89.0 percent.

I think it goes without saying that typically (there might be exceptions) firms with less than 20 or even less than 100 workers simply wont make it to the S&P 500 or NASDAQ 100 or most of them might not even be listed.  Consequently, stock market is not really 'the economy', it is just a part of it.  Stock market is not a representative sample of firms, but sample of the large firms that decide to go public (e.g. small pop and mom shops would never get listed at stock exchanges because a) they do not need to raise that much capital; b) getting listed is expensive and publicly listed firms have higher regulatory burden so it does not make sense for most small firms financially). What even more, indexes such as S&P 500 and NASDAQ 100 look only at the largest firms from already sample of large firms (e.g. really the cream of the crop of the top crop). Hence these indexes are primarily affected by shocks to the profitability and expectations of the profitability of the listed large firms and there might be many macroeconomic shocks that can severely impact the economy as a whole but not as much largest firms or sometimes a macroeconomic shock that is negative for small and medium size business can be positive for large business (e.g. Covid-19 effect on restaurants vs technology giants).
Additionally,  some economists as Paul Krugman** argued that stock market is getting more disconnected from the real economy because of savings glut and because wealthy individuals have not that many other good alternatives to invest in since bonds offer little return due to low interest rates.
Lastly, Fed's job is not to take care of the stock market prices. Fed pursues loose monetary policy (e.g. low interest rates, QE and so on) because that is the textbook 101 advice for central bank to purse in recessions (e.g. see Mankiw Macroeconomics). That this leads to asset price inflation is more of an side effect of the 'cure' for the real economy not the primary intent of Fed.

 * Although an important caveat is that most of this research is preliminary as Covid-19 is recent event.
 ** However, this is opinion piece, consequently even if it is made by Nobel Prize winning economist please take it with extra grain of salt.


What are the implication of state-run oil production vs. private production from leases?

Like many international markets, the crude oil market is a mix of private and public participants.

From most perspectives, it really doesn't matter whether oil production is state-run or run by private leases. Both, ultimately, are combinations of public and private forces - just different blends. Both are trying to optimise the outcomes for their owners. They're all suppliers that respond to demand.

They're all largely price takers; the larger they are, the more they contribute to being a price maker - same as any other market. Nominally, Saudi Arabia as the main force in OPEC carries the market force of the entire cartel: but as Saudi is by far the largest producer in OPEC, the supply levels of the other cartel members don't carry anywhere near as much weight, and there is some evidence that they haven't always implemented fully proportional cuts when Saudi has called for cartel-wide production cuts.

So Saudi Aramco is just another market player, albeit the single largest one, and has some additional leverage via other OPEC members.

Many myths surround Saudi's oil policy; many widely-held assumptions are wrong. The most accurate thing I've heard said of it, credited to various folks is: the people who talk about Saudi intentions, know nothing; and the people who do know the intentions, say nothing.

Footnoe: the products are not completely fungible: different crudes have different chemical compositions (sweet crude has low sulphur content, sour crude has high)


I am not aware of any definition, and believe me, lots of people would like to have one - mostly for regulatory purposes. There are attempts in many countries to regulate digital platforms, but without good definition, you can't create a good legal regulation.

Any definition I have seen so far is a list of features, that is, however, not exhaustive - it would be a good definition if we could say, "it is a platform if it has those 8 features," but then you will find one that doesn't have all of them, and you find something that has all of them, but you won't call it a platform...


What are production limiting programmes?

Subsidies are often primarily aimed at increasing the income of the agricultural sector

When subsidies lead to increases in production, this can cause the market price to fall with negative effects on agricultural incomes, with much of the benefit flowing through to consumers rather than farmers.  This then leads to calls for further subsidies

To avoid a spiral of increasing subsidies, governments will often move to a stage which restricts production, and one possibility is to tie the subsidies to given levels of production (often based on previous levels of production or on fixed assumptions about areas and yields), either not paying subsidies on production above some stated level or, in more extreme cases, withdrawing subsidies if production exceeds a quota

The aim of such production-limiting subsidies is to maintain subsidies at a level which is sustainable both in terms of market prices and government expenditure.  It is in fact a double distortion of the domestic market, both through the subsidies and through the production-limiting elements, but was deemed to be less distorting of international trade than unlimited production-based subsidies


Two Endogenous Variables and Two Instruments

I don't think you need anything like a 3SLS. Just run the first stage for the two endogenous variables separately:

$$\tilde{X}=\alpha_1+\beta_1X'+\beta_2Y'+\beta_3C+\epsilon$$
$$\tilde{Y}=\kappa_1+\gamma_1X'+\gamma_2Y'+\gamma_3C+\delta$$

Then run a regression of Z on the predicted value of $\tilde{X}$, $\tilde{Y}$ as and $C$. If the assumptions are all satisfied, this should give a consistent estimate of the coefficients.

I also found this thread online. It might be helpful.
https://www.stata.com/statalist/archive/2012-08/msg01238.html


What is Q in the quantitative theory of money?

Assume there are two agents in an economy, $A$ and $B$, (and some costless transaction mechanism). Per time period, agent $A$ produces alone quantity of intermediate good $q_A$. Agent $B$, thorugh a company where it is shareholder, buys this quantity, the company inputs also some other intermediate good , say $q_B$, and the two together through a production function result in a final good quantity $(q_A,q_B) \to Q$. $Q$ is then bought by agent $A$ and agent $B$ as consumers, at price $P$. $PQ > p_Aq_A$ since it embodies a larger amount of productive resources (assume no inflation, which is not essential here).

Now, for the first transaction to take place, the buying of intermediate good $q_A$, we need quantity of money $M_A=p_Aq_A$. This quantity of money is now held by agent $A$. How much money we want, in order to facilitate also the purchasing by both consumers of the final good?

It will depend on what kind of transactions we envisage to the end of the cycle.  Agent $A$ already has $M_A = p_Aq_A$ with which he can buy part of the final good. If $B$ buys the good pior to any other interaction with $A$, then $B$ must also hold quantity of money equal to $M_B = PQ - p_Aq_A$.  So the total quantity of money needed, in this scenario, is $PQ$, and the intermediate goods do not enter the final equation of the quantitative theory of money.

Assume now that the only quantity of money available is $p_Aq_A$. What can we do to facilitate all transactions? Well, agent $A$ can buy first a part of the final good from company belonging to $B$, then the company will give the money to $B$ as dividends, and then $B$ will give the money to the company as consumer buying the rest of the final product. But in order for the transactions to happen sequentially as described, and be completed within the same time period as previously, money must "move faster". So in such a situation, the velocity of money will be higher than unity.

$$PQ = \left(\frac{PQ}{p_Aq_A}\right) \cdot M$$

where the term in the parenthesis is $\left(\frac{PQ}{p_Aq_A}\right)\equiv \mathbf v$, the velocity of money. But again, the "product" side of the equation will be $PQ$. No counting of intermediate goods.


An alternative approximating approach you could use as as check might be to say there are $X$ job offers in total and $u$ unemployed.

So the probability that an individual does not get a particular job offer is $\left(1-\dfrac{1}{u}\right)$ and so the probability the individual does not get any job offer is $\left(1-\dfrac{1}{u}\right)^X$ which is $\left(\left(1-\dfrac{1}{u}\right)^u\right)^{x/u} \approx e^{-X/u}$ much as one might expect from a Poisson distribution. You have an error in the sign of $e^{-X/u}$.

That makes the probability the individual does get a job offer about $1-e^{-X/u}$. It makes the expected number of individuals receiving job offers about $u(1-e^{-X/u})$.

The probability that $y$ individuals get job offers (rather than $x$ as you have already used $X$) is then roughly the binomial $\displaystyle {y \choose u}\left(1-e^{-X/u}\right)^y e^{-(u-y)X/u}$ and or if you wanted a Poisson approximation to this then about $\dfrac{e^{-u\left(1-e^{-X/u}\right)}u^y\left(1-e^{-X/u}\right)^y}{y!}$ and you could apply further approximations to this but it would not get much tidier


Is it correct to say that wealth is "destroyed" when an asset bubble bursts?

We need to distinguish two things here: the intrinsic value of the assets, and their fair value, or market value.

No matter how much the price of an asset changes on financial markets (say, stock exchanges), its intrincic value stays the same, all other things equals. Typically, you could use the cost of the asset to assess its intrinsic value: for a house, the price of its construction. For a share, the book value per share of the underlying firm. For bonds, the nominal (or face value) amount.

But assessing cost is sometimes tricky. The book value might incorporate things that are valued at fair value. For real estate, the price of the land is a large part of the total cost, and it changes all the time. For bonds, you could incorporate interests and discount them, but you need to choose a discount rate.

Thus we typically assess wealth using fair value. It's a concept precisely defined by IFRS (IFRS 13 - Fair value measurement). Basically, the fair value of an asset is the price at which you could sell it under normal market conditions ('at arm's length'). It is true 'value', as you could trade the asset against that amount of cash.

To come back to your question: wealth is measured at fair value, or market value. Therefore, the $34.4bn were indeed destroyed, or disappeared, even though the intrinsic value may have been constant. Your second suggestion is correct.


If PPP Exchange rate is 0.291, what does that mean?

S =  P1 / P2  ; P1=Price at country A's currency for good X,

P2=Price at country B for the same good X and S is equal to the exchange rate of currency #1 and currency #2.

This formula measures the difference between the two countries’ rates of inflation and the cost of commodities will drive changes in the exchange rate between the two countries.

this relates the change in two countries’ inflation rates to the change in their exchange rate.

for every 1.00 dollar spent on the good X in country B, it takes  .29 cents to obtain the same good X in in country A buying it with country A's currency.

So if S doesn't match the exchange rate between the two countries for good X then one of the country's currency's is devalued.

As countries with higher levels of inflation are likely to end up with their currencies devalued

https://www.investopedia.com/updates/purchasing-power-parity-ppp/


Alternative to Intermediate-macroeconomics book by Mankiw

I would highly recommend Blanchard's Macroeconomics, especially for the short and medium run.
Another book to look at, especially for the long run and growth is Burda and Wyplosz (2012)


What happens to aggregate C, K, and Y when TFP increases permanently?

Take a look at the dynamics of the capital:
$k_{t+1}=sA_ty_t+(1-\delta-n)k_t$. A sudden positive shock to TFP in period $t$ increases the capital stock of the next period $k_{t+1}$. So, there is no contemporaneous effect on $k$, convergence to the new steady state will be gradual.

The other variables have contemporaneous relationship with TFP.

EDIT: A google search for a good explanation lead me to lecture notes from MIT which you can find here. Start reading slide # 62+. You will see the graphical explanation as well.


Repeated game with stage game has more than 1 NE

Though a bit unconventional to assume an infinitely repeated game without discounting, the short answer to your question is: yes. Any sequence of stage game Nash Equilibria is supportable as a SPNE. See, for example slide 11 of this deck, or these notes from Levin, starting on page 6. This is more directly addressed by the definition of SPNEs, found on page 137, Definition 132 of these notes. Or where those Columbia notes suggest on page 142: "For example if there are multiple Nash equilibria in any of the final subgames then there are multiple sub game perfect Nash equilibria in the game itself"

Though you should note that, given your problem's construction (with no discounting), the set of SPNEs grows substantially, as you can see.


Economics is a science and hence the objective of economics is the same as of any science just applied to the specific discipline. Thus this objective depends on how you want to define what an objective of science is and on the demarcation of what economics is.
On the objective of science:
There are many different approaches to the philosophy  of science and some of them differ in their view of what the objective of science is. Unfortunately, SE format is not suited to broad answers so it is beyond the scope of this answer to provide an overview of all possible philosophical views on what the objective/purpose of science is (although I encourage you to have a look at some textbook on the philosophy of science such as Rosenberg, Philosophy of Science: A Contemporary Introduction; Okasha Philosophy of Science etc.)  but it is generally agreed by the philosophers of science that the main purpose of science is to explain or understand and predict various phenomena (see Purtill 1970).
Under this predominant view, in the same way, as it is the objective of physics to explain gravity, not to use it to improve people's lives, it is also objective of economics to explain various economic phenomena (trade patterns let's say) not necessary to help anyone by explaining them (however, people are not just scientists they are also moral beings, and as such many scientists might be motivated by helping others, but these are personal goals not goals of the science itself, and also as mentioned above there might be philosophers of science that would disagree with this view - but this view is quite general).
Demarcation of economics:
Currently the mainstream definition of what economics is the one that was provided by Robbins (1935),

the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses.

Again, this is not the only possible way of defining the field of economics but it is the mainstream one that you would find in almost any undergraduate general economics textbook (for further discussion on this issue consult some philosophy of economics textbooks such as Ross, Philosophy of Economics or Hausman The Philosophy of Economics: An Anthology).
Conclusion:
Consequently combining the mainstream view of what science and economics is the goal of economics is to explain/understand and predict economic phenomena.  Where economic phenomena should be understood as phenomena pertaining to human behavior in situation when humans have some ends and scarce means with alternative uses to pursue those ends.
Hence, in reference to the topics you mentioned above, the goal of economics would not be to

define laws so that basic human needs are fulfilled and people do not die of hunger due to lack of money

but to study how people fulfill their needs, what choices they make, how they can best optimize their resources in various situations - but it is not the role of economics to define laws that ensure human needs are satisfied. 'Law' in economics does not even have necessarily the same meaning as in common speech, rather in economics we try to formulate scientific laws.  You can use the knowledge of economics to produce famine as well as to help reduce impact of one.
Similarly, it is not the role/purpose of economics to:

Find out laws so that inequality does not increase only

but rather to study what drives inequality, how different policies can affect inequality etc. and then it is up to policymakers to decide if inequality is a problem and how to solve it.


Effect of specific tax on demand and supply

The answer has to do with the fact that (under typical assumptions) producers experience increasing marginal costs as output rises. This makes the supply curve upward-sloping.

Suppose firms simply increased the price by the amount of the tax. Consumers would then buy fewer units of the good, resulting in less profits for producers. However, marginal costs are also lower at the lower level of production, meaning that firms could profitably produce at a lower price than that. So they lower the price and increase production a bit to partially offset the losses caused by the tax (note, the price remains above where it would be without the tax, and production still less than without the tax).


Market Theory: Upper limit of capital/weatlth accumulation?

A market is an social institution with various attributes and works under complex conditions. Under some conditions a pure monopoly is the only known limit.

Markets are creations by human intelligence and will not exist if not used for merchand. The market mechanisms are used in our society to enable a complex, diverse, work-sharing economy. Not the other way around.

Side note: The question is not how big a government is. A small government can have a bigger impact on a market than big government.

  My question is simple; would a laissez-faire approach to the market always result in inequality?

The reason why markets are so useful, they lower the costs to synchronize the various supplies with the societal limitless demands (individual demands can be limited). Markets can react to  technological, cultural and knowledge changes in the societies. But they depend on the kind of economy wherein they function. When the economy prefers bigger enterprises than many small firms, markets can not change much on these structures. Nevertheless the answer to your question is no. When a market starts under the condition of fair distribution and some other factors like transactions costs, market entry costs, costs for research and development and others don't play an important role, markets can stabelize the fair distribution of wealth.