It appears that you are conflating a "closed interval" with a "closed set". Certainly, (multidimensional) closed intervals are closed sets, but the concept of a closed set is wider.

A closed set is, well, only closed, and not necessarily bounded.

I understand that the everyday-meanings of the terms "closed" and "bounded" can indicate that "being closed is a stronger property than being bounded" (i.e. that being closed implies being bounded), but this is not true in mathematics. Neither property implies the other.

Yes, a set can be bounded and yet it may not be closed. Standard example: Consider the set of rational numbers  in $[1,3]$ . This is a bounded set. Now consider the sequence of rational numbers

$$x_{n+1} = \frac {x_n}{2} +\frac {1}{x_n},\;\; x_0 = 3$$

The sequence remains in the $[1,3]$ interval, but its limit is $\sqrt 2$, which is not rational. So this set of rationals in $[1,3]$ is bounded but it is not closed, because it does not contain a limit point of a sequence that can be constructed in it, stay in it, except its limit point.

So, what "closed set" means then in mathematics? There are various ways to look at it:

A closed set is a set whose complement is open.
A closed set is a set that coincides with its "closure".
A closed set is a set that contains all its limit points.

Try them out mentally and stick with what is more intuitive to you.


In your last paragraph you write "high interest rate in Country A". There is no such thing. There is higher interest rate, per some given criterion. Say higher than historical average. In out case, it is "higher interest rate in Country A compared to the interest rate in Country B".

So your last paragraph says that if returns (from nominal interest rates) in Country A are higher than returns (from nominal interest rates) in Country B, that would put upward pressure to the exchange rate (units of currency B per one unit of currency A): people will demand more currency A, etc.

Now a different but equivalent  way to say "higher returns in A than in B" is "excess returns", in the specific context we are talking about. Of course the term "excess returns" is very general and can be used in any situation. Sometimes it may used to mean "abnormal" returns (in some sense and again compared to some base). But in the specific situation we are talking, it has the exact same meaning with "higher returns from nominal interest rates in Country A compared to returns (from nominal interest rates) in Country B".

So moving to your penultimate paragraph you say that "if returns in A are higher than returns in B, that should put downward pressure on the interest rates in country A (because funds flow in), and it should leave the exchange rate unaffected".

Although there is logic to the mid-term effect of an increase in the flow-in of funds, still it appears that your last paragraph already answers the question "why should exchange rate be affected?", stated  in your penultimate paragraph.

ADDEDNUM
Responding to the OP's comment, let's write the Uncovered Interest Rate Parity expression

$$(1+i_{A,t}) = \frac {S^e_{A|B, t+1}}{S_{A|B, t}} (1+i_{B,t})$$
where $S_{A|B}=$ units of currency A per unit of currency B.
This is the claim.  The numerator is the expected future exchange rate

Assume that $i_{A,t} > i_{B,t}$. Standard economic logic (and a few assumptions on ability to invest over borders), says that then investors would want to invest in country A. To do that they will increase demand for currency A. This should lead to appreciation of currency A. Does the UIRP say something like that?

If $$i_{A,t} > i_{B,t} \implies 1+i_{A,t} > 1+i_{B,t}$$
and so the UIRP asserts that we must have

$$\frac {S^e_{A|B, t+1}}{S_{A|B, t}} > 1 \implies S^e_{A|B, t+1} > S_{A|B, t}$$

Given how $S$ was defined the last inequality means that we expect that the currency will depreciate. Indeed, in the future. Because, what is affected in the first place by the reaction of the investors is $S_{A|B, t}$, not $S^e_{A|B, t+1}$. The reaction of the investors will decrease $S_{A|B, t}$ (which reflects the appreciation of currency A), creating expectations of future depreciation, i.e. leading to $S^e_{A|B, t+1} > S_{A|B, t}$, as the UIRP asserts. Why should we expect depreciation of currency A in the future, compared to the present? Because the inflow of funds in country A will eventually pressure $i_{A,t}$ downwards, as the OP remarked, investors will stop loving currency A so much, demand for it will fall, hence depreciation.

The UIRP therefore compacts two effects from a discrepancy in interest rates: a first effect of currency appreciation and a second (expected) effect of currency depreciation in the future.


If you run a vector autoregression you could also follow up by testing for Granger causality. I don't know whether this is implemented directly in SPSS, but once you have an estimated model it is easy to calculate the statistic as it is basically just an F-test. In R you can do it directly via for example the package vars.


Question on Fama-French Three Factor model

Here is the link that helps you replicate Fama-French 1993 paper.


Obstfeld and Rogoff (Foundations of International Macroeconomics) or Romer (Advanced Macroeconomics) both cover a lot and present all the math.


The PPI (Producer Price Index) measures the domestic output of raw goods and services.

The GDP deflator measures the changes in all goods and services produced in an economy.

For your purpose the US Bureau does release indexes that drill down further into the PPI in the form of industry level classifications and commodity classifications.


How to solve a linear forward-looking equation $x_t = \beta E_t[x_{t+1}] + k$ where $\lim_{t \to \infty} x_t = 0$ and $0<\beta < 1$?

(I suspect that $E_t[x_{t+1}] = E(x_{t+1}\mid I_t)$, where $I_t$ is the Information Set, i.e. $E_t[x_{t+1}]$ is translated as "the expected value of random variable $x_{t+1}$ conditional on Information available at time $t$").

@FooBar answer implemented the "standard solution algorithm", arrived at a constant solution, and remarked that such a solution violates the limiting condition.
Strictly speaking, a constant solution would satisfy the limiting solution also, if it was equal to zero (which in our case implies $k=0$): If indeed we had as solution

$$x_t = \frac k{1-\beta}$$
then

$$\lim_{t \to \infty} x_t = \lim_{t \to \infty}\frac k{1-\beta} = \frac k{1-\beta} \neq 0$$

except if $k=0$. But this is trivial (or at least, uninteresting).

So I ask, if $k\neq 0$, what would it take to obtain a solution? The law of evolution must hold also at the limit so

$$\lim_{t \to \infty} x_t = \lim_{t \to \infty}\Big[\beta E_t[x_{t+1}] + k\Big]=0$$

$$\Rightarrow \lim_{t \to \infty}E_t[x_{t+1}] = -\frac k{\beta} \tag{1}$$

In words, as the random variable tends to zero, its expected value conditional on information available in the previous period, must equal a non-zero constant, while the variable itself tends to zero. Consider some cases, to see whether and when such a thing can happen:

A) If $x_t$ was totally unobservable, (not observed directly, and not amenable to indirect estimation though other variables)? In such a case, the information set would not contain any information on $x_t$, meaning that the conditional expected value would equal the unconditional one. So we would have,

$$\lim_{t \to \infty}E_t[x_{t+1}] = \lim_{t \to \infty}E[x_{t+1}] \tag{2}$$

This is economics, meaning that the properties of the mathematical entities should take into account what do they represent in the real world. So it would be meaningless not to assume that $x_{t+1}$ is not bounded (more over, this would mean that the $x$ random variable goes to infinity in finite time and then "comes back" to tend to zero in infinite time -and for those who may think "hyperinflation", hyperinflation is a very very small number compared to infinity). If it is bounded then the Dominated/Bounded Convergence theorem holds and we have that

$$\lim_{t \to \infty}E\left(x_{t+1}\right) =E\left(\lim_{t \to \infty}x_{t+1}\right) =0 \neq -\frac k{\beta} \tag{3}$$

So this is not a case in which we can obtain the required relation $(1)$.

B) The limiting condition is part of the information sets for finite $t$. Then immediately

$$E\left(\lim_{t \to \infty}x_{t+1}\mid I_t\right) =0$$

In words, at any point in time we know that the variable will tend eventually to zero. Can this be compatible with relation $(1)$? We could invoke again the boundedness of $x_t$, and a generalization of the Dominated Convergence Theorem, since here the (conditional probability) measure is also time-varying, and again obtain a zero expectation at the limit. But intuitively also, if we know that eventually the variable will tend to zero, at some point in time this must be reflected in $E_t$, most certainly at the limit. But then again, the required relation $(1)$ won't hold.

C) We do observe perfectly or imperfectly the value of the variable, but "we don't give proper notice": our Expectations are not Rational. In that case, the symbol $E_t$ no longer represents the conditional expectations operator, but it is just a generic symbol for the "anticipation at time $t$" (in Adaptive Learning literature this is usually symbolized by $E^*_t$) and it acquires the value it needs to acquire for the limiting condition to hold.  In other words, given the law of evolution and the limiting condition, we conclude that expectations must not obey the Rational Expectations Hypothesis, not even at the limit.

In such a case, a solution is obvious: define
$$E^*_t(x_{t+1}) = -\frac {(1-c\beta^t)k}{\beta}$$

for some constant $c$, This expectations formation rule satisfies $(1)$, and  transforms the law of motion into

$$x_t = \beta \Big[-\frac {(1-c\beta^t)k}{\beta}\Big] + k = ck\beta^t$$

which in turn satisfies the limiting condition.

So we conclude that: the specific equation, with the specific limiting condition, has a solution if expectations formation does not obey the Rational Expectations Hypothesis, but instead follows the above rule, or some other to the same limiting effect.


What are the economic disadvantages of a single payer system in the USA?

I don't know whether you'll count this as an economic argument, but it's about the role of competition in the allocation of scarce resources, so I think you should:

Suppose that some disease emerges that disproportionately affects the members of some ethnic group, or is concentrated in some geographic area, or is concentrated in some population that is either in or out of political favor with the current administration.

Suppose also that there's some pretty expensive procedure that helps to alleviate symptoms of this disease.

Under a single-payer system, one decision will be made about whether to cover the costs of that procedure.  Quite plausibly, that decision will be either made or influenced by somebody with less than pure motives.  Maybe an out-and-out bigot.  Maybe a politician looking out for the people who  support him.  Et cetera.

I would much prefer that that decision be made independently by multiple competing insurance companies, so that people can choose up front to affiliate with insurance companies that they believe will look out for their interests.

If you think this is all just  speculation, look at what happened when the government took temporary partial control of General Motors.  Politicians immediately started thinking up ways to get General Motors to make decisions that would benefit their own constitutents.

General Motors is one firm, and the takeover was partial and temporary.  Now imagine a takeover of the entire insurance industry, both total and permanent.

You might or might not believe the attendant risks are worth taking --- but you asked what the downside is, and that's a big part of the downside.


Elasticity of intertemporal sustitution with composite CRRA function

Suppose $c_t$ is some composite function of interest rate $r$, e.g. $c_t(G(r_t))$.
First the intertemporal elasticity of substitution is actually (IES) given by $\frac{\partial \ln(c_{t+1}/c_{t})}{\partial r}$ (or also $\frac{\partial \ln(c_{t+1}/c_{t})}{\partial \ln( u'(c_{t+1})/u'(c_t))}$).
You can take the derivative above using chain rule for composite functions which says that $dF(G(x))/dx = \frac{dF}{dG}\frac{dG}{dx}$.
So in the above case we would get:
$$\frac{\partial \ln(c_{t+1}(G(r))/c_{t}(G(r)))}{\partial r} = \frac{\partial \ln(c_{t+1}(G(r))/c_{t}(G(r)))}{\partial G(r)} \cdot \frac{\partial G(r)}{\partial r} $$
You can directly apply this to the problem you mention in your question.


Internalising an externality for a free merit good

Even if the vaccine is free it is possible to additionally internalize the externality with subsidy. That is government can just pay people to take the vaccine.  Note, that even with a flu vaccine just providing voucher for a free vaccine might not be enough to fully internalize the positive externality (or it could be too much this is always case dependent).
For covid-19 specifically, there is an interesting recent paper on optimal covid-19 vaccine subsidies (among other things) by Goodkin-Gold et. al. (2020).  Their model shows that optimal subsidy will depend on the reproduction rate of the disease. Higher reproduction rate makes stronger case for subsidies, while if the reproduction rate it sufficiently low there might be no need for the subsidies at all. Moreover, the authors show that optimal level of subsidies also depends on competition in the industry. Specifically, they show that generally the more competitive industry is the greater case there will be for per unit subsidy.


Is the European Single Market a Free Trade Area?

Free trade area means there's no customs checking goods traded within that area.

From Wikipedia:

  The European Single Market, Internal Market or Common Market is a single market which seeks to guarantee the free movement of goods, capital, services, and labour – the "four freedoms" – within the European Union (EU).

So goods are traded freely within the EU, and hence EU is an FTA.


How to properly interpret logged interaction variables

You have to add them even though they are in $\ln$ the model is still linear in the parameters. To see this more clearly you can rewrite your equation as (I use $D$ for demand, $c$ for constant, $F$ for fare and $M$ and $T$ for Monday and Tuesday):
$$\ln(D) = c  + \left(- 0.45  - 0.15(T)  + 0.05(M) \right)\ln(F) + 10(M) + 15(T)$$
Hence when both $T=0$ and $M=0$ the elasticity for Wednesday is -0.45, when $T=1$ and $M=0$ (i.e. its Tuesday) the elasticity is $-0.6$, and finally when $T=0$ and $M=1$ (i.e. its Monday) the elasticity is $-0.4$.
If you want to learn more Wooldridge discusses this in his introduction to econometrics in chapter 7.


The formula for calculating the real value of an amount $x$ in a given year using dollars from a particular reference year is

$$
\frac{x_{baseyear}}{x_{current year}}=\frac{CPI_{base year}}{CPI_{current year}}
$$

So in your hypothetical example (where the deflator is higher in 1980 than in 2000, which indicates deflation over the period), the value of \$5 in 2000 is equal in 1980 to:

$$
x_{current year}=x_{base year}*\frac{CPI_{current year}}{CPI_{base year}}
$$

or

$$
$5*\frac{1.6}{1}=$8
$$

So in your example, a basket of goods costing \$5 in 2000 would cost \$8 in 1980.


Are newspaper articles more liberal than their readership?

This strikes me as a straightforward industrial organization anomaly question. Like, why do hot dogs and buns come in different counts to a package or why are the finest apples shipped out of apple producing regions?

Gentzkow and Shapiro (2010) that this paradox is only an apparent contradiction. They find slant is mostly a product of readership bias.

  By contrast, we find much less evidence for a role of newspaper owners
  in determining slant. While slant is somewhat correlated across
  co-owned papers, this effect is driven by the geographic clustering of
  ownership groups. After controlling for the geographic location of
  newspapers, we find no evidence that the variation in slant has an
  owner-specific component. We also find no evidence that pressure from
  incumbent politicians or the tastes of reporters are important drivers
  of slant.

But jmbejara already mentions knowing about Gentzkow so it isn't obvious what he's looking for beyond this insight.

Timothy Taylor,  Managing Editor of the JEP and blogger at CONVERSABLE ECONOMIST also has an article on this subject that may be of interest,
The Economics of Media Bias.


Arbitrage Pricing Theory (APT), orthogonal factors

If you want to describe excess returns in terms of exposure to common risk factors, you want the risk factors to be orthogonal. However, if you have $k$ factors with no perfect collinearity, you can always orthogonalize them and use those. You then call the orthogonalized factors the risk factors.


Stag Hare inefficient Nash Equilibrium and level-K thinking

Level-k reasoning in the stag hunt game is analyzed in
Gracia-Lázaro, Carlos, Luis Mario Floría, and Yamir Moreno. "Cognitive hierarchy theory and two-person games." Games 8.1 (2017): 1.
The idea that playing $s$ guarantees its payoff is discussed in
Aumann, Robert "Nash equilibria are not self-enforcing, in ‘‘Economic Decision-Making: Games, Econometrics and Optimization’’(JJ Gabszewicz, J.-F. Richard, and LA Wolsey, Eds.)." (1990).
The stag hunt game is due to Aumann.
Related to that is the idea of risk dominance of Harsanyi and Selten in their 1988 book "A general theory of equilibrium selection in games." The stag hunt is pretty much the leading example. It formulates the idea that this Pareto inefficient equilibrium is less risky to play.
It should be noted that the game has a third equilibrium in mixed strategies.


How will the Brexit affect the Euro and the Pound?

It depends. (As all things do.) I'm personally of the opinion that this will not result in any major long-run changes in the value of the Euro. This will, however, negatively affect the pound.

Most of the currency volatility that we're currently seeing is a result from investor panic rather than actual changes in economics. The UK and EU still need to discuss exactly how the UK will leave before investors can accurately assess the economic impact.

Brexit will negatively impact the EU by:

Decreasing the amount of trade between the EU and UK, due to
    increased trade barriers (this is going to affect the UK more than
    EU, mostly due to size)
Increased instability from political changes; may result in a domino effect on other troubled Eurozone countries attempting to
    leave (see: Greece)
Lack of job opportunities; the UK has a lot of foreign workers who will eventually have to go back to Europe when their
visas
    expire

Brexit will positively impact the EU by:

Forcing a possible move of the financial industry to European
centers rather than London

In my opinion, this currency change has more to do with a shift in politics than economics. The UK and EU will probably agree to some form of trade policy that will mitigate most of the economic damage.


In Blackwell's condition for T to be a contraction mapping, we require that satisfies discounting. What is the intuition of discounting?

Without discounting, you cannot show either

$$
T(g + || f - g||) \leq Tg + \beta || f - g||
$$

or

$$
T(f + || g - f||) \leq Tf + \beta || g - f||
$$

and thus you cannot demonstrate that $T$ is a contraction mapping in the traditional proof.

For intuition, note that Blackwell's sufficiency conditions for a contraction mapping is a fixed-point theorem used to establish the convergence of value function iterations. In the simplest case possible of a collapsed state space with the linear operator $T(W) = \sigma + \beta W$, it is clear that $\beta \in (0,1)$ will always lead to an intersection of $T(W)$ (from above) with the 45-degree line $T(W) = W$ and thus converge at a fixed point independent of the initial guess. The discounting condition that pins down such a $\beta$ assures this.

In systems with fully specified state spaces, however, discounting is no longer sufficient to establish uniqueness. Monotonicity is additionally required in these cases to establish the sufficiency of Blackwell's conditions because of additional concerns over dynamics.


Completeness of the strict binary relation

Your counterexample looks correct.

But an even simpler example would be $X=\{a\}$. We have $a \succsim a$, so $a \not \succ a$.

More generally, we usually define

$\succsim$ to be reflexive (i.e. $a \succsim a$ for all $a$); and
$\succ$ by $a \succ b$ if $a \succsim b$ and $b \not \succsim a$.

By the above definitions, $\succ$ cannot be complete on any non-empty set $X$ because for any $a\in X$, we have $a \not \succ a$.


Research at the county-level: Macro or Micro?

I would consider county-level research macroeconomics.

Microeconomics focuses on individuals and firms acting as decision makers. I think, looking at county-level aggregates, you are too high level to look at decision makers.


Illustrating Karesh Kuhn Tucker with two non-nonnegativy constraints binding

Only two or three papers spontaneously come to my mind, they are not very recent (but quite seminal papers, useful to find further newest contributions):
Lee, Lung-Fei, and Mark M. Pitt, 1986, "Microeconometric Demand System with Binding Nonnegativity Constraints: The Dual Approach," Econometrica, 54, 1237–1242.
Lee, Lung-Fei and Mark M. Pitt, 1987, "Microeconometric models of rationing, imperfect markets, and non-negativity constraints," Journal of Econometrics, 36, 89-110.
Here is a more recent contribution:
Golan, Amos, et al. “Estimating a Demand System with Nonnegativity Constraints: Mexican Meat Demand,” The Review of Economics and Statistics, vol. 83, no. 3, 2001, pp. 541–550.


What graph theory textbook should I use?

Social and Economic Networks, by Matthew O. Jackson, and its associated MOOC at https://www.coursera.org/course/networksonline (videos from the 2014 session available, new live session starting September 2015).


Even at the break-even inflation rate, isn't the YTM of a TIPS greater than that of a normal treasury bond?

Your analysis is incorrect.
The quoted market "breakeven inflation" is the spread between quoted yields. It is an approximation of the true economic breakeven from a total return perspective (possibly ignoring the par put on principal) to maturity, but under normal circumstances, it is not far off the economic breakeven. (Since the exact pattern of future CPI would matter, even the economic breakeven has technical calculation issues.)
Take an imaginary world with a 5% nominal par coupon, and a 3% inflation-linked par coupon (2% breakeven).

The conventional bond pays a 5% coupon.
The TIPS starts out with a 3% coupon, with the 3% of a par value that is expected to grow at 2% a year. It takes a long time for 3% to catch up to 5% at a 2% growth rate. (The principal future value increases by 2% per year, but that increase needs to be discounted by the nominal discount rate of 5%.)

The lower coupons are compensated by growth of the principal, keeping total returns equal.
(For an easier example, imagine a 2% nominal bond, and a 0% inflation-linked.)


What's the role of initial endowments in an edgeworth box?

It's crucial to understand that only initial endowments and preferences matter. Everything else (utility functions, prices, equilibria, core etc) follows from them.

Your initial endowment shows how strong is your negotiation position.

Situation I. Imagine that the wealth is equally distributed (50/50) and the 1-st player slightly prefers good A to good B and the 2-nd player slightly (to the same extent) prefers good B to good A  than something like this happens.

i.e. you get extra 10 units of A and give 10 units of B (1 to 1 exchange).

Situation II. Now, consider that by the same initial endowment, the 2-nd player prefers B to A much more than you prefer A to B. Then, something like this happens:

i.e. by the same initial endowment and preferences you get more, now e.g. 35 units of extra giving away 20 units of B (better conditions of exchange as before).

Situation III. Now, imagine that you have only 10% of wealth. Then your negotiation position is weak and by the preferences as in situation I you get much less after exchange.

p.s. Calculations and plotting were made with pyEdgeworthBox package (I wrote it).


Difference in interpretation of log-lin coefficients with binary variable

As you can see from the plot, they are not the same. Infact, the 'rule of thumb', i.e. your first formula, only holds in approximation. The error gets larger for large coefficient values.


Independence of latent price and market microstructure noise

I guess you're asking three inter-relatated though conceptually separate questions:

What is the latent or efficient price?
What is market microstructure noise (as opposed to non-market-microstructure noise)?
What does it mean to assume that the efficient price process and the microstructure noise process are independent?

I won't attempt to provide a full examination of the issues raised by these questions: entire articles could (and have been) written about them. I'll only attempt a summary review.

The efficient price process is, by assumption, unobserved. Nevertheless, under the maintained assumptions of frictionless markets, perfect and symmetric information, and rationality of all market participants, one can apply some basic theorems to infer some statistical properties of the efficient price process. Most importantly, the efficient price process must generally be a semimartingale with respect to some filtration. In a continuous-time setting, it is also straightforward to show that the continuous (i.e., non-jump) component of the efficient price process must be local Brownian motion with local drift and variance components.

If the observed price process isn't a semimartingale, it's either because there's an additional component (usually, but not always, called market microstructure noise) or because the maintained assumptions (rationality, perfect and symmetric information, etc) aren't valid to begin with. Let's pursue the first reason.
To make market microstructure noise more than just a catch-all (and tautological) component, it's important to relate any observed or known deviations from the semimartingale property to actual properties of the ways in which trading is conducted. I'll mention three such phenomena (there are several more, of course):

Trading is known to produce not just one price, but two prices: the bid and the ask price. The bid-ask spread is in general non-zero even in a competitive market, for reasons well-explored in the literature of the past 25 years. (In econometric practice, one frequently calculates the average of bid and ask prices and tries to model this average as some kind of martingale but, strictly speaking, that's not always valid.)
Because of inventory and liquidity effects, there's often predictability of price movements at the very highest sampling frequencies. E.g., if prices are observed every second, it may be possible to predict price movements out to a few seconds. This predictability is at the heart of what algorithmic trading rules are all about.
There may also be some mean-reversion of prices and volatilities at the very highest sampling frequencies due to the so-called "bid ask bounce": Empirically, it's been found that if a given trade happens at the bid, the very next trade is more likely to happen at the ask than at the bid, and vice versa. Because there's a spread between bid and ask prices, there's a bit of extra volatility that arises at the very highest sampling frequencies. This usually shows up in so-called volatility signature plots, which plot the intraday sampling freqency $\nu$ against the associated realized volatility $\sigma_\nu$: At the very highest sampling frequencies, $\sigma_\nu$ is frequently an increasing function of $\nu$.

Note that these "noise components" are generated as a consequence of the fact that trading takes place in real places among real people. As such, these noise components are not akin to those generated by computer programming errors. I.e., flash crashes in certain stocks are not an example of market microstructure noise.
To assume, then, that the latent or efficient price process and the market microstructure noise process are mutually independent really means that they're generated by two underlying processes which don't interact with each other. E.g., if new information enters the market, it should affect the value of the semimartingale but not meaningfully alter the noise process. (The interaction between the two processes is usually assumed to be additive, but I suppose one could generate a statistical theory that works with multiplicative rather than with additive noise.)

Naturally, it is not possible to verify directly that the independence assumption is correct, as one of the two underlying processes isn't observed. However, if it were the case that some observed departures from the semimartingale property are such that they cannot be explained usefully/credibly as being related to various market imperfections (such as the bid-ask bounce), one would have reason to doubt the empirical usefulness of the entire approach to modelling observed prices.


What are the 'best' variables out there to forecast accurately GDP and inflation according to macro literature?

You likely want to do this in real time which means you want to do a nowcasting. This is tricky because many economic series are revised, which means you also need real time data (what were the figures when released rather than what is known with hindsight).

The Atlanta Federal Reserve Bank uses this approach in their GDP Now product. You should check out their methodology and inputs to get started on your attempt.

  The growth rate of real gross domestic product (GDP) measured by the
  U.S. Bureau of Economic Analysis (BEA) is a key metric of the pace of
  economic activity. It is one of the four variables included in the
  economic projections of Federal Reserve Board members and Bank
  presidents for every other Federal Open Market Committee (FOMC)
  meeting. As with many economic statistics, GDP estimates are released
  with a lag whose timing can be important for policymakers. For
  example, of the four scheduled 2014 release dates of an “advance” (or
  first) estimate of GDP growth, two are on the second day of a
  scheduled FOMC meeting with the other two on the day after the
  meeting. In preparation for FOMC meetings, policymakers have the Fed
  Board staff projection of this “advance” estimate at their disposal.
  These projections—available through 2008 at the Philadelphia Fed’s
  Real Time Data Center—have generally been more accurate than forecasts
  from simple statistical models. As stated by economists Jon Faust and
  Jonathan H. Wright in a 2009 paper, “by mirroring key elements of the
  data construction machinery of the Bureau of Economic Analysis, the
  Fed staff forms a relatively precise estimate of what BEA will
  announce for the previous quarter’s GDP even before it is announced.”

  The Atlanta Fed GDPNow model also mimics the methods used by the BEA
  to estimate real GDP growth. The GDPNow forecast is constructed by
  aggregating statistical model forecasts of 13 subcomponents that
  comprise GDP. Other private forecasters use similar approaches to
  “nowcast” GDP growth. However, these forecasts are not updated more
  than once a month or quarter, are not publicly available, or do not
  have forecasts of the subcomponents of GDP that add “color” to the
  top-line number. The Atlanta Fed GDPNow model fills these three voids.

  The BEA’s advance estimates of the subcomponents of GDP use publicly
  released data from the U.S. Census Bureau, U.S. Bureau of Labor
  Statistics, and other sources. Much of this data is displayed in the
  BEA’s Key Source Data and Assumptions table that accompanies the
  “advance” GDP estimate. GDPNow relates these source data to their
  corresponding GDP subcomponents using a “bridge equation” approach
  similar to the one described in a Minneapolis Fed study by Preston J.
  Miller and Daniel M. Chin. Whenever the monthly source data is not
  available, the missing values are forecasted using econometric
  techniques similar to those described in papers by James H. Stock and
  Mark W. Watson and Domenico Giannone, Lucrezia Reichlin, and David
  Small. A detailed description of the data sources and methods used in
  the GDPNow model is provided in an accompanying Atlanta Fed working
  paper.

  As more monthly source data becomes available, the GDPNow forecast for
  a particular quarter evolves and generally becomes more accurate. That
  said, the forecasting error can still be substantial just prior to the
  “advance” GDP estimate release. It is important to emphasize that the
  Atlanta Fed GDPNow forecast is a model projection not subject to
  judgmental adjustments. It is not an official forecast of the Federal
  Reserve Bank of Atlanta, its president, the Federal Reserve System, or
  the FOMC.


Here are some recent books:

Taming Capital Flows: Capital Account Management in an Era of Globalization (2015)
Capital Flows and Exchange Rate Management (2013)
Managing Capital Flows: The Search for a Framework (2010)
Capital rising: how capital flows are changing business systems all over the world (2010)


Price measurement in most national and international agencies is based on the “statistical approach” to price indexes developed by Dutot (1738), Carli (1764), and Jevons (1865). The methodologies developed in these papers form the foundation of 98% of all consumer price indexes generated by government statistical agencies (Stoevska, 2008). Following this approach, statistical agencies are more likely to select a good for inclusion in the sample with a past high sales share. Because agencies typically choose products based on their historic sales shares, this explains why "the selection of G&S within the CPI is always favorable to what it has been for years".

Economic theory has largely rejected the "statistical approach" to price measurement in favor of the "economic approach," which asserts that all price indexes should be derived from consumer theory and correspond to the unit expenditure function. See Redding and Weinstein's paper "A Unified Approach to Estimating Demand and Welfare", they provide the argument and the above  references.


Does 'ethical investing' have any effect?

In a world where capital markets are infinitely deep there shouldn't be any price response to capital and so we wouldn't expect any consequences on firm behavior. There is some good empirical evidence for slow moving capital so that's probably not true. Ethical investing is likely to drive up capital costs at least some. In which case your question boils down to two questions, "can an non-ethical company become an ethical one" and one that may seem quite different, "what is the elasticity of demand for capital with respect to the price of capital?" Tobacco and munitions companies can shift into other business but they can't get ethical investment funds without abandoning that business. Other businesses like fair-trade coffee and certified conflict-free diamonds, may allow ethical investing in the current business with higher costs. The former group really only have one choice, to shrink in response to expensive capital. The latter group have a second option, to shift into ethical investing business models that may be more expensive but less expensive than the increase in the cost of capital.

Let's ignore that first channel for a minute. Let's just consider a world with some ethical industries and some non-ethical ones. One day some people wake up and become ethical investors and this causes a change in the demand for non-ethical securities. If firms have an inelastic demand for capital (lower right graph, moving from $S_2$ to $S_1$) then the quantity of capital demanded changes relatively little from the ethical preference shock and so the  total capital invested in the non-ethical industry changes very little ($Q_2$ to $Q_1$). Instead returns go up a lot ($P_2$ to $P_1$) for those investors who remain invested in non-ethical industries.

On the other hand, if capital demand is quite inelastic (lower left graph, moving from $S_2$ curve to $S_1$)), the return on capital changes very little ($P_2$ to $P_1$) but the total capital invested in  non-ethical industries falls a lot ($Q_2$ to $Q_1$). In that world the ethical industries crowd out the non-ethical ones because the profits aren't there to preserve their capital supply.

Now consider production shifting. If substitution into ethical practice is more expensive than paying a higher cost of capital, this won't be an important channel. Firms are better of taking their licks in the capital markets. However, if substitution is cheaper (like when capital demand is inelastic and and substitution easy) then behavioral adaption will become the dominant channel.

In practice, according to Does Ethical Investing Impose a Cost Upon the Firm? A Theoretical Perspective by James J. Angel and Pietra Rivoli (1997), the actual effects on capital costs and quantities are quite small, suggesting that neither industry shrinking nor industry transformation are important under prevailing circumstances. If more money shifts into ethical investing this could easily change.


Marshalian and Hickisian Demands and Slutsky Equation

Although my teacher has yet to verify my solution for d) (which I believe is incorrect), he shared his answer and the condition $x_2<k$ derives from the positivity of $V(p_1,p_2,b)$ and from the budget inequality $p_1x_1+p_2x_2≤b$. The other answers are correct.


maximize untility AND minimize disutility

I believe what you're getting at is a concept in economics called "Loss Aversion". It has been used quite extensively in recent economic research, especially in behavioral economics and finance. Another important key word here is "Prospect Theory", which encompasses loss aversion as well.

The basic idea is that people suffer more from a loss of x\$ than they enjoy a gain of x\$ compared to a reference point. The utility function has a different (smaller in absolute value) slope when moving in a positive direction from the reference point than when moving in a negative direction. This is illustrated by the graph below (source: https://upload.wikimedia.org/wikipedia/commons/4/4e/Valuefun.jpg).

Under these conditions,  utility and disutility are different. Otherwise, maximizing untility or minimizing disutility is the same, as you mentioned.

This insight goes back to Amos Tversky and Daniel Kahneman.Kahneman received the Nobel Prize in Economics in 2002 for this work.

A summary of the most important literature can be found for example here: http://loss-aversion.behaviouralfinance.net/


Is there a formal definition of opportunity costs, economic costs, and/or economic profits?

Economic profit
I think there are some problems with the formulation in your question, first you heavily focus on opportunity cost but note accounting profit does not even properly capture all revenue firm gets.
I will first focus on economic profit more broadly and at the end go back to opportunity cost.
Following Varian Microeconomic Analysis pp 24 (emphasis mine),

Economic profit is defined to be the difference between the revenue a firm receives and the costs that it incurs. It is important to understand that all costs must be included in the calculation of profit. …
Both revenues and costs of a firm depend on the actions taken by the firm. These actions may take many forms: actual production activities, purchases of factors, and purchases of advertising are all examples of actions undertaken by a firm. At a rather abstract level, we can imagine that a firm can engage in a large variety of actions such as these. We can write revenue as a function of the level of operations of some $n$ actions, $R(a_1,...,a_n)$, and costs as a function of these same $n$ activity levels, $C(a_1,...,a_n)$.

The above is already rigorous definition for economic profit, economic revenue and economic cost.
Note accounting operates under its own sets of rules. Accounting, is not econometrics. Point of accounting is not to accurately estimate or measure economic relationships. So what you are doing above, defining economic profit at simply accounting profit minus opportunity cost is somewhat misguided (it would only hold in an unrealistic special case where we assume accounting only missed opportunity cost but accounting misses so much more).
First, following Varian we can start with a set of occurring actions $a$ that create revenue and costs. However, when it comes to accounting, not all of these will be reported. For example l, consider selling goods without issuing receipt to avoid paying taxes, which will not get reported as accounting profit. The same holds for many costs other than opportunity cost. For example, consider how depreciation costs are treated by accountants. From purely economic perspective most assets definitely do not depreciate in linear fashion yet accounting allows firms to use linear depreciation for most assets regarding what the true depreciation cost each period is. So the accounting depreciation costs will too be different from economic depreciation cost. Or consider the fact that many countries allow small business to just in their accounting declare portion of their profit as cost without actually even proving they actually incurred these costs (this is done in some countries to ease administrative burden for sole traders and similar one person businesses).
Consequently, it makes more sense to rather define accounting profit in terms of economic profit and then work backwards by inverting those functions if you would want to get back to economic profit from accounting one.
Call $A=(a_1,...,a_n)$.  set of all economic actions that actually occur, and $R(A)$ would be economic revenue from those activities. Now apply some function $\rho$ which tells you based on $R(A)$ what actually accounting revenue is.
So accounting revenue $R_{acc}$ would be:
$$R_{acc}= \rho(R(A))$$
Similarly accounting cost $C_{acc}$, we will have actual economic cost firm incurs $C$ based on set of actions it takes $A$ and then we will have some function gamma that tells us what enters into accounting and what not so we would have that
$$C_{acc} = \gamma (C(A))$$
Finally the accounting profit would be difference between the above so:
$$\Pi_{acc} = \rho(R(A)) -\gamma (C(A))$$
Note what $\rho(.)$ and $\gamma(.)$ are will depend on local accounting regulations and laws. in some country government could declare that any firm has to put  always just 1000€ as their cost (yes it is unrealistic example but I am just showing it to prove a point) and in such case $\gamma(C(A))=1000$ regardless of anything that happens.
Whereas the true economic profit would just difference between economic revenues and economic costs:
$$\pi=R(A)-C(A)$$.
If you would want work your way back from accounting definitions then you could use inverse functions and economic profit calculated from accounting one would be:
$$\pi= \rho^{-1}-\gamma^{-1}$$.
Note the above does not really require definition of opportunity cost since here simply the function $C(A)$ is just some function where we assume that $C(A)$ assigns true economic cost to every action $a_1,…,a_n$, for standard profit optimization it’s completely okay to treat $C(.)$ as a black box and thus you will not really require separate definition of opportunity cost. As a matter of fact note in situation where there are no alternatives like having mana (from the biblical story where it could only be immediately consumed not stored or used for anything else) opportunity cost would not even be part of $C$ but still there would be distinction and disconnect between accounting and economic profit.
For example if we have $C(A)=c_1 a_1+c_2 a_2… c_n a_n$ and if we define $C$ as a function that always assigns true economic cost then you definition $c_1$ includes all economic costs including opportunity cost, but the economic profit, economic revenue and economic costs are still rigorously defined (even if the constituents of costs such as opportunity costs aren’t).
opportunity cost
Next tackling the opportunity cost separately (which can be later used to build function $C$), good places to start are Buchanan (1987) or Alchian (1968). The authors do not necessarily derive rigorous opportunity cost functions but give rigorous verbal definitions on which we can build.
For example, according by to Buchanan (1987) :

Opportunity cost is the anticipated value of 'that which might be' if choice were made differently. Note that it is not the value of 'that which might have been' without the qualifying reference to choice. In the absence of choice, it may be sometimes meaningful to discuss values of events that might have occurred but did not. It is not meaningful to define these values as opportunity costs, since the alternative scenario does not represent a lost or sacrificed opportunity. Once this basic relationship between choice and opportunity cost is acknowledged, several implications follow.

First, if choice is made among separately valued options, someone must do the choosing. That is to say, a chooser is required, a person who decides. From this the second implication emerges. The value placed on the option that is not chosen, the opportunity cost, must be that value that exists in the mind of the individual who chooses. It can find no other location. Hence, cost must be borne exclusively by the chooser; it can be shifted to no one else. A third necessary consequence is that opportunity cost must be subjective. It is within the mind of the chooser, and it cannot be objectified or measured by anyone external to the chooser. It cannot be readily translated into a resource, commodity, or money dimension. Fourth, opportunity cost exists only at the moment of decision when choice is made. It vanishes immediately thereafter. From this it follows that cost can never be realized; that which is rejected can never be enjoyed.

since the opportunity cost is the value of the next best alternative you can rigorously define as follows.
Suppose that for each action that actually was taken in the set $A=a_1,…,a_n$, there are alternative sets of alternative actions (e.g. each $a_i$, has some alternative $b_i$, $z_i$ and so on, and let’s assume that all the other alternatives are mutually exclusive with $a$. Well then opportunity cost of each action $a$ is just value of the next best alternative so if we have that:
$a_i\succ b_i \succ z_i … $
Then opportunity cost will simply be $V(b_1)$ where $V$ is simply just some function that assigns monetary value (or in consumer problem utility to $b_i$).
If you want to then put this value into profit function explicitly then simply the cost function will be some composite function given by $C(a_i, V(b_i))$ (note this is ultimately just function of $a_i$ because the next best alternative necessarily depends on what you choose as $a_i$, so in the end it’s just more complex way of writing $C(a_1))$
I think the above is rigorous even though it’s very simple (of course perhaps the notation I use could be made more precise).


How does one derive the elasticity of substitution?

How to derive elasticity of substitution
The first step is to recall the definition of a differential. If you have a function $f: \Bbb R^n \to \Bbb R$, say, $f(x_1,\cdots,x_n)$, then: $${\rm d}f = \frac{\partial f}{\partial x_1}{\rm d}x_1 + \cdots + \frac{\partial f}{\partial x_n}\,{\rm d}x_n. $$
For example,  $$d\log v = \frac{1}{v}dv$$
Now suppose $v = \tfrac{y}{x}$, then we have $$ d\log(y/x)=\frac{d(y/x)}{(y/x)}$$
and for $v = \tfrac{U_x}{U_y}$
$$ d\log(U_x/U_y)=\frac{d(U_x/U_y)}{(U_x/U_y)}$$
In other words, if you reduce the problem to (1) understanding the definition of a differential and (2) use a simple change of variable, the problem becomes very straightforward.
You then get
$$\sigma \equiv \frac{d\log\left(\frac{y}{x}\right)}{ d\log\left(\frac{U_x}{U_y}\right) }= \frac{ \frac{d(y/x)}{(y/x)} }{ \frac{d(U_x/U_y)}{(U_x/U_y)} } $$
ASIDE:
Note, it is important to recognize that $ d(y/x)$ is a meaningful concept. You simply apply quotient rule and you find
$$ d(y/x)= \frac{xdy-ydx}{x^2}$$
This makes sense because
$$ d\log(y/x) = d\log(y) - d\log(x) = \frac{dy}{y}-\frac{dx}{x}$$
And if you compute
$$ d\log(y/x)=\frac{d(y/x)}{(y/x)}=\frac{ \frac{xdy-ydx}{x^2}}{y/x} =  \frac{xdy-ydx}{xy} = \frac{dy}{y}-\frac{dx}{x}$$
Same logic applies to $d(U_x/U_y)$.
Thus, all of $\sigma$ is well-defined in the sense we are using the calculus tools correctly / legally.

What is elasticity of substitution?
Elasticity is by how much % one thing changes relative to a % change in another. Therefore, in this case, it is % change in ratio of two goods  relative to a single % change in the $MRS$ for those two goods.


Is the capital-output ratio in developing countries low or high?

The key to the answer is good data on Capital. There is a project (KLEMS), which is computing harmonised (i.e. comparable) information on capital, labour, energy, etc for many countries. At the moment it has information mainly on developed countries, but data for more developing countries are coming up.

For example, this is a calculation of the capital-output ratio for the case of the UK:

For the case of developing countries, here is a figure:

(source here)

It seems that developing countries might have lower capital-output ratios. However, this is ultimately an empirical question which requires a proper analysis. I suggest you use the KLEMS data to make your own, harmonised study.


What is the difference between Common Goods and Common Pool Resources?

Common pool resources can be depleted by overuse.

For example, an area of communal grazing land would be a common pool resource because grazing too many animals decreases the amount of grass available to each animal.

Wikipedia is a common good but are not a common pool resource - reading a lot of articles on Wikipedia does not reduce the number of articles available for everyone else to read.


I do not quite get a phrase that I read in the Financial Times today.

When the share price has fallen 90% from some price P, the current price is P $\cdot$ 10%.

When it has fallen 95% from some price P, the current price is P $\cdot$ 5%..

P $\cdot$ 5% is half of P $\cdot$ 10%.

As the share price falls from P, it will first pass through P $\cdot$ 10% before reaching P $\cdot$ 5%. So if we obsere the time period from when the price was P until it was P $\cdot$ 5%, it has hallen 95%. But if we observe the time period from when the price was P $\cdot$ 10% until it was P $\cdot$ 5% the share price has fallen by half.


The first statement is a result from an old economic model.

The two views can be reconciled by the following argument: the central bank is constrained by the structure of the economy as to its choice of the policy rate. If persistently too low, inflation rises, and if persistently too high, the economy risks falling into deflation.

In this case, the central bank allegedly needs to keep the real policy rate near the growth rate of real GDP. Arguably, not many people are convinced that is precisely the case, rather that there is a “natural rate if interest” (now preferably referred to as $r^*$) that the policy rate needs to revert to.

The argument is then that bond investors will be aware of this tendency, and thus bond yields will revert to the long-term average.

However, one would need to search for the modern literature on $r^*$ to get a better handle on preferred ways to estimate it. I am in the camp that $r^*$ does not exist, so I am not the person to offer such references.


Finding "own" price elasticity of demand

Your work for 1 and 2 is correct.

For 3, look at each segment individually. There are two points which have an elasticity of $-1$, one on each segment.


What would happen if corporations can only make a maximum profit of 10% per product

I also think that international corporations like Coca Cola, Nestlé definitely make more than 200 % profit for each of their products (does anybody know how much).

How are you defining profit?

Coca Cola, first quarter 2018
Net income:  \$1.37 billion
Revenue:  \$7.6 billion
Profit:  18%
Percentage markup:  22%

Nestle was higher in the second half of 2017, around 49.9% profit or 98% markup.

I don't know which, but if you are getting 200+%, I think that you are leaving out a lot of costs.  If you enforced that standard, companies would go out of business, because you aren't leaving enough for operating expenses.

If all companies, even the risky ones, could only have 10% profit, their product prices would tend to swing greatly.  Because when their costs are high relative to their revenues (e.g. in a recession), their prices would go up.  When costs are relatively low, their prices would be restrained.  So they would be cutting prices in good markets and increasing them in bad markets.

Companies would also have trouble financing through equity, which is rewarded by profits.  They would have to borrow, as that would show up as costs.  But it's hard to borrow unless a company is already profitable.  So no venture capital startups.

Companies would also tend to become bigger.  Because it is easier for a company that does many things to make sure that some are not so profitable.  And of course they wouldn't have to worry about startups stealing their business.

Companies would take few risks.  Because the maximum return is 10%, they have no incentive to pick the riskier choices.  They might just as well pick the safe choice, as it has the same return.  Progress would slow.

To engage in things like pharmaceutical development or oil discovery, companies would basically write off the upfront costs (to minimize profit).  Then if they actually succeeded, they'd have to engage in a flurry of additional costs so as to get full value for their products.  So the companies engaging in such behavior would expand revenues quickly.  And only companies that are already profitable could undertake the most profitable activities.

This encourages companies to out and out waste money.  Higher costs means a higher base, so the 10% is bigger.


Here is what I thought could be a simple example. Suppose there are two periods ($t_0, t_1$) and two possible states of nature ($s_1,s_2 $) that arise in period $t_1$. Suppose there are two agents. The first agent has an income stream that pays off only in state $s_1$ while, second agent's income stream pays off only in state $s_2$. Consumption happens only in $t_1$ and both agents prefer to consume in both the states than in just one (Inada condition as consuption goes to 0 is not satisfied) and more is always better. If at $t_0$, neither agent knows the state at $t_1$ then they might enter into a trade that ensures them positive consumption across both the states.

On the other hand, if, in $t_0$, state is already revealed publicly (in this example even privately to one agent) an agent who has a positive pay-off in $t_1$ would not be interested in any trade with the unlucky agent.


Market Value of a publicly traded company

One possibility would be to use Wolfram Alpha.

If I type the search phrase market capitalization msft 3rd December 2010 then the search engine returns the following:

If you use Wolfram Mathematica on your PC then you can import the data directly for processing using the FinancialData command (see here).


What are some critiques to the principle of comparative advantage?

One of the most common critiques is that the standard model is a static analysis.  While a country might be better off today by focusing on its comparative advantage, its longer term prospects depend on the future of that industry, particularly on how quickly productivity can be improved.

For instance, in David Ricardo's original example, comparative advantage recommended that Portugal should produce wine while England produced cloth.  Maybe that's okay as a static analysis, but over time the productivity of wine production improved slowly whereas for England,

  exports of cotton cloth led to accumulation, mechanisation and the whole spiralling growth of the industrial revolution.

The idea that a country should temporarily distort free trade in order to establish industries in which it wants to develop a comparative advantage is often known as "dynamic comparative advantage".  The East Asian economies are frequently referred to as a good example of countries using these ideas successfully.  South Korea didn't start out with a comparative advantage in steel, for instance -- it created one through extensive state intervention.


Dealing with Missing Values in Diff-in-Diff Estimation

Just to make sure I understand:

You have a daily panel (with missing values), probably weekday only, running from 2013 to mid-2017 with $n=3$ cross-sectional units.
You believe that after 2015, there is a stronger incentive to do something different at quarter end, so the average basis should be different on month end days during that period.

The typical difference-in-difference regression compares two types of units, some that are treated and some that are not, before and after the start of treatment. You don't have that here, but I think that's okay, because you are instead looking at the difference between quarter end and the rest of the quarter, pre- and post-.

As a general matter, this setup is fine. As long as the missing-ness of the data is "missing completely at random" or "missing at random". The problem occurs when the pattern of missing data reflects selection on the LHS variable, omitted variables causing changes in the LHS, or the value of the variable that's missing is related to the reason it is missing.

Another concern would be empty cases. You want to make sure that you have quarter end and quarter not end observations for the pre- and post-period. If you use of unit FE, you want to make sure that you have data pre-/post-, quarter end / not, for each unit.

Make sure you are also satisfying the requirements of the difference-in-difference setup. Some are un-testable, but at a minimum you should make a graphical or statistical argument that your data satisfy the common trend assumption.

Here is some sample Python code that shows that you can recover the initial parameters in your setup. I did it with month end, not quarter end, because it makes it easier to have a large enough sample of missing data for both difference variables.  It has an example of MAR and MCAR. It shows it works for both.

    import numpy as np
    import pandas as pd
    import statsmodels.formula.api as smf
    import itertools
    np.random.seed(9876789)

    startyear=2013
    endyear=2017
    daysinmonth = 21
    nummonths = 12
    numgroups = 3
    numyears = (endyear-startyear+1)
    dropfraction = 0.20
    # np.sort(np.tile(np.arange(2013,2018), 12*21))
    years   = np.repeat(np.arange(startyear,endyear+1), nummonths*daysinmonth*numgroups )
    days    = np.tile(np.arange(1,daysinmonth+1), nummonths*numyears*numgroups )
    months  = np.tile(np.repeat(np.arange(1,nummonths+1), daysinmonth),  numyears*numgroups)
    groups  = np.tile(np.repeat(np.arange(1,(numgroups+1)),nummonths*daysinmonth), numyears)
    df = pd.DataFrame({'year': years, 'month': months, 'day': days, 'group': groups})

    # Last weekday is monthend or the 21st
    df['endofmonth'] = df.day == 21
    df['QendW'] = 1*df['endofmonth']
    df['Post15'] = 1* (df.year > 2015)
    df['random'] = np.random.rand(daysinmonth*nummonths*numyears*numgroups)
    df['missing_completely_at_random'] = df['random'] <= dropfraction
    df['missing_at_random'] = (df['random'] - 0.05 * df['Post15'])  <= dropfraction
    df['QendWxPost15'] = df['QendW'] * df['Post15']

    alpha0 =  0.2
    gamma1 =  0.05
    beta1  = -0.10
    beta2  =  0.15
    df['error'] = np.random.randn(daysinmonth*nummonths*numyears*numgroups) / 10

    df['x1'] = alpha0 + gamma1 * df['Post15'] + beta1 * df['QendW'] + beta2 * df['QendWxPost15'] +  df['error']

    df['x1_missing_completely_at_random'] = df['x1'].copy()
    df.loc[df['missing_completely_at_random'] == True, 'x1_missing_completely_at_random'] = np.nan

    df['x1_missing_at_random'] = df['x1'].copy()
    df.loc[df['missing_at_random'] == True, 'x1_missing_at_random'] = np.nan

    print("No missing data")
    model = smf.ols(formula='x1 ~ Post15 + QendW + QendWxPost15', data=df)
    results = model.fit()
    print(results.summary())

    print("Missing Completely At Random (MCAR)")
    model_missing_completely_at_random = smf.ols(formula='x1_missing_completely_at_random ~ Post15 + QendW + QendWxPost15', data=df)
    results_missing_completely_at_random = model_missing_completely_at_random.fit()
    print(results_missing_completely_at_random.summary())

    print("Number missing quarter end (MCAR)")
    print(np.sum(df.loc[df['missing_completely_at_random'] == True, 'QendW'] ))
    print("Number missing quarter end post 2015 (CAR)")
    print(np.sum(df.loc[df['missing_completely_at_random'] == True, 'QendWxPost15'] ))

    print("Missing At Random (MAR)")
    model_missing_at_random = smf.ols(formula='x1_missing_at_random ~ Post15 + QendW + QendWxPost15', data=df)
    results_missing_at_random = model_missing_at_random.fit()
    print(results_missing_at_random.summary())

    print("Number missing quarter end (MAR)")
    print(np.sum(df.loc[df['missing_at_random'] == True, 'QendW'] ))
    print("Number missing quarter end post 2015 (MAR)")
    print(np.sum(df.loc[df['missing_at_random'] == True, 'QendWxPost15'] ))


Meaning of a paragraph related to economics

This means that interest rates rose in India at the same time that they stayed relatively low in other areas of the world. The fact that investors could get a higher yield in India led to an increase in demand for their currency (if I can get a 5% yield on Indian bonds, why would I get a 2% yield on US bonds?) This led to an increase in demand for their currency so that investors could buy Indian bonds/Invest in India at higher yields. The appreciation of the rupee (increase in value) led to India’s goods costing more (it took more dollars to buy Indian goods for example due to the currencies rise in value). This led to fewer goods being demanded (higher prices decrease demand). So Indias exports declined.


Should mean wages increase with inflation? If not, why?

No. Inflation is not the only factor that determines wages. Inflation has to do with the nominal value of wages, but these also entail components pertaining to the real value of work.

Here are three examples that tend to push mean wages down,

(1) technology, to the extent that it replaces human labor;

(2) foreign markets may lure high skilled workers (or, for what matters, workers with the highest salaries) to emigrate, causing -caeteris paribus- mean wages to decrease;

(3) public policy (whether it stems from unions or from policies aimed at reducing the number of unemployed) might prompt -otherwise unneeded- entry of more people in the labor market.

Other factors would have the effect of pushing mean wages up. For instance, the converse of (2) (that is, emigration only of workers earning the lowest salaries), unavailability of workers with high productivity, and so forth.

  Has this profound consequences in the redistribution of wealth?

That depends on the particularities of the economy being analyzed: what fiscal policy is implemented; whether households own significant portions of capital; the sensitivity and preponderance of different sectors in the economy; etc.


Shouldn't national Stock Indices proportional to GDP of the country?

One of likely many contributing factors is the definition of GDP.  Consider the humble iPhone, produced by Apple contractor FoxConn. FoxConn is headquartered in Taiwan, while Apple is headquartered in Cupertino, California, USA.

Apple owns the intellectual property that goes into the iPhone, and so iPhone sales to American citizens support growth in Apple stock.  But because the iPhone is produced offshore, when an American citizen buys a cell phone that buoys the stock of an American-domiciled company, it does not contribute to American GDP - it contributes to Chinese, or Taiwanese, or wherever the factory happens to be, and in fact subtracts from American GDP.


If $\succsim$ is transitive but irreflexive, then it is asymmetric, proof

A relation that is not asymmetric need not be symmetric. But if $\succsim$ is not asymmetric, there must be elements $y$ and $x$ such that both $x\succsim y$ and $y\succsim x$ hold and these elements are all you need to arrive at a contradiction.

Also, $x\succsim y\implies y\succsim x$ can hold for an asymmetric relation, namely when $x\succsim y$ does not hold.

I'd like also to point out that "any" is ambiguous here and might be read as "all" instead
of the intended "some."


Weak stationarity and weak dependence are complementary conditions.
A weakly stationary time series $y_t$ has an underlying statistical process which is time-invariant.
This is characterised by three conditions:

$$E(y_t)= \mu$$
$$E((y_t-\mu)^2)=\sigma^2 <\infty$$
$$Cov(y_t,y_{t+s})=Cov(y_t,y_{t-s})$$

Condition (1) indicates that a weakly stationary series has an underlying (or long-run equilibrium) unconditional average.
Condition (2) indicates that the unconditional variance of a weakly stationary series does not change through time and remains within a certain range/finite.
Condition (3) indicates that a weakly stationary series has stable (symmetrical) dynamics between periods, which only depend on the number of periods, rather than time itself.
That is to say, Condition (3) states that a weakly stationary process we can assume a stable relationship between certain periods, regardless of whether it's the year 2000 or the year 3000. Time plays no role, only the relationship across periods.
Weak Dependency:
Weak dependency ensures mean reversion back to an underlying process.
Weak dependency states that the correlation between $y_t$ and future periods $y_{t+h}$, tends to zero sufficiently quickly. And with periods increasing to infinity, dependency should tend to zero.
$$Corr(y_t,y_{t+h})\rightarrow 0, h\rightarrow\infty$$
Importance in the Context of Time Series
With time series data, our central objective remains to estimate the magnitude of change in Y terms of linear unidirectional causation from X.
That is to say, our goal is to take a sample time series and fit a line which best explains the change in Y given X characteristic of the underlying population.
Yet, in the time series context, conducting estimation representative of the underlying population is undermined by a failure of i.i.d conditions
An evolving (non-stationary) time series, entails that observations will not be drawn from an identical distribution and parameter estimates will not convey a constant linear (time-invariant) relationship on Y given X;
Furthermore, given that time series data is necessarily non-independently distributed (i.e. current realisations dependent on past realisations), then again parameter estimates will no longer convey information on the underlying population.
It follows that two mutually dependent properties must be upheld to ensure the i.i.d conditions known as:

Sufficiently fast weak dependency – random variables should exhibit
finite memory/asymptotic independence;

Weak stationarity – the underlying statistical properties (mean,
variance, and covariance) should be time invariant as if drawn from
an identical distribution.

Note that both conditions are mutually dependent, given that sufficiently quick finite memory is a requirement for a process to exhibit mean-reversion back to some underlying population.
Moreover, the combination of weak dependency and weak stationarity ensure that a property known as ergodic stationarity is upheld.
Essentially, ergodic stationarity implies that with a quick return to an identical distribution, then realisations are representatives of the greater process, and averaging is relevant.
Further, weak stationarity is important in terms of the exogeneity condition to control for a causal relationship, given that if contemporaneous exogeneity holds in one period; then by weak stationarity, it should hold for all periods.
Also, ergodic stationarity ensure that the process mimics a random independent sample required to uphold LoLN/consistent estimation/CLT are valid.
Finally, weak stationarity overall avoids the problem of so-called spurious regression in which two mutually trending series are wrongly identified to have a causal relationship.


Regression of Real GDP on Intellectual Property Products Problem

It is never a good idea to regress something so complex as GDP on a single variate. There may be myriad confounding factors.

I can think of two suggestions here:

1) Lookup Barro's and Sala-i-Martin book on Growth, the empirical part, to get an idea of what kind of factors seem to appear to influence growth (or what kind of factors growth can be decomposed into). They can be used as control variables, in order to check whether in their presence IntProp remains a statistically significant variable, and even if it does, what is the magnitude of its effect (so that one can assess its economic significance also).

2) Consider detrending your variables, either linearly or non-linearly using for example the HP-filter, and regress the cycle components and perform the usual correlation analysis of real business cycle models, in order to see whether IntProp leads or lags the GDP cycle. If it leads, an argument could be made that in the short-run at least IntProp pushes the growth cycle out of its trough.


What economic factors should be considered in attracting suitable staff and determining pay for temporary project work?

Characteristics of a "project", as per the Project Management Institute, are that it is temporary and unique (ie. non-routine).

For a person employed to work on a project, the temporary nature of the work is likely to be a source of additional cost and risk, as compared with a person in a permanent job. This is because, as the project comes towards its end, the person is likely to have to find another job.  That involves both the transaction costs of finding a job (searching, assessing whether jobs are suitable, negotiation, etc., implying time and effort if not monetary costs) and the risk of not finding a suitable job when needed and therefore experiencing a loss of income. These factors suggest that to recruit to a project role will require somewhat higher pay than to recruit to a permanent job with a similar skill level.

An exception to the above is where a current staff member in a permanent job is seconded to work on a project. Provided they can be sure that at the end of the project they can return to their original job, they will not be subject to the sort of risk described above. On the other hand if a current staff member is to be asked to work on a project while also keeping their regular work going, they may decline unless offered higher pay (for reasons of workload, not risk).

A further issue is how pay for project work might be structured. Suppose A is working on a project which is 90% complete. The employer will want A to stay and complete the project.  But A knows that staying until the end of the project will increase the risk that they will find themselves without a job. So they will be tempted to apply for other jobs before the project is complete. Hence there is a risk for the employer, ie that the departure of a key person will mean that completion of the project becomes problematic.  To address this, an employer might consider a pay structure consisting of a regular monthly salary plus a bonus payable on successful completion of the project.

But that in turn raises the issue of asymmetric information. Because of the non-routine nature of project work, it may be that only those working on the project really know how near (or far) the project is from successful completion. This is especially likely where the output of a project is not very visible or tangible, an ICT system implementation for example. So even if the project is only 90% complete, and A knows that, there is a risk for the employer that A will leave to take another job while also persuading management that the project is complete and receiving the bonus.


Maximisation problem in a multiproduct firm

The question "how are the first order conditions" seems very unclear to me, and I am providing a set-up for finding and writing them out, while explaining the Kuhn-Tucker conditions that are easy to struggle with.

Though we try to avoid giving away basic study question answers, it's a positively voted question without an answer, and I still think these questions have value for future users. Enough time has also passed where there's no risk of helping someone get through a class, if that was the intention.

Usually in cost minimization problems where the production function is a constraint, there is a set quantity to be produced that then leads to minimizing cost with respect to that. Here in this profit maximization problem, you have a minimum amount of production required of both goods, which I find interesting.

We can note the objective function (profit) is linear, and therefore concave and differentiable. The constraints are more interesting because the Cobb-Douglas form is not necessarily concave, namely when the exponents sum to more than 1. So you can ask yourself when the constraints will be convex (they are certainly differentiable). These ideas are related to the sufficient conditions for an optimum. A solution exists by the Weierstrass Theorem (constraint is compact and the objective function is continuous).

To look at the first order necessary conditions, we can omit your last two equalities since they don't add anything. We take:

\begin{equation}
\begin{array}{l}
\max_{q_{x}, q_{y}, l_{x}, l_{y}, k_{x}, l_{x}} \quad p_{x}q_{x}+p_{y}q_{y}-w(l_{x}+l_{y})-v(k_{x}+k_{y}) \\
\text{s.t.} \\
q_{x} - k_{x}^{\alpha}l_{x}^{\beta} \leq 0 \\
q_{y} - k_{y}^{\gamma}l_{y}^{\delta} \leq 0 \\
\end{array}
\end{equation}

Form the Lagrangean:

$$\max \quad \mathscr{L} = p_{x}q_{x}+p_{y}q_{y}-w(l_{x}+l_{y})-v(k_{x}+k_{y}) - \mu_x (q_{x} - k_{x}^{\alpha}l_{x}^{\beta}) - \mu_y (q_{y} - k_{y}^{\gamma}l_{y}^{\delta})$$

(note the signs in front of the Lagrangean multipliers)

Then, in place of the derivative with respect to the multipliers, we have our constraints with complementary slackness conditions. We take the other constraints as well.

$$\frac{\partial \mathscr{L}}{\partial \ ( \cdot )} \leq 0 \quad ; \quad ( \cdot ) \geq 0 \quad \forall ( \cdot )$$
$$\text{constraint} \quad ; \quad \mu_{(\cdot)} \geq 0$$

And from there you can move on to solve the system, but you may find yourself in a weird spot because we haven't been able to establish uniqueness.


Is my logic on taxation for this question legit?

The burden of taxation is shared among suppliers and demanders according to the price elasticities of supply and demand. The more elastic side carries less of the tax burden.

To understand this, note that the tax effectively increases the price demanders pay and decreases the price suppliers get. Elasticity tells us how demanders and suppliers react to this price change. For demanders this means buying less and for suppliers it means supplying less. If you are more elastic then you react to the tax more (by definition of elasticity), which means you avoid it more.

To illustrate this, imagine a tax on cigarettes. We assume that smokers who are addicted will have to buy cigarettes at any price (as they are addicted). This means they do not react and have 0 elasticity of demand. In that case, imposing a tax on cigarettes (a price increase) will not affect how many cigarettes are bought. So the producers of cigarettes are making just as much profit as before, while the consumers are paying higher prices, hence the consumers carry the full burden of the tax.

You could imagine a similar argument, where the supplier produces less because of the tax, thereby causing prices to increase and hurting the consumer.

In your case the producer is totally inelastic, while consumers have some elasticity and hence carries the burden of the tax (as he cannot react and reduce the number of seats).

Alternatively imagine a tax put on producers, where the producers are inelastic, but consumers are elastic. The producers when trying to push the tax onto the consumers (raise the price) the consumers will buy less (are elastic) and thus reduce the suppliers profits, so the producer cannot push the full tax onto the consumers if he's maximizing profits. He can only push a part of it, which means his profits will partially fall and buyers will pay higher prices. Hence both carry the tax burden. The producer can push less of the tax increase the more elastic the consumers are compared to him.

We can also see this general point graphically. The figures I have for this are when the tax is on the producers so affects the supply, but the result is equivalent to a tax on the consumers affecting demand. The blue region represents the surplus lost by the consumer that is captured as tax revenue as a result of the tax and the red region represents the surplus lost by the producer that is captured as tax revenue as a result of the tax. The grey region is deadweight loss resulting from the tax -- it is surplus lost, but not captured as tax revenue. In the first plot below, supply is relatively inelastic (i.e. not very steep) and demand is quite elastic (i.e. steep). In the second, supply is relatively elastic (i.e. steep) and demand is quite inelastic (i.e. not very steep).


Isn’t a zero-coupon perpetual bond not worthless if the relevant government has some probability of buying them back?

If I understand you correctly, the bond pays out nothing whatsoever. Such a bond must be worthless. The value of the bond is equal to the net present value, i.e. the sum of the discounted value of all future cash flows, which are zero in your case. If the government would buy it back, it would necessarily buy it back at a price of zero. Therefore, the bond will remain worthless in any case.


Macroeconomics differences of 2008 financial crisis and 2020 economic crisis

There are a lot of differences between the two crises + how these differences manifest in the stock market.
First, on the point of the stock market- it's "kinda" an indicator of the overall economy. However, it's not a perfect metric; really, it reflects the weighted sentiment of those who participate. Furthermore, the stock market is more akin to a leading economic indicator, in that it predicts the future. In contrast, GDP is a lagging indicator and has taken a severe hit. Thus, people who participate in the markets are overall confident that the economy will rebound after COVID ends, in spite of hits to GDP.
This optimism is driven by a few factors that differ from the 2008 crisis:

Fiscal policy has been strong and immediate - PPP + other stimulus has passed rapidly and in huge volumes. The CARES act was almost \$2 trillion (compared to the 2008 fiscal support of $800 billion).
Expanding role of monetary policy - while monetary policy has greatly supported banks, the Fed has also directly supported businesses with loans and injections into specific capital markets. This is a huge expansion compared to the 2008 financial crisis
There are no obvious systemic failures in the financial system - yes, some banks are under stressed by non-performing loans, but the monetary and fiscal policy has both helped banks. Furthermore, stimulus has mostly been saved, meaning that banks have even more capital.

Even though the stock market is high, this is not necessarily a guarantee of future macroeconomic stability - there are rising concerns of exploding corporate and sovereign debt, trade tensions, collapsing supply chains... there are still many risks that will need to be addressed. Additionally, some characterize Wall Street as being "disconnected" from Main Street, again related to the fiscal/monetary policy described above.
Main takeaway: the performance of the stock market does not mean households and individuals are also doing well during the crisis.


Given a utility function $u(c)$, $c>0$, with $u'(c)>0, u''(c) <0$. Regarding the sign of the second derivative, if it is zero, then both measures are zero, if it is positive, it would imply increasing marginal utility, and I don't remember having seen these attitude-towards-risk measures for such a utility function.

Denote Absolute Risk Aversion ($ARA$) and Relative Risk Aversion ($RRA$) correspondingly by

$$A(c) = -\frac {u''(c)}{u'(c)},\;\;\; R(c) = cA(c), \;\; A(c) = \frac 1c R(c)$$

1) Monotonicity of $A(c)$ in relation to $R(c)$

$$\frac {\partial A(c)}{\partial c} = \frac {\partial [(1/c)R(c)]}{\partial c}= -\frac 1{c^2} R(c) + \frac 1{c}\frac {\partial R(c)}{\partial c} $$

So if
$$\frac {\partial R(c)}{\partial c} \leq 0 \Rightarrow \frac {\partial A(c)}{\partial c} < 0$$

$RRA$ weakly decreasing  $\Rightarrow ARA$ is strictly decreasing in $c$.

2) Monotonicity of $R(c)$ in relation to $A(c)$

$$\frac {\partial R(c)}{\partial c} = \frac {\partial [cA(c)]}{\partial c}=A(c) + c\frac {\partial A(c)}{\partial c} $$

So if $$\frac {\partial A(c)}{\partial c} \geq 0 \Rightarrow \frac {\partial R(c)}{\partial c} > 0$$

$ARA$ weakly increasing $\Rightarrow RRA$ is strictly increasing in $c$.


Are there historical cases of country A printing the currency of country B for the main purpose of economic warfare?

Very obscure historical example:
From 1287 to 1295, the Danish nobleman Stig Andersen Hvide was leading a band of outlaws from the island of Hjelm supported by the king of Norway against the king of Denmark. Stig managed to kidnap expert coin makers and bring them to Hjelm, where they produced counterfeit Danish coins.
This allowed Stig and his supporters to buy up all the supplies they wanted in Danish merchant cities. The heavy inflation and the problems it caused for the Danish economy made it difficult for the Danish king to afford assembling an army against Stig.


When a demand function is split, how do we algebraically change function

The Bertrand case you mention is a little special because it induces a discontinuity in the demand function. Suppose that market demand is $D(P)=1-P$, zero marginal cost, and that all consumers buy from the low priced firm (and split equally in the event of a tie).

If the two firms try to collude around a price $p=0.5$ (with quantity $0.25$ each) then each firm's demand curve will look like this:

Mathematically, a firm's demand will be: $Q_i(p_i,p_j)=0$ if $p_i>p_j$; $Q_i(p_i,p_j)=D(p_i)/2$ if $p_i=p_j$; and $Q_i(p_i,p_j)=D(p_i)$ if $p_i<p_j$.

More generally, it is possible for the slope of a firm's residual demand curve to differ from that of the market demand curve. It all depends on how consumers are allocated across the two firms. The relevant material on rationing rules can be found in Section 5.3 of Tirole's The Theory of Industrial Organization.


Competition and welfare - empirical evidence

Here is a fact sheet by the OECD about the effects of competition. It is mostly concerned with growth and productivity, but also discusses the effects on poor consumers and prices. It cites a wide range of empirical studies that you might find of interest.

https://www.oecd.org/daf/competition/2014-competition-factsheet-for-print-en.pdf

OECD, Factsheet on how competition policy affects macro-economic outcomes, 2014.


What are some books for pricing theory with heavy math?

Darrel Duffie is the graduate school standard for asset pricing:

Gary Becker and Milton Friedman also have titles that fall within your prompt and are very accessible given your background


What impact did monetary policy actions taken during the Great Recession (2007-09) have on U.S. economic performance?

The consensus appears to be that the effect was large and positive. The IGM Forum asked a panel of economists to assess the statement

"Informed postmortems of Ben Bernanke’s Fed chairmanship will judge favorably the Fed's creative and aggressive policy initiatives from autumn 2008 through early 2009."

Not one disagreed. Weighted by their confidence, 90% of those polled agreed or strongly agreed with the statement.

http://www.igmchicago.org/igm-economic-experts-panel/poll-results?SurveyID=SV_1GF6NyHSVWhEtN3


Why does the violation of a preference axiom not invalidate the discipline of economics?

The violation of the preference axioms is frequent in the real world, which questions the usefulness of neoclassical model results. This is something that most neoclassical economists will say that it is just "noise", or has a minimum effect, while there is limited empirical data to sustain this - maybe some lab work (controlled experiences), but not real world observations that I know of. The consensus in most policy making institutions and economic universities it is, still (and unfortunately), that neoclassical assumptions are what should be used.

You can read Transaction cost economics for a different perspective which doesn't assume the preference axioms, from Oliver Williamson, Nobel Prize in 2009. It refutes the preference axioms, stating that there is bounded rationality in every choice: humans have limited reasoning and searching capabilities.

Here's a summary of his work on TCE:

  Williamson’s contributions to the field of Transaction Cost Economics complement, and extend, those of Coase. First, Williamson started with an explicitly behavioral assumption of human behavior (bounded rationality). Second, he recognized that transacting parties sometimes behave opportunistically and take advantage of their counterparties. Finally, he identified features of transactions (e.g., specificity, uncertainty, frequency) that cause markets to fail; and hence, are likely to lead certain transactions to be organized within firms (hierarchies) rather than markets.

Source


What does it mean if a economic model is internally consistent?

In theoretical modeling the consistency is applied in the same way as in philosophy/logic. Internal consistency simply means that the argument is consistent with itself and has no contradiction within itself (as opposed to external consistency where its not enough for argument to be valid on its own but it should also not contradict other facts).

A simple verbal example of internal inconsistency would be statement: "I always lie". The statement is internally inconsistent because if I always lie then the statement above must be also lie. But if the statement is lie that implies I do not always lie so it is not correct and it contradicts itself.

Math can reveal if a model is internally inconsistent. If you have model relaying on following system:
$$a+b =1$$  $$5a+5b=15$$

You would get that $5 = 0$ which means that the model is clearly  inconsistent with itself. However, sometimes the model might be internally inconsistent because the assumptions made in the model are not consistent with each other but the model might still solve and even have equilibrium (if the logical fallacy is made in a step where you build equations based on the assumptions - or make assumptions that are not consistent with each other but the consistency of assumptions is not checked) and finding the inconsistency is harder.

Moreover, note that internal consistency of model has also some special meaning in statistics, but from your question I assume that was not the meaning you were looking for.


Who holds the Reverse Repo and Repo respectively in a Repurchase Agreement?

A repurchase agreement (or repo) is the sale of securities for cash, but with a commitment to repurchase them later at a specified price at a future date.

This is practically a collateralized loan, and I find it conceptually helpful to think of it that way.

If you are taking out the loan, it's a repo for you. You are agreeing to repurchase the security (usually U.S. Treasuries) that you posted as collateral at a later date.

If you are giving out the loan, it's a reverse repo.

Note that this terminology is in terms of the dealer's viewpoint. If the dealer borrow money, it's a repo. If the dealer lends money, it's a reverse repo.

Edit #1 -----------------

Let me respond to the first comment. Although I agree with you that it is confusing, I'm not sure there is an inconsistency per se, in what the lecturer is saying.

Other people can jump in, but I believe what's going on in the video is the following.

Suppose I'm taking out a loan from you that I will pay back in a year. I am "promising" you that I will pay back the principal plus interest. As a proof of this promise, you have a debt contract that enters into your balance sheet as an asset (for me it's a liability). You are the holder of this debt contract.

Similarly, say I enter into a repo transaction with you, posting as collateral a watch (just like in the example given in the video). The "promise" here is that I am going to pay back cash. This promise, along with the watch, enters into your balance sheet as an asset. You are the holder of the repo contract.

But unlike a simple debt, there's another side to this "promise": if I give you your money, you have to give me my watch back! The reverse repo contract essentially represents this side of the promise, and hence I am the holder of the reverse repo contract.


I assume that you found Firm 3's best response to be
\begin{equation}
q_3^*(q_1,q_2)=\frac12(16-q_1-q_2).
\end{equation}

The next step would be to solve for Firm 2's best response. Since Firm 2 observes Firm 1's output and correctly anticipates Firm 3's best response, its profit maximization problem is
\begin{equation}
\max_{q_2}\;(16-q_1-q_2-q_3^*(q_1,q_2))q_2=
\max_{q_2}\;\left(16-q_1-q_2-\frac12(16-q_1-q_2)\right)q_2.
\end{equation}
Solve this problem, you should get Firm 2's best response as a function of $q_1$ only. Denote this best response $q_2^*(q_1)$.

Then, you solve Firm 1's profit maximization, with Firm 1 correctly anticipating the responses of the subsequent two firms:
\begin{equation}
\max_{q_1}\; \bigl(16-q_1-q_2^*(q_1)-q_3^*(q_1,q_2)\bigr)q_1.
\end{equation}


Is there an intuitive explanation for the tax incidence formula from elasticity?

Ah, never mind, got there.

Both supply and demand share the same change in quantity as a result of the tax, so

$$\Delta Q_S = \Delta Q_D$$

The amount by which the quantities changed is the derivative of that quantity wrt price, times the change in price that side experienced

$$\frac{\partial Q_S}{\partial P_S}\Delta P_S = \frac{\partial Q_D}{\partial P_D}\Delta P_D$$

Multiply both sides by the equilibrium P/Q

$$\frac{\partial Q_S}{\partial P_S}(\frac{P^*}{Q^*})\Delta P_S = \frac{\partial Q_D}{\partial P_D}(\frac{P^*}{Q^*})\Delta P_D$$

which of course is the formula for elasticity at the equilibrium

$$\varepsilon_S \Delta P_S = \varepsilon_D \Delta P_D$$
$$\frac{\varepsilon_S}{\varepsilon_D}\Delta P_S = \Delta P_D$$

So then the producer's share is

$$Incidence_S=\frac{P^*-P_S}{T}=\frac{|\Delta P_S|}{|\Delta P_S|+|\Delta P_D|} = \frac{|\Delta P_S|}{|\Delta P_S|+|\frac{\varepsilon_S}{\varepsilon_D}\Delta P_S|}$$

the price change cancels out and you get producer's share as

$$Incidence_S=\frac{1}{1+|\frac{\varepsilon_S}{\varepsilon_D}|}=\frac{|\varepsilon_D|}{|\varepsilon_D|+\varepsilon_S}$$

So it all comes from the fact that this is the unique formula that produces the price-change ratios that generate identical quantity changes in both sides of the market. Probably not any intuition that can come down to a principles course level but makes a lot more sense than it did before!


Why might it make sense to buy bonds in a company facing restructuring?

There are experts in distressed debt. They buy the debt at a deep discount to face value, with the belief that the salvage value is greater than what they paid.
To do this successfully, knowledge of how bankruptcy works is required. Different types of debt have different priorities in bankruptcy. In most cases, bonds are deeply subordinated, and so often have extremely low recovery rates versus senior debtors.


We solve the utility maximization problem of the individual whose utility function is $u(c, l) = c - \frac{l^{1+\gamma}}{1+\gamma}$ to get the supply function. The problem can be written as:
\begin{eqnarray*} \max_{c,l} & & c - \frac{l^{1+\gamma}}{1+\gamma} \\ \text{s.t.} && c \leq wl\end{eqnarray*}
In this problem, we are assuming that the only source of income of the consumer is his wage income. When we solve the problem we get the labor supply function as:
\begin{eqnarray*} l(w) = w^{1/\gamma}\end{eqnarray*}
The elasticity of labor supply curve is this case is constant and equal to $\frac{1}{\gamma}$. Supply will be elastic if $0 < \gamma < 1$ and inelastic if $\gamma > 1$.


Why would higher expected inflation increase corporate earnings and dividends' rate of increase?

I don’t have the book, but the passage hints that it will be covered in Chapter 13.

The passage has “reasonably project” as a disclaimer, which does imply that it would not hold in all cases. It is not that hard to find counter-examples. If we define inflation as a change in the CPI, oil price spike will normally raise CPI. The higher energy costs are bad for firms with energy input costs, while the oil producers that benefit may be foreigners.

However, if we stick to an idealised scenario, where all prices are rising at roughly the same rate, we can get the result that profits rise in line with inflation. If we make two assumptions:

real activity is not affected by the change in inflation rates;
profit margins are stable.

We then are in a situation where we are just projecting current variables forward with the same real growth rate, while nominal variables are growing by the sum of real growth and inflation.

If profit margins are stable, then that implies that nominal profits are proportional to nominal revenues, and so would scale with inflation.

The passage implies that the author thinks those two assumptions are reasonable, and presumably detailed in Chapter 13. (I don’t have any references handy, but I’ve certainly seen similar arguments many times. The main difficulty is whether higher inflation is associated with higher or lower profit margin.)


Good economic indicators for presidential performance

Get rid of the gasoline and the home value. Home value can only be used as an indicator of net worth and gasoline is dependant in most cases on too many outside factors. An important one you missed is the trade deficit, and I would add us bond interest rates too. Because they are a good indicator of how investors think about the us. I would also add the average household income, cause the difference between the average and the median is also interesting.


Does the government ever pay back its debt from the OMO to the Fed?

Yes, it does. Maturing bonds held by the Federal Reserve are repaid by the Treasury, just as they repay any other creditor.

However, the cost to the Treasury of doing this is a policy decision of the Fed and can be effectively zero. It’s critical to understand that the Fed’s excess profits are paid to the Treasury. Last year, they paid $54.9 billion to the Treasury. As a result, the vast majority of the money that the Treasury “pays back” to the Fed flows right back to the Treasury.

To dig a little more into the dynamics, the Fed often decides to reinvest the repaid principal on a bond, so as to not reduce the aggregate money supply (you can think of this as choosing to keep rates lower). When they do this, the actual impact on the Treasury is approximately zero. If they choose not to do so (if they choose to allow borrowing costs to rise), the demand for Treasury debt is reduced accordingly, and the Treasury’s borrowing costs will rise.


In a competitive equilibrium, can price of a commodity ever be zero?

It depends on the assumptions you make.

If you assume that preferences are locally insatiable then no.

Without this assumption, it is easy to construct counterexamples.  Consider an economy with two consumers ($A,B$) and two goods ($x,y$). Both consumers have preferences represented by their utility function
$$
U_i(x_i,y_i) = \min(x_i,1) + y_i.
$$
If the initial endowments are such that the total supply of $x$ is larger than 2 then the equilibrium price of $x$ is 0. (The numeraire is the good $y$.)


Should the "value function" be "utility function" in prospect theory?

The terminology in microeconomics is not completely unified but typically differs slightly from the mathematical one. For a real-valued random (outcome) variable $X$, the mathematical expected value $\mathbb{E}(X)$ would rather be called the expectation of $X$. The utility $u(x)$ of an outcome $x$ is understood to be given by a (Bernoulli) utility function in the sense of EUT, i.e. a function $u(.)$ such that the expected utility $\mathbb{E}(u(.))$, sometimes written as a (von-Neumann-Morgenstern) utility function $U(.)$ on lotteries on outcomes, represents the preferences over lotteries on outcomes, while in behavioral economics the term value or valuation $v(x)$ typically represents some kind of subjective valuation of $x$, as e.g. in prospect theory. (So here, value is not to be understood as the mere numerical value of $x$, as in the usual mathematical definition of a "function value" or an "expected value".)
Confusingly, however, $v(.)$ is sometimes also used to denote an alternative (Bernoulli) utility function. For example when explaining that a positive linear transformation of a utility function is again a utility function representing the same preferences, one of them is usually denoted by $u$ and the other by $v$. Moreover, in a utilitarian framework, $v(.)$ is often used to denote the willingness to pay for $x$, i.e. a monetary valuation of $x$.


Why was 'structural' chosen to mean 'structural unemployment'?

Cyclical unemployment is named after the business cycle, as in the part of unemployment that varies with the business cycle. Structures are things that last a long time, so structural unemployment is the part of unemployment that is driven by features of the economy, like structures, that last a long time.

But as you intuit, surely the length, frequency, and other details of business cycles are driven by such structural features. That's almost certainly true, and major reason why structural change and good policy are thought by many to explain the great moderation in business cycles.


Derive the market demand function and market supply function

Hidden on the third page (left column, middle) you can find that the markets they set up experimentally comprised 8 buyers and 4 sellers. If you multiply the individual demand and supply values from Table 1 by the corresponding number of individuals, you get the linearly approximated demand and supply functions stated on the second page.