Principal Components Analysis in Economic analysis

I know this is a old post, but since I found it and I know the answer I might as well share it.

I would go further than the first component, but it also depends on how much of the variance is actually explained within the component. There is two approaches that I use when determining which components I am using.

The rule of thumb is to use and component with a eigen value greater than one.
Keep an eye on the cumulative variance, for example I have a data set of 150 macro indicators and the first ten can explain 96% of the variance for a particular problem I was attempting to forecast, if I extend the components out to 20 it explains 96.7% of the variance. Is the ten additional factors really relevant?   In this case I would stick to the first 10, but again it all depends on your individual analysis.

To answer your second question, yes..

PCA is process commonly used in Factor Investing, or smart beta.. Google Smart beta and see how many firms are approaching it. As far as strait economics, the answer is still yes. I have seen many working papers published by the Fed that references PCA. It certainly does not hurt to include other methods for reducing high dimensional data. Generalized boosted models comes to mind, it uses a tree based approach. Another would be the minimum redundancy maximum relevance model. Probably the best method would be a combination of models using different machine learning and ensemble modeling.

Either way, I hope this helps even if I am answering the question 3 years after it was posted. Good luck solving the worlds problems with econometrics.

Cheers,
Aaron


I'm not a VAR specialist, but it could go along the following lines.
First write the VAR in companion form:
$$
X_{t+k} = A + B X_{t+k-1} + E_{t+k-1}.
$$
Where $t$ is the last observation, $E_j$ are the errors and $A$ and $B$ are fixed coefficients.
Then solving recursively:
$$
\begin{align*}
X_{t+k} &= A + B X_{t+k-1}+ E_{t+k-1},\\
&= A + B(A + B X_{t+k-2} + E_{t+k-2}) + E_{t+k-1},\\
&= (I + B)A + B^2 X_{t+k-2} + B E_{t+k-2} + E_{t+k-1},\\
&= (I + B)A + B^2(A + BX_{t+k-3} + E_{t+k-3}) + B E_{t+k-2} + E_{t+k-1},\\
&= (I + B + B^2)A + B^3 X_{t+k-3} + B^2 E_{t+k-3} + B E_{t+k-2} + E_{t+k-1},\\
&= \ldots\\
&= A \sum_{j = 0}^{k-1} B^j + B^{k} X_t + \sum_{j = 0}^{k-1} B^j E_{t+k-j-1}
\end{align*}
$$
Then taking conditional expectations and using $\mathbb{E}(E_j|X_t) = 0$, gives:
$$
\mathbb{E}(X_{t+k}|X_t) = A \sum_{j = 0}^{k-1} B^j + B^{k} X_t.
$$
If we let $k \to \infty$ and if $B^k \to 0$ (i.e. the process is stable) we have that:
$$
\mathbb{E}(X_{t+k}|X_t)\to \mathbb{E}(X_\infty) =  A \sum_{j = 0}^\infty B^j = A(1 - B)^{-1}
$$
Substituting $A = \mathbb{E}(X_\infty) (1- B)$ gives:
$$
\mathbb{E}(X_{t+k}) = \mathbb{E}(X_\infty)(1- B) \sum_{j = 0}^{k-1} B^j + B^{k} X_t.
$$
This is a weighted sum of $\mathbb{E}(X_\infty)$ and $X_t$. In fact a weighted average as the weights add to the unit matrix:
$$
(I - B) \sum_{j = 0}^{k-1} B^j + B^k = \sum_{j = 0}^{k-1} B^j - \sum_{j = 1}^{k} B^j + B^{k} = B^0 - B^{k} + B^{k} = I
$$


What's the Difference Between Being Pro-business and Being Pro-market?

In a nutshell, pro-business policies help currently existing businesses and their shareholders. Pro-market economies are those that follow the tenets of the free market - competition, no barriers to entry and so on. Pro-business policies may not imply a pro-market environment.

This is discussed at length for the case of India in Rodrick, D. and Subramanian, A., 2004. From “Hindu growth” to productivity surge: the mystery of the Indian growth transition, NBER Working Paper Series, WP 10376.


Finding demand function given a utility min(x,y) function

No, you should not use Lagrange multipliers here, but sound thinking. Suppose $x\neq y$, say for concreteness $x<y$. Let $\epsilon=y-x$. Then $\min\{x,y\}=x=\min\{x,x\}=\min\{x,y-\epsilon\}.$
So the consumer could reduce her consumption of good 2, without being worse off. On the other hand for all $\delta>0$, we would have $\min\{x+\delta,y-\epsilon/2\}>x=\min\{x,y\}$, so the consumer could be better of by reducing the consumption of the second good and spending the freed money on the first good. In an optimum, a consumer cannot improve so optimality requires $x=y$. It is also clear that consumers improve along the $x=y$ 45° ray. So you can simply use $x=y$ as an optimality condition to be substituted into your budget constraint and bypass Lagrange multipliers.


UK: Is there a list published by the government for the amount of tax paid by companies?

For publicly own corporations, you can access their financial statements, where they record the amount of taxes paid (e.g. for Apple here). Notice that where the tax is paid depend on the location of the firm, and whether it is a multinational business. The latter type of firm is particularly problematic, because multinational companies use transfer pricing methods to optimise (i.e. lower) their tax burden. Therefore, it is difficult to determine where those taxes should be paid. Apple and other multinationals seem to be particularly keen to do this (e.g. here).

The reason that sometimes you see headlines stating that big companies do not pay enough taxes locally is because many of these analysis is done by those with access to detailed information like HRMC or Inland Revenue (e.g. here; NB: excuse the doggy source; could not find a better one).

Outsiders to these institutions can have access to this data only in a anonymous fashion (see data available for UK here), so you won't be able to trace Apple's taxation in the UK with such data (or at least you should not be able to do so).


Why does the incentive compatibility constraint bind (moral hazard)?

The idea behind the incentive compatibility constraint is that the expected utility when doing the high effort level needs to be as least as large as the one the agent would obtain from doing the low effort level. There are many contracts that satisfy the IC. Given that the agent is risk averse, i.e., has strictly convex indiference curves, the further the asymmetric information contract deviates from the symmetric information contract, the higher will be the expected payoff the agent receives. Of course, the higher the expected payoff for the agent, the lower the expected payoff for the principal. Hence, the principal will choose that contract that a) provides the agent with just her reservation utility and b) the IC is satisfied as an equality.

Perhaps what you have in mind is that, since the expected utility for the agent is the same irrespective of the effort level she exerts, she may as well end up doing the low effort level anyway. But recall that the agent is indifferent between the two effort levels, and, therefore, we can assume that she chooses to do the high effort level.


Relationship between short-run marginal cost(SRMC) and long-run marginal cost(LRMC)

First, fix an output level $y^*$. Now you solve the (long run) cost-minimization problem conditional to this output level. We obtain the cost function $c(y)$ and the conditional demand function $k(y)$.

Suppose we are at a cost-minimizing size of plant, $k(y^*)=:k^*$, for this output level. If costs decreased when you increase plant size, then it would pay to increase plant size. This contradicts our choice of plant size. Likewise, if cost increases when you increase plant size, then it would pay to decrease plant size. Again, this contradicts our choice of plant size. So, $$\left(\frac{\partial c(y)}{\partial k}\right)_{y=y^*,k=k^*}=0.$$

The calculus treatment is already given in the mentioned page. Let me know if there is still something that bugs you.


Solow model and economic interpretation of $\dot{K}=sY(t)$

One interpretation:
Think about the equation above in GDP accounting terms. We know that in an economy, Total Savings ($S(t)$) = Total Investment ($I(t)$) (standard GDP accounting exercise).
How is this relevant here?
The right hand side is the Total savings in the economy: $\underbrace{s}_{\text{savings rate}}\times\underbrace{Y(t)}_{\text{total output}}$.

The left hand side is related to the total investment in the economy: The increase in capital stock ($\dot{K}$) in the economy is the difference between investment ($I$) and depreciation ($\delta K$). That is, $\dot{K} = I(t) - \delta K(t)$.

Replacing the identity $S=I$ then yields:
\begin{align}
\dot{K} &= S(t) - \delta K\\
&=sY(t) - \delta K
\end{align}


Nash Equilibrium (N.E) is a general solution concept in Game Theory. N.E is a state of game when any player does not want to deviate from the strategy she is playing because she cannot do so profitably. So, no players wants to deviate from the strategy that they are playing given that others don't change their strategy. Thus, it is a mutually enforcing kind of strategy profile.

'Pareto optimality' is an efficiency concept. So no state will be Pareto Optimal if, at least one of the players can get more payoff without decreasing the payoff of any other player. There are many many examples of Nash Equilibria which are not pareto optimal. The most famous example could be the N.E in prisoner's dilemma.


Why are Hicksian demand curves unobservable

Observable just means directly measured by data. In that sense, Marshallian demand is observable because we can see people's optimal consumptions of a certain good for different prices and the same income.

In contrast, the Hicksian demand is the optimal consumption after a change in price and after adjusting income so that the same utility is achieved. Given that utility cannot be directly measured (i.e. is not observable) the Hicksian demand is also not observable. Anything we conclude about the Hicksian demand is due to the Slutsky equation that relates it to measurable things: the slopes of the Marshallian demand and the Engel curve ($\frac{\partial x_i(\cdot)}{\partial y}$), and the quantity demanded.


Estimating $TFP$ using Cobb-Douglas production function

If you want to control for some deterministic trend you could add a trend term to your equation. For example, if you think there is a linear trend you can add $\gamma t$:
$$\Delta \ln Y_t= \mu+ \gamma t + \alpha \Delta \ln K_t+ \beta \Delta \ln L_t+\epsilon_t.$$
However, this being said since the equation is already estimated in first differences you have to assume there is a trend in the growth rate of TFP not just in TFP.
Regarding the interpretation, the TFP itself is unitless and there is no agreed upon measurement but higher is better. What studies usually do is not to look at TFP itself but at its growth. So in this case if TFP in 2020 is 2 and TFP in 2019 is 3 the growth rate would be $\frac{2-3}{3}=\approx -0.3$. That would mean that economy technologically regressed somewhat between the two years.
This being said one has to be very careful when interpreting TFP as its estimates can also partially reflect changes in returns to scale, markups due to imperfect competition, or gains from sectoral reallocations (see this World Bank brief). This is especially problem when you look at an aggregate data rather than at firm level panel data.


The monopoly markup with inelastic demand?

The expression $p=\frac{c}{1-1/\epsilon}$ is derived on the assumption that an interior solution is feasible, i.e. FOC is set to equal zero. But inelastic demand curve may not yield an interior solution at all, i.e. it may be impossible to set FOC equal to zero.

Perhaps this example will help. Consider a demand curve that features constant price elasticity: $q=p^{-\epsilon}$, or its inverse $p=q^{-1/\epsilon}$, where $\epsilon$ captures the price elasticity of demand. We set $\epsilon<1$, so that the demand is inelastic everywhere.

Assuming constant MC at $c$, let's maximize the profit of a monopolist facing this demand:
\begin{equation}
\max_q\; pq-cq=q^{1-1/\epsilon}-cq.
\end{equation}
The FOC is
\begin{equation}
\left(1-\frac1\epsilon\right)q^{-1/\epsilon}-c<0.
\end{equation}
Note that since $\epsilon<1$, the entire LHS is negative, and therefore we cannot have an interior solution.

On a more intuitive level, when a monopolist is operating at a point on the demand curve that is inelastic, it could always increase profit by raising price (or lowering quantity):

since demand is inelastic, 1% increase in price will cause a less than 1% reduction in quantity, so total revenue will go up;
with less quantity, total cost will also go down.

With more revenue and less cost, the monopolist will simply not be content at operating on the inelastic segment of the demand.


How to solve Bertrand Equilibrium with a non-constant MC?

The main issue seems to be that you assume that under Bertrand competition a firm is free to set a price and a quantity as well. But under Bertrand competition firms set prices and then have to meet the demand whatever level it takes. Below is a detailed discussion.

I will assume that cost functions are the same for all firms and also that the marginal cost function is non-decreasing. (You do not explicitly state these conditions but seem to assume them as well.) I will also assume that demand is non-increasing in price.

In this case, your candidate for the equilibrium is when all firms set the price $p^*$, where all firms produce
$$
q^* = \frac{D(p^*)}{5}
$$
and where
$$
p^* = MC(q^*).
$$
Is this really an equilibrium, or could a firm profit by deviating to a slightly lower price $p'$?

If say firm 1 were to deviate to $p' < p^*$ while the other firms still charge $p^*$ then all consumers would seek out firm 1. Since $p' < p^*$ and demand is non-increasing in price
$$
D(p') > D(p^*).
$$
Firm 1 has to fulfill this alone, so
$$
q_1 = D(p') > D(p^*) = 5 \cdot q^*.
$$
Because marginal cost is non-decreasing you also have
$$
MC(q_1) > MC(q^*).
$$
So now firm one charges a price lower than $p^* = MC(q^*)$ but it has to produce so much that the marginal cost is larger than $p^*$, hence it earns negative profits, hence deviation from $p^*$ was not profitable.


Why is overlapping generation model not a main model in macroeconomics?

Complexity issues are always a factor in choosing model frameworks.

In addition, my feeling is that historically the impressive possibility brought to light by the OG models was that an economy could be "dynamically inefficient" (=being characterized by over-accumulation of capital). After some empirical studies provided evidence that the main western economies did not appear to over-accumulate capital, there was perhaps an impression that the model, while structurally more realistic, had not "that much to offer" to compensate for its added complexity...

Not so, according to Philipe Weil, who celebrated the model's first 50 years of existence in the paper
Weil, Philippe. 2008. "Overlapping Generations: The First Jubilee." Journal of Economic Perspectives, 22(4): 115-34.  (freely downloadable).

The paper is an enjoyable non-technical read, full of compacted information, knowledge and insights.  A quote (bold my emphasis):

  "Samuelson’s stroke of genius was to construct a model that makes economies in which the first welfare theorem always holds, absent
  externalities and distortions, look like quite a special case. It is
  hard to escape the conclusion that the features of the overlapping
  generations model are the norm, rather than the exception: after all,
  can we seriously argue, once we understand what “new” means, against
  the realism of a model that rests entirely on the assumption that “we
  live in a world where new generations are always coming along?” In
  that respect, it is not the overlapping generations model, with the
  wealth of interesting issues it raises and its rich welfare
  properties, that is a simple toy model, but rather the competing
  workhorse of modern macroeconomics, the Ramsey–Cass–Koopmans model
  that assumes that no “new” generation ever comes along as future
  agents are all part of pre-existing families. Barro’s (1974) famous
  paper on debt neutrality and Weil (1987) make it clear that such a
  model emerges only if parents love their children (or future
  immigrants) enough to leave all of them positive bequests. This
  condition is very restrictive, and therefore the Ramsey model, with
  its long-run interest rate that always exceeds the growth rate, is
  literally quite extraordinary compared to the overlapping generations
  model."

This is the rallying cry -in his paper Weil details many aspects and issues raised by the model, and the insights that it can provide.


Sequential games and Perfect Bayesian Equilibrium

As Fudenberg and Tirole note, the fact that they assume players move simultaneously does not matter:

  This specification is more general than
  it may appear, because the set of feasible actions can be time and history
  dependent, so that games with alternating moves are included.

Moreover, all actions are observable by all players, so the definition would be unchanged by the assumption that moves are alternating.


Liquidity Trap Question? Return to equilibrium

However will that not in turn cause the bond prices to drop

Low yields provide incentives for people to sell bonds in exchange for cash. And they do. However, bond prices will not drop if there is a counter-acting amount of buying. Who is that buyer? The various central banks that are causing the liquidity trap in the first place. If that sounds counter-intuitive, remember central banks are trying to stimulate the economy instead of running a trading operating that tries to maximize profits, and so it is not always in their interest to do the same thing other market participants do. If you're skeptical that central banks could do enough bond buying to counter-act the selling, take a look at the central bank of Japan (Bank of Japan). According to this Reuters article, they own 45% of the Japanese government bond market.


Short run Aggregate supply curve is horizontal or positively sloped?

Explaining the shape of the horizontal range

In the very short run, the AS curve is perfectly price-elastic (i.e. on the diagram, it is a horizontal line). It is also referred to as the Keynesian range. In this time period, firms respond to a rise in demand for their product without considering the effects of the rising demand, such as higher prices. This can be generalised for most firms, but not all.

We can look at the example of a book shop or a food stall. It is highly unlikely for these business owners to revise their prices on daily basis based on their past sales records. Prices are more likely reviewed on a yearly basis, or at some longer time intervals. Regular price revision could complicate simple business accounting and may also dispel customers, especially if your price revisions keep increasing the price.

There are some firms that are able to change prices based on instantaneous demand. These would include the ride-hailing service providers, such as Uber and Grab. Their computer algorithms are programmed to respond their market conditions of the moment and price according to the level of demand at that instant. Firms that are able to do this are more of an exception rather than the norm.

Another way to view it would be from the resource employment viewpoint. In this time period, the real national output/income is much lower than the full employment level. Thus, there is an abundance of un-utilised or under-utilised resources. Should there be an increase in aggregate demand, the spare capacity will allow producers to increase output production easily without incurring high costs. Hence, there is no pressure on the general price level to increase.

Explaining the shape of the vertical range

In the long run, prices do not keep rising. The AS curve is simply perfectly price-inelastic (i.e. on the diagram, it is a vertical line). This is also called the classical range of the AS curve. In the LR, all firms in the economy reach their full employment level (i.e. resources have been fully-utilised to their maximum capacity) and there is no demand-deficient unemployment.

  Price levels are fully flexible as all firms will make adjustments to their prices in response to changes in demand.

This is what the vertical line indicates, not what you have mentioned.

The model I have learnt for the AS curve is a hybrid model, which is also depicted by the Wiki page for AS. The "short run" here essentially corresponds to your so-called "very short run". The "long run" here is the same as yours.


Learning Suggestion to better understanding on Advanced Microeconometrics (Master Degree Level)

This youtube channel has excellent videos for metrics:

https://www.youtube.com/channel/UC3tFZR3eL1bDY8CqZDOQh-w


Differences between Slope of budget line and MRS?

Not really, you're right in that (loosely speaking) the MRS is the amount of one good someone is willing to give up in order to get an additional unit of another good. However, the slope of the budget line measures the amount of one good someone has to give up in order to get an additional unit of another good.

In the first case, only preferences matter, whereas in the second only prices are relevant.


Why are Americans at the top of nearly every field?

A thought experiment
Let us for a moment suppose that all of the tractors, trucks, fertilizer plants, irrigation systems, shovels and other production equipment (capital goods) used in the US agriculture instantly disappeared and that no food could be imported from abroad. Under these hypothetical circumstances, do you think that all of the expert ballet dancers, sushi chefs, pole vaulters, computer scientists and basket weavers would continue to practice their niche talent? A more likely outcome is that they will revert to subsistence farming like everyone else. Now suppose that all tractors, trucks, fertilizer plants, etc. became twice as efficient, allowing food to be produced at even lower costs. Do you think this will result in an increase or decrease in the number of people who can make a living pursuing their niche talent?
The point is that a country with a large amount of capital goods has an easier time satisfying the basic needs of more people. This allows these people to engage in other activities. We can therefore rephrase your question in the following way:
"Is there any logic to the market economy that makes the accumulation of capital goods likely, or even inevitable?"
As an illustrative example, suppose that you and a couple of friends are stranded on a deserted island and are forced to survive by catching fish with your bare hands. If one day you happen to catch so many fish that you have enough for a week, you may decide to eat them while you construct a fishing rod. This "investment" will allow you to improve your "fish productivity". If you catch so many fish that you have more than you can eat yourself, you might trade some of them to a friend if he promises to build a hut for you. Maybe you loan out 10 fish to another friends so he can build a fishing rod of his own. In return he promises to give you 15 fish at some later date. If the project is a success you might sell off your old rod and fashion a musical instrument out of some coconuts. You can entertain your friends in exchange for fish.
Saving and investing resources in capital goods and voluntarily exchanging goods and services allows you to solve your most urgent problems and to raise your standard of living. You can get away with living from playing coconut music because valuable consumer goods are made readily available by this process.
Now suppose that I declare myself Chief of the island and mandate that fishing rods can only be owned if form 1304-b is filled out and delivered to my hut atop the volcano before summer solstice. Also, they may only be constructed by and bought from my best friend who holds an exclusive patent. Furthermore, anyone catching more than 20 fish per day is required to hand 35% of the excess over to me for my personal use. The rest will be handed to those who did not catch as many that day (no matter how hard they tried). Moreover, to protect hut-builders from exploitation, it is forbidden to pay them less than 100 fish per day to construct housing. Charging "interest" for loaning out fish is also exploitative and forbidden. To enforce these rules, some islanders who could have been engaged in production are recruited to my court and paid with part of the fish-tax I collect. Anyone caught engaging in illegal "black market activities" are put in prison for at least one month.
The example is admittedly cartoonish, but hopefully illustrates why Adam Smith might believe that an unhampered market has an easier time facilitating the build-up of capital goods than one subject to intervention.


The Prime Minister of Australia seems to earn around AUD $500,000 according to Wikipedia

This would give him a marginal tax rate of 45% plus supplements

The average for full-time employees seems to be close to AUD $80,000 which would make typical marginal rates either 32.5% or 37%

So the Prime Minister of Australia pays a higher marginal tax ate


What is the term for workers who outputs more than other workers at the same wage thus depressing wage for others?

There are two aspects to your question. One is indeed (marginal) productivity of workers. The other is the "depressing wages" [for others] angle. (In more neutral terms, the latter would be "wage adjustment".)
Wages are generally downward sticky, so lowering everyone else's wages when one more productive worker shows up is a bit mythical (or at least uncommon), I think, in a salaried environment anyway. For wages to get lowered due to worker competition, it would have to be a more sustained effect, like finding cheaper labor "in bulk" due to some systemic reasons (e.g. finding labor abroad, or there being substantial unemployment etc.)
If there really is term for this pertaining to an individual worker... it might be:

The term Stakhanovite (стахановец) originated in the Soviet Union and referred to workers who modeled themselves after Alexey Stakhanov. These workers took pride in their ability to produce more than was required, by working harder and more efficiently, thus strengthening the socialist state. [...]  The movement eventually encountered resistance as the increased productivity led to increased demands on workers.

But note it assumes some kind of political motivation on behalf of the worker... In a more "normal" capitalist setting, that could be substituted with other hopes/goals, e.g. not being (the one) fired or maybe even being promoted... but I am extrapolating.
I'll also say that the term is somewhat unrelated to how Stakhanov actually improved
productivity (not only of his own, but of his mine). His innovation was no different in what it involved than what Ford or McDonalds did for other kinds production: a combination of
task sequencing (pipelining) [more] specialized workers and using new technology, although the Stakhanovite movement in the Soviet Union later involved training workers in highly specialized roles, in view of increasing their individual productivity. Stakhanov himself was promoted to mine management, so even in the Soviet Union you can't say there were no individualized incentives.
It's even been claimed that Stakhanovism was merely the Soviet form of Taylorism, however this
ignores that Taylor proposed a clear process for deriving quotas, piece-rate (with a subunitary bonus for over-quota), and firing the least
productive workers continuously, while Stakhanovism in practice had a haphazard approach to these. As far as terminology goes, Taylor only named the opposite behavior, i.e. "soldiering", which he defined as working at the slowest pace that goes unpunished.
There may actually be some terms in Japanese that more or less equate "Stakhanovite" in a different context, namely

The victims of
karoshi are known in their companies as moretsu shain (fanatical
workers) and yoi kigyo senshi (good corporate soldiers).

cf. Organizational Behavior and Management, 9th ed., p. 252. (I'm not really familiar with Japanese work slang and a bit of searching finds that at lest "moretsu shain" can have some positive meanings as well, so take that with a grain of salt.)  Another (English) term given in this book (p. 288) that's probably more clear is

rate-busters (Producing above the group norm of acceptance).

Oxford Reference actually links this term with Stakhanovite

An employee who is highly productive and exceeds the formally agreed rate of output for the particular task. Whilst this is advantageous for management, rate-busters are usually disliked by their colleagues because their action provides managers with the excuse to raise the rate of output for all the other employees. Typically, there is informal social regulation of work in most workgroups where rate-busting is deemed antisocial behaviour and potential rate-busters are brought into line by their work colleagues through a mixture of persuasion and coercion. [See Stakhanovite.]

The Merriam-Webster definition for the same term more clearly links it to a reduction in income for co-workers, albeit in a per-piece rather than salary setting:

a pieceworker who produces to the utmost of his ability despite opposition by his fellows who fear that his high earnings may cause a reduction in the piece rate.

A similar term that in even more specific circumstances may be "strikebreaker".
If you go just by dictionary defs, you'd think rate-busting is always
viewed negatively by co-workers, but that's not really the case, see e.g.
a case study done at a warehouse, which involved a gamification aspect to a certain extent, but which also had loads of "standard" Taylorism elements.


Name of phenomenon where migrant workers receive decent wage but absolutely no chance of getting any government benefits

There's a lot going on in this question, so I'll focus on the labor-economics part of the question rather than the political-economy part of the question.

If it is the case that these migrant workers are being paid considerably better than their alternative labor options, but they are at the same time being deprived of rights they might value (access to social services, citizenship), then this is an example of a compensating differential.

The theory of compensating differentials says that, in order to get people to take a job that they'd rather not take (perhaps because taking it means they'd have to live as a second-class citizen), you have to pay them a little extra in order to convince them to take the job, "compensating" them for accepting the downsides of the job.

This explains, from a labor-econ perspective, why the workers might accept this arrangement. It doesn't explain the political system that deprives the workers of rights.


How should the government react to 50% of the population being wiped out?

Let's break the problem into two parts.

Part one is the negative shocks to supply and demand from half the consumers and half the producers never coming back. This is a permanent negative real shock, and it isn't clear that the government can do much about those with fiscal or monetary policy. The Avengers Endgame movie also suggests widespread and long lasting mental health issues that the government might have a helpful role in mitigating. One could also imagine changes in immigration policy now that a substantial number of homes are now vacant. With an infrastructure base built for twice as many people resources could be redirected to other purposes. However, in the long run things should be largely neutral. The US had half as many people in 1950 and the economy functioned.

Part two is the transitory shocks from disruption from some companies or teams losing members and being unable to produce. It is possible that the government could mitigate the disruption by lowering interest rates, cutting taxes, and resorting to other forms of emergency transfers (like perhaps opening the strategic petroleum reserves or changing federal law enforcement priorities to allow federal police to assist with local matters). They might also try to stabilize a banking system where almost half of consumer loans were never to be repaid. Some resources might also be redirected to handle the abandoned housing, cars, and other infrastructure and millions of abandoned pets and livestock would need assistance.


Explaining the development of income inequality

First off, dollars are not the sole determination of wealth.  People can be wealthy by having a large house and a nice car and still have \$0 in the bank.  Dollars hold wealth only because they can be exchanged for goods or services that actually do have value.

With that being said, let us take a look at what you have asked:

  I understand how it's possible where one person creates a product that everyone else purchases (creating a concentration of wealth). But, then, that person invests or saves the money, which ultimately gets circulated back into the economy. If total money is zero-sum (all saved money is spent, all received money is spent), how does any one actor in the system end up with more money than others?

The two phrases that I have emphasized above, hold the answer to your question.  In fact, you have almost answered your question in the first phrase.

The only reason that wealth has concentrated in Person A (who creates a product that everyone else purchases) is because everyone else has determined that they would rather have that product than the money that they have.  The benefits of having that product outweigh the benefits of keeping the money.  Many economists would argue that everyone is better off because the seller has improved his life by getting more money and the buyers have improved their lives by getting the better product.

Now, what does Person A do with the money?  Person A does the exact same thing that everyone else did originally: he looks for products that he would rather have than money.
In the marketplace, Person B, Person C, and Person D are all selling unique products of their own.  Person A sees value in purchasing the products from Person B and Person D, but values his money more than the product from Person C.  So, Person A purchases from Persons B and D but not from Person C.  This means that the money flows from Person A to Person B and Person D.

This cycle will continue to repeat indefinitely.
As long as Persons A, B, and D continue to produce products that people value more than money, they will continue to gain wealth.
On the flip side, as long as Person C does not produce a product that people value more than money, he will not gain wealth.


Quarterly GDP growth terminology: YoY vs QoQ

As long as you clearly label what you are doing, there’s probably a lot of ways of phrasing the change in the year-on-year growth rate. However, anyone with experience will take those changes with a large grain of salt, as a result of “base effects.”

For relatively low growth rates, the annual rate of change is roughly equal to the sum of non-annualised quarterly growth rates over the last four quarters (or the average of annualised growth rates for 4 quarters). The change in the annual growth rate on quarter is therefore driven by the difference between the latest quarter, and the quarter that dropped out of the annual average. If that previous period was abnormally low, the current annual growth rate will jump higher, even if the quarterly growth rate is unchanged.

Since that hardly represents new information about the economy, that jump is not providing a lot of new information - we knew about that weak quarter for a long time.

Year-on-year changes are a great for time series plots, but you need to dig further if you want to discuss acceleration/deceleration.


What are the applications of complex numbers in modern economics?

You might refer to this answer to a question similar to yours.

Financial economics often use the complex plane to visualize duration and the internal rate of return, to which this resource proves very helpful.


I assume you meant $u(x_1,x_2)=x_1+x_2+\min\{x_1,x_2\}$. To check if a function of two or more variables is convex, you need to check if you can differentiate it twice, calculate its Hessian and check if it is positive semi-definite. On top of that the domain needs to be a convex set but since we talk about utility in economics - you don't need to worry about that too much - Can A Utility Function Take On Negative Values?

In your example:
$$\mathbf{H}=\begin{bmatrix}
 \frac{\partial^2u}{\partial x_1^2} & \frac{\partial^2u}{\partial x_1x_2}  \\
 \frac{\partial^2u}{\partial x_1x_2} & \frac{\partial^2u}{\partial x_2^2}  \\
\end{bmatrix}=\begin{bmatrix}
 0 & 0  \\
 0 & 0  \\
\end{bmatrix}$$

so it is?

The approach described above is a general way of showing a function being convex but as pointed in the comment section, you need to be careful - make sure that all the assumptions are met before jumping to conclusions. (Or else, you end up being mistaken as me right now.)

EDIT.

Minimum function mislead me, I agree.

Let's get back to basics:

$u(x_1,x_2)=x_1+x_2+\min(x_1,x_2)=x_1+x_2 + \frac{x_1+x_2-|x_1-x_2|}{2}=\begin{cases}x_1+2x_2 \;\; \mathrm{for} \;\; x_1 \geq x_2\\ 2x_1+x_2 \;\; \mathrm{for} \;\; x_1<x_2 \end{cases}$

It looks like that (on horizontal axis $x_1$, on vertical axis $x_2$:

It is neither convex, nor concave. Due to the absolute value function that is in a core of minimum function:
Why is the absolute value function not differentiable at x=0?

I thought that eliminating point where $x_1=x_2$ would solve the issue but as shown in the other answer - it does not.

EDIT. 2 (final)

It is, in fact, concave as the user @afreelunch was telling me. You can see it directly from the definition. Or (what I think is even easier) from the way our utility function is built.

We have $u(x_1,x_2) = \min\{2x_1+x_2,x_1+2x_2\}$. Now, let's take it apart: $f_1(x_1) = 2x_1$ and $g_1(x_2) = x_2$, they are both linear Is linear function convex or concave? - hence, concave (and convex but we care about concavity here) and therefore their sum (property of concave function) $h_1(x_1,x_2)=2x_1+x_2$ is also concave. Similarly, $f_2(x_1) = x_1$ and $g_2(x_2)=2x_2$ and $h_2(x_1,x_2)=x_1+2x_2$ is concave as well. Knowing that, we get $\min\{h_1(x_1,x_2),h_2(x_1,x_2)\}$ - minimum of two concave functions - which is then concave (also a well known fact).

In short, minimum preserves concavity (and maximum preserves convexity).


Could a carbon tax be a substitute for VAT?

No, taxing externalities is not a good substitute for VAT for several reasons.

Pigovian taxes (taxes on externalities) have to be set up in an optimal way to actually reduce externalities. However, it is astronomically improbable that the same level of tax that mitigates the negative externality is exactly the same level that government has to raise for optimal provision of public goods.
It can be shown that Pigovian taxes can be imposed without deadweight loss on economy (which almost all taxes have) but such result hinges on a fact that they are made revenue neutral - hence to use Pigovian taxes in most efficient manner you should used them in a way that does not raise any additional revenue.
By discouraging the activity that creates negative externality the tax basically by substitution effect erodes its own tax base. Every tax does this to some extent - VAT for example encourage substitution of savings for consumption or substitution away from market goods to non market goods but since VAT is very broad this should reduce tax base much less than just selective tax on goods that create negative externality.

Consequently, Pigovian taxes are ill suited for  raising government revenue. Government should still pursue Pigovian taxes everywhere where there are externalities as correcting the externality is always optimal from social perspective. However, no modern government can realistically raise the kind of money it needs to run itself without VAT - in fact it is no accident that modern governments expanded precisely when modern VAT and income taxes were developed, if you check most government budgets VAT and income tax are always the biggest sections on revenue side of budget.


Structural shocks in economic models are unobservable and country-specific?

If $oil_t$ is the worldwide price of oil, neither $\rho$ nor $e_t$ are determined in a country specific model (unless the country's economy is so large that it affects global oil demand, which is not the case here).
Because there is no feedback from domestic variables to the worldwide oil price $e_t$ is the structural shock if AR(1) is an appropriate model of the oil price.The residual of the regression of $oil_t$ on $oil_{t-1}$  $\widehat{e_{AR(1),t}}$ is already an estimate for the structural shock $e_t$. So you can assess the volatility of the structural oil price shock $e_t$ based on $\widehat{e_{AR(1),t}}$. For an AR(1)-process it is possible to do this by looking at a plot of the original series.


Your question reversed the order... If the central bank issues the bond, its gets money (thus decreasing the amount of money in the system),
it pays the money back throughout a certain period of time, then money goes back to circulation
But by that time, the central bank can issue new bonds and repeat the process

Essentially, central banks manages a balance
Of course the catch is it pays money it *invents *...


Why does capital initially reduce and then rise with an anticipated future decrease in taxes in the RCK model?

When the economy transitions between time $t_1$ and $t_2$ (the red line in the phase diagram), capital reduces (because of the leftwards movement). That is why the green line showing the path of capital over time goes down initially. Because capital reduces, the marginal product of capital increases (as seen by $r$ increasing in the time path between $t_1$ and $t_2$.

Then, the economy is on the saddle path, and gradually returns to the new balanced growth path, which is a point with the permanently lower tax rate (and thus permanently higher consumption spending). When this gradual change happens (shown as the black arrow in the phase diagram), capital increases (movement to the right), so in the time path, the green line goes back up to the initial level. Because capital returns to its initial level, the return on capital also decreases to its initial level.

Consumption initially jumps up because disposable income over time is discounted, according to the formula $$\int_{t=t_1}^\infty e^{-R(t)}(W(t) - T(t)) \mathrm dt $$ where $R$ is the discount factor, $W$ is the wage and $T$ is tax, so even though disposable income doesn't decrease immediately, the anticipated decrease in the future affects current consumption. Other than that, the anticipated permanent change in government spending puts consumers on the new saddle path.


Should housing be considered a form of capital for the purpose of capital gains taxation?

I think there are a few different ways one can go about examining the topic and general question you bring up. For that reason, I'd like to highlight that this isn't my primary field, and so my answer might not align with a more expert view from that field. However, to get the ball rolling...

First, we should probably take a short digression and consider the intuition behind what their results suggest; why exactly is taxation on capital income so problematic? Generally, it's because it creates a comparatively large distortion in future consumption. As Banks and Diamond (2010) shows starting on page 31, capital taxation leads to increasingly large wedges in consumption as the number of periods increases. Unlike labor taxation, which creates large, but contemporary, distortions in consumption, capital taxation creates intertemporal distortions.

From that point, I think there's a growing literature questioning whether the traditional Chamley-Judd result is actually as substantial in application (as opposed to theory) as some consider it to be. For example, there are several, different papers which get non-zero optimal taxation rates on capital if there's endogeneity in wages, or if wages from different types of labor cannot be taxed at their individually optimal rates. In fact, one of these potentially problematic areas is entrepreneurship: as Reis' dissertation (2007) points out in the third chapter, "when entrepreneurial labor income cannot be observed separately from capital income, then it is optimal to have positive capital taxation in the long run." A pretty comprehensive analysis by Straub and Werning (2015) does a wonderful job of thoroughly analyzing both models presented by Chamley (1986) and Judd (1985) and highlighting which assumptions end up leading to their landmark result. Perhaps a useful workthrough of the math can be seen in Moll's lecture notes, where it's relatively straightforward to get a positive optimal taxation rate, depending on the intertempoiral elasticity of substitution considered in utility.

Second, I'd point out that there are reasons to be skeptical, even inside either Chamley or Judd's frameworks, that housing interest payments should be considered as capital. Most prominently, both models (I believe) assume no financial constraints (they only require no-Ponzi scheme conditions). Therefore, any agent would be able to purchase a house "out of pocket" (or through costless single-period bonds where the only interest payment is the one imposed by discounting). This in and of itself suggests that mortgages (as we observe them in real life) shouldn't ever be used, since you never ought to pay for liquidity beyond the natural rate of interest. Since mortgages are effectively assumed away in the model, I'd be cautious of applying the model to the optimal taxation of mortgage interest in the first place.

This, I hope, is a bit instructive for the third point- I agree with your skepticism that housing should be viewed as a capital good in this context. Not only because it isn't clear in the general debate of whether housing is capital or not, but also because it doesn't seem to create the same distortionary effects that lead to the zero-capital taxation result even if we accept the Chamley-Judd result. Given that the interest payments aren't an infinite horizon stream of payments (you'd hope), the wedge wouldn't necessarily lead to the same "intertemporal snowballing" that we observe in both models (and that serves as the basis of their zero-capital tax rate finding to begin with).

If I think of any more, I'll be sure to come back and update this answer. And I hope that this just gets the ball rolling for others (which more specialized knowledge in either public economics or optimal taxation theory) to provide more insightful comments. And let me know if this answer was just totally non-responsive to the question you were really interested in to begin with- it's pretty early in the morning here, so I can't guarantee my mind is working at capacity yet!


Are rational expectations in economics purely forwards looking, or can they have backward looking components?

Depends on what exactly you mean by backward looking component. When people talk about agents being 'backward looking' they often mean adaptive expectations, if that is what you have in mind then answer would be no.
However, rational expectations really just require agents to have model consistent expectations (see Snowdon, Vane, & Wynarczyk,(1994). A modern guide to macroeconomics). So if by 'backward looking components' you mean that agents utilize past information in some way answer is yes. This is especially true if the underlying model predicts that expected variable has some structural relationship to its past realizations. This being said there are not many situations where this occurs.
There can also be mixed expectations models that assume that portion of agents follows adaptive expectations for example and rest rational expectations, but these would not strictly be called rational expectations models.


Having difficultly to understand two indices with regard to inflation

Headline Inflation:

Headline inflation is the raw inflation figure reported through the Consumer Price Index (CPI)

Raw means it is not adjusted in any way (like core inflation that excludes energy and food prices). Provisional headline inflation just means the numbers are not final and subject to change.
Imputed Index
If some prices are missing statisticians will impute them (figure out what they likely should have been) using statistics, like in this report.


Incentive Compatibility Construction Rule

As @brunosalcedo suggests in the comment, the agent's payoff, $\pi(a,s)$, should depend on i) the realized state $s$ and ii) the announced state $a$. Note that the announcement $a$ should in general be a function of the realized state $s$, i.e. $a:S\to S$. Incentive compatibility should thus be
\begin{equation}
\pi(s,s)\ge \pi(a',s),\quad\forall a'\ne s.
\end{equation}
In words, incentive compatibility requires that the agent's payoff when his announcement agrees with the state (i.e. the state is $s$ and his announcement is also $s$) be no lower than his payoff when his announcement disagrees with the state (i.e. the state is $s$ but his announcement is $a'\ne s$).

Check out the following book chapters for more detail:

Mas-Colell, Whinston, and Green (1995) (aka MWG) chapter 14.C discusses IC in the context of moral hazard
MWG chapter 23.B introduces IC for truthful implementation
Krishna (2010) chapter 5 is similar to the previous MWG chapter, but draws out more implications of IC.


Intuition of Shephard's Lemma in Reverse

Expenditures on good $2$ are, indeed, included and can vary.
We know from Shephard's lemma that whenever the marginal change in expenditure for good $1$
with respect to its price varies with the price of good $1$, the Hicksian demand for good $1$ must vary too. But since utility is fixed, changes in the Hicksian demand for good $1$ require changes in the Hicksian demand for good $2$, a move along the indifference curve. And since we do not change the price of good $2$, the expenditure on good $2$ must vary too.


MICROECONOMICS: Optimal quantity produced in a Perfect Competition Market

A typo seems very probable; more importantly, your procedure and concepts of setting MC=MR to solve for profit maximisation is definitely correct, regardless of the numbers being plug into the question.


What Does "Quantity" Represent When Looking at Supply and Demand?

Each curve simply shows the amount of goods that producers would supply at given prices and how many units of goods consumers would demand at all the different prices. Let's say the price of wooden chairs would be 5 million Euros. Producers would want to supply a large quantity of wooden chairs, since they're so profitable. Consumers would likely demand very little though. However since no one suppliers want to sell more than people want to buy the price will have to change and will not stay there in equilibrium. We only have an equilibrium when all parties agree, otherwise transactions don't take place. That's one intuitive way of thinking about it if it helps you.

The quantity at which suppliers want to supply exactly as much as consumers demand and consumers want to demand exactly as much as suppliers want to supply is given by the intersection of the two graphs. In this case neither party is unhappy with the outcome (both agree on the quantity to buy and sell). This means we are in equilibrium (i.e. no one wnats to change behavior). That one point is the equilibrium quantity.

Note that consumers cannot buy more than producers are willing to sell and vice-versa. If that would be the case, then they would have to change somthing, which means we wouldn't be in equilibrium.

So in short:

The Demand curve shows the quantity demanded.
The Supply curve shows the quantity supplied.
For the two curves we have several not easily observed hypothetical quantites. One quantity for each possible price.
The intersection is the observed equilibrium quantity.


Monetary Policy adjustments as a result of Brexit

I agree that Money supply/Money stock is negatively correlated with short-term interest rates.

Since Great Britain has been using the Sterling before and after Brexit, there is no fundamental change in Monetary Policy caused by Brexit. Your (implied) assumption that Brexit causes a change in interest rates is therefore highly speculative.

Rather, the Bank of England's Monetary Policy Committee decides on interest rates based on (forecasted) growth and inflation rates, and a big number of other factors.

The economic performance (and therefore, interest rates) of the UK in the aftermath of Brexit will depend heavily on the negotiated deals between the EU and the UK.

If the UK economy indeed slows down as a response to Brexit, as many assume, the BoE will most likely keep the interest rates down. However, this is not at all guaranteed to happen.


Intertemporal utility maximization through consumption

Hint : You can write down your Lagrangian and simply take derivatives according to $c^{o}$ and $c^{m}$. So, you will mainly have two first order conditions with a lagrange multiplier. After, you will combine these two equation to eliminate your lagrange multiplier, which will give your intertemporal optimality condition (so-called Euler equation). I don't write the whole maximisation problem but only Euler condition.

$$bu^{'}\left(c^{o}\right)=\frac{u^{'}\left(c^{m}\right)}{1+r}$$

The role of interest rate is quite evident. If your interest rate increases, the marginal utiliy of consuming when old decreases, which means that it is optimal to consume more when old. Intuitively, it is plausible, if interest rate increases, the opportunity cost of consuming today increases.

For the second question, when you try to find optimal level of education, you should take the derivative according to $e$. So, the lagrange multiplier cancels out, which mean that you find a somewhat static problem. You will have ;

$$h^{'}(e)=\frac{1+r}{w}$$

If wage is higher, marginal utility of capital is less, which means that the level of education is higher. So, if wage is high, there are more incentives to invest on human capital. You can make the inverse similar interpretation for the interest rate.


What's a good alternative to Romer's Advanced Macroeconomics?

I like Introducing Advanced Macroeconomics: Growth and Business Cycles by Sorensen and Whitta-Jacobsen. The text provides intuitive explanation and exposition of models. The book is neither too mathy nor descriptive, I would say, they offer somewhat balanced introduction.

A similar question was submitted before, have a look here.


Would it be accurate to say that the economies of most of the countries are capitalistic?

A nice way to check this is by looking at the economic freedom index
As we can see, this is a scale, there is no perfect capitalistic country and most of the countries are not very free economically


As a caveat, disclaimer, I am not too familiar with Evolutionary Game Theory and it is quite outside my research area/main areas of expertise. However, I can show here that range of $(a,b)$ for which $B$ is a best response is strictly greater than for either of the other two strategies. To that end,

You can characterize the range of $(a,b)$ for which $B$ is a (strict) best response by solving the system

$$\begin{split}
4−2b−3a &\> b+2-2a\\
4−2b−3a &> 1+a-b\\
\end{split}$$

Or,

$$\begin{split}
2 &> 3b + a\\
3 &> 4a + b\\
\end{split}$$

Or,

$$\begin{split}
a &> 7/11, b<3-4a\\
a &\leq 7/11, b<\frac{2-a}{3}\\
\end{split}$$

We can solve for the area of this quadrilateral:

$$\begin{split}
B &= \int_{0}^{7/11}\frac{2-a}{3}da + \int_{7/11}^{3/4}(3-4a)da\\
&= \frac{259}{726} + \frac{25}{968} = \frac{101}{264} \approx .382\bar{57}
\end{split}$$

Through similar analysis for $M$, we obtain

$$\begin{split}
a &> 7/11, b>\frac{3a-1}{2}\\
a &\leq 7/11, b>\frac{2-a}{3}\\
\end{split}$$

Hence,

$$\begin{split}
M &= 1 - \int_{0}^{7/11}\frac{2-a}{3}da - \int_{7/11}^{1}\frac{3a-1}{2}da\\
&= 1 - \frac{259}{726} - \frac{32}{121} = \frac{25}{66} \approx .3\bar{78}
\end{split}$$

Finally,

$$T=1-\frac{25}{66}-\frac{101}{264} = \frac{21}{88} \approx .238\bar{63}$$

Since $$\begin{split}
\frac{101}{264} &> \frac{25}{66}\\ \frac{101}{264} &> \frac{21}{88}
\end{split}$$
the result is shown.


On the interpretation of the coefficients in a log-log model as an average

The short answer is no, this is not so when the dependent variable is in log, unless you define the $\%\Delta$ as (100 times) log-difference. The reason is that $E(\log y) \ne \log E(y)$.

We don't need a multiple regression model to show this so consider $\log(y)=\beta_0 + \beta_1 x + u$. It is the dependent variable in log that matters, so let us not take log to $x$ for notational brevity.

It is true that $\%\Delta y \approx 100 \beta_1 \Delta x$ if $\Delta u = 0$ (ceteris paribus) when $\beta_1 \Delta x \approx 0$. But we cannot interpret it in terms of $E(y|x)$. The reason follows.

As you can see, we have $y = \exp(\beta_0 + \beta_1 x + u)$. When $x$ increases from $x_1$ to $x_1+\Delta x$, $y$ increases from $y_1 = \exp(\beta_0 + \beta_1 x_1 + u_1)$ to $y_2 = \exp \{ \beta_0 + \beta_1 (x_1+\Delta x)+u_2 \}$. Importantly, $u$ can also change because $u$ is not held fixed, which is why I wrote $u_1$ and $u_2$.

Now, what is the change rate of $y$? We have
$$
\frac{y_2}{y_1} -1 = \exp(\beta \Delta x) \exp(\Delta u) -1 \approx (1+\beta \Delta x) \cdot \exp(\Delta u) -1,
$$
where $\Delta u = u_2-u_1$. Here, the left-hand side is the change rate of $y$. You can verify that the change rate of $y$ is approximately $\beta \Delta x$ if $u_1 = u_2$, i.e., if $u$ is held fixed so that $\exp(\Delta u) = 1$. But the problem is that $u$ is not held fixed so $\exp(\Delta u)$ can be very different from 1. When $x$ and $u$ are independent, the average growth rate is $\exp(\beta \Delta x) \cdot E(e^{\Delta u}) -1$.

How different would this be from $\beta \Delta x$? If $u\sim N(0,\sigma^2)$ and $u_1$ and $u_2$ are mutually independent, then $u_2-u_1 \sim N(0,2 \sigma^2)$, and thus $E(e^{u_2-u_1}) = \exp(\sigma^2)$. For example, if $\beta_1 \Delta x_1 = 0.01$ and $u\sim N(0,1)$, then the average growth rate of $y$ is $\exp(0.01) \times e - 1 \approx 1.75$, that is, approximately 175%, which is hugely different from 1%. Interesting, isn't it?

Simulations: Use R to do the following simulation.

set.seed(1)
n <- 1e6
x1 <- rnorm(n)
x2 <- x1+1
y1 <- exp(1+0.01*x1 + rnorm(n))
y2 <- exp(1+0.01*x2 + rnorm(n))
100*mean(y2/y1-1)
# 175.2406
mean(log(y2)-log(y1))
# 0.01004419

When $\beta_1 \Delta x = 0.01 \times 1 = 0.01$, the average growth rate is not 1% but 175%. But $\Delta E(\log y)$ is approximately 0.01.

Note 1: The above discussion is for the log-level model. Little is changed for the log-log model.

Note 2: If you define the growth rate as the log-difference, the expected growth rate equals the expected log-difference = $\beta \Delta x$.


Bayesian Nash Equilibrium in a Duopoly Cournot Competition

If the capacities $q_1$ and $q_2$ are the same, say $\overline{q}$ you can rule out some cases.
The best response functions are given by:
$$
\begin{align*}
\hat q_1 &= \min\left\{\frac{M - (1-\theta) \hat q_2^L - \theta \hat q_2^H - m}{2}, \overline{q}\right\},\\
\hat q_2^L &= \min\left\{\frac{M - \hat q_1 - m}{2}, \overline{q}\right\},\\
\hat q_2^H &= \min\left\{\frac{\alpha M - \hat q_1 - m}{2}, \overline{q}\right\}
\end{align*}
$$
First we can show that $\hat q_2^H \ge \hat q_2^L$.
Proposition: $\hat q_2^H \ge \hat q_2^L$

Proof: If $\hat q_2^H = \overline{q}$ the proof is obvious. Else, from $\alpha > 0$, we have:
$$
\hat q_2^H = \frac{\alpha M - \hat q_1 - m}{2} \ge \frac{M - \hat q_1 - m}{2} \ge q_2^L.
$$

Next, we can also show that $\hat q_2^L \ge \hat q_1$.
Proposition: $\hat q_2^L \ge \hat q_1$.

Proof: if $\hat q_2^L = \overline{q}$ the proof is obvious. If not, then:
$$
\begin{align*}
\hat q_1 - \hat q_2^L &\le \frac{M - (1-\theta) \hat q_2^L - \theta \hat q_2^H - m}{2} - \hat q_2^L,\\
&= \frac{M - (1-\theta) \hat q_2^L - \theta \hat q_2^H - m}{2} - \frac{M - \hat q_1 - m}{2},\\
&= \frac{-(1-\theta) \hat q_2^L - \theta \hat q_2^H + \hat q_1}{2},\\
\to 2 \hat q_1 - 2 \hat q_2^L &\le -(1- \theta) \hat q_2^L - \theta \hat q_2^H + \hat q_1 \\
\to \hat q_1 &\le \hat q_2^L - \theta(\hat q_2^H - \hat q_2^L) \le \hat q_2^L.
\end{align*}
$$
where the last line follows from $\hat q_2^H \ge \hat q_2^L$.

The two propositions show that we only have to consider the following cases:

$\hat q_2^H < \overline{q}$ and $\hat q_2^L < \overline{q}$ and $\hat q_1 < \overline{q}$
$\hat q_2^H = \overline{q}$ and $\hat q_2^L < \overline{q}$ and $\hat q_1 < \overline{q}$.
$\hat q_2^H = \hat q_2^L = \overline{q}$ and $\hat q_1 < \overline{q}$.
$\hat q_2^H = \hat q_2^L = \hat q_1 = \overline{q}$.

For each of them you can compute the equilibrium values and then you need to check if they indeed correspond to the best responses.
This reduces the number of cases from 8 to 4. I don't think you can do better.


References to learn continuous-time dynamic programming

For continuous-time stochastic dynamic programming, the small, nontechnical Art of Smooth Pasting by Dixit is a wonderful option. It does a very effective job of conveying the basic intuition.

Stokey's more recent The Economics of Inaction is also decent, but for a practical-minded person it probably underperforms Dixit - its much greater length and somewhat heavier notation do not yield commensurate rewards.

If the underlying stochastic processes are not Itō diffusions, then I'm not sure what the best reference is. The most common such case I've seen (and that I use myself) is the case of discretely many exogenous states, where if we are currently in state $s$ there is some constant hazard rate $\lambda_{s,s'}$ of a switch to state $s'$. Fortunately, this is a pretty simple case in practice: one can just modify the HJB equation to account for the flow probability of switching from $V(\cdot,s)$ to $V(\cdot,s')$. (You can see this, for instance, in equations (1)-(5) in this Acemoglu and Akcigit paper. Conceptually it's no different from setting up the HJB equation when we have an Itō diffusion as the driving process, except that it's simpler because we just get a system of linear equations and we don't need to think about Itō's lemma, etc.)

Of course, maybe there is a good textbook reference for this too - but unlike the potentially much more complicated cases involving stochastic calculus, this is straightforward enough that a text has never seemed necessary to me.


Are FedWire transfers and volume tracking exchanges of reserves between banks that have accounts at the Federal Reserve?

When sending a payment order to the Fedwire Funds Service, a Fedwire participant authorizes its Federal Reserve Bank to debit its master account for the amount of the transfer. If the payment order is accepted, the Federal Reserve Bank holding the master account of the Fedwire participant that is
  to receive the transfer will credit the same amount to that master account. Each payment order sent through the Fedwire Funds Service is processed individually upon receipt through an encrypted electronic network.

The above came from the Fedwire product sheet.

So yes, it is a transfer of a balance held at a Federal Reserve bank.


Fractional reserve banking and the value of someone else's currency

When the retail bank in country B multiplies the money supply, it does so by creating an asset (money owed by the borrowers) and a liability (the obligation to honor the loan it gave to the borrowers) on its books, and an asset (money to spend) and a liability (the loan to pay back) on the books of each of the borrowers.

Now, if these borrowers are all spending their money amongst each other, this probably isn't going to be a problem. It by no means affects inflation in country A, because there's no additional demand on country A's resources.

But lets say instead all of the borrowers then go to use this money to import some goods from country A. This will be, on the face of it, inflationary. Suddenly you have all this extra demand, for no additional supply.

One of two things happens:

The borrowers say to country A's exporter, 'I would like to buy some widgets, get the cash from my bank' and exporter (or rather, the exporter's bank, who is based in country A) says to the bank 'Cash please!'. Country B's bank doesn't have the cash to honor every loan they've made, so they're unable to honor the payments and the loan they gave the borrowers is essentially worthless. The exporter doesn't send the goods.
The exporter in country A's bank is ok with the promise of payment from the bank in country B. Afterall - there are also exporters in country B selling goods to people in country A, and this will balance out the loan.

In this second scenario - the there is also an increase in supply because country B is also exporting to country A, so it's not necessarily inflationary.

Bottom line: The inflationary effect of multiplying the money supply will be tempered by the bank's ability to honor the loans. A bank can make all the on paper loans it wants, but unless it's able to actually honor them, they are meaningless. In our modern banking system, there's a high degree of trust which means when a bank lends us money, and we see that money in our bank account,  we assume that that money is actually going to be able to buy us goods.


In itself, a price index for single period does not tell you much, as any index is, by definition, without dimension. Assuming that 3.41 is the annual change in the CPI (= Consumer Price Index), this would imply an inflation rate of 3.41%. To be specific, it implies that the total price of a representative commodity basket changed by 3.41%. The WPI (= Wholesale Price Index) covers different commodities and hence yields a different inflation rate. Which one to prefer depends on what you are interested in. Consumer Prices are more relevant to private households, but businesses care about wholesale prices.


Auction Theory: Proving that the found equilibrium is indeed optimal

$$
G(z) (z-x) =
\int_x^z G(z) dy
$$
and since $G$ is increasing on $[x,z]$, the right hand side is larger than
$\int_x^z G(y) dy$.


Are there studies that empirically validate results in contract theory?

An excellent (if slightly old) survey on the empirics of contracts can be found in

  Pierre-André Chiappori and Bernard Salanie (2003) "Testing Contract Theory: A Survey of Some Recent Work", in Advances in Economics and Econometrics, vol 1, M. Dewatripont, L. Hansen and S. Turnovsky eds, Cambridge University Press.

It is mostly non-technical. Bernard Salanie (literally) wrote the book on contract theory. An ungated pdf of the chapter is available here.


The Federal Reserve is a system, consisting of the reserve banks, which are "owned" (in a limited fashion) by their member banks (members select the leadership of the reserve banks), and the Federal Reserve Board of Governors, which oversees the member banks and is independent within the government (that is, the Governors are nominated by the President and confirmed by the Senate, but neither has direct control over the Board beyond its composition and the legal authorities assigned to it). All of this is covered in the law that gives the Fed its authorities, the Federal Reserve Act.


Nash equilibrium for Bertrand Model with Spatial Differentiation

This is a version of Hotelling's Model of Spatial Differentiation

The easiest solution to this is to find an "X" in-between [0,2] such that the customer is indifferent between going to Firm A or Firm B.

This will happen when
$ P(A) + Distance(X-A) - Utility = P(B) + Distance(B-X) - Utility $

Then demand [0, X] would go to A, and demand [X, 2] will go to B. Then using Best Response Functions for this Simultaneous game we can compute prices, quantity, and profits.

I have attached my solution below. For more information, you can see -  SNYDER NICHOLSON HOTELLING'S BEACH EXAMPLE


Regarding the units of the shift produced by subsidies on supply curve

The confusion stems from the fact that the price is also a $\frac{\$}{units}$


CES v. Leontief Aggregator in Production

I guess it depends on the application in question. Leontief function presumes that there is no substitution between the arguments, i.e. no amount of increase in one argument can compensate the decrease in another to keep output at some original level. In contrast, the general CES does allow some degree (as captured by the parameter $\gamma$) of substitution between the arguments. Note also that Leontief is obtained by taking $\gamma\to\infty$, thereby rendering the elasticity of substitution between the arguments $0$.

If you were doing a simulation exercise, you should usually "let the data speak" first, namely, estimating the value of $\gamma$ (and $\sigma_j$) from some existing data. However, if you have some a priori reason to believe that $\gamma$ is of a particular value, then you would assume that value and carry on with the analysis.


How does the ban on surge pricing affect the companies, customers and the economy?

It depends.

If customers are currently making informed decisions when they book a surge-priced car, then banning surge pricing punishes customers, drivers, intermediaries, and the wider economy. It deliberately introduces an economic inefficiency.

If, however, a lot of customers are making uninformed decisions and are effectively being conned by surge pricing that they didn't expect or didn't understand, then banning surge pricing stops economically inefficient activity, protects consumers, and can be a net contributor to the economy.

Or if surge pricing increases negative externalities, then banning it could give a net benefit: for example, if the worst-polluting cars only come onto the road when surge pricing is available.


Supermarket selling seasonal items below cost?

Is it simply for saving warehouse costs?

Probably yes, holding onto inventory is very expensive. You have to pay for warehousing of the good, it takes the spot of some other inventory that might be in high demand. Food is also perishable so it cannot be stored indefinitely.
Stores have to always guess what demand for their products will be, sometimes they get it wrong and order excess inventory. It can often be more profitable to sell excess inventory below the production cost or many times even just to throw it away to make space for new products. Holding onto it would be a sunk-cost fallacy. The costs are already incurred in the past, even if they were a result of a mistake it does not make sense to try to double down on the mistake and hold onto them. It is more profitable to ignore the sunk-costs of getting them and make new decisions of what to do with the inventory based on new information.


Do reserves at Bank of Canada count as money?

1) Does "currency and bank deposits owned by the government of Canada"
  in (b) mean currency and deposits at the Bank of Canada? If not, are
  these deposits just deposits by the Canadian government at some other
  bank? And where does the government of Canada keep its currency?

Yes. The Canadian Federal Government consolidates its deposits in a single account (managed by the Receiver General) at the Bank of Canada. It does not bank elsewhere.

"Currency" in this context refers to Canadian bills and coins (and not foreign currency reserves). These are liabilities of the Canadian Federal government itself. Normal accounting practice is to net to zero any liabilities "owned" by the issuer on its own balance sheet.

  2) If the above ^^ is correct that these are currency and deposits at
  the Bank of Canada, does "M2 consists of M1 plus all other deposits"
  in (c) imply that while deposits at the Bank of Canada are part of M2,
  currency held by Bank of Canada (ie its reserves) are not part of M2,
  ie Bank of Canada's reserves are not money?

Once again. "currency" here means Canadian dollar bills, and they are liabilities directly issued by the Bank of Canada. This currency represents bills and coins that have been minted, and not distributed to private banks (or returned by banks). These are not considered "reserves" under any definition of "reserves." These notes/coins only enter the money supply when they are shipped to private banks.

(Note that Canada abolished bank reserve requirements -- the requirement for private banks to hold deposits at the central bank based on a percentage of their deposits. As such, "reserves" is no longer a meaningful term, you would need to be more specific - e.g., foreign currency reserves. If that is what you mean, foreign currency reserves are not normally considered part of the domestic money supply, except perhaps in cases where the country is "dollarised.")


How to set up the payoffs properly for a division of labor game

If you want "both don't clean" to be a Nash equilibrium (NE), then your payoff matrix doesn't work. With those payoffs, cleaning is a strictly dominant strategy and "both clean" is the only NE.
A more systematic approach would be to assume that a clean house has utility $b>0$ for both partners, but cleaning alone generates a disutility of $c>0$. Cleaning is likely to have increasing marginal disutility, such that sharing the cleaning work costs only, say, $c/4$. Then the payoff matrix is

Husband Cleans
Husband Doesn't Clean

Wife Cleans
$b-c/4$,$b-c/4$
$b-c$,$b$

Wife Doesn't Clean
$b$,$b-c$
0, 0

If $b>c$, then "only wife cleans" and "only husband cleans" are two strict NE (and there's also a symmetric mixed one).
If $b<c$, then "wife doesn't clean" and "husband doesn't clean" are strictly dominant strategies and "no one cleans" is the unique NE. If $b<c$ but $4b>c$, then "both clean" is Pareto better than "no one cleans" (and uniquely maximizes total payoffs), which implies that your game is a Prisoner's Dilemma.
With this approach, of course, "both clean" is never a NE, since saving the costs of cleaning while enjoying the benefits of a clean house is always preferred to participating in the cleaning. But this is what you seem to assume ("they would both prefer it if the other person did all of the cleaning"). This assumption contradicts your wish for having a NE where both clean.
However, you could go on to argue that the game they actually play is the infinitely repeated version of this 2x2 game. This opens up the possibility that "both always clean" is a NE-outcome of the repeated game, e.g. based on Trigger strategies, even for the Prisoner's Dilemma case (with small enough discount rate).


Would this alternative to fractional reserve banking work?

How, in practise, would that be different from what is happening now? ..or what you, as an "investor" could already do?

No one is forcing you to deposit all your liquid assets into a bank account - most banks offer some sort of investment-opportunities, where you can buy into different funds with different risk-profiles.

Next thing; suppose banks take all deposits and lend out to different customers for various purposes and they receive interest on those lendings - some of those interest are transfered back to you for allowing the bank to use your money. This would actually be the same, as putting your money into a fund. The bank "invests" your money with other customers and recieve some sort of yield on that investment (interest). Difference will be the risk-profile. By "investing" in the bank you recieve an extremely diversified portfolio, funding everything from consumer goods, cars, houses to actually backing large corporations. Most of your investment is being placed high up in the hierarchy of priotization if the lenders go into default, which makes your investment "safer" than if you go and buy bonds in that same company (but obviously also recieves a lower yield).

Your proposal will put peoples money in a different place in the hierarchy of debt if customer/lenders go bankrupt, actually increasing the general risk-profile of the bank.

In all everything comes down to your risk profile - which exposure do you want your investment to have. The risk of a AAA rated sovereign bond yielding -0.5%, receive 0% on your deposits in a government backed bank, 5% on a BB- mutual fund or 25% on an emerging markets bonds?


Proof that for increasing AC MC(Q)>AC(Q) at any Q>0?

Let $C(y)$ denote the cost function and $y$ the quantity.

The average cost function is $AC=C/y$. This is increasing if its derivative is positive. Let $C'$ denote the derivative of $C$. The marginal cost function is $MC=C'$. We have:

$\frac{\partial AC}{\partial y}= \frac{C'y - C}{y^2}>0$ by using the quotient rule. As $y^2>0$, the condition becomes:

$C'y - C>0$.

Rearranging yields:

$C' > C/y$

Hence: $MC>AC$.


I am using the answer function as I do not have enough rep. to comment.

Take note that gold, or most of the commodities available for trading, are priced in USD. By saying yen is down because of gold is therefore a rather speculative statement in my opinion.

Why I say this is because when the price of gold moves, it should have no effect on the JPY/USD quote. I think the journalist saw that there was a rise in gold prices, as well as appreciating USD/ depreciating JPY which led him to conclude JPY has depreciated in terms of gold prices. But how strongly are the two events correlated should be up to your own research and conclusions.


Interpretation of a differenced regression

If the levels specification is deemed acceptable,

$$y_t = \beta_0 + \beta_1 x_t + \beta_2 x_t^2$$

then it follows that the first-difference specification should not include a constant term in order to be methodologically consistent,

$$\Delta y_t = \beta_1\Delta x_t + \beta_2\Delta x_t^2$$

since $\Delta \beta_0 = \beta_0 -\beta_0 = 0$.

I suggest you run the specification of first-differences without intercept and see what happens.

Note: the slope coefficient estimates you should obtain in the first-difference specification without intercept should be close to the corresponding estimates you obtained in the levels-estimation. Otherwise, the maintained hypothesis of constant slope coefficients becomes questionable, or the model has other misspecification issues.

(this is not to be confused with the case where we run a regression without constant term but with the dependent variable and the regressors centered on their sample means -there we would obtain the exact same slope estimates with the levels specification that includes a constant term, as a matter of algebraic property of least-squares estimation).


Share of domestic and foreign monetary flows between sectors of a country

The exports and imports in an input-output analysis should correspond to imports and exports as components of GDP so you can use those or some measures derived from them.
For example, trade as a $\%$ of GDP is the sum of exports plus imports over GDP (i.e. $\frac{E+M}{Y}$, where $E$ is export, $M$ import and $Y$ GDP/output).
Hence to calculate any comparable measure from input output table you would have to take the import and export variables from your study (your input-output table should have import row and export column), sum them together and then divide them by output.
You can also find data for $E$, $M$ and $Y$ separately and compare those to the measure you constructed. Based on the question I guess you came to $20\%$ by perhaps doing something like $\frac{E-M}{Y}$?
In addition input-output analysis is not necessary accurate and you should not be surprised to find discrepancies between what your input-output model shows and what holds in real world. Especially, if your input-output model has too many sectors small errors introduced by rounding and linearization can add up and produce results that will not be accurate.


This is a general problem with anyone who comes face to face with official statistics for the first time.
Official statistics of a particular country comes from an institution of that country. However, there is something called the System of National Accounts that contains guidelines on pretty much every aspect of national accounts data. This is maintained by UN, IMF and World Bank, among others. Thanks to this you will find data definitions and compilation methods across countries to be largely consistent.
Rule 1: So if you have to use cross country data make sure that you select comparable datasets, i.e. choosing same variables.
Rule 2: Unless you are doing a deep data data analysis, it is okay to chose the variable which is seems most common. What is most common? The variable that you see easily available for any country.
Rule 3: Just keep in mind the difference between real and nominal variables. In most scenarios, you'd need real variables, especially, if you are doing cross-country comparison.
If you still get confused about what to use, the best way is to just read about the variable on wiki and you'll get some sense. Going into more details than what is easily available on wiki is only going to confuse you further as official statistics get fairly complex as you start peeling the layers.
Finally I'd recommend to give CEIC database a try, if you have access to it (through your institution). It is one of the best place today to find any data related to official statistics (I have seen even the officials from IMF using it).


How does a liquidity guarantee for Asset-Backed Commercial Paper used to finance a Conduit work ?

Unlike what is often assumed the in the realms of theoretical macroeconomics there is not always an infinitely deep market with constant buying and selling. In fact, one of the reasons Investment banks are so important is in the role they play in making markets for buyers and sellers of stocks and financial products of many kinds.

Lets think about liquidity in the current context - the liquidity of a, say, firm's shares is determined by the available quantity, indications of interest &C. We could construct a working difenition of liquidity as the difference between bid price and selling price (in the context of a firm's shares although the intuition holds).

In practice, the liquidity guarantee means that an entity involved in the exchange acts as the guarantor for there being bid prices and selling prices, a guaranteed quantity of shares available, and especially ensuring that the discrepancy does not become too great between bid price and selling price. This is of importance for, say, investors, since a smaller spread significantly affects the transaction costs, risks etc.

It is essentially someone (post-financial crisis it has often been the government) guaranteeing that a market will be made (with buying and selling prices in good faith) for a given time period (or indefinitely). It is a form of prevention of stocks becoming illiquid, almost like a form of insurance.

The application of a liquidity guarantee is far from limited to buying/selling shares in a firm of course (although I believe the intuition is the same) - a notable example has been the US government's temporary liquidity guarantee programme which was constructed in the aftermath of the 2008 financial crisis with the purpose of getting banks to begin lending to each other again - the purpose of the programme was "to decrease the cost of bank funding so that bank lending to consumers and businesses will normalize."  and "to strengthen confidence and encourage liquidity in the banking system by guaranteeing newly issued senior unsecured debt of banks, thrifts, and certain holding company, and by providing full coverage of non-interest bearing deposit transaction accounts, regardless of dollar amount."

Banks who made out liquidity guarantees are essentially insurance salesman and hence are exposed to the downside if things catastrophize. Often this insurance money was being kept in markets that would also be affected by market forces that made the liquidity guarantee they were insuring catastrophize in the first place! (These problems were exacerbated by many other factors including very high effective leverage rates).


What would happen to the world GDP if we run out of minerals to mine but not of sources of energy?

You're correct that just because 1% of the economy is in mineral extraction, doesn't mean that stopping it would reduce global GDP by 1%. Similarly, roughly 10% of global GDP is on energy expenditure, and about the same on food, but if we stopped either of those, global GDP drops to pretty much zero.

Do bear in mind that we will stop using coal, oil and gas quite shortly - probably within 2-3 decades. Nevertheless, we've got no shortage of energy - we've got far more solar energy available than we can use.

However, minerals aren't going anywhere: very little of them leaves the planet. Which means almost all the elements we've ever mined, are still around.  And retrieving valuable elements from waste is largely just a matter of energy, and a bit of know-how.

Let's think about what might happen if an element starts to become scarce.

First, its price goes up.

As a result, there's investment in substitutes, as they're now more competitive.

There's also investment in recovering the scarce material from waste streams, as this investment now has a higher payback -  a higher financial return on investment.

The short-term effect on global GDP would be a small reduction. The longer-term effect could go in either direction. But, ultimately, the global effect on humanity will be positive: sooner or later, we're going to need to move a circular economy, and this transition will be part of it.


The formula $t = \frac{e^2}{T^2 + e}$ does not scale linearly:

$\frac{(ke)^2}{(kT)^2 + ke} \ne c \frac{e^2}{T^2 + e}$ for any constants $k$ and $c$. (Linear scaling is OK as this is just a denomination of money. Nonlinear scale is a kind of nonsense as it introduces the need to add arbitrary coefficients, without a hint how to choose particular numerical values.)

So I choose the formula $t = \frac{e^2}{T + e}$.


Sequential game equilibrium strategy question

I think the question has a typo. Whoever wrote it was trying to make you think about the fact that a, c and d are not even properly defined strategies for player B. This is because a strategy must specify what player B would do in all possibly scenarios he might face.

However, the typo comes from the fact that b is not part of a NE. If the first payoff instead of being (3,5) it was (5,3) then b will be an equilibrium strategy for B.

Your intuition is partially right, for it to be an NE strategy it had to choose the highest payoffs on the path of play. If it was a SPNE it would have to choose the highest payoffs in all subgames, not only the ones that are reached in equilibrium.


Mixed strategy Nash equilibrium in 3x3 game

The general procedure to solve for a MSNE in a 3-by-3 (or larger) game is always a bit tricky and involves some trial and error

Step 1: Conjecture (i.e. guess) a subset of strategies that will be used in equilibrium
Step 2: Calculate their probabilities using the indifference condition
Step 3: Verify that the equilibrium payoff cannot be unilaterally improved upon; that is, no player has a strict incentive to deviate to another strategy

Suppose your conjectured strategies are $\{B,C\}\times\{A,B\}$ (it doesn't really matter what the basis for your conjecture is; you're going to find out one way or another whether that's correct). Next, calculate the probabilities using players' indifference conditions. Let $p=\sigma_1(B)$ and $q=\sigma_2(A)$, we have
\begin{align}
-3p&=-1&&\Rightarrow\quad p=1/3\\
3q+1-q&=1-q&&\Rightarrow\quad q=0.
\end{align}
[This suggests that your calculation for $q$ was incorrect.]

Lastly (this is the most easily forgotten step), check that no one has an incentive to deviate from this equilibrium. In this case, player 1's payoff is $1$, which is already highest given player 2's strategy of choosing $B$ with probability 1. He'd be indifferent between mixing in other proportions over $B$ and $C$, and his payoff is strictly lower if he plays $A$ with positive probability.

Player 2's expected payoff in this equilibrium is $-1$, which is also the highest given player 1's mixed strategy. She's indifferent between mixing over $A$ and $B$ with any other proportions and is strictly worse off if $C$ is played with positive probability.

So, one MSNE is $((0,1/3,2/3),(0,1,0))$. This is only borderline consistent with your initial conjecture because $\sigma_2(A)=0$. But it's nonetheless a MSNE. In fact, there are infinitely many MSNEs of this form: $((0,p,1-p),(0,1,0))$ where $p\ge1/3$. This is a full description of all equilibria (including the pure one) in this game.


$1.25 PPP per day poverty threshold - What precisely does it mean?

A partial answer : I m not sure what measurement the World Bank would recommend, but I suspect that in practice, the expenses that are taken into account to measure whether one is below or above the 1.25$ threshold vary from one study to another.

For instance in their famous The Economic Lives of the Poor, Banerjee and Duflo write

  "From each of these surveys we identified the extremely poor as those living in households where the consumption per capita is less than \$1.08 per person per day, as well as the merely “poor” defined as those who live under \$2.16 a day using the PPP in year 1993 as benchmark.3 The use of consumption, rather than income, is motivated by the better quality of the consumption data in these surveys (Deaton, 2004)."

The use of "consumption per capita" suggest that capital income, such as the benefits one get from owning her own house, would not be taken into account.
Because data quality considerations seem to enter the picture, it opens the door to much variation from one study to the other.

The above quote however suggests that at equal data quality, income data would be preferred.


Comparative statics with multiple equations using calculus

Since this looks like a homework and there was no attempt made, in order to avoid giving full solution I will just give you conceptual explanation using equation 1 and 2 (you can later add equation 3 using steps below).
The standard method how you approach this is by taking total derivative and solving the system for $dY$. Total derivative is similar to regular derivative but you will take it with respect to every variable in the system.  Here you will have set of equations:
$$ dY = C_Y'dY + C_t'dY + I_r'dr + dG + dNX \\
 \frac{PdM-MdP}{P^2}= M^{d'}_Y dY +  M^{d'}_r dr $$
The same logic can be also applied to equation 3 that I purposefully omitted.
Now you can employ any method for solving system of multiple equations you want. The most common methods of solving sytems of equations are substitution, linear combination method (i.e. the method where you add multiple equations), various matrix methods such as Cramer's rule. However, no matter what method you use they will all work but get progressively more difficult the more equations you have. There is really no good shortcut here. Even matrix methods that work reasonably well when you have multiple equations can quickly become unwieldy as number of equations grow. In practice you would not solve difficult systems by hand but just set up the system in pyhton or matlab and let it solve it for you (however, in classroom setting it is unlikely you will encounter systems with more than 4 equations).
In this case the substitution is the quickest method so I will us that one. We can solve the equation 2 for $dr$:
$$ dr = \frac{PdM-MdP}{P^2M^{d'}_r} - \frac{M^{d'}_Y}{M^{d'}_r} dY  $$
Now substitute this equation back into the first one to get:
$$dY = C_Y'dY + C_t'dY + I_r' \left( \frac{PdM-MdP}{P^2M^{d'}_r} - \frac{M^{d'}_Y}{M^{d'}_r} dY \right) + dG + dNX$$
solve for $dY$:
$$ dY =  \frac{1}{\left(1 -  C_Y' -  C_t' - I_r' \frac{M^{d'}_Y}{M^{d'}_r} \right)} \left( I_r'  \frac{PdM-MdP}{P^2M^{d'}_r} + dG + dNX\right)$$
This expression should already be familiar to you from your macro 101 class - the first fraction is the multiplier and the second brackets are the exogenous sources of spending.
Now if you want to examine what happens when let's say government spending increases ceteris paribus you will just set all other $dX=0$ save for $dY$ and $dG$ (because other things will not change) giving you in this case:
$$ dY =  \frac{1}{\left(1 -  C_Y' -  C_t' - I_r' \frac{M^{d'}_Y}{M^{d'}_r} \right)} dG $$
And then we can find the partial derivative of $\frac{\partial Y}{\partial G}$ simply as:
$$ \frac{\partial Y}{\partial G} =  \frac{1}{\left(1 -  C_Y' -  C_t' - I_r' \frac{M^{d'}_Y}{M^{d'}_r} \right)} $$.
And you are done this expression tells you by how much the $Y$ changes when $G$ changes. You can redo the above for any other variable you want, and you can easily add the third equation as well (just take its total derivative and do 1 extra substitution that is required). The above method will work for arbitrary system. Sometimes it is easier to use different method of solving the system of equations but that is very situational and there I don't have better advice than just to practice various ways of solving systems and in the end you will develop 'knack' for knowing which method is quickest in particular case (although they will all work just some are sometimes quicker/easier to apply).


Is this equivalent to the game of chicken?

As you stated the strategy set in the traditional chicken game has two elements and in the Akagi version it is an interval, so there will be no bijective mapping between the two.
If I misunderstood and you are using a different notion of equivalence please clarify.

Also as you state in the Akagi version there are weakly dominant strategies but in the traditional version there are none. I believe the story of the traditional version is rather about two cars driving towards each other and you have to decide whether to swerve at the last moment or trust that the other driver will. Even if you were to extend the strategies to allow swerving before the last moment this game would differ from the Akagi version, because here not swerving at all is better than swerving at the last moment if the other driver swerves at last moment whereas swerving at the last moment is better if the other does not swerve at all, hence there are no weakly dominant strategies.


What are the known / alleged problems against using energy as currency?

tl;dr: Energy is one of the worst ideas for a currency that anyone's had.
It's not fungible, it's not a store of value, and has absolutely no scarcity. Let's look at the details.
Commodity currencies are problematic
jmbjera has listed core problems with the gold standard, and with commodity currencies generally: its supply is managed not by a democratically-elected government, but by one industry, so you remove a hugely useful tool from the policy-makers' toolbox; and you expose yourself to very high short-term volatility.
But even amongst commodities, energy has unique issues that make it worse than useless as a currency.
Energy is ubiquitous and bountiful
There's absolutely enormous quantities of it lying around everywhere. One kilogramme of anything contains mind-boggling amounts of energy - $E=mc^2$, right? c, the speed of light, is huge. Squaring it makes it a lot bigger (when we're in SI units). $3\times10^8m/s$ squared becomes $9\times10^{16}$ - that's how many joules of energy in 1kg of matter. Any matter. Soil, platinum, flesh and blood, it doesn't matter, it's all the same amount of energy: 90 Petajoules; aka 25 Terawatt-hours. That's a few weeks worth of electricity supply in Britain. So an energy currency means one kilogramme of sand is equivalent value to sufficient electricity to power that G7 country for a few weeks.
Additional energy arrives on Earth every second of the day
So not only is there immense quantities of energy lying around for free, but vast amounts arrive every second from our friendly neighbourhood fusion reactor, the sun. Irradiation on the earth is about 130 Petawatts. That's about four orders of magnitude more than human energy consumption. Or, to put it another way, sunlight on one ten-thousandth of the Earth's surface brings energy in at about the same average rate as global energy demand.
Energy's value varies hugely in time and space
But it gets worse, as far as currencies are concerned. See, energy's value differs constantly in time, in space and in form. You must have noticed that wind and sunlight are free - no one's charging you for them - and they're energy. Whereas you have to pay for natural gas. You probably pay even more, joule for joule, for diesel or petrol (gasoline). And you will pay a lot more for electricity. On top of that, one unit of electricity has a very different price depending on where and when you buy it, and the quantity you're buying it in.
Energy's form drives its economic value. 1 Joule is rarely worth 1 Joule
Electricity is a really high quality vector of energy: it can do huge amounts of work. But you can only use it in the instant it's generated (ok, you can store it briefly in a capacitor, but that will cost you, too). Heat is a really low quality vector of energy: and the closer the temperature is to ambient temperature, the less work it can do. i.e. the less economic value it has. Now, we've got enough problems with variable marginal values of money as it is, without this scale of problem. The other thing is, if you're storing your currency of energy as heat, it's losing its value every second as it cools off. Yet, at the same time, cooling has economic value, in aircon and in refrigeration. So that's a lack of energy that has economic value.
Many energy transformations add economic value but consume energy
Refining crude oil costs you energy and other resources. And the end product is more valuable, which is why people do it, and make a profit at it. But that would be absurd if energy were the currency.
On sophistry
I've seen the argument for an energy currency before. Usually, when you dig deeper, they're not really talking about an energy currency. They're actually trying to sell something. Typically, but not always, fossil gas. So their proposed energy currency turns out to be something very like gas futures contracts (which have the usual commodity-currency problems: control of supply lies with one industry, not democratically-elected governments; and the whole economy is exposed to high short-term volatility). Now, as we know, we've got far more gas than we can possibly burn (see my colleagues' paper in Nature on this), and the gas industry is starting to get nervous. And so it should. This desparate bid for a "gas currency" is just one more bit of special pleading and rent-seeking from what must become a dying industry.