First, two clarifying points:

The capital injection program, known as the Capital Purchase Program (CPP), was authorized by Congress as a part of the Troubled Asset Relief Program, and run by the US Treasury, not the Federal Reserve. However, the Federal Reserve Board and in particular the Federal Reserve Bank of New York (FRBNY, primary regulator of the major dealer banks) were both involved in the design of the program.
The program was voluntary, not mandatory, though the heads of the nine largest institutions were certainly strongly encouraged to participate, as described by Timothy Geithner, then-President of the FRBNY, in Stress Test:

  As we developed our scripts, I pushed hard to make the capital investments sound as close to mandatory as possible. We couldn't force participation, but we could make it sound inevitable. I expected some firms to resist taking new capital with government strings attached. But in a financial crisis, no firm is ever as well capitalized as it thinks. Lehman thought it had plenty of capital until it didn't.

Having made that clear, we can turn to your question, which is about the "government strings" that Geithner mentions. These took three primary forms:

Dividends and warrants. These were in the authorizing law. Quoting the Treasury's CPP page:

  Most financial institutions participating in the CPP pay Treasury a five percent dividend on preferred shares for the first five years and a nine percent rate thereafter. In addition, Treasury received warrants to purchase common shares or other securities from the banks at the time of the CPP investment. The purpose of the additional securities was to enable taxpayers to reap additional returns on their investments as banks recover.

The ability to nominate up to two members to the board of directors of a participating institution, should the institution miss six quarterly payments to the Treasury, per the CPP contracts.
Limits on executive compensation. These were in the authorizing law that Congress passed (titled the "Emergency Economic Stabilization Act of 2008"), and they're well-described on the law's Wikipedia page:

  If the Treasury purchases assets directly from a company, and also receives a meaningful equity or debt position in that company, the company is not allowed to offer incentives that encourage "unnecessary and excessive risks" to its senior executives (that is, the top five executives). Also, the company is prohibited from making golden parachute payments to a senior executive. Both of these prohibitions expire when the Treasury no longer holds an equity or debt position in that company. The company also is given "clawback" permission; that is, the opportunity to recover senior executive bonus or incentive pay based on earnings, gains, or other data that proves to be inaccurate.

  If the Treasury purchases assets via auction, and that purchase exceeds $300 million, any new employment contract for a senior officer may not include a golden parachute provision in the case of involuntary termination, bankruptcy filing, insolvency, or receivership. This prohibition only applies to future contracts; golden parachutes already in place will remain unaffected.
  In either scenario, no limits are placed on executive salary, and existing golden parachutes will not be altered.


How is Regression different from Econometrics?

To put it simply regression modeling is a technique, a tool, used by econometricians. Regression deals with dependence amongst variables within a model. But it cannot always imply causation.

As economists, we are biased towards establishing causal relationships. So, this explains our focus on endogeneity issues and strategies of identification to establish causality.

In other words, as Ragnar Frisch (one of the founders of the Econometric Society) would say

  Econometrics by no means the same as economic statistics

See also the answers to What is an econometrician? where the key idea is that economic theory is crucial to econometrics and understand measurement. Neither theory nor measurement on its own is sufficient to further our understanding of economic phenomena. We need both and measurement without theory is unlikely to provide a satisfactory explanation of the way economic forces interact with each other.

In summary, regression is super useful but not sufficient to make causal inference and to understand economic phenomena.

Stock and Watson's Introduction to Econometrics and Jeffrey Woolridge's Introductory Econometrics are both excellent textbooks.


Identification strategy for measuring impact of employee turnover on project quality

This looks challenging, and the following would only work in circumstances where:

Reliable estimates can be made of how long each completed project should have taken to complete.
Good records are maintained of the progress of individual projects, enabling assessments to be made of the form "Project A was X% complete after n years".

Suppose two projects that were estimated to require 3 years were actually completed in 5 years, and in both cases a project manager had moved on after 2 years.  One might then focus on how complete each project was at the time when the project manager left.  If one project was 70% complete at that time, and so took a further 3 years on the last 30%, that suggests Scenario 1.  If the other project was only 40% complete after 2 years, and subsequently progressed at the same speed taking a further 3 years on the remaining 60%, that suggests Scenario 3.


Does mathematical education imply a better salary?

Math is easier if you are smarter. As such, math education is a costly and therefore credible signal of general intelligence. Below are two experiments that try to get around this selection issue by looking at exogenous variation in worker mathematical ability on labor market outcomes. However, a word of caution. They do not present evidence that mathematical education makes you more productive. Even if all mathematical education is is a signal of intelligence, it can be valuable to individual workers to get more mathematical education, and they will on average benefit from interventions that lower the cost of getting a mathematical education

  We examine the link between math skills and labor-market outcomes
  using a resume-based field experiment. Specifically, we send
  fictitious resumes in response to online job postings, randomly
  assigning some resumes to indicate stronger math skills, and measure
  employer responses. The resumes that are randomly assigned to indicate
  stronger math skills receive more interest from employers than the
  comparison resumes. Our findings add to the body of evidence showing
  that stronger math skills positively affect labor-market outcomes.

Math skills and labor-market outcomes: Evidence from a resume-based field experiment by Koedel and Tyhurst (2012)

  Outsourcing  of  jobs  to  low-wage  countries  has  increased  the
  focus  on  the  accumulation  of skills – such as Math skills – in
  high-wage count ries. In this paper, we exploit a high school pilot
  scheme  to  identify  the  causal  effect  of  advanced  high  school
  Math  on  labor  market outcomes.  The  pilot  scheme  reduced  the
  costs  of  choosing  advanced  Math  because  it  allowed for at more
  flexible combination of Math with other courses. We find clear
  evidence of  a  causal  relationship  between  Math  and  earnings
  for  the  students  who  are  induced  to  choose  Math  after  being
  exposed  to  the  pilot  scheme.  The  effect  partly  stems  from
  the  fact  that these students end up with higher education.

Is there a causal effect of high school math on labor market outcomes? by
Joensen and Nielsen (2009)


Weak axiom of revealed preference and choice coherence - how to show they are equivalent

Yes, they are equivalent. Here is a formal proof by contradiction.

WARP $\Rightarrow$ Choice coherence

Suppose that WARP holds but that choice coherence is not true. There exists $B,B'$, $x,y \in B \cap B'$ such that $x \in c(B)$, $y \notin c(B)$ and $y \in c(B')$.

But WARP applied to the conditions $y \in c(B')$ and $x \in c(B)$ implies $y \in c(B)$. This is a contradiction.

Choice coherence $\Rightarrow$ WARP

Suppose now that choice coherence is true but that WARP is falsified. There exists $B,B'$, $x,y \in B \cap B'$ such that $x \in c(B), y \in c(B')$ and $x \notin c(B')$.

But choice coherence applied to the conditions $y \in c(B')$, $x \notin c(B')$ yields $x \notin c(B)$. This is a contradiction.


How does a division of a company spinning off from it's 'parent' company change it from a cost center to a profit center?

The parent company pays (roughly) the same amount for IT services as it did before, but those payments are now counted as income for the spinoff company.  It's mostly an accounting trick, although as Aidan O'Gara pointed out, the new spinoff may take on other clients and start to generate additional income that way.

Note that some companies use a form of internal currency to track inter-departmental services.  In such situations, departments providing critical infrastructure (such as IT) can often build up a hefty surplus of this internal currency, which can sometimes be spent by managers on internal "perks" for employees (free meals in the company cafeteria, free transportation home from work in a company limo, etc.)  I enjoyed such perks (working in a software support department as an otherwise low-paid intern) at a Wall Street firm in the 1990s.  However, if such internal accounting is denominated in actual currency — as it was when I worked at a public university — then it can lead to flawed apples-to-oranges comparisons by some departments, in which they (incorrectly) believe it is cheaper to use an outside vendor rather than the institution's internal department, due to local optimization that ignores economies of scale or the needs of other departments.


This is a huge question and I doubt any SE answer could address this fully, but I would like to elaborate on a few points.

Effects of the Alaska Permanent Fund Dividend

I would argue that the primary success of the Alaska Permanent Fund Dividend is its positive effect on decreasing poverty. The effect on employment seems to be more neutral. Here is an exerpt from the latest research paper by the National Bureau of Economic Research related to the subject,

  The employment to population ratio in Alaska after the introduction of
  the dividend is similar to that of synthetic control states. On the
  other hand, the share of people employed part-time in the overall
  population increases by 1.8 percentage points after the introduction
  of the dividend and relative to the synthetic controls. The
  unconditional cash transfer thus has no significant effect on
  employment, yet increases part-time work.

  Given prior findings on the magnitude of the income effect, it is
  somewhat surprising for an unconditional cash transfer not to decrease
  employment. General equilibrium effects could explain why we do not
  find a negative effect on employment. Indeed, in our unique setting,
  the whole population in the state receives the dividend. Therefore, it
  is plausible that the dividend increases labor demand through its
  effects on consumption. And indeed, we find that the non-tradable
  sector shows more favorable effects than the tradable sector. In the
  tradable sector, employment decreases and part-time work increases,
  while in the non-tradable sector the effects on both employment and
  part-time work are close to zero and insignificant. Overall, the
  evidence is consistent with positive macro effects offsetting any
  negative micro effects, and leading to an overall null effect of an
  unconditional cash transfer on aggregate employment in the long-run.

In my opinion, that employment didn't decrease is surprising and positive. But I have heard some people spin its effect on jobs negatively by mentioning how it didn't increase employment. My guess is as time goes on people will spin this data in whatever way they see fit.

On the 4th industrial revolution

Yang asserts that millions of manufacturing jobs have disappeared in the Midwest and that is how Donald Trump got elected. As the following chart shows manufacturing jobs have declined in number since 2000 without much recovery,

But stagnant wages have been an issue since 1970, and I think this has caused some push back among many economists that argue the effects of automation have been over-amplified over the effects of de-unionization. That said UBI could act as an expensive large scale form of unionization.

On the framing of payments

I hear a lot of people talk about how the government would "pay for something" and personally I don't like the framing of this question. It often weakly implies that the government can't run a budget deficit. In the Fifteen Fatal Fallacies of Financial Fundamentalism, William Vickery eloquently addresses this as follows:

  Deficits are considered to represent sinful profligate spending at the
  expense of future generations who will be left with a smaller
  endowment of invested capital. This fallacy seems to stem from a false
  analogy to borrowing by individuals.

  Current reality is almost the exact opposite. Deficits add to the net
  disposable income of individuals, to the extent that government
  disbursements that constitute income to recipients exceed that
  abstracted from disposable income in taxes, fees, and other charges.
  This added purchasing power, when spent, provides markets for private
  production, inducing producers to invest in additional plant capacity,
  which will form part of the real heritage left to the future. This is
  in addition to whatever public investment takes place in
  infrastructure, education, research, and the like. Larger deficits,
  sufficient to recycle savings out of a growing gross domestic product
  (GDP) in excess of what can be recycled by profit-seeking private
  investment, are not an economic sin but an economic necessity.
  Deficits in excess of a gap growing as a result of the maximum
  feasible growth in real output might indeed cause problems, but we are
  nowhere near that level.

Note that while the above was written in 1996, the last sentence still applies in the year 2019. Notably, inflation is still below the 2% target. Instead we should always be asking does this expenditure make sense? "Make sense" is of course a broad term but vaguely we are concerned with whether it would increase GDP, increase life expectancy, decrease child mortality, etc. Or to put it differently, is this the best way to spend trillions of dollars?

On the effects of UBI

    Bad environment? Lower class people will buy more drugs, alcohol, and stuff like that, creating a negative atmosphere or a bad stereotype in America

I think this is an unjustified fear. None of the UBI pilots so far have seen increased drug rate use in participants.

    But the positive side is that America's economy can now renew itself since people are buying more goods

UBI implemented through a VAT tax and debt monetization is effectively a redistribution mechanism from those who consume more and those who hold more cash. Broadly, poor people should get more money to consume, which should boost GDP:

  The IMF report said the way income is distributed matters for growth. "If the income share of the top 20% increases, then GDP growth actually
  declines over the medium term, suggesting that the benefits do not
  trickle down. In contrast, an increase in the income share of the
  bottom 20% is associated with higher GDP growth."


What are the current economic theories of product advertising?

There is a vast literature in both economics and marketing on advertising. Some highlights relevant to your specific question:

A classic paper (Grossman and Shapiro 1984) looks at competitive advertising and indeed finds that advertisers tend to send too many ads. Intuitively, some customers attracted by an ad aren't new demand, but rather simply stolen from a rival firm. Sending the ads these consumers saw wasted resources without creating any genuinely new trade, and are therefore sub-optimal.
More recent work (such as Anderson and de Palma, 2012, and van Zandt, 2004) considers information overload: the problem that ads crowd out other ads when consumers have limited attention. Again, this leads to sellers sending too many ads.

More generally, Bagwell (2007) has a nice survey of the economic literature on advertising.

Incidentally, it is also conceivable that there might be too few ads in equilibrium for some kinds of product. For example, if you and I both sell products that are very similar and the main function of ads is to inform consumers about the characteristics of those products then advertising is a public good. I would prefer to free-ride and let you bear the expense of advertising to inform consumers. I can take the money I save on advertising and use it to offer lower prices or better quality and thereby capture the demand of the consumers you so generously informed. The fact that arrangements such as vertical integration or franchising help to mitigate this free-riding problem is one reason why such arrangements are sometimes viewed more favourably by competition authorities.


Repeated games with decreasing marginal returns

It seems to me that this will not make a difference if you only consider pure strategies. Consider that player $i$'s winnings is $x$ and from this he gets utility $U_i(x)$. As long as $U_i()$ is increasing in $x$ player $i$ will prefer outcomes with larger winnings to lower ones, which is exactly what he would do if he had constant marginal returns.

Another way of forming this argument is that as long as $U_i()$ is increasing another utility function representing the same preferences over outcomes would be $\hat{U}_i(x) = x$. (You can perform a monotone transformation that transfers $U_i()$ to this.)


How does buying/selling houses affect the GDP?

There is only a change in GDP to the extent there are market goods and services used in the sale and only those goods and services are counted. The actual sales revenue are irrelevant. For example, the home inspection, appraisal, brokerage fees, and, I believe mortgage closing costs, would be in GDP. If you and your sister swapped houses in as-is condition without getting the market, bank, or tax authority involved, there would be no change in GDP.


What you're doing when you calculate $dQ/dP$ is assuming that all forms change their process by $dP$. This is the same as saying that the market price changes by $dP$ and is why you get the same number.

The main difference here is that when the book says price elasticity of demand is higher for an individual firm, it means that, holding the prices of other firms constant, a 1 percent change in a firm's price would result in a greater change in quantity demanded than when all forms change the price.


Can the Certainty Equivalent be negative?

If you start out with €0, then the certainty equivalent of losing €2.5 with probability 1 is -€2.5.
Your exercise basically asks you to calculate what difference winning the lottery with a small probability makes. Given this utility function, not much.


Stock Valuation - Interest rates, Dividends, and Shares Repurchases

We value stock if no dividends are paid, exactly for the reason you mention, we can sell them at a higher value later on. The value added to the stock if we invest the profits (would be dividends) into the company should be around the same value the dividends would pay off. However instead of paying taxes each time we get a dividend we can pay them just once when the stock is sold potentially saving on taxes.

P.s. Blanchard is a great book and how I first started learning macro :).


Check if a utility function represents a monotone preference

Using this definition of monotone preferences (Thanks to User Art). Let $X\subseteq \mathbb{R}^n$ be the set of possible consumption bundles, and $x,y\in X$. We have weak monotone preferences if $y>>x \implies y\succ x $, and strict preferences if $y\geq x \implies y\succ x$. If we have a utility function $u: X\to \mathbb{R}$, then $y\succ x$ is the same as $u(y)> u(x)$.

Regarding notation for two bundles $x,y\in X, x=(x_1,x_2,...,x_n), y=(y_1,y_2,...,y_n)$, we say $y>>x$ if $y_i>x_i$ forall $i=1,2,...,n$ and we say $y\geq x$ if $y_i\geq x_i$ forall $i=1,2,...,n$ and there is a $j$, such that $y_j>x_j$. (This notation is not universal)

Now, see if you can use this to answer your question


In demand and supply chart, why is it assumed that import will only happen if domestic suppliers are unable to supply?

The standard supply and demand model is a model. Models by definition are distorted version of reality not reality by themselves. The model also ignores transportation costs and other real life issues as well.
However, think about the main message of the model that tariffs decrease overall welfare. Assuming that consumers randomly decide to either consume imported product or domestic product would indeed affect possible size of government tax revenue but not in any way change the main outcome that overall welfare has decreased as when you calculate area of all losses even in corner case where everything is imported taking into account higher government revenue there would still be net welfare loss. Hence adding this bit of 'realism' (I am using '' as randomly picking product is still not realistic either) would complicate math, make your exams more difficult and would not teach you much more. The point of every model is to be as simple as possible in order to be general but still complex enough to allow us to analyze the issue of interest in a way that exposes the underlying economic mechanism.
Of course, different people can disagree about where the dividing line between being too simple to be useful is - that’s an opinion based question, but it’s common for undergraduate models to be oversimplification. In undergraduate physics you will be constantly assuming perfect vacuum, massless points, zero friction etc. all of which is unrealistic in order to gain enough knowledge to understand more nuanced models. All fields use this method of ‘teaching beginners by oversimplified models’ for better or worse.
Later in your academic carrier at PhD or Master level you will encounter far more complex and also more realistic trade models. If you want to see them before you get there you can read Advanced International Trade by Feenstra or Applied International Trade by Bowen, Hollander or Viaene. Or published papers on these topics. Some of those models deal with geography, monopolistic competition and other issues that address international trade in more nuanced way than a Econ 101 could. For the sort of question you are asking here some industrial organization (IO) model with explicitly modeled competition would be more appropriate than assuming 50/50 split for example.
PS: Your last picture is actually not correct. The Government revenue, in this model, will always be a rectangle since it is by definition $Q_{import}*t$.  A correct way of visualizing government revenue if you assume half of the quantity would be imported no matter what would be this:


Will major economies be able to repay their debt?

Whether a country's debt is sustainable is a difficult question to answer. Bohn developped a framework for answering this question and the cited paper is a summary of much of its findings.

Henning Bohn, 2005.
"The Sustainability of Fiscal Policy in the United States,"
CESifo Working Paper Series
1446, CESifo Group Munich.

Bohn proposes to estimate a fiscal reaction function to model a country's fiscal policy. He concludes that for a country's fiscal policy to be sustainable, it is sufficient that its primary balance is a (linear) increasing function of its lagged debt-GDP .

$pb_{it}= \alpha_i + \beta_i d_{it-1} + \delta_i X_{it} \ \ \ \ \  \beta_i>0$

($X_{it}$ is a vector of control variables)

There are however some problems with this framework. If for example does not exclude an ever increasing debt limit, which seems counter-intuitive. Some of these problems are resolved in Ghosh et. al. 2011 and non-linearities in a fiscal reaction function are added and they introduce the concept of fiscal space.

Atish R. Ghosh & Jun I. Kim & Enrique G. Mendoza & Jonathan D. Ostry & Mahvash S. Qureshi, 2011.
"Fiscal Fatigue, Fiscal Space and Debt Sustainability in Advanced Economies,"
NBER Working Papers
16782, National Bureau of Economic Research, Inc.

Yet, there are still some drawbacks to this approach. Japan is a country that has almost reached its debt limit but financial markets doe no seem to response to this fact. Therefore this framework as evolved from estimating these fiscal reaction functions to stochastic debt simulation. Celasun, Debrun and Ostry wrote a seminal paper on this procedure.

Oya Celasun & Xavier Debrun & Jonathan David Ostry, 2006.
"Primary Surplus Behavior and Risks to Fiscal Sustainability in Emerging Market Countries: A "Fan-Chart" Approach,"
IMF Working Papers
06/67, International Monetary Fund.

They for example calculate the probability that a country's debt-GDP ratio increases more than 10% in comparison to a baseline-year. But again, this does not answer some fundamental questions. If for example the indebtedness of a country does increase with 10% in 5 years, does this mean that fiscal policy was insustainable?

It should be clear now that determining the sustainability of a country's debt is difficult. In the end, it really all comes down to the fact that a country's fiscal policy is sustainable as long as institutions and individuals are willing to buy their debt instruments.

edit:
To answer your comment, I think the easiest concept to understand, it the one of fiscal space so I will try to explain that:

Assume a country's fiscal policy looks the following:

$pb_{t}= \alpha + \beta_1 d_{t-1}+ \beta_2 d_{t-1}^2 + \delta X_{it} $

In which $\beta_1>0>\beta_2$. This thus implies that a country's primary balance initially increases as debt increases to offset the debt accumulation but will decrease after a certain point, called fiscal fatigue. This fiscal fatigue implies a limit to the debt ratio a country could pay back. To see this, you have to look at the debt dynamics, (r-g) equals the interest rate-growth differential:
$\Delta d_t= (r-g) d_{t-1} -pb_t$

This equation explains how an increase in primary balance could counter debt accumulation. The debt limit, lets call it $d^m$ is the highest debt ratio for which the primary balance sufficies to hold the debt ratio constant. ($X_t$ is ignored):

$(r-g) d^m = pb= \alpha + \beta_1 d^m + \beta_2 d^{m2}$

After $d^m$, the primary balance will never be high enough to hold debt constant due to fiscal fatigue. Lenders to the government know this and thus will never buy debt securities after this point, causing a country to default.
 Fiscal space is now simply the difference between a country's current debt ratio and the debt limit and thus is a measure of the amount of fiscal maneouvrability a country has left before it will default.


Why doesn't Aggregate Demand mean National Income whereas Aggregate supply is equal to national income?

The existence of aggregate demand does not mean that aggregate demand is fulfilled. For instance if the aggregate planned demand is 100 billion dollars and due to crisis or war output is very low at just 10 billion dollars. Does the national income equal 100 billion dollars?


Keat is mistaken. Most economists agree with Mankiw. The term oppportunity cost includes all costs, including explicit out-of-pocket ones and any other implicit ones.
As Henderson states:

The word “opportunity” in “opportunity cost” is actually redundant. The cost of using something is already the value of the highest-valued alternative use. But as contract lawyers and airplane pilots know, redundancy can be a virtue. In this case, its virtue is to remind us that the cost of using a resource arises from the value of what it could be used for instead.

Here are some definitions from introductory economics textbooks:

The next-best good that is forgone represents the opportunity cost of a decision. (Samuelson & Nordhaus, Economics, 2010, p. 13)
Opportunity cost is the benefit that you might have gained from choosing the next-best alternative. (Colander, Microeconomics, 2017, p. 9)
We refer to this best alternative activity as the opportunity cost. (Acemoğlu, Laibson, & List, Microeconomics, 2016, p. 41)
Economists call that kind of cost—what you must give up in order to get an item you want—the opportunity cost of that item. ... The opportunity cost of an item—what you must give up in order to get it—is its true cost. (Krugman & Wells, Economics, 2015, p. 7).

Of course, as mentioned by Henry, you can define any term as you please. So the question might be, "What is the standard or most commonly used definition among economists?" And the answer is, "Mankiw's and not Keat's."


Is it right to derive social marginal benefit by adding individual prices instead of quantities?

Vertical summation of the individual marginal benefit curves is the correct way to find social marginal benefit if the camera system, so far as the two stores are concerned, is a public good. Horizontal summation is the way to find total market demand for a private good.
A good is a public good if it satisfies two conditions.  One is non-excludability: that would be satisfied if neither store can prevent the other from benefiting from the camera system.  The other is non-rivalrousness: that would be the case if neither store's use of the system limited the other's use of it.
Even if the camera system covers both stores, it could be that it does not fulfil both these conditions. For example, one store might control its use and limit or charge for the other's access to photos / film taken.  In that case the non-excludability condition would not be met.
Let's suppose however that the conditions for a public good are fully met. Let's also suppose that "quantity" of the camera system can meaningfully be defined (this does not seem straightforward). In that case the quantity must be the same for both stores, and the social marginal benefit at that quantity is the sum (graphically the vertical sum) of the marginal benefits to each store. The principle of vertical summation is implicit in the Samuelson condition for optimal provision of a public good, according to which the quantity provided is optimal where the sum of the marginal benefits equals the marginal cost of provision (Samuelson 1954, see equation 2 p 387 and explanation p 388).
It possibly makes vertical summation seem paradoxical when the vertical axis of the diagram is labelled P for price.  It would be more correct I suggest to label it MB for marginal benefit, as in this diagram illustrating the Samuelson condition.


Is a country in economic downturn if its growth rate is negative, but improving?

I'm no macroeconomist, so other will correct me if I am wrong, but I would use the following definition:

More formally, if $Y(t)$ is GDP at time $t$, one could express this as

Recession: $Y'(t)<0$ for 6+ months

downturn: $Y''(t)<0$

recovery: $Y''(t)>0$.

The definition of recession (2 quarters of negative growth) is widely standardised. I think the definition of downturn and recovery is less standard, and therefore more subject to debate.


Interpretation of sub-unity money velocity

Hypothetically, if the US was to fall into sub-unity velocity, would there be a clear interpretation

$V$ is by definition the average number of times 1 unit of currency is used (See Fed explainer on that). A clear interpretation of $V<1$ is that there are some units of currency that were never used during the measurement period. If an hypothetical economy consists of 2 1\$ bills, 1 will be buried  in the garden and never used and the second one only used once to buy goods and services worth of 1\$ the velocity of money would 0.5 (you could either calculate this directly as $V=\frac{1+0}{2}$, or indirectly in this scenario $M=2$ and $PY=1$ so you will have $V=\frac{1}{2}$.
Furthermore, note the equation for velocity actually calls for nominal output not GDP. That is the equation is given by $V=\frac{PY}{M}$, nominal GDP is just gross nominal output (in fact even this is a derivation the relationship actually is $V=PT/M$ where $T$ is number of transactions (see Mankiw Macroeconomics 8ed pp 108) but under very light assumptions one can derive that $PT=PY$ so we can run with it). This does not mean you can't use gross nominal output as a proxy, gross nominal output tries to capture $PY$, but you should be aware that there will be measurement error and GDP excludes gray/black market activity and so forth.

how much significance would the act of crossing below one have?

There would be no special significance to that per se. This would be like asking what would be special significance of CPI index crossing some arbitrary threshold like 1.05.

Optionally, what if V was negative?

$V$ proper cannot be negative because the smallest amount of times you can use unit of currency is 0 times. Its like asking if number of pies that you eat can be negative. That is simply impossible. This being said you could probably measure negative $V$ if you use $GDP/M$ to measure it but then you are running into a problems that arise from a fact that GDP is not exactly the same as nominal output.
For example, consider some state that completely disintegrated into Hobbesian anarchy, where all activity is done in grey/black markets with exception of 1 importing company, yet for some reason there are still statisticians going around and recording output in formal sector. Now let us assume that people get 1000 nominal income earned at grey and black markets, use half of that to import goods and services. Since GDP only tracks legal activity and since imports count negatively toward $GDP=c+i+g+(x-m)$ one would record negative nominal GDP and then $V$ will be negative as well, but that's rather due to limitations of measurement technique that you use not because $V$ as a concept can be negative.


You should quote directly from the book you're citing, because I think you might be mis-reading it.

At the Econ101 level, there are two important frames for thinking about fixed costs: one is that in the long run, the contribution of fixed costs to average cost falls to zero.  You can see this in the standard textbook graph, which will typically look something like this:

Source: Wikipedia

Note that Average Fixed Cost tends to zero as more units are produced.  True to the circular nature of much economic reasoning, often "the long run" is defined as the period for which average fixed costs fall to zero.

The other frame is around what constitutes fixed costs in the first place.  In the short run - which, in your example, is the contract period - rents are fixed costs because they are contractually unavoidable. When faced with the decision about whether to operate, in the short run a factory treats fixed costs as "sunk" and cares only about recovering average variable cost (if I can produce widgets and sell them at cost, I am indifferent to producing or not producing).

Where it gets confusing is that in the "long run", defined as the period exceeding the term of the contract, the firm is free to negotiate rents and so these are now treated as variable costs.  This isn't really the same as "fixed costs falling to zero" because it's not as if the firm can somehow escape paying rent for buildings.  Rather, it reflects the fact that in the long run, the firm has some ability to control what it pays for rents.

It's pretty normal to be confused by this at the entry level.  In my opinion, it's not great nomenclature. The takeaway is that you need to be very clear about the context you're working in, before you can be clear about the meaning of the terms you're dealing with.  In one context, "fixed costs" are "one-time sunk costs" whose impact on average cost of production falls to zero in the long run because that one-time cost is distributed over a growing number of widgets.  In the other context, "fixed costs" are unavoidable costs in the short run that, in the long run, don't fall to zero but rather are redefined as "variable costs" because the firm gains the ability to control them in some limited sense.

In neither case does the firm avoid fixed costs by not paying them, unless it really does shut down.


Economic and Statistical Significance Of Coefficient

Statistical significance refers to situation where we can say the coefficient is statistically significant at some level based on test statistics (in regression $t$-statistics given by $\hat{\beta}/(s.e.(\hat{\beta}))$) and corresponding $p$-value.
Economic significance indeed depends on the magnitude of the coefficient. As pointed out in this question by KennyLJ sometimes rule of thumb when measuring economic significance is to look at if one standard deviation change in $x$ would result in 'sufficiently large' (which could be 1/2 of standard deviation or also 1 standard deviation change in $y$ - this is very context and field dependent - what is economically significant simply requires good knowledge of wider economic literature).
For example, if we would estimate regression of  daily wages ($w$), measured in ${\\\$}$, on education $(E)$, measured in years of education:
$$w_i = \beta_0 + \beta_1 E_i + e_i $$
and if we would estimate that $\hat{\beta_1} = 0.01$, and this estimate would be statistically significant at $1\%$ due to $t\text{-stat}=10$, we would say the coefficient is statistically significant but economically insignificant because if one extra year of education on average increases daily wages just by ${\\\$}0.01$ and that is nothing. Even if the coefficient is statistically significant from economic perspective it might as well be just zero. However, if $\hat{\beta_1} = 20$ meaning that on average one year of education increases daily wage by  ${\\\$}20$, it would be both statistically and economically significant because having extra $20$ dollars per day thanks to one year of education is non-trivial.
Next the estimated wage $\hat{w_i}= \hat{\beta_0} + \hat{\beta_1} E_i$ would be just that  - an estimated/predicted value by the model given what $E_i$ is. Since the predicted value is derived from estimates of $\beta_0$ and $\beta_2$ it can tell you about statistical significance of the result only as much as just looking at the betas individually.
Lastly $R^2$ is measure that tells you what amount of variation in the sample can the model explain. For example, $R^2=0.25$ would tell you that your model can explain. This is important to take into consideration but again does not tell you about economic significance of the result. For example, it is possible to have results where  betas are for all practical perspective very close to zero, but there is so little variation in the data that they are significant and you get excellent fit (high $R^2$), but that would not necessarily made the result economically significant if the magnitude of the effect is relatively miniscule.


Is there any investment model with time-varying depreciation rate?

A type of model you might be interested on is one where the depreciation rate is endogenous. This means, it depends on the value of other variables in the model, instead of being exogenous and constant (naturally, you can have a variable depreciation rate but exogenous, but that is as ad-hoc as a constant depreciation rate).

One example is this model. Equation 19.4 shows the formula of the depreciaton rate (which the authors also coll net utilization rate). Then, they state:

  Thus the adjustment path for the capital utilization rate is a flexible accelerator.
  This result rigorously establishes the empirical finding obtained by Nadiri and Rosen (1969) that the adjustment' path of the
  capital utilization rate is similar to the paths of capital and labor growth rates which are governed by flexible accelerators. Along the
  dynamic path, as the capital intensity decreases the net utilization rate declines for two reasons. First, there is the direct effect of the
  capital intensity on the net utilization rate. A decrease in the capital intensity leads the firm to reallocate resources towards capital output
  which decreases the rate. Second, there is the indirect effect, which arises because the decrease in capital intensity causes the price of
  unutilized capital to increase. Since the marginal value of unutilized capital increases, the firm then utilizes less of its capital and so the
  utilization rate falls. Thus $k^° > k^*$ implies that $\delta^° > \delta^*$. These results mean that along the dynamic path the net capital utilization rate
  and the rate of capital investment are inversely related while the net capital utilization rate and the labor investment rate are directly
  correlated.

This paper presents another model. They show that "embodied" technical change (i.e. technical change which comes with new machines rather than occur amid current machines) decreases the lifetime of capital, thereby increasing the depreciation rate.

This paper finds empirical support for a variable depreciation rate.


Basically structural unemployment is caused by a mismatch in the skill of the workers available in the labor market and the skill demanded by the "buyers" in the labor market (firms). These workers don't have the skills that the market needs, so no one is willing to hire them.

This happens because the industry in which these workers used to be employed in is in the decline, and I think this is the more correct definition.

Generally, a decline in an industry is caused by a technological change, but not necessary.

Example of tech change -> decline:

The decline in demand for scribes (people who manually write texts in books) when the printing press was invented. This leads to the "decline of the scribe industry" and structural unemployment.

Example of industry decline without tech change:

Lumber industry... after 100 years all trees are gone. This would lead to a decline in demand for lumberjacks and a structural unemployment without technological change.

The second question might be better asked as a separate question per the guideline :)


When JNK appeared on the market on Dec 04 2007 it started at approximately the same price level at SPY (143 vs 146). It is however not true that after this they moved together. I downloaded the price data and calculated the correlation of the price differences* for the first 180 days. The correlation coefficient was -0.0526.

They seem to move together because

both fell during the financial crisis
in Dec 2007 the prices were roughly the same
later the two prices drifted apart, this increases the scale and makes it hard to discern visually what happened in the early days.

So in essence this is an optical illusion.

*I did not run statistical tests for integration, but this seems very likely. Calculation was done in R using cor(diff(SPY$Close[1:180]),diff(JNK$Close[1:180])).


During QE what prevents the banks from selling government debt to invest in the stock market instead of lending out the money?

Nothing. Depending on which country we are talking about there might be a regulation preventing/limiting some banks investing in a stock market, in order to make banking sector less prone to collapse. For example, Basel rules mandate that banks have to keep certain level of capital buffer for their investments in stock market in order to make them resilient (see discussion about that here). This is very context dependent, there is actually no such thing as just a bank, there are commercial banks, investment banks, retail banks etc - all are regulated slightly differently and it is beyond scope of an SE answer to provide detailed overview rules for all individual institutions.
Besides that the only thing that would restrict them from doing so is that at some point it would not be profitable, especially if there is sufficient demand for loans.  Both equity and credit prices are subject to supply and demand forces.

But what if the banks instead of lending out money put most of their money in the stock market?

Ceteris paribus, this will stimulate economic activity the same way as issuing a loan would. From a macro perspective there is a little difference between bank buying from a firm A a bond with face value 1000USD with 10% coupon rate, versus bank buying 1000USD worth of stock expected to pay 100$ yearly dividend. What stimulates economic activity is firms getting the money (no matter whether as a loan or in exchange for equity) which allows them to increase investment.
If anything buying equity could potentially stimulate economy more, since there is some evidence that debt overhangs make firms invest less (e.g. see this Fed St. Louis explainer). Consequently, in situation where firms are already overleveraged such action would likely stimulate economic activity even more than just increase in borrowing.


Will inflation rise or fall when the Fed increase their interest rate?

Hint:

If you're keeping real interest rates constant in the second scenario, you should keep real interest rates constant in the first scenario.

If you do this, lowering the nominal interest rate will decrease inflation off the bat (by the Fischer's equation) by the same amount as the change in nominal interest rates.


Do welfare payments for children incentivise having children?

For example, "Assessing the Impact of the Maternity Capital Policy in Russia" (2014):

  Starting in 2007, the federal government has pursued an ambitious pro-natalist policy. Women who give birth to at least two children are entitled to "maternity capital" assistance ($11,000). In this paper we estimate a structural dynamic programming model of fertility and labor force participation in order to evaluate the effectiveness of the policy. We find that the program increased long-run fertility by about 0.15 children per woman.

Of course, the impact varies by country and incentive. See references and citations of this paper for more research.


How to solve a pricing competition for complementary goods

Here are some hints in the spirit of helping you prepare for the exam by thinking about the solution yourself:

You are right about the two players.

Your formulation of strategies is incorrect. Think of a strategy as a complete set of instructions that a player writes down before the game begins. These instructions are handed to a lawyer and must contain the information that the lawyer needs to play the game on the player's behalf and in accordance with his wishes. In particular, the instructions can't ask the lawyer to use information he doesn't have access to and must tell the lawyer how to deal with any circumstances that might conceivably arise. Put yourself in the shoes of the lawyer: you are supposed to choose a price on $R$'s behalf. What is the set of instructions that you would understand and have enough information to implement? Note that "$p_R<p_L$" doesn't pass this test because

it would require that the lawyer knows the value of $p_L$ before the game starts.
it doesn't pin down which $p_R$ the lawyer should choose. Even if he knew that $p_L=10$, there are many choices of $p_L$ that satisfy $p_L<p_R$. The poor lawyer would be very confused!

The payoffs from a game are a function that takes as an argument (i.e. as an input) a strategy for each player and spits out as the result the corresponding payoff. So $L$'s payoff function will look something like this $$\underbrace{(\text{L's strategy},\text{R's strategy})}_{\text{input}}\mapsto \underbrace{\text{payoff for $L$}}_{\text{output}}$$.

Now, imagine that I have asked $L$ and $R$ what their strategies are and have them written down here in front of me: $p_L=\ldots,\ p_R=\ldots$. Your job is to give me a rule that I can use to take this (and only this) information and calculate what each player's payoff will be. That rule is the payoff function.

Once you found the right payoff function, the question of what happens when the two prices are not equal will answer itself because the function will output the payoffs for any valid input strategies.


How to measure quality of education across countries?

A good reference is the Barro-Lee Dataset wich gives educational attainment
for 146 countries from 1960 to 2010. The data are disaggregated by sex and by 5-year age intervals. Their estimates of educational attainment provide a reasonable proxy for the stock of human capital.

An older reference is hosted by the World Bank.

In case you need more recent data, you may apply their methodology.


Why does an increase in money in circulation increases price levels?

new money is created; it belongs to somebody.
that somebody can either save/invest the money, or spend it on consumption.
provided at least some of the money is spent on consumption, the demand for consumer goods increases.
in a competitive market, this increase in demand results in an increase in the price level. (Note that the sellers are not "willingly rising prices" because, in this simple model of competition, sellers take the price a given natural consequence of the interaction of supply and demand).

You might also find it interesting to read about the quantity theory of money.


Lecture note recommendations for the endogenous growth model (EGM)

I think that the notes closely resembling what you are looking for - a gradual explanation of how the growth theories develop - are those from Acemoglu. Check his web page for slides and look in your school library for Intro to Modern Economic Growth. The slides are very detailed I think.

Growth models are, I believe, presented very rigorously. Additionally, in my opinion, it is discussed how they are connected.


Does unit elasticity has to be at exactly the middle of the demand curve?

No, that is only true in the linear case. For a simple counterexample consider
$$
D(p) = 1 - \sqrt{p}.
$$
$$
\epsilon(p) = \frac{d D(p)}{dp} \cdot \frac{p}{D(p)} = -\frac{1}{2  \cdot \sqrt{p}} \cdot \frac{p}{1 - \sqrt{p}} = \frac{1}{2}  \cdot \frac{\sqrt{p}}{1 - \sqrt{p}}
$$
\begin{eqnarray*}
\epsilon(p) & = & 1 \\
\\
\frac{1}{2}  \cdot \frac{\sqrt{p}}{1 - \sqrt{p}} & = & 1 \\
\\
\sqrt{p} & = & 2 - 2 \cdot \sqrt{p}  \\
\\
\sqrt{p} & = & \frac{2}{3} \\
\\
p & = & \frac{4}{9}.
\end{eqnarray*}
This is not in the middle of the demand curve in any sense.


Are White's Robust standard errors robust to clustered errors?

No. White's robust errors are only robust to errors that vary linearly based on X(t).  If you have a type of serial correlation or clustering you will want to use Hansen method to adjust standard errors.


That won't really happen in reality. A sales promotion which runs losses, as in your example, will not mean that the gross value added is negative. The final selling price of the item is the gross value added and that will not typically be negative. So, if an item is sold for 5 dollars then the gross value added is 5 dollars only. No matter if someone in the value chain made losses.
So, gross value added will never be negative unless and until the final selling price of something is negative.


Depends on what you mean by money supply: M0, M1 etc (see here for example). If M0, then printing money is literally issuing new coins and bills.

The Fed can certainly "add zeroes" and use the extra money to buy other assets. The frequency of this is an empirical question.

Ask your professor what she/he meant.


Why doesn't Nintendo fire up the old factories and re-produce *exact* copies of many of their most popular games, controllers and consoles?

I am unsure whether this qualifies as economics, but it would be something that might be discussed in business school. Furthermore, the answer is almost entirely engineering. As such, I will do this briefly.

Those “old” factories are probably in the same physical space as their current factories. They would need to build new facilities.
They would need suppliers to retool to manufacture obsolete electronics.
The cost of producing the old hardware would be much greater than using a modern CPU and emulator software, where they can package the entire library in one piece.
The video output would probably be useless on most modern TV’s.
Who is going to be willing to warehouse and retail these games? If they use emulator software, they can stick the entire library in one package, vs. one package for each old game.

These engineering problems imply a need for a large market, which is uncertain. Would collectors be willing to pay the same amount for copies of old games, or would they demand the “authentic” versions?


A variance is an incomplete measure of risk in a sense, that it measures uncertainty in security payoffs, rather than uncertainty in holder's welfare. In the simplest way we can demonstrate this point as follows.

Suppose that agents want to marginally increase her holding of an asset by $\xi$ and a unit of asset provides a payoff of $x$, which is a random variable. Now we model agent as caring about volatility in her consumption $c$, which is more reasonable, than assumption that she cares about volatility in the payoff directly. Then $$Var(c+\xi x) = Var(c) + 2\xi\cdot cov(c,x) + \xi^2Var(x)$$ and the last term is negligible as $\xi$ is marginal change.

So, as marginal changes are considered in equilibrium, it is more reasonable to take covariance between consumption and payoff as measure of riskiness of the security, rather than just variance of the payoff. In practice the aggregate consumption is often used as a proxy for individual consumption.

Yet another more accurate measure of riskiness can be introduced if we were to presume not direct care for volatility in consumption, but consider an agent with a utility dependent upon his consumption stream. It is a rather lengthy technical exercise and those who are interested should look it up in an excellent textbook "Asset Pricing" by John Cochrane. I will limit myself with presenting the result for multiperiod discrete time model here.

If we were to defined a random variable $m_t = \beta\frac{u'(c_{t+1})}{u'(c_t)}$ called stochastic discount factor for agent with von Neumann-Morgenstern utility and discount factor $\beta$ then by the similar argument as above we may derive that $cov(m_t,x_t)$ should be considered to be even a better measure of riskiness of the security.

Obviously, any nonrandom transformation of the above may be considered to be measure of riskiness, namely, standard deviation and correlation instead of variance and covariance are quite popular. Also different nonrandom normalizations may be applied to $m_t$. For example $u'(c)$ may be used, as $\beta$ is nonrandom and $c_t$ is known at time $t$


Why would a CBDC (Central Bank Digital Currency) be good for the economy?

There is good summary (with references) of the benefits of CBDCs on Wikipedia.
With respect to banking sector disruption see this 2018 report by the Sveriges Riksbank. It finds that

while a given outflow of retail deposits into e-krona
reduces banks’ liquidity portfolios and worsens their funding
profiles, banks can normally control this outflow via deposit rates.
Banks can also issue more market funding to restore their liquidity
and funding profiles.

That said,

The exact features of an e-krona can, however, be controlled by the
policy maker. In sum, we do not find any decisive argument against the
issuance of an e-krona when studying financial stability effects on
banks.


The dictionary definitions of capitalism (currently the most upvoted answer) have been critiqued as inadequate, in the sense that they are too broad, by Geoffrey Hodgson in his treatise, Conceptualising Capitalism (University of Chicago Press, 2015).

  Definitions proliferate. Most dictionaries stress private
  ownership and markets; many add the profit motive. The entry in The
   New Encyclopaedia Britannica (1998, 2:831) is typical: “Capitalism, also
  called free market economy, or free enterprise economy: economic system
  . . . in which most of the means of production are privately owned
  and production is guided and income distributed largely through the operation
  of markets.”

  But, if we consider systems based largely on private ownership, the
  profit motive, and markets, we can find many examples before the medieval
  era. [...] trade has existed for tens of
  thousands of years, markets may have appeared as long as five millennia
  ago in China, and they were evident in several locations in the eastern
  Mediterranean and the Middle East around 600 BC. Private ownership,
  the pecuniary calculus, and the profit motive have similarly played
  a prominent role in leading economic regions for thousands of years.
  Hence, prominent dictionary definitions of capitalism make it a system
  that has lasted for multiple millennia. If we adopt such definitions, then
  we need to find another word to describe the institutional structures that
  were consolidated in the eighteenth century and gave rise to spectacular
  economic growth and ongoing technological innovation.

And he does have his own definition (which I will not try to argue that is mainstream); its premise is that:

  Capitalism [...] requires special forms of state that cannot confiscate property arbitrarily at will, that are effectively restrained by laws, that have internal
  checks and balances, and that are faced with countervailing (democratic
  or other) powers that help protect a relatively autonomous legal system.
  Such states are necessary to legitimate and protect property rights and
  to enforce contracts. They required peculiar circumstances and a long
  time to evolve. Foreshadowed in the Italian city- states, they did not appear
  on a national scale until the seventeenth century, in Britain and the
  Netherlands. [...]

  I argue that, while markets are central to capitalism, capitalism is not
  simply a market system: unavoidably it contains different subsystems of
  governance, production, distribution, and exchange. Furthermore, capitalism
  cannot in principle have markets for everything or bring everything
  within the orbit of commodity exchange.
  In particular, under capitalism there can never be a complete set of
  markets for future labor power. For there to be full futures markets for
  labor, all workers must enter into contracts for their expected working
  life. This would be tantamount to voluntary bondage, limiting the freedom
  of workers to quit their employment. Paradoxically, pushing markets
  to their limits would mean the return of slavery for the workforce.

He goes on to semi-formalize a list of tenets that constitute capitalism first according to [his interpretation of] Marx and then similarly for Schumpeter

  M[arx]-capitalism

  A legal system supporting widespread individual rights and liberties to own,
  buy, and sell private property
  Widespread commodity exchange and markets involving money
  Widespread private ownership of the means of production by firms producing
  goods or services for sale in the pursuit of profit
  Much of production organized separately and apart from the home and
  family
  Widespread wage labor and employment contracts

  Most dictionary and encyclopedia definitions emphasize conditions
  (1), (2), and (3), albeit often without emphasis on the legal system. More
  than Marx, the notion of M-capitalism emphasizes the legal system and
  individual rights. Regarding condition (4), the separation of much production
  from the family and the domestic sphere was emphasized by Weber
  (1968) and is implicit in Marx’s writings. Weber rightly argued that
  this separation was important to subject production to systems of rational
  accounting and pecuniary motivation. Marx emphasized condition
  (5) because it flows from the nature of capitalism as generalized commodity
  production and encapsulates the driving antagonism between
  the employing class and the working class. [...]

  A problem [...] is that the addition of
  condition (5), widespread wage labor, to the preceding four does not effectively
  demarcate modern industrial capitalism from the four preceding
  centuries when wage labor was widespread in rural England.
  M-capitalism, as specified above, gives us a better-than-average definition,
  but not one that is sufficiently sharp in terms of historical specificity.

  Further problems of historical demarcation arise with the first four
  conditions. As noted in section 4.4 above, the idea that secure property
  rights were first established in England in the seventeenth century is a
  myth. Furthermore, money, markets, and private ownership have existed
  for millennia. The fourth condition is helpful but hardly sufficient as an
  explanation of massive expansion of capitalism around 1800. The five
  conditions are inadequate to demarcate historically the explosion of capitalist
  productivity even if they are necessary for that system.

As for S[chumpeter]-capitalism, he argues that it contains all of the above, except for #5, but that additionally it has

  A developed financial system with banking institutions, the widespread use
  of credit with property as collateral, and the selling of debt

  The addition of condition (6) takes seriously the monetary definition
  of capital and institutions promoting collateralization and the salability
  of debt. Capitalism is thus marked by “the predominance of ‘capital’”
  (Sombart 1930, 196). This definition points to the development of institutions
  of clearer historical specificity. According to this definition, the
  emergence of capitalism in England is marked by developments in financial
  institutions in the eighteenth century.

  Note that condition (1) in the definitions of M-capitalism and
  S-capitalism rules out (at least widespread) slavery. Consequently, both
  M-capitalism and S-capitalism involve missing futures markets for labor.
  With S-capitalism, there may be self-employment or worker cooperatives
  but not extensive slave labor. M-capitalism rules out an economy
  with predominant self- employment or worker cooperatives.

As you can probably guess, by synthesis, he argues that the definition of capitalism should include the superset of 1-6 from both M- and S-capitalism (which already have a lot in common).

  But there is a cost to the exclusion of condition (5) from this definition. [...] the advantage of retaining
  the employment contract in the definition of capitalism is that it demarcates
  the system from economies involving self-employed producers or
  worker cooperatives. Furthermore, the introduction of wage labor gave
  important incentives for labor-saving innovations and led to increases in
  productivity. Consequently, the definition of capitalism that is favored
  here— and described simply as capitalism, without any prefix— involves
  all six of the above conditions. [...]

  Conditions
  (5) and (6) serve as fuzzy historical bookends, together demarcating
  capitalism from preceding systems and possible successors. [...]

  Explicit reference to social class is also absent from the definition offered
  above. In his review of different definitions of capitalism, Michael
  Merrill (1995) argued that capitalism is much more than simply a market
  economy. He was right about that. But he wanted to define capitalism as
  “a market economy ruled by, or in the interests of, capitalists” (322). A
  problem [...] is that the basis of class itself has to
  be defined, and Marx and Engels were obliged to use legal terms such as
  ownership to do so. Being a capitalist or a worker is not a congenital attribute:
  it is the occupation of a social role that is determined in part by
  legal relations. Class-related legal terms appear in all points above, except
  condition (4).

  Marxists have abused class-centered definitions of capitalism. They
  see class struggle, where the destiny of one class is to seize control of the system from another. But, notwithstanding the power of capitalists over
  workers, capitalists do not control capitalism: it is more the other way
  round. Such large and complex systems cannot be fully controlled, as if
  from a single chamber of power.

And while he disses that idea that capitalists control capitalism itself as a caricature, he nevertheless argues that it is an integrated economic and political system (and in this he is not that original, as he acknowledges):

  The central role of the state within capitalism means that we must
  address politics as well as economics. [...] Douglass C. North, John J. Wallis,
  and Barry R. Weingast (2009, 269) argued: “The seeming independence
  of the economic and political systems on the surface is apparent, not
  real. In fact, these systems are deeply intertwined.” I also concur with
  Bruce R. Scott (2009, 4) in his claim that capitalism is both “a political
  phenomenon” and “an economic one” and that “specifically it requires
  the visible hands of political actors exercising power through political
  institutions.” Capitalism always involves legal and political institutions:
  pure “anarchocapitalism” is an unrealizable fantasy.

And since the Austrians (Hayek actually) got mentioned in the question, Hodgson has this to say about their approach to defining capitalism:

  The Austrian economists had
  little to say about the sphere of production—conditions (4) and (5)—
  including the employment contract. Although they mentioned money
  many times, they paid insufficient attention to the use of property as collateral
  or to the selling of debt—condition (6). Their theory of money
  downplays the vital constitutive role of the state. At best, they stressed
  three of the six conditions only. The first three conditions are insufficient to identify the institutions of the capitalist system that were responsible
  for the explosive production of wealth in the last three hundred
  years.

While this may not be any sort of consensus statement, it is at least a comparative analysis of various definitions (from fairly prominent sources).


If I had a new economical theory, how could I share it with academical environments? I call it “algorythmic economy”

Might be better for Academia:SE, but I'll give my two thoughts anyway.
The primary means of academic discourse is through peer-reviewed journal publications. Therefore, the simple advice is to write a paper and try to publish it in a (serious) economic journal. Here is a list of journals in decreasing order of reputation. There are different rankings used in the profession, but the broad classification of what constitutes a top 5 or an A-journal is not overly disputed in my opinion. Please do try to publish in an Economics journal. A lot of physicists or computer scientists publish papers claiming to revolutionize or reinvent Economics in non-economic journals like Nature of Science, probably get non-economists as reviewers and publish their claims without reference to the previous similar work in Economics.
As side-conversations during the publishing process you may also think of presenting at conferences, workshops, have private conversations / e-mails with academics in the field, or discuss your research on Twitter.


I have electrical engineering training, and the focus on CPUs creates a technical issue. Before the early 1990s (when I did my degree), CPUs would have largely been only embedded in what are recognisably digital computers. Electronics would otherwise be implemented with simpler digital/analog circuits. Currently, the path of least resistance is to just use a CPU for practically everything. For example, cars have plenty of embedded CPUs, whereas back in the 1970s, many might have been almost entirely analog circuits (the radio being the only possible exception).

So any measure of CPUs per unit of GDP should be exploding higher. This is just a reflection of electrical engineers’ design preferences.

If we just try to confine ourselves to CPUs in digital computers, we could try to compute some normalised computing capacity. (This raises problems: do we include phones? Game consoles? Tablets?)

The next problem is: so what? My computer CPUs have been getting faster over the decades, but at the same time, the bloated operating system chews up more of the resources. I end up doing roughly the same tasks in the same amount of time. Even without having any data, knowing what we know about computing power, my experiences are obviously be widely replicated. That is, real GDP is growing at around 2% per year in the developed countries, but installed CPU capacity has to be growing way faster than that, even if we take a narrow definition of “computer”.

Realistically, you might need to identify some industries that might show a correlation between their growth and CPU usage. It’s hard to see how adding server farms can increase the production of restaurants and barber shops.


interpretation: linear regressions with both unit dummies and time dummies

what exactly is the variation used in model 3 with both unit and time
  dummies?

The variation that is being used to identify $\beta_3$ is basically the individual level deviations away from both the individual mean and average across individuals for the year. So to the degree that your variable of interest is varying over time but does not vary in a differential manner across individuals, you will fail to detect its effect.

  Economists using model 3 often loosely say they "have controlled for unit fixed effects and time fixed effects", but "controlling for x" usually have the ceteris paribus interpretation, meaning we are comparing within groups with same values of x. See this answer for a nice presentation. I'm looking for intuitive and verbose interpretations that are more precise than "controlling for unit fixed effects and time fixed effects".

To be clear, when we say a unit fixed effect in econometrics, we are referencing any time invariant observed or unobserved determinant of the dependent variable. It's easy to show that all of these are "wiped out" by the individual level demeaning. The individual level demeaning also controls for the average differences in observed and unobserved independent variables across individuals. Individual fixed effects in the model mean that any source of bias must be time varying. Thus, if someone argues that your variable is endogenous on account of some  variable that is constant in your sample, you have already controlled for that by only using variation over time to identify your point estimate.

So with the inclusion of these individual fixed effects you can focus on determining time varying covariates that determine your independent variable. Time fixed effects will strip away any changes in variables that are the same for all individuals in a given period of time. For instance, if "individuals" are grouped together in the same state or municipality, and there are some changes in state or municipal policies for all individuals in that year,then the time fixed effects could strip away these effects without a need for measuring them. This only leaves concern over variables that have different time paths within different individuals.

So with both time and panel fixed effects, to identify the effect of your variable of interest, exogeneity concerns aside, that variable of interest must

be time varying

and

have variation in within-individual time paths across individuals (i.e. individual heterogeneity, must be varying at $it$ not just homogeneous across $i$ within $t$.)

So we are indeed "controlling for" those time invariant unobserved confounding factors, as well as average differences in observed factors across individuals with panel fixed effects. We are indeed "controlling for" covariates that have common variation across individuals within a given year with time fixed effects.

Let $y_{it}=\beta_i+\lambda_t+X_{it}\beta+\epsilon_{it}$ be the specification with $i=1\dots N$ and $t=1\dots T$.

Initially you do the panel demeaning, which creates the transformed variable

$\overset{..}{y}=(y_{it}-\overline{y_i})=\underbrace{(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)}_\text{panel demeaned time dummies}+\text{other demeaned terms unrelated to year FE}$.

This removes the $\beta_i$, along with any other time invariant confounding factors.

Then you demean with year FE, doing the transformation $\overset{...}{y}=\overset{..}{y_i}-\overline{y_t}$ where $\overline{y_t}=\frac{1}{N}\sum_{j=1}^{N}y_{jt}$.

$=\underbrace{(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)-\frac{1}{N}\sum_{i=1}^{N}(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)}_\text{want to show this = 0}+\text{other demeaned terms unrelated to year FE}$

.
.

since $\lambda_t$ is common to all $i$ in a given year, $\sum_{i=1}^{N} \lambda_t=N\lambda_t$ and similarly for $\sum_{i=1}^{N}\frac{1}{T}\sum_{j=1}^{T}\lambda_j=N \frac{1}{T}\sum_{j=1}^{T}\lambda_j$

.
.

$=(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)-\frac{1}{N}(N \lambda_t -N \frac{1}{T}\sum_{j=1}^{T}\lambda_j)+\text{other demeaned terms unrelated to year FE}$

.
.

distribute the $\frac{1}{N}$

.
.

$=\underbrace{(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)-(\lambda_t-\frac{1}{T}\sum_{j=1}^{T}\lambda_j)}_\text{=0}+\text{other demeaned terms unrelated to year FE}$

$=\text{other demeaned terms unrelated to year FE}$


Could the Feds objectives be adjusted to decrease or at least not increase wealth inequality?

Trivially, answer is yes. Congress could just give Fed a mandate for lowering inequality, the same way as congress could pass a law giving Fed mandate to bring world peace or solve world hunger or conduct space exploration. US is sovereign state and Fed is government institution and technically speaking US government can assign Fed whatever mandate it in its infinite wisdom chooses to.

Would changing the objectives of the Feds to: low unemployment, stable prices, and moderate long-term interest rates decrease or at least not increase wealth inequality?

No.  Wealth inequality does not just depend on interest rates. We actually do not yet fully understand what is driving wealth inequality, since data on wealth (as opposed to incomes) is harder to come by and only very recently we became able to reasonably measure it (see Zucman 2019), but there is virtually no doubt that it depends on more than just interest rates.
For example, economic growth tends to reduce wealth inequality between countries but does increase economic inequality within countries (Zucman 2019). Next wealth inequality depends on design of tax system, capital incomes (Saez 2016) and other factors as well. So just keeping moderate interest rates would not reduce inequality or prevent its increase in itself.
Fed could probably do something about wealth inequality, if it would have mandate, but it would not be as simple as keeping interest rates 'moderate'.  However, this would likely often conflict with its other two mandates and thus lead to worse economic performance, and hence the lower inequality might come at a cost of worse economic outcomes.
To illustrate this point, there is some evidence showing loose monetary policy during Great Recession increased relative wealth inequality in the US (Domanski, Scatigna, & Zabai, 2016). However, it is virtually unanimously agreed by macroeconomists that loose monetary policy is necessary during recessions. The reason why Great Depression became "Great Depression" is precisely because Fed decided to follow contractionary monetary policy (see Friedman & Schwartz Monetary History of the US). Great Recession of 2008/9, as bad as it was, was not even a half as bad as Great Depression if we look at metrics such as unemployment (during GD it was $\approx 25\%$ during GR $\approx 10\%$ see Petrosky-Nadeau & Zhang 2016)  or fall of output (see here discussion of drop in US output during GD and here  Fred stats for GR).
Lastly, there is generally a consensus among policy economists that technocratic economic institutions, run not by elected officials but experts and technocrats, such central banks, various regulatory bodies such as Securities and Exchange Commission (SEC) or  Federal Trade Commission (FTC) should deal with technical economic matters and leave matters of redistribution of either wealth or income to elected officials who still have plenty of policy tools to, at least in principle, achieve any level of relative inequality their constituency desires. In principle arbitrary GINI coefficient could be achieved through arbitrary taxation and redistribution (although the same does not hold for absolute welfare and material standards). This being said there is no economic reason why technocrats should not be in charge of inequality this comes down more to moral philosophy and to belief that equity (as opposed to efficiency) related objectives  should be decided democratically.


Rationale and Methodology for Discounting the Investment in an NPV Calculation

To be clear, we are assuming that the nominal investment value is a constant $K$. As such, if we calculate the NPV in the future, that future NPV will rise.
Of course, we do not work with future NPV’s, we are making a decision now, and so we need to discount that future NPV by the discount rate, to get an apples-to-apples comparison.
I will now look at the sub-questions.

An investment $K$ in the future is not the same thing as an investment $K$ now. We can buy a zero coupon bond with maturity $t$, and meet the future outflow of size $K$ with an bond that costs $Ke^{-r_bt}$, where $r_b$ is the mathematical discount rate on the bond. (Quoted yields follow different conventions, so you need to adjust quoted yields for the calculation.)
The reason I would discount future outflows at the bond rate is because that is the mechanism to translate current cash to future cash flows in a “guaranteed” fashion. You could use the project discount rate, but you are assuming reinvestment at the project discount rate, which is awkward if the project is only implemented in the future. The exact answer will depend on conventions for the NPV calculation, but there are pros and cons of each definition.

Which bond to use? If one is cautious, one sticks to “risk-free” sovereign bonds. (Disclaimer: all bonds have risks.) If one feels lucky, one can take credit risk with the investment, raising the bond yield. However, default risk then needs to be taken into account in the NPV analysis.
Since one might not find a zero coupon bond, one needs to calculate a zero coupon curve from observed bond yields. This is non-trivial, but descriptions would be found in most textbooks of fixed income analysis.


Question on Human Capital Accumulation Model by Acemoglu/Autor

The FOC from (1.9) is (I'm skipping the content of the $\exp()$ expression for brevity):

$ \frac{1}{r+\nu+g_h + g_w} \left(\eta'(S)w(0) \exp(\cdot) + \eta(S) w(0) (r + \nu - g_w)\exp(\cdot)\right) \overset{!}{=} 0$

This makes use of product rule.
Since the first term can't be 0 due to the last equation on p.~11, it must be second one

$ \eta'(S)w(0) \exp(\cdot) = \eta(S) w(0) (r + \nu - g_w)\exp(\cdot) $

$ \eta'(S) = \eta(S) (r + \nu - g_w) $
yielding Eq. (1.10):

$ \frac{\eta'(S)}{\eta(S)} = r + \nu - g_w $


This is going to be a long answer, and I'm not completely sure if it is going to answer your questions as I'm mostly going to focus on the derivations of the own and cross price elasticities. Most of the derivations can also be found in Houthakker (1960), "Additive Preferences"
TLDR:

If the utiltiy function is additively separable and if the share of the goods are small, then indeed own price elasticities are proportional to income elasticities. The ratio of cross price elasticities (for different goods) are proportional to income elasticities.
If the indirect utility function is additively separable, I could not show that the own price elasticity is proportional to the income elasticity. The ratio of cross price elasticities (accross different goods) is constant.
If both the direct and indirect utility function is additively separable, then Engel curves are linear.

Additive separability
Let start from a utility function that is additively separable. For the utility function $u$ let's write $u_i = \dfrac{\partial u}{\partial x_i}$ and $u_{i,j} = \frac{\partial^2 u}{\partial x_i \partial x_k}$. From additivity, we have that if $i \ne k$ then $u_{i,k} = 0$.
The following conditions gives the first order condition and the budget constraint:
$$
u_i = p_i\lambda,\\
\sum_i p_i x_i = \mu
$$
Differentiating with respect to $p_k$ ($k \ne i$), $p_i$ and $\mu$ gives the following 5 conditions:
$$
\begin{align*}
&u_{ii} \frac{\partial x_i}{\partial p_k} = p_i \frac{\partial \lambda}{\partial p_k}, \tag {1}\\
&u_{ii} \frac{\partial x_i}{\partial p_i} = \lambda + p_i \frac{\partial \lambda}{\partial p_i}, \tag{2}\\
&u_{ii} \frac{\partial x_i}{\partial \mu} = p_i \frac{\partial \lambda}{\partial \mu}, \tag{3}\\
&\sum_i p_i \frac{\partial x_i}{\partial p_k} = - x_k, \tag{4}\\
& \sum_i p_i \frac{\partial x_i}{\partial \mu} = 1. \tag{5}
\end{align*}
$$
Condition $(4)$ and $(5)$ are the cournot and Engel aggregation. Then substituting $(1)$ and $(2)$ in $(4)$ and $(3)$ in $(5)$ gives:
$$
\begin{align*}
&\sum_i p_i p_i \frac{\partial \lambda}{\partial p_k}\frac{1}{u_{ii}} + p_k p_k \frac{\lambda}{u_{kk}} = - x_k,\\
\to &\frac{\partial \lambda}{\partial p_k} \sum_i \frac{(p_i)^2}{u_{ii}} = - x_k - (p_k)^2 \frac{\lambda}{u_{kk}}. \tag{6}\\
&\sum_i p_i p_i \frac{\partial \lambda}{\partial \mu} \frac{1}{u_{ii}} = 1,\\
\to & \frac{\partial \lambda}{\partial \mu} \sum_i (p_i)^2 \frac{1}{u_{ii}} = 1. \tag{7}
\end{align*}
$$
Combining $(6)$ and $(7)$ together with $(3)$ gives:
$$
\begin{align*}
\frac{\partial \lambda}{\partial p_k} &= \frac{\partial \lambda}{\partial \mu}\left(-x_k - (p_k)^2 \frac{\lambda}{u_{kk}}\right),\\
&= \frac{\partial \lambda}{\partial \mu}\left(-x_k - \frac{\lambda}{\frac{\partial \lambda}{\partial \mu}} p_k \frac{\partial x_k}{\partial \mu}\right),\\
&= \frac{\partial \lambda}{\partial \mu}\left(-x_k - \chi \frac{\partial x_k}{\partial \mu}\right)
\end{align*}
$$
where:
$$
\chi = \frac{\lambda}{\dfrac{\partial \lambda}{\partial \mu}}
$$
which is called the money flexibility (according to Houthakker).
Substituting this into $(1)$ and using $(3)$, we get that for $i \ne k$:
$$
\begin{align*}
\frac{\partial x_i}{\partial p_k} &= \frac{1}{u_{ii}} p_i \frac{\partial \lambda}{\partial p_k},\\
&=\frac{1}{u_{ii}} p_i \frac{\partial \lambda}{\partial \mu}\left( - x_k - \chi \frac{\partial x_k}{\partial \mu}\right),\\
&= \frac{\partial x_i}{\partial \mu}\left(-x_k - \chi \frac{\partial x_k}{\partial \mu}\right) \tag{a}
\end{align*}
$$
Then from $(4)$ (and using $(5)$) we can compute the own price elasticity:
$$
\begin{align*}
p_k \frac{\partial x_k}{\partial p_k} &= -x_k - \sum_{j \ne k} p_j\frac{\partial x_j}{\partial p_k},\\
&= - x_k - \sum_{j \ne k} p_j \frac{\partial x_j}{\partial \mu}\left(-x_k - \chi \frac{\partial x_k}{\partial \mu}\right),\\
&= - x_k + \left(x_k + \chi \frac{\partial x_k}{\partial \mu}\right) \left(\sum_{j \ne k} p_j \frac{\partial x_k}{\partial \mu}\right),\\
&= - x_k + \left(x_k + \chi \frac{\partial x_k}{\partial \mu}\right)\left(1 - p_k \frac{\partial x_k}{\partial \mu}\right),\\
&= -x_k + x_k - p_k x_k \frac{\partial x_k}{\partial \mu} + \chi \frac{\partial x_k}{\partial \mu} -\chi p_k \frac{\partial x_k}{\partial \mu} \frac{\partial x_k}{\partial \mu},\\
&= \frac{\partial x_k}{\partial \mu} \left(-p_k x_k + \chi\left(1 - p_k \frac{\partial x_k}{\partial \mu}\right)\right)
\end{align*}
$$
Dividing both sides by $x_k$, we can express this In elasticity terms:
$$
\begin{align*}
\varepsilon^k_k &= \varepsilon^k_\mu\left(-s_k + \kappa \left(1 - s_k \varepsilon^k_\mu\right)   \right),\\
\end{align*}
$$
Here we use the notation $\varepsilon^k_i$ for the $p_i$ elasticity of $x_i$ and we use $s_k = \frac{p_k x_k}{\mu}$ to denote the share of good $k$ in the budget. Also:
$$
\kappa = \left(\frac{\partial \lambda}{\partial \mu}\frac{\mu}{\lambda}\right)^{-1} = \left(\frac{\ln \lambda}{\ln \mu}\right)^{-1}
$$
which is the inverse of the income elasticity of the marginal utility of income.
If there are many goods, then $s_k$ is very small, so:
$$
\varepsilon^k_k \approx \kappa \varepsilon^k_\mu
$$
So indeed, own price elasticities are close to approximate proportional to income elasticities.
Let's now look at the cross price elasticities. Writing $(a)$ in elasticity terms gives:
$$
\varepsilon^i_k = \varepsilon^i_\mu(-s_k - \kappa \varepsilon^k_\mu)
$$
So cross price elasticities are proportional to the income elasticity. Also for $i, j \ne k$, we have:
$$
\frac{\varepsilon^i_k}{\varepsilon^j_k} = \frac{\varepsilon^i_\mu}{\varepsilon^i_\mu} \tag{A}
$$
As such, the ratio of the cross price elasticities are equal across goods.
Additive indirect utility
Let $v$ be the indirect utility function, which depends on prices $p_i$ and income $\mu$. Similar to before, write $v_i = \frac{\partial v}{\partial p_i}$, $v_\mu = \frac{\partial v}{\partial \mu}$ and we use 2 indices for the second order derivatives. If $v$ is additive, then for $i \ne k$: $v_{i,k} = \frac{\partial^2 v}{\partial p_k \partial p_i} = 0$.
We start from the following equations (the first one is Roy's identity):
$$
x_i = -\frac{v_{i}}{v_\mu},\\
\sum_i p_i x_i = \mu.
$$
Taking derivatives with respect to $p_k$ ($k \ne i$), $p_i$ and $\mu$ gives:
$$
\begin{align*}
&\frac{\partial x_i}{\partial p_k} = \frac{-v_{i,k} v_\mu + v_i v_{\mu,k}}{(v_\mu)^2} = -x_i \frac{v_{\mu,k}}{v_\mu}, \tag{9} \\
&\frac{\partial x_i}{\partial p_i} = \frac{- v_{ii}v_\mu + v_i v_{\mu, i} }{(v_\mu)^2} = -\frac{v_{ii}}{v_\mu} - x_i\frac{v_{\mu,i}}{v_\mu}, \tag{10}\\
&\frac{\partial x_i}{\partial \mu} = \frac{-v_{i,\mu}v_\mu + v_i v_{\mu,\mu} }{(v_\mu)^2} = -\frac{v_{i,\mu}}{v_\mu} - x_i \frac{v_{\mu,\mu}}{v_\mu}, \tag{11}\\
&\sum_j p_j \frac{\partial x_j}{\partial p_k} = - x_k, \tag{12}\\
&\sum_j p_j \frac{\partial x_j}{\partial \mu} = 1. \tag{13}
\end{align*}
$$
Use $(9)$ into $(12)$:
$$
\begin{align*}
\sum_j p_j (-x_j) \frac{v_{\mu, k}}{v_\mu} - p_k \frac{v_{kk}}{v_\mu} = -x_k,\\
\to -\mu \frac{v_{\mu,k}}{v_\mu} = - x_k + p_k \frac{v_{kk}}{v_\mu}
\end{align*}
$$
Substituting this back into $(9)$:
$$
\begin{align*}
\frac{\partial x_i}{\partial p_k} &= -x_i \frac{v_{\mu, k}}{v_\mu},\\
&=-x_i \left(\frac{x_k}{\mu} - \frac{p_k}{\mu} \frac{v_{kk}}{v_\mu} \right) \tag{b}
\end{align*}
$$
Then use $(12)$ again to compute the own price effect:
$$
\begin{align*}
p_k \frac{\partial x_k}{\partial p_k} &= -x_k - \sum_{j \ne k} p_j \frac{\partial x_j}{\partial p_k},\\
&= -x_k + \sum_{j \ne k} p_j x_j \left(\frac{x_k}{\mu} - \frac{p_k}{\mu} \frac{v_{kk}}{v_\mu}\right),\\
&= -x_k + \left(\frac{x_k}{\mu} - \frac{p_k}{\mu} \frac{v_{kk}}{v_\mu}\right)(\mu - p_k x_k) \tag{14}
\end{align*}
$$
Where the last line follows from the budget constraint: $\mu = \sum_j p_j x_j$.
Also from $(10)$ and $(11)$:
$$
\begin{align*}
\frac{v_{kk}}{v_\mu} &= -\frac{\partial x_k}{\partial p_k} - x_k \frac{v_{\mu,k}}{v_\mu},\\
&= -\frac{\partial x_k}{\partial p_k} + x_k \frac{\partial x_k}{\partial \mu} + (x_k)^2 \frac{v_{\mu,\mu}}{v_\mu} \tag{c}
\end{align*}
$$
Substituting this back into $(14)$:
$$
\begin{align*}
p_k \frac{\partial x_k}{\partial p_k} &= -x_k + \left(x_k - p_k \frac{v_{k,k}}{v_\mu}\right)(1 - s_k),\\
&= -x_k + \left(x_k + p_k \frac{\partial x_k}{\partial p_k} - p_k x_k \frac{\partial x_k}{\partial \mu} - p_k (x_k)^2 \frac{v_{\mu,\mu}}{v_\mu}\right)(1- s_k),\\
\to p_k \frac{\partial x_k}{\partial p_k} s_k &= - x_k + \left(x_k - p_k x_k \frac{\partial x_k}{\partial \mu} - s_k x_k  \mu \frac{v_{\mu, \mu}}{v_\mu} \right) (1 - s_k),\\
\to p_k \frac{\partial x_k}{\partial p_k} & = - \frac{x_k}{s_k} + \frac{1 - s_k}{s_k} \left(x_k - s_k \mu \frac{\partial x_k}{\partial \mu} - x_k s_k \mu \frac{v_{\mu,\mu}}{v_\mu} \right).
\end{align*}
$$
If we divide both sides by $x_k$, we can express this in elasticity terms:
$$
\begin{align*}
\varepsilon^k_k &= -\frac{1}{s_k} + \frac{(1-s_k)}{s_k}(1 - s_k \varepsilon^k_\mu - s_k \delta),\\
&= -1 - (1 - s_k)\varepsilon^k_\mu - (1 - s_k)\frac{1}{\kappa}. \tag{14}\\
\end{align*}
$$
where
$$
\mu \frac{v_{\mu,\mu}}{v_\mu} = \frac{\partial \ln v_\mu}{\partial \ln \mu} = \frac{\partial \ln \lambda}{\partial \ln \mu} =  \frac{1}{\kappa}
$$
Expression $(14)$ gives the own price elasticity

There's a small difference here with Houthakker's derivation as I have a minus sign for the last term while he has a '+' sign so I might have made a mistake.

If $s_k$ becomes small we have:
$$
\varepsilon^k_k = -1 - \varepsilon^k_\mu - \frac{1}{\kappa}.
$$
So, here we do not really obtain that $\varepsilon^k_k$ is proportional to $\varepsilon^k_\mu$. On the other hand, if $1/\kappa$ is very small then in this case:
$$
1 - \varepsilon^k_k \approx \varepsilon^k_\mu
$$
So one minus the own price elasticity equals the income elasticity.
For the cross price elasticities, we can write $(b)$ in elasticity terms (and using $(c)$):
$$
\begin{align*}
\varepsilon^i_k &= -\left(s_k - \frac{(p_k)^2}{\mu} \frac{v_{kk}}{v\mu}\right),\\
&= s_k\left(-1 + \varepsilon^{k}_k + s_k \varepsilon^k_\mu + s_k \frac{1}{\kappa}\right)
\end{align*}
$$
Taking ratio for $i, j \ne k$ gives:
$$
\frac{\varepsilon^i_k}{\varepsilon^j_k} = 1. \tag{B}
$$
Both direct and indirect additivity
Let's see what happens if both the direct and indirect utility functions are additively separable,
Take $i,j \ne k$  Then $(A)$ and $(B)$ give:
$$
\frac{\varepsilon^i_k}{\varepsilon^j_k} = \frac{\varepsilon^i_\mu}{\varepsilon^j_\mu} = 1.
$$
This shows that all income elasticities are the same. Using $(5)$ in elasticity form, i.e.:
$$
\sum_i s_i \varepsilon^j_\mu = 1,
$$
shows that all income elasticities are equal to 1, so we have linear Engel curves.


How do you determine the strict core in a matching game?

When preferences are strict, the strong core is unique and the top-trading cycle procedure (TTCP) is the unique mechanism yielding the strong core allocation. Here are the highlights.

Lemma 1: Let $x$ be the allocation generated by the Top-Trading Cycle Procedure. Then $x$ is in the strong core.

Lemma 2: Let $P$ be the set of players, $\Omega = \{ \omega^{i} = i : i \in P \}$ be the set of initial allocations, and $\succ^{i}$ be the set of strict preferences. We have $(P, \Omega, (\succ^{i})_{i=1}^{|P|})$ as the instance of the house allocation problem. The strong core is unique.

Proof: Suppose to the contrary that there exist two allocations $x, y$ in the strong core and suppose $x$ is the allocation returned by the TTCP. As $x \neq y$, $S = \{ i : x_{i} \neq y_{i} \}$ has at least two elements. Observe that if there is a player $i \in S$ such that $x_{i} \succ^{i} y_{i}$, then $x_{i}$ can form a weakly improving coalition with the other players in $x_{i}$'s cycle whose allocations are the same in $x$ and $y$. As preferences are strict, every player in $S$ strictly prefers either $x$ or $y$. We construct a weakly improving coalition to block $y$.

Let $i \in S$ such that $y_{i} \succ^{i} x_{i}$. Let $i$ be matched in iteration $k$ of the TTCP. Since $y_{i} \succ^{i} x_{i}$, we have that player $i$ prefers some other house $j$ to $x_{i}$ and that $j$ was assigned in a previous iteration by the fact that $x$ belongs to the strong core. And so using the allocation $y$ displaces the player who received house $j$ under the TTCP. As preferences are strict, each player who received a house when $j$ was allocated under the TTCP can form a weakly improving coalition to block $y$. And so the fact that $S \neq \emptyset$ implies that $y$ is not in the strong core. QED.

I actually have a blog entry going into more detail about this: https://michaellevet.wordpress.com/2015/06/01/algorithmic-game-theory-house-allocation-problem/


Submodularity property in congestion games?

This proposition is in general not true. One can show that it is true in the case $n=2$ and $m=2$. Here, I exhibit a counter example when $n=3$ and $m=2$.

A brief comment. We can rephrase the question in words: does a Nash equilibrium that is "more random" ($e'$ versus $e$) is less efficient? Intuitively, as more mixed strategies are played, the realized outcome is more random and it can be very inefficient due to a lack of coordination between agents. When agents play pure strategies, we can think that we reduce the coordination problem given that we consider Nash equilibria. This intuition does not hold if the proposition is false, as I will show when $n=3$ and $m=2$.

Denote $A$ and $B$ the two possible actions. The delay functions are defined as follows:
$d_A(1)=5$, $d_A(2)=7$, $d_A(3)=10$ and $d_B(1)=1$, $d_B(2)=6$, $d_B(3)=7$.
It means that when $x$ agents play $A$ (resp. $B$), they receive the payoff $-d_A(x)$ (resp. $-d_B(x)$). This is a (symmetric) congestion game as long as the delay functions are increasing.

Define $e$ as the equilibrium when 1 agent plays $A$ and 2 agents play $B$. Define $e'$ as the equilibrium when 1 agent always plays $B$, and the 2 others plays $A$ with probability $\mu=2/3$ and $B$ with probability $1-\mu=1/3$. It satisfies the property  $sup(e) \subseteq sup(e')$.

First, we show that $e$ is a Nash equilibrium. The agent who plays $A$ is maximizing her payoff given the two other players' strategy when choosing $A$ is better than choosing $B$, $d_A(1)<d_B(3)$ (i.e. $5<7$). Both agents who play $B$ are playing optimally if $d_B(2)<d_A(2)$ (i.e. $6<7$). $e$ is thus a Nash equilibrium and its social cost is $d_A(1)+2d_B(2)=17=\frac{153}{9}$.

Second, we show that $e'$ is a Nash equilibrium. On one hand, the agent who plays $B$ is maximizing her payoff when the two others play mixed strategy if she is better off playing $B$ than $A$,
$$(1-\mu)^2d_B(3)+2\mu(1-\mu)d_B(2)+\mu^2d_B(1)<(1-\mu)^2d_A(1)+2\mu(1-\mu)d_A(2)+\mu^2d_A(3)$$
i.e. $\frac{1}{9}5+\frac{4}{9}7+\frac{4}{9}10<\frac{1}{9}7+\frac{4}{9}6+\frac{4}{9}1$, which is true.
On the other hand, each one of the agents playing the mixed strategy is indifferent between choosing $A$ or $B$ if
$$\mu d_A(2)+(1-\mu)d_A(1)=\mu d_B(2)+(1-\mu)d_B(3)$$
i.e. $\frac{19}{3}=\frac{19}{3}$.
$e'$ is then a Nash equilibrium and its social cost is
$$(1-\mu)^2 [3d_B(3)]+2\mu(1-\mu)[d_A(1)+2d_B(2)]+\mu^2[2d_A(2)+d_B(1)]$$
 which is equal to $\frac{1}{9}21+\frac{4}{9}17+\frac{4}{9}15=\frac{149}{9}$.

Finally, we have shown that $sup(e) \subseteq sup(e')$ but $SC(e) > SC(e')$. The mixed-strategy Nash equilibrium results in a lower social cost than the pure-strategy one.


How exactly does QE affect the value of the USD?

QE actually can increase money supply in the economy. We have different measures of money supply ranging from $M0-M4$. QE is not physical printing of banknotes so it does not expand $M0$ but it can expand wider money supply.

Following, the St. Louis Fed definition the QE is defined as:

  large-scale
  asset purchases—in the hundreds of billions of dollars range—of, for example, mortgage-backed securities
  and Treasury securities.

To the extent that these securities are bought with newly created bank reserves (which you even in your question assume) they will actually increase money supply measures beside the most narrow one which only includes physical notes and coins. In addition QE is supposed to also encourage banks to lend more which also further expands money supply.

In fact increase in the money supply is what is supposed to bring interest rates down as interest rates are determined at money market as you can see on the graph below taken from Blanchard et al. Macroeconomics an European Perspective.

Furthermore, when we talk about devaluation we are talking about currencies with fixed exchange rate. Dollar is not under fixed exchange rate regime.  Here it is more correct to talk about depreciation not devaluation (see the explanation of differences here).

The value of currency on foreign exchange depends on both money supply and interest rate. A simple basic classroom monetary model of exchange rate can be expressed as:

$$S= m-m_f -\phi (y-y_f)+\lambda(i-i_f) $$

Where $S$ is the exchange rate $m$ is the log of money supply, $y$ log of real output, $i$ interest rate and $f$ denotes foreign country. So QE can lead to depreciation through both its direct effect on money supply and through its effect on interest rates, however whether dollar depreciates or appreciates depends also on what the other economy, relative to which we value dollar, does.


What are some important mathematics results that were first developed in Economics?

The Envelope theorem is arguably one, https://en.wikipedia.org/wiki/Envelope_theorem . I would say Blackwell's papers on comparisons of experiments are a significant mathematical contribution--extending Hardy,Littlewood and Polya's earlier results, and paving the way for further investigation.

Econometricians have made statistical contributions as well; see e.g. https://en.wikipedia.org/wiki/Heckman_correction .

More recently, I saw a paper presented last year on how a certain type of stochastic PDE could be solved by reinterpreting it as a (stochastic) differential game. I would expect to see more related to this (the relationship between differential games and stochastic pdes) in the coming years, especially given the recent surge of interest in mean-field games.


What changes is the macroeconomic identity of the model: from
$$F(K,L) = S + C$$
it becomes

$$F(K,L) + V = S+ C = (1+v)F(K,L) = S+C$$

where $V$ denotes the value of reparations per period, and the last expression reflects an assumption made in the question, that reparations are a constant proportion of domestic output.

"Using the Solow model" means I guess, using the standard Solow model, and so it implies that you don't change its basic assumptions. So there is no distributional issues related to this transfer payment from abroad, while also the savings rate remains fixed, and it is applied to the whole available income (irrespective of how it becomes available). I guess you can take it from here.

PS: It appears interesting to also consider a fixed level of $V$, which would reflect a situation where as time passes and the domestic economy grows, reparations become less important in relative size (which is a "reasonable" scenario, something like "we will help more when you are weak, and as you get economically stronger, helped also by the reparations, our contribution will diminish in relative importance")