If you ask yourself how much a potential employer would have to pay you to convince you to work for him, the answer is probably something like "at least as much as I could earn by doing the same job for another employer". So, provided there are several employers competing to hire workers, you can think of employers as bidding against each other for the best employees. But how much will firms bid? Each employer would be willing to pay up to the difference between its profit if it hired this particular worker and its profit if it took the next best alternative.

This is why the supply of alternative employees matters. If there are many people capable of doing the same job then a firm's profit if it hires you will be almost the same as the profit if it hire someone else. So why should it 'bid' a lot to hire you? An unskilled factory worker who asks his employer for a significant pay rise will likely be refused because virtually any worker doing the same job would have equivalent productivity.

Celebrities are a different story. Just as factory workers are hired to produce goods for the factory owner, actors are hired to produce movie ticket sales for the studio. But viewed in this light, Tom Hanks or Angelina Jolie are not close substitutes for as yet unknown actors at all. Indeed well-known actors are considerably better at producing ticket sales than unknown talent. Studios are willing to bid a lot to attract top actors because having a star in your cast is worth millions of dollars in extra ticket sales.


Even if the measure calculated is total debt at a point in time as a percentage of annual GDP, the frequency with which it is calculated could be quarterly. Taking quarter 1 2018 as an example, debt would be the total at end of that quarter.  GDP might relate to the last available full year, ie 2017, or to the year ending in that quarter, ie from quarter 2 2017 to quarter 1 2018.

A chart of such quarterly data for the US may be viewed here.


Why do housing and parking cost more in urban than in rural areas, but road access doesn't?

Land values tend to increase with population density, because it is typically possible to put the land to more productive use.  A store operating in a dense urban core will have more potential customers, more access to public infrastructure (e.g. electricity, communications networks, those roads you mention, public transportation that makes it easier for more customers to visit the store, etc.) and any number of other external factors that contribute to such land being more desirable.  Since more high-wage jobs also tend to be available in such places, there is more access to desirable public and private places (e.g. museums, theaters, upscale businesses) people are also generally willing to pay more to live in such places.

Roads, on the other hand, are generally built and paid for by the government.  The government almost never collects the full use value for the things they make, and so roads are drastically underpriced, in terms of the charge consumers must pay (if there is any such use fee at all.)


ratio independent time-invariant variable

Given that you have one value for this $M/N$ allocation variable per individual, it can be considered a time-invariant explanatory variable.  If you estimate this model without including individual fixed effects (which control for all unobserved time-invariant differences between individuals), you will be estimating a pooled cross-section regression, which will allow you to get an estimate of $a_{1}$ but which will ignore the panel dimension of your data.  As a result, the estimate of this coefficient and all other coefficients will be based on the assumption that the model applies in the same manner to every individual (individuals are not unique) and that there are no effects across time.  (You could include dummy variables for each year in the model and do a joint test on those dummies to see if the assumption about the effect of time is valid, though you would not be able to include any other variables that vary only over time.)

If you, however, wanted to estimate a fixed effects model because you thought that there are unique attributes of individuals that do not vary across time (in other words, you thought that outcomes of the same individual were correlated in different periods), you would not be able to use this kind of time-invariant explanatory variable in the model.  This is because in a fixed effects model, coefficients are identified using only variation over time within each individual.  So if a variable varies only across individuals, but not over time, then its effects cannot be identified at all in a fixed-effects model.


Why is the law of diminishing marginal returns justified?

Marginal product can be constant, increasing or diminishing depending on Returns to Scale (also called Economies of scale). If Returns to Scale is constant, it means that each extra unit of product/output takes one extra unit of input. In this case, marginal product will be constant. In other words, the efficiency of production is constant.

However, take the case of increasing Returns to Scale, which is usually seen in the real world in a typical factory. In this case, one extra unit of product takes less than one extra quantum of cost inputs. In such a case, the firm has high fixed costs as compared to variable costs and said to possess an operating leverage (in financial terms). And, the efficiency of production increases. Same inputs produce higher outputs.

Example: a software company selling products has high fixed costs and close to zero variable costs. So, selling one extra copy of software can happen at a much lower increase in inputs. This is however, till you reach the ceiling where you have to employ additional fixed cost resources to generate the next item of product. Say, when the next version/iteration of the software has to be produced, it would need additional fixed cost resources to develop it.

So, yes it isn't necessary that marginal product will always have a diminishing curve.


First of all, just because there is an economic cycle in a data that does not imply that data is non-stationary. For example, consider the completely stationary process simulated in R below, which is based on $x_t = \phi x_{t-1} +e_t$ with $\phi=0.9$ which is stationary by construction yet it exhibits something like economic cycle.

So the first thing you should do is to make sure there actually is non-stationarity in your data and not just assume that because data exhibit some sort of cycle. Either look in the literature on the given aggregate whether in general it is considered to be integrated of some order or perform your own unit root tests (assuming there is enough observations to carry them out - you dont mention if your data is on monthly, quarterly or yearly frequency. Depending on frequency there might be plenty of data points to carry out these tests even within the given years).
Second, taking logs  is not a solution for non-stationarity even if you have long time series (and I am not even sure what you mean by change percentages). The way how you solve non-stationarity is by taking differences of your data.
For example, a simple non-stationary process is given by
$$x_t= x_{t-1} + e_t$$
The process is non stationary because $\phi=1$  which implies that present variables are fully determined by initial conditions of the system and sum of shocks. If that is the case just taking logs of variables $\ln x_t = \ln x_{t-1} +e_t$ does not solve the unit root problem as you will have the same dependence and it does not matter how long your time series is. Taking logs of your variables can be desirable for many different reasons but not related to non-stationarity per se.
What actually solves the non-stationarity/unit-root problem is to take first differences. For example, in the example above we could transform data as:
$x_t -x_{t-1}= e_t$
Which would become stationary series. You can apply the same procedure to $\ln x_t$ but what ultimately gets rid of the non-stationarity is the differencing. In a worse case scenario you could have series that is integrated of order 2 in which case you would have to make one more second difference. However, eventually taking difference of the series will always produce stationary series. According to Verbeek (2008) guide to modern macroeconomics most economic series that are non-stationary are I(1) and in some rare cases I(2), so even with extremely short time series you should be able to make it stationary.


What are the advantages and disadvantages of second price auctions?

A lot of questions here. Might be helpful to split them into different posts, but let's take a crack at them.

The reason why the second price sealed auction is set up this way is simple. Imagine you have a set of individuals/bidders, thinking of what $b_i$ to bid. They all have independent values, $v_i$ of a single object up for auction (their enjoyment does not depend on others). The optimal strategy in a second price auction is to bid however much you value that object, that is,

$$b_i = v_i$$

Since you won't pay as much as you bid if you win, you have a chance to get a positive benefit from the auction, or if someone bids the same as you, you also get a payoff of zero, which is an, "oh well", but no big deal.

If you unilaterally change your bid so that $b_i > v_i$, then you increase the chance of winning, but if you're still the top bid, then you still pay the same amount, so it seems kinda useless. If you happen to beat out someone's bid, $b_j$, where $b_i > b_j > v_i$ then you win! But now your payoff is negative. You can think similarly for the case of underbidding. It decreases the chance of winning AND it doesn't increase the chance of paying less for your object, so that's two bad things.

Basically, in theory, second price auctions induce truthful bidding.

Compare that to the first price auction. If you simply bid $v_i = b_i$, then your expected payoff is exactly zero. Very unfun. So instead you "underbid" your "true value". It decreases the chance of winning, but if you do, then you actually have a positive payout. The amount you underbid is obviously based on what you think other people's values are. So bidding here in first price auctions is strategic, rather than truthful.

But the big result you should know is that, in theory:

  First price and second price auctions are revenue equivalent for the auctioneer, given independent private values for a single object without budget constraints.

In theory. But in real life, people's bids are interdependent. You can imagine a bunch of big-wig types in a room for an auction of a priceless piece of art, where part of the value of the object is winning it and beating out the competition because you're the coolest kid on the block (with lots of cash to throw around). In this case, you can imagine a first price auction would be better suited for the house than a second price sealed auction.


If negative externalities are priced into the market via a pigouvian tax, then those responsible for the negative externalities pay.

As a whole, the public is better off. Firstly because the market will now move to a more efficient situation, and the amount of negative externality will decrease (assuming something other than perfect inelasticity). And secondly, it will also benefit in other ways, through improved public services paid for by the taxes; and/or decreased deadweight loss, achieved by lowering non-pigouvian taxes.

Some market participants will be worse off - the polluter will be paying. Exactly how the burden is spread among those responsible, will depend on the specifics of the situation. Consumers, equity-owners, managers, employees, land-owners, in short anyone involved in the relevant polluting supply chain - each of them might bear some, none or all of the burden, and between them, they'll bear it all.


How did Cowen and Tabarrok calculate the probability of Hewlett-Packard sales?

It is this probability that is consistent with shares trading for their expected value. The expected value is given by:
$$E(S_1+S_2)=E(S_1)+E(S_2)=p_1 \times1+p_2 \times1=p_1+p_2$$
where $S_i$ is the payment of share $i$, $p_i$ is the probability of payment of share $i$ (due to the sales figure falling in some range). So if the expected value is to equal the price of the security, then:
$$E(S_1+S_2)=P_1+P_2 \Rightarrow p_1+p_2=P_1+P_2 $$
Where $P_i$ denotes the price of share $i$. So the probability of one of them paying off equals the sum of their prices (note that the events of share 1 and 2 paying off are mutually exclusive).
In your case
$P_1=$$\\\$$$0.20$
$P_2=$$\\\$$$0.10$
Of course these calculations totally disregard the many complications that go into asset pricing (like discounting cash flows, accounting for having to wait for payment), but I assume the authors want to keep it simple to prove a point. The bigger picture you should take away from this answer is that share prices can be interpreted as reflecting information about (what investors believe to be) probabilities of future events.


Pareto efficient allocation and core allocation in 2 person exchange economy with discontinuous utility

Here are two pertinent observations:

If A has a positive amount of the $x$-good, and $B$ a positive amount of the $y$-good, both can be made strictly better off (why?).
There is no efficient allocation in which $A$ has no amount of the $x$-good, but a positive amount of the $y$-good (why?).

Remaining are the allocations in which $B$ gets everything and allocations in which A has a positive amount of the $x$-good and all of the $y$-good. Verify that these are indeed Pareto efficient allocations.

Finding core allocations is not much different. There is a small difference if you define blocking so that all parties involved might be strictly better off. In that case, you can have allocations in which $A$ has none of the $x$-good but some of the $y$-good.


Interest Rate vs. Borrowing Rate - Difference

The Interest Rate is a more general term, and can cover all shorts of financial transactions where one expects by a contractual clause to get their money back plus "a little something" (I am not referring to the expected profit from an investment, where the "little something" is not a contractual obligation).

So you can talk about "the interest rate of a loan" but also of the "interest rate of a deposit in the bank".

The Borrowing Rate specifically refers to the Interest Rate that the entity under consideration (a business, a public sector, or an individual) has to accept to pay as this extra little something if they want to borrow funds.

So the "borrowing rate" is an "interest rate" - one among many kinds of interest rates.


Independently and Identically distributed random variables

Suppose $X$ and $Y$ are iid random variables. The 'identically distributed' part means both random variables have the same distribution function (cdf). Formally this can be stated as
$$F_X(z)=F_Y(z),$$
where $F_X(\cdot)$ and $F_Y(\cdot)$ are the marginal cdfs of $X$ and $Y$, respectively. The 'independently distributed' part means the joint cdf of $X$ and $Y$, denoted $F_{XY}(x,y)$ is simply the product of the marginal cdfs $F_X(x)$ and $F_Y(y)$, i.e.
$$F_{XY}(x,y)=F_X(x)F_Y(y).$$


What does the area defined by two different values for quantity under a supply curve represent?

Your intuition is right, but you are not accounting for fixed costs.

The supply curve is the part of the marginal cost curve above its intersection with the average variable cost curve. Marginal costs are defined as the derivative of the cost function (or the variable cost function as the derivative of fixed cost is zero).

To get the area you integrate. That is, you get back the variable cost.
So the area below the supply curve between $a$ and $b$ is the difference of the variable costs.


Regarding the assumption of Classical Linear Regression Model

I  assume that x is a scalar.( but argument is similar if it isn't).

They both violate condition 4) because they both say that x, the regressor, is not independent of $\epsilon$. Case 1 a theoretical statement about the regressor and the noise term. Case 2 is an empirical statement about some of the  actual regressor data and the noise term.   If you want a nice intuitive explanation of econometrics ( not a standard textbook exposition ), I highly recommend Peter Kennedy's text. I think it's called "econometrics"but I'm pretty sure he only wrote one book so you can google for him on amazon.


Marginal productivity of aggregate labor relation to wage

So the firm's problem is, letting $L=Nl$

$\max_{L} Y-Lw\implies w=(1-\alpha)AL^{-\alpha}, \pi(w)=\alpha AL^{1-\alpha}$

Consumer's problem: substitute out c and only choose l, taking $\pi$ and $w$ as given

$\max_l u(l, wl+1/(N\pi))\implies -\gamma/(1-l)+w/(wl+1/(N\pi))=0$

Now substitute in the values for w and profit from the firm into the consumer's FOC and you will solve for labor supply (=labor demand as we are in eq) in terms of exogenous parameters ($\gamma, N,\alpha, A)$.


How do we determine if wealth will be created or destroyed (and how much) when we want to provide (quasi)public goods?

You are right, given that it is not possible for markets to provide public goods this is often done by the state and surveys are one way of establishing the amount desired. Such surveys could establish individual demand curves. Note that, due to the non-rivalry of public goods constructing a demand curve for society as a whole requires vertical addition of the individual demand curves. This is known as the Samuelson rule.

There are however all kinds of problems associated with such surveys. Given that tax levels depend upon your answer you may want to understate your true willingness to pay. Survey respondents may also find themselves in a public goods game where it is individually rational to understate WTP even though as a group they would want the public good.

Of course we are not surveyed all the time in practice, but we approve or disapprove of further increases or decreases in public goods provision by voting for certain parties (of course that is not the only consideration when voting but it may be part).

Depending on the public good in question sometimes other tools are available. For example a large recreational spot or natural area that is freely available for recreation is also a public good, provided it doesn't become congested. The demand curve can then be constructed by estimating the travel costs that people incur to travel and recreate in that spot. That is called the travel cost method.

Also, sometimes we can construct demand curves for public goods by observing the premium people are willing to pay in terms of house prices for a nice view in their house, or the reduction because they live in a highly polluted area (pollution is often an example of a public bad). Those are hedonic methods.


Constant marginal cost for a firm and perfect competition

This is somewhat unrealistic, but so be it...
If $p=MC$, then for each additional unit you get as much as you pay, therefore your profit is constant in quantity, either 0 or negative (because of fixed costs). In the latter case you would exit the market in the long run, but in the short run your profit maximization problem is indeterminate (every quantity is optimal). In the former case your profit maximization problem is indeterminate in the short and in the long run.
If $p<MC$, then you would shut down to minimize your loss.
If $p>MC$, then your profit maximization problem has no solution in the real numbers. (You want to produce an "infinite quantity".)


Real-life applications of repeated games theory

I believe insights from repeated game theory have been used in designing the assignment scheme for security patrols in many occasions.

In this matter, the main research projects with close connections to applications that I know of are

(a) PROTECT for the US Coast Guard

(b) TRUSTS for the Los Angeles Sheriff's Department

(c) IRIS for the Federal Air Marshal's Service

(d) LAX for the Los Angeles Airport Police

(e) GUARDS for the Transportation Security Administration

which all revolve around the teamcore research group at the University of Southern California.

Examples of papers involving members of the group and repeated games (usually repeated Stackelberg security games) include:

Defender Strategies In Domains Involving Frequent Adversary Interaction
Learning Bounded Rationality Models of the Adversary in
Repeated Stackelberg Security Games
Conducting Longitudinal Experiments with Behavioral
Models in Repeated Stackelberg Security Games on
Amazon Mechanical Turk
A Game of Thrones": When Human Behavior Models Compete in Repeated Stackelberg Security Games


Yes, it is acceptable. Consider a Mincer wage equation. Let's define a dummy variable Female taking on the value one for females and the value zero for males and a dummy variable Married to equal one if a person is married and zero if otherwise. Then, you can estimate a model that allows for wage differences among four groups: married men, married women, single men, and single women. For this you need to interact the dummy variables, for instance Female*Married. But be cautious, you must select a base group to avoid the dummy trap.
A good reference is the chapter "Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables" (Chapter 7) in Wooldridge Introductory Econometrics.


Solve for the steady state with CRS Cobb-Douglas, problem with the system of equations

1) Instead of $(9)$ use

$$C = K^{\alpha}L^{1-\alpha} \tag{9*}$$

2) Equation $(10)$ is not correct. For constant population $N$ and identical agents,

$$c^{\sigma} l^{\gamma} =  w \implies (C/N)^{\sigma} (L/N)^{\gamma} =  w \implies C^{\sigma} L^{\gamma} = N^{\sigma +\gamma}w \tag{10*}$$

Insert $(9^*)$ in $(11)$ and solve for $C$ as a function of $K$, using $(8)$ for $r$. Substitute back in $(9^*)$ and solve for $K$ as a function of $L$. Insert into $(12)$ and you will get $w$ as a function of parameters. Etc.


Is the development of "developing countries" evidence against the idea that former colonial powers owe their headway to exploitation?

The recent development of many "developing countries" and "emerging markets" does not mean that Western colonial powers did not exploit them in the past.

Emerging economies owe their modern GDP growth to cheap labor, as (former) Western colonial powers switched their (expensive) manufacturing bases to countries like China, India, etc. In the past, they were exploited mainly for their natural resources moreso than labor.

The continued GDP growth of Western colonial powers is due to a shift from manufacturing (labor-intensive, low-education professions) to "high tech" industries which require significantly more education (medical research, computer programming, etc.).

The book "Capital" by Thomas Piketty has a far more detailed history of how capital flows shifted from the 18th century to modern times, and the economic interplay between Western colonial powers and the developing nations. https://en.wikipedia.org/wiki/Capital_in_the_Twenty-First_Century


How to determine correct fixed effects in linear regression model using unbalanced paneldata

It seems that you have firm-level panel data over multiple industries in multiple countries.

Year fixed effects control for trends. Do not omit them. For others, a general rule is: ___ FE's control for time-invariant heterogeneity across different ____, but do not control for heterogeneity within the same ____. You replace ____ with "firm", "industry", "country", "industry-country", etc.

Firm FE's successfully control for all firm-level time-invariant effects that are possibly correlated with the RHS variables. So you compare different years within the same firm. In contrast, industry FE's control for time-invariant effects (that are possibly correlated with the RHS variables) across different industries, but do not control for firm-level heterogeneity (fixed effects) within the same industry. And so on.

I personally think a model with firm FE (and year effects) is easiest to defend using. If you can explain why firms within the same industry are homogenous, you can use industry FE, etc.


What is used more: economic growth rate compared to previous quarter (GPSA), or growth rate compared to the same quarter of the previous year (GYSA)

QOQ (quarter on quarter) GDP growth is more common, at least in the US and the UK.

See the press releases by the US BEA and UK ONS. For both, the headline rate is the QOQ rate.


Why Chinese exports increased even though Yuan appreciated after the Chinese removed the peg against USD?

Simple answer: Drawing on Ricardian model I can say the Chinese have not lost their comparative advantage in producing many types of manufactured goods. Yuan, on its way to appreciation, has to walk a few more miles to affect the comparative advantage that the Chinese possess.


Roy's identities for Stone-Geary utility functions

Monir is correct to point out in the comments that $\ln$ is not a linear transformation. It is an increasing transformation, however, and so should not matter as long as

$$ \sum_{i=1}^n p_i \gamma_i < I$$

In any case, your mistake is in the derivatives you find. Here are the correct expressions:

$$\frac{\partial W}{\partial p_i} = -\gamma_i \sum_{j=1}^n \frac{\beta_j}{I - \sum_{k=1}^n p_k \gamma_k} - \frac{\beta_i}{p_i} = -\frac{\gamma_i}{I - \sum_{j=1}^n p_j \gamma_j} - \frac{\beta_i}{p_i} $$

$$ \frac{\partial W}{\partial I} = \sum_{j=1}^n \frac{\beta_j}{I - \sum_{k=1}^n p_k \gamma_k} = \frac{1}{I - \sum_{j=1}^n p_j \gamma_j} $$

Roy's identity then gives you the Marshallian demand you've initially, and correctly, derived.


Monetary policy takes medium run time to take effect. Why study it in short run?

In an IS-LM model, expansionary monetary policy increases output and decreases the interest rate. Expansionary fiscal policy increases output and increases the interest rate, so the two types of policies are not equivalent.

Furthermore, monetary policy does not work in the long-run. See an AS-AD model for more details on that. The only reason monetary policy actually works, is because of the fixed prices and their slowly changing (increase) in the medium run. Otherwise inflation would rise immediately in response thereby making monetary policy not have any effect on real GDP.

To really get to the medium run you have to understand the short run first. Effects observed in the actual economy often are not as quick as in our models and take time. I wouldn't consider 4-8 quarters the medium run.

Hope this helps.


Aggregate efficient production and division of labour

I would say, it is more Proposition $5.\text{F}.1$ on page 150, in combination with $5.\text{E}.1$ that leads to, as the authors write discussing the two,

  "...if a collection of firms each independently maximizes profits with respect to the same fixed price vector $p >>0$, then the aggregate
  production is socially efficient. That is, there is no other production plan for the economy as a whole that could produce more output using no additional inputs."

Note that no assumption of identical suppliers is made here, or any other restrictive assumption about technology or the production set. The crucial assumption is the "price-taking" behavior assumed for all firms. Or, "no-market-power". Or, "all firms are equal": doesn't Perfect Competition sound a lot like the market version of Communism (not Socialism)?

P.S.: Externalities could also create issues here, of course.


Will a shift in the average total cost curve mean that the marginal cost curve will also shift up?

No, a shift in the average total cost curve does not necessarily mean that the marginal cost curve will also shift up.

Your intuition is correct. Here is a simple numerical example:

Consider a shift from the total cost function $C_1(q)=q$ to $C_2(q)=1+q$.

Then the average cost functions shifts upward from $AC_1(q)=1$ to $AC_2(q)=1/q+1$.

But the marginal cost function which was $MC_1(q)=1$ remains unchanged at $MC_2(q)=1$.


Deriving First Order Condition of Hansen and Singleton (1982)

This has nothing to do with GMM, which is a statistical method for estimating parameters. This is a macroeconomics model or say a dynamic optimization problem.
From FOCs for $C_t$, $C_{t+1}$, and $Q_{jt}$, youg can get $$\mathbb{E}\frac{\beta U'(C_{t+1})}{U'(C_t)} = \frac{\lambda_{t+1}}{\lambda_t}$$, and $$\lambda_t P_{jt} = \lambda_{t+1} P_{jt+1}$$.
Then by replacing $\lambda$ you have $$P_{j t} U^{\prime}\left(C_{t}\right)=\beta \mathbb{E} \left[P_{j t+1} U^{\prime}\left(C_{t+1}\right)\right]$$.


How to calculate marginal utility with two goods?

This is quite simple to answer given you know a bit of multivariable differential calculus. You're looking for partial derivatives of the utility function.

So, given $$U(x,y) = x(y+1)$$ we have $$\frac{\partial U}{\partial x} = y+1$$ and $$\frac{\partial U}{\partial y} = x$$.

These are the goods' marginal utilities.


Are there two different meanings of the discount rate?

It is standard approach by now to acknowledge the existence of a "rate of pure time preference", denoted usually by $\rho$, that characterizes individuals. This is a fundamental aspect of preferences - a "primitive" parameter. It is not a proxy for the existence of uncertainty (this is why it is found also in deterministic models), neither does it reflect "economic opportunity costs" (this is why it discounts future utility also).

In continuous-time model this discount factor takes the form $e^{-\rho t}$, while in discrete-time models, this discount factor takes the form $\beta = 1/(1+\rho)$.

In the individual's intertemporal maximization problem, the interest rate does not enter in the utility/objective function, but in the law of motion of personal wealth. So a negative real interest rate does not reflect on the personal discount rate.

Traditionally also, firms are considered to discount the future only with respect to the opportunity cost which in turn is assumed to be reflected in the (real) interest rate $r$. So when modelling the intertemporal problem of a firm we usually see $e^{-r t}$ and $\beta = 1/(1+ r)$.


I think your solution is on the right direction. But allow me to make a few observations.

The question talked about income gap and wanted you to use a Ramsey model, with exogenous technology growth, and no endogenous growth mechanism through public spending. Given these, I would think that the focus should not entirely be on the growth rate of technology. Using your example, if the two countries have the exact same set of parameters, then the country with higher x will have lower long run $k^*$.
See the starting point of my (1) and your analysis: we assumed that the countries have identical parameters. For Rich(R) and Poor(P) countries, this is not a reasonable assumption. Furthermore, I am not sure if assuming that technological progress in R is faster than P is correct.

Might I suggest the following. Pick a parameter (say the tax rate). Make the tax rate higher in one of the countries. Which automatically divides the countries into R and P. Then even if you do not have different x, the income gap will remain.

I assumed the following parameter values (L=Low, H=High):

$\delta = 0.1; n = 0.01; \rho = 0.05; \alpha = 0.3; \theta = 1/0.3$

$t_H = 0.2; t_L = 0.1; x_H = 0.5; x_L = 0.2;$

My Utility function is $\frac{C^{1-\theta}}{1-\theta}$

The corresponding long run $k$ values are:

(i) $ k^*_{t_H x_H} = 0.583336 $

(ii) $k^*_{t_H x_L} = 0.943351  $

(iii) $k^*_{t_L x_H} = 0.69023 $

(iv) $k^*_{t_L x_L} = 1.11622 $

Now you can pick whichever pair among (i)-(iv) as your R and P. Suppose you say (ii) is P and (iv) is R. Then say both countries have the exact tech improvement and x rises from $x_L $ to $x_H$. (ii) moves to 0.58 and (iv) moves to 0.69. The gap (R-P) before the shock was 0.172869. After the shock income gap (R-P) is 0.106894. The gap went down.

My code (Mathematica):

kdot = (1 - t)*k^a - c - (n + d + x)*k;

Ham = c^(1 - \[Theta])/(
   1 - \[Theta]) + \[Lambda]*((1 - t)*k^a - c - (n + d + x)*k);

(*Costate1=\[Lambda]dot/\[Lambda]-(\[Rho]-n*(1-\[Theta])-x*(1-\[Theta]\
))==-D[Ham,k];*)

FOCc1 = D[Ham, c] == 0;

FOCc2 = FOCc1[[1]] + \[Lambda] == \[Lambda];

FOCc3 = FOCc2 /. {c -> c[t], \[Lambda] -> \[Lambda][t]};

FOCc4 = D[FOCc3, t][[1]]/c[t]^-\[Theta] ==
   D[FOCc3, t][[2]]/c[t]^-\[Theta];

FOCc5 = FOCc4 //. {c[t]^\[Theta] -> \[Lambda][t]^-1} /. {\[Lambda]'[
      t]/\[Lambda][t] -> \[Lambda]dot/\[Lambda]};

Euler = -((\[Theta] Derivative[1][c][t])/
     c[t]) - (\[Rho] - n*(1 - \[Theta]) - x*(1 - \[Theta])) == -D[Ham,
       k]/\[Lambda];

Euler2 = chat /. Flatten[Solve[Euler /. {c'[t]/c[t] -> chat}, chat]];

kstar = k /. Flatten[Solve[Euler2 == 0, k]];

csol = c /. Flatten[Solve[kdot == 0, c]];

cstar = csol /. {k -> kstar};

(*Parameters*)

d = 0.1; n = 0.01; \[Rho] = 0.05; a = 0.3; \[Theta] = 1/0.3;

Ht = 0.2; Lt = 0.1; Hx = 0.05; Lx = 0.02;

kstarHtauHx = kstar /. {t -> Ht, x -> Hx}
kstarHtauLx = kstar /. {t -> Ht, x -> Lx}
kstarLtauHx = kstar /. {t -> Lt, x -> Hx}
kstarLtauLx = kstar /. {t -> Lt, x -> Lx}


Book recommendation: agent based modeling?

One book that comes to mind is "Economic Dynamics" by John Stachurski found here.

This book discusses programming methods that are applied to economics. All of its accompanying code is kept on this github repository.


Housing Supply Elasticity: Proxy for Exogenous House Price Movements

The idea is that all else equal, areas with inelastic housing supply (e.g., areas that are surrounded by water, so you can't just build some more houses "over that way") will see greater house price growth, which increases household wealth, and that these differences in supply elasticity are exogenous and uncorrelated with other things that matter. In Saiz (2010), he uses terrain and water data to determine the degree to which the housing supply in different metropolitan areas is constrained by geographic features.

It's a method that really only works (to the extent that it does) when house prices are rising, which would induce housing construction. If house prices are falling, supply elasticity tends to be pretty low— we don't tend to bulldoze homes at a particularly high rate.

With respect to the degree to which house price changes are anticipated, I think the best work on expectations is that done by Case and Shiller over the years, most recently Case, Shiller, and Thompson (2012).


What are some good Phd or research programs for Economic History / Economic Thought History?

The RePEc/IDEAS website publishes various rankings in economics. I would suggest checking out their ranking of authors working in Business, Economic & Financial History or History & Philosophy of Economics and see if there are people that you'd like to work with. Usually you can find a list of recent publications or working papers on their website/CV. See if any of those are to your interest. The ranking of departments in those fields (HIS, HPE) may also be relevant.
In addition, there are specialized journals for the fields you mention, such as the Journal of Economic History and Journal of the History of Economic Thought.
Their recent publications should give you a sense of who's currently active in those fields and what they are working on. Also, papers on economic history do get published in mainstream journals. You should be able to find them by searching for articles with the appropriate JEL codes. In your case, codes "B" (History of Economic Thought, Methodology, and Heterodox Approaches) and "N" (Economic History) are relevant.
There is also "A Guide to Graduate Study in Economics: Ranking Economics Departments by Fields of Expertise". Table 3 in this article ranks economics graduate programs in the US by JEL codes. Keep in mind, however, that this is a 13-year-old paper. So information therein may not be up to date, since people may have moved/retired/etc.


Help understanding definition of expected utility function

That's true, but it should be constructed the other way round: The assignment of numbers $u_i$ to outcomes is not given by the Bernoulli utility function but defines it. The terminology of Bernoulli and Von Neumann-Morgenstern utility functions is common, but not universally used in the context of EUT.


Best econometric models to fit when there is no control group?

If you dont want to used DID, a simple cross sectional analysis would do with the addition of a dummy variable to indicate whether the individual was treated in period $t+1$ or not (implying they were treated in period $t$

for example if you were trying to see if  test scores of students change depending on when they have their class on monday or tuesday. your data set would be:

$$
\begin{array}{c|lcr}
n & \text{Test score} & \text{IQ} & \text{Class on monday} \\
\hline
1 & 60 & 80 &0\\
2 &80 &65 & 1 \\
3 & 90& 70 & 0
\end{array}
$$

By the incorporation of dummies you can just use regression methods for cross sectional data.


intertemporal utility function usage : calculating consumption

This is the two-period budget constraint:

  C1 + C2/(1+r) = Y1 + Y2/(1+r)

Derivation is straightforward. On the LHS, you have the present value of consumption (considered during period 1), and on the RHS you have the present value of income.

Intuitively, think about 1/(1+r) on the LHS as the price ratio between Good 1 and Good 2. Now you can solve the u-max problem as per usual.


Constant absolute risk aversion and certainty equivalent

This is perhaps not the best way to prove the statement. But notice that a CARA utility is equivalent to having the general functional form:
\begin{equation}
u(x)=-\alpha \mathrm e^{-ax-b}+\beta,\quad\text{where $a,\alpha>0$ and $b,\beta\in\mathbb R$}.
\end{equation}
Knowing this, the rest should be straightforward.


Are there Recent Studies of Price Elasticity of Demand for Housing in England?

I found a working paper  by Auterson (2014) which estimates that the price elasticity of demand for housing in UK was $-0.92$, but it is an auxiliary result as the paper's main intent was to forecast the housing prices in UK not to estimate elasticity - might be still useful though.


How can this be proved? (Convex optimization)

Take $\Omega_1=\Omega_2=[0,0.5]$.
Let $p(x)=x^4$, so that $p''(x)=12x^2>0$ on $(0,0.5)$.
Let $q(x)=0.5x$, which is linear in $x$, and $K=0.2>0$.
For $\bar\gamma=1$, both objective functions attain their respective maximum at $x=0.5$. As the following figure shows, objective function $(1)$ (blue curve) has a higher maximum than that of objective function $(2)$ (red curve).

But for $\gamma=5>\bar\gamma$, objective function $(1)$ attains a maximum value of approximately $0.276$, while objective function $(2)$ attains a higher maximum value of approximately $0.309$, contradicting the claim that $(1)$'s maximum would always be greater than $(2)$'s maximum for $\gamma>\bar\gamma$.


I'm not sure I fully understand what you mean by computing it endogenously? The nominal interest rate would be endogenous in the NK framework because the output gap and inflation are endogenous. There'd be feedback between nominal interest rates and inflation, which both feed into each other for example. So in that sense I'd consider the approximation of interest rate policy using a Taylor rule as entailing that interest rate policy WAS endogenous wrt. the output gap and inflation.

As for why we'd use the Taylor rule as opposed to some more complex but accurate approximation for interest rate policy... well I can only suppose that the Taylor rule is a sufficiently good approximation such that the marginal benefit of adding more complexity / accuracy is minimal and perhaps less than the computational cost of doing so. There comes a point where you start to prioritize parsimony more and more. Perhaps this is the motivation.


How do I test if a given policy was successful?

Let us assume a balanced panel of companies $i=1,...,N$ observed in time periods $t=1,...,T$.
Let $s_{it}$ denote R&D expenses for company $i$ time $t$ and let
$$\bar s_t :=\frac{1}{N} \sum_i s_{it},$$
denote the time $t$ average.
Consider now a simple regression model
$$(1)\ \ s_{it} = \beta_0 + \beta_1I[t\geq 2016] + e_{it}$$
where $I[t\geq 2016]$ is indicator for the time being after policy implementation. I assume the policy is intended to stimulate R&D investments by companies and hence we assume $\beta_1>0$ if the policy has effect.
Using the normal equations for POOLED OLS estimator it is easy to show that
$$\hat \beta_0 = \frac{1}{NT_1} \sum_i \sum_t s_{it}I[t< 2016],$$
where $T_1$ is number of time periods before treatment. Hence $\hat \beta_1$ is roughly the average R&D expense level BEFORE treatment/policy implementation.
If model (1) is true then $plim \ \ \hat \beta_0 = \beta_0 = \mathbb E[s_{it} \lvert t<2016]$ which is the expected level of R&D expenses for any company for any year before treatment.
You can also show that
$$\hat \beta_1 = \frac{1}{NT_2} \sum_i s_{it} I[t \geq 2016]-\frac{1}{NT_1} \sum_i \sum_t s_{it}I[t< 2016]$$
If model (1) is true then $plim \ \ \hat \beta_1 = \beta_1 = \mathbb E[s_{it} \lvert t\geq 2016]- \mathbb E[s_{it} \lvert t<2016]$ which is the expected level of R&D expenses for any company for any year AFTER treatment.
You could plot the series $\{s_t\}_{t=1}^T$ and get - if model (1) is true - something like this

However something else may be happening over time and while you do not have any control group of companies not affected by the policy implementation you can still potentially do better than estimate model (1).
Consider the following plot

where in each time period going from $s_t$ to $s_{t+1}$ there is a small increase in R&D let us call it $\lambda$. Then in the year of treatment the increase is slightly smaller.
The blue and read line are found using the estimators above and it is clear that because they do not take the time trend into consideration they overestimate the effect. The data used for the plot was generated after the model
$$s_{it} = \beta_0 + \lambda \cdot t + \beta_1 I[t\geq 2016] + e_{it},$$
which includes the time trend.
Here in this model the effect is immediate. But the effect could also happen in the years after the treatment instead of only in the year of treatment hence you could have a model like
$$s_{it} = \beta_0 + \lambda \cdot t + \beta_1 I[t\geq 2016] + \beta_2 I[
t\geq 2017]+ e_{it}.$$
The take away is that you should do as the professor says and calculate the time averages $s_t$ and plot them and look at them. Paying particular attention to what goes on around in the year treatment and in the years close to but still after treatment (based on the principle that effect comes after cause and takes some time be fully realized)
I R to simulate data and generate the plot by means of the following lines of code:
# simulate panel of RD
library(data.table)
set.seed(1)
N <- 50 #Number of firms
T <- 16     #Number of time periods
TT <- 9 #policy implementation year (treatment year)

b_0 <- 5
b_1 <- 2

# Simulate model (1)
s <- b_0 + b_1 * (1:T >= TT) + rnorm(N*T)

# Make dataset
id <- rep(1:N,each=T)
time <- rep(1:T,N)
I <- as.numeric(time>=TT)
mydata <- data.table(id,time,s,I)

# Compute time averages
time_agg <- mydata[,.(s_t=mean(s)),by=time]

#Plot time averages
plot(time_agg$s_t,ylim=c(0,10),pch=20,xlab="time",ylab="R&D")
points(1:TT,rep(b_0,TT),type="l",lwd=1,col="blue")
points((TT):T,rep(b_0+b_1,T-TT+1),type="l",lwd=1,col="red")
savePlot("plot1.jpg",type="jpg")

# Simulate model (2)
s <- b_0 + b_1 * (1:T >= TT) + 0.5*1:T + rnorm(N*T)

# Make dataset
mydata <- data.table(id,time,s,I)

# Compute time averages
time_agg <- mydata[,.(s_t=mean(s)),by=time]
model <- lm(s~I,data=mydata)
hat_b_0 <- coef(model)[1]
hat_b_1 <- coef(model)[2]

#Plot time averages
plot(time_agg$s_t,ylim=c(5,16),pch=20,xlab="time",ylab="R&D")
points(1:TT,rep(hat_b_0,TT),type="l",lwd=1,col="blue")
points((TT):T,rep(hat_b_0+hat_b_1,T-TT+1),type="l",lwd=1,col="red")
savePlot("plot2.jpg",type="jpg")


Conversion of GDP PPP International \$ 2005 into GDP PPP International \$ 2011

I do not know where you got the forecasts for GDP, but it is very likely they do not include a forecast on how PPPs are changing. As such, a computation of the rate of growth of GDP in 2005 PPP is a mreasure of real GDP growth.

Hence, what you want to do is to concatenate two series. For this, take the latest GDP level of each country from the World Bank series in 2011 prices, and extend it using the forecasted GDP growth rates in 2005 prices, calculated from your database of forecast. Naturally you only want to forecast GDP for the years that are missing in the World Bank database. The reliability of this exercise depends on how outdated the forecasts from the 2005 database are.


The New Keynesian Philips Curve (NKPC) may be derived using price settings frictions introduced by Guillermo Calvo. The NKPC can under such conditions be written as you write it, i.e. $$\pi_t=\beta E_t\pi_{t+1}+\kappa x_t,$$ and where $\kappa=0$ if no firm change their prices, and $\kappa=\infty$ if all firms change their prices.  (Technical note: Here I allow myself to work with the extended real line, viewing $\pm\infty$ as entities and not concepts.)

The background given above, then, does not seem to say anything about the output gap $x_t$. If $\kappa=0$, then inflation follow its trend, i.e. $$\pi_t=\beta E_t\pi_{t+1},$$ and if $\kappa=\infty$ then $\pi_t=\infty$ (given that the expected inflation is bounded below).

This, at least, answer your first question: If fully flexible price setting is interpreted as the case when all firms change their prices, then $\pi_t=\infty$, which is understandable, but not realistic. What we may say, though, is the more firms there are who change their prices, the closer to hyperinflation we are.

To me it seems that we can not from the NKPC alone say whether the output gap will be zero if $\kappa=\infty$. Often we have to supplement the NKPC with the DIS $$x_t=E_tx_{t+1}-\frac{1}{\sigma}(r_t-r^n_t),$$ where $r_t$ is the real interest rate and $r^n_t$ is its natural level, and add some additional information on whether we have random shocks in e.g. the output gap.


Can be the duality theorem applied to not locally non-satiated utility functions?

This system doesn´t have solution, because the consumer want to maximize their utility, so there is not a possible solution because is better consume x=0 that any other bundle


If X and Y are normal, then the distribution of X conditional on Y is:
$$ X | (Y = y) = N(\mu_x + \rho \frac{\sigma_x}{\sigma_y}(y - \mu_y), (1-\rho)^2 \sigma^2_x )$$

Therefore, $E[X|Y = y] = \mu_x + \rho \frac{\sigma_x}{\sigma_y}(y - \mu_y) = (\mu_x - \mu_y (\rho \frac{\sigma_x}{\sigma_y})) + (\rho \frac{\sigma_x}{\sigma_y})y $ which is linear in $y$.
Symmetrically:
$$E[Y|X = x] = (\mu_y - \mu_x (\rho \frac{\sigma_y}{\sigma_x})) + (\rho \frac{\sigma_y}{\sigma_x})x   $$
which is also linear in $x$


What is the difference between identification, calibration and estimation?

Identification and estimation are often used interchangeably (at least that’s my observation from attending conferences and reading papers) but according to econometric literature there is a subtle difference.
For example, in the John Stachurski "A Primer in Econometric Theory" the identification is a process of finding out if the parameters are identifiable and identifiability is defined as

“Identifiability means that the parameter vector associated with the unknown distribution can eventually be distinguished from the data.”

I will omit all the other formalism that follows but in essence we can say that parameter is identified if the map from the parameter space to the space of probability distributions of observables can be inverted.
The estimation is actually the process of calculating what the actual parameter $\hat{\beta}$ is.
When it comes to calibration in forecasting it is the comparison of model estimates to actual data with an aim to improve the model fit.
In macroeconomics it is a strategy for finding numerical values for the parameters of artificial economic worlds. A model is calibrated when its parameters are quantified from casual empiricism or unrelated economic studies or are chosen to guarantee that the model mimics some particular feature of the historical data. In other fields it might have different uses as well.


Attributing Labor Productivity Gains to Capital Deepening vs. Total Factor Productivity Growth

The increase in output in Solow-Swan model can come both from total factor productivity and capital deepening. Let’s step back from the real life example you have given because there many factors come to place and it’s hard to satisfy the ceteris paribus.

Consider simple Cobb-Douglas production function - that is actually almost always used in standard Solow-Swan model:

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

Now here the total factor productivity is represented by the $A$ parameter.

If you look at this this way you should quickly see that growth can come both from capital deepening (if K increases and L stays constant) or from increase in total factor or even both at the same time.

In your farmer example, the new technology increases $A$ so output should increase even if the investment in the new capital would be exactly equal to depreciation of old capital (and hence no net increase in K).

However, this of course abstracts from markets for factors of production where now demand for new technology increases because the new capital is perceived as more valuable as old one.

So both things can happen at the same time in Solow-Swan model, you can have both change in output due to change in A or K. Unfortunately due to this just from the raw change in output $\Delta f$ you can’t directly deduce which part of change is due to capital deepening and which part due to increase in $A$. You either have to keep one of them constant or you need to get data on the changes in either total factor productivity or capital accumulation to be able to deduce the other (assuming labor is always fixed constant).

In the second example if there is no change in total factor productivity (not every new technology has to change it) then you could attribute changes in output solely to capital deepening.

However, the problem there is that here you have another confounding variable and that is that the quality of car probably changed in the eyes of consumer so the output will have higher value. Here you would have to modify the Solow-Swan model to account for that the new output is not same as old and you basically switch from production $f$ to some $f^*$ - the problem is here that you are also trying to fit the macro model that can easily abstract from quality differentials to microeconomic problem where those things matter, but if you would hold the quality constant you could still use it. Or you could include the quality change explicitly with some pice wise production function or assuming multiple plants with different production functions etc.


How does one asses the impact of a proposed policy?

Being an economist, I'd say that carrying out a quantitative forecast ("the policy will create/cost x jobs") would require setting up a model, feeding appropriate data and applying appropriate estimations. You would require decent data on the firm level and a detailed model dealing with lots of production factors. In other words, lots of work and probably way beyond your team's resources.
What might be feasible is to do a research on the relevant scientific literature in that realm. In particular, you should look for theoretical work on the expected qualitative effects from such a policy. Second, you may look for empirical studies on comparable policies in other countries and contexts. This might give you a rough picture of what one can expect. Assessing the external empirical results might be hard if you're not an expert in the field, but the following guidelines might be of help:

Prefer authors from academic institutions over e.g. international organizations or (way worse) lobby groups.
Prefer peer-reviewed publications to non peer-reviewed publications. Use Google Scholar as search engine. For any research item, click on "all versions". If the item is published in some scientific journal (say, Journal of Environmental Economics), this might give you an indication for quality.
prefer more recent studies and studies that deal with countries similiar to yours over others.


Are there internationally published sources of trade sanctions?

The following site might help you out: https://www.globaltradealert.org


Measuring inequality at the zip code or county level in the United States?

I'm not sure specifically what data you have but it sounds like you have enough data to calculate a rough gini estimate.

Gini is equal to: (The average variance for incomes) / (2 * the mean)

Since Gini is a standardized indicator and not dependent on the level of income, just the variance, you can construct a rough gini metric using your data.

If you assume 0-25K = 0, 25K-50K = 1, 50K-75K = 2 etc. You can treat the buckets as incomes. You can then calculate the average variance between the incomes weighted by the number of individuals in each bucket and divide it by 2 X the average income (bucket). In doing so you've created a rough gini coefficient for each zipcode.

You should note that this statistic is rough and not perfectly accurate since it ignores any variation in distribution of income within the bucket. It may be that the 25K-50K bucket has most of its individuals at 40K income and few in the 25K income which would negatively bias your gini estimate.

Assuming this is for a research paper this methodology should be discussed and the possible flaws with this estimate should be disclosed. It does however provide the best approximation for inequality given the information you have.


Does it make economic sense for only some universities in a certain area to pay their sports scholars?

I mean, even in the US, the ability of a school to offer sports scholarships is determined by what athletic conference(/division) they belong to: Division III schools are forbidden from offering athletic scholarships. (So in that regard- it can vary even at the city level. Boston has some schools that give athletic scholarships, and some that don't). In fact, in that example, Northeastern University provides athletic scholarships, but three blocks away is Wentworth Institute of Technology that doesn't.

As far as why would someone play a sport without the possibility of scholarship? For some, I'm sure it's the potential for a future athletic scholarship (for example, many US football players will play their first couple of seasons for a community college in the hopes of attracting attention from NCAA Division I programs). For others, I'd guess the potential of going pro- if you didn't get offered a spot on a team that provides athletic scholarships, this might be your only chance to improve and try out for a pro team later.

But for most, I'd guess it's probably because they enjoy it. Many US schools have club teams in many sports, which compete against local schools and offer no real chance for financial returns. Moreover, why do so many students participate in other competitive extra curricular activities? I'd guess the same reason.


What prevents a bank from simply going into their computer system and adding some zeros to their bank account?

Double entry book keeping.

If we take the example given here, of the Greek Central Bank (bank nerd trivia - interestingly the GCB is a commercial bank, listed on the Greek Stock Exchange), arbitrarily adding 000's to its deposit account.

This cannot be done as stated. The double entry book keeping accounting system on which all banking is based, requires that two ledgers are simultaneously updated, one with a credit and one with a debit. So if the GCB credits its own deposit account (which in practice would be a liability income account of some kind), there has to be a matching debit on another ledger. Those are the rules.

Typically deposit accounts increase their value either by people depositing cash in them (see below), by direct transfer from another deposit account, or by receiving a loan.

For example, if the government prints physical cash then this can be deposited at the central bank:

[debit cash account, credit GCB bank deposit account]

and this is how the US TARP intervention was performed. The federal reserve then used the money to buy loans from the US Banks. Note that a debit to an asset account increases its value, and a credit reduces it, whilst on the liability (right hand side), a debit reduces the account, and a credit increases it.

If the European central bank makes the Greek Central bank a loan then on the Greek side, the book keeping is:

[debit cash, credit interbank loan]

The GCB has received cash, and now has a debt that it owes the ECB among its liabilities.

The one thing it can't do is just arbitrarily increase the value of a single account, one of the main reasons for double entry book keeping is in fact to prevent that.

Bank fraud and the various forms of banking system abuse that do occur typically involve manipulations of banking operations within the framework of double entry book keeping. So while DEB. prevents accounts from having zero's tacked on the end, it doesn't prevent Mr. Smith the bank manager, lending his very good friend Mr. Brown a large sum of money that Mr. Brown has no intention of repaying.


Why do national governments not rent money from central banks directly?

Note: This answer was posted 4 months before the OP clarified what it really wanted to ask (see comments below the answer).

I will accept @Ubiquitous view of the question, which in summary is:

Why not having a publicly owned monopoly in the banking sector? Instead of a, however regulated, private banking sector?

It would be naive to counter "then why not nationalize all production activities?". This is not a "socialism or capitalism?" socio-philosophico-ideological issue. Strictly from the point of view of how the Economics discipline analyzes things, we must recognize the critical and special role the banking system plays in modern economies (because it intermediates savings and investment, consumption smoothing and intertemporal allocation of resources), as well as its peculiarities.
And whenever the words "critical","special", "peculiar", arise in economic analysis, the merits of a private-market structure against a publicly controlled one must be carefully analyzed and assessed.

First let's define what is the main business-line of a bank, in the most basic, abstract sense: sublessor of perfectly liquid capital. It borrows capital from the savers and lends it to the debtors -for a fee, which is the interest, and which depends on the length of the lease. And as with any lease, the lessee must also return what has been lent.

Let's move now to some of the main reasons that have made societies produced a good/service through a state monopoly (obviously I won't go through all the taxonomy and intricacies of the various categories of public goods):

Non-excludability  pertains to public goods like national safety: we cannot exclude those citizens that don't pay for it. This creates disincentives to pay for it. But the society as a whole needs it. So everybody pays whether they want it or not (through taxes that are backed by the monopoly of violence that the state holds), and the service is offered to all by a state monopoly. Does non-excludability holds for what the banking sector offers? No, because the bank gets to decide who gets the service or not. And the "service" is then at the control of the debtor.  So this cannot be an argument in favor of a public monopoly in the banking sector.

Decreasing Average Costs with the scale of production, is what traditionally defines a "natural monopoly": it is in the nature of the production process to have decreasing average costs as quantity produced increases, and so the state steps in so that the benefits to the monopolist accrue to all society through the state. Is the business of a bank characterized by "automatically" decreasing average costs as credit expands?

What are the "production costs" here? Various administrative costs certainly (branches, overheads etc), which are unlikely to exhibit a downward trend through out -rather they are very well likely to have the familiar U-shape, falling for initial small levels of credit volume, then perhaps a stage of constant returns to scale, then decreasing returns to scale (and so increasing average costs). A distinct and peculiar cost here, is outright loss of the capital. Surely, when one lets an apartment, it may suffer various damages from the use of it by the tenant -but it won't disappear with him. These costs will almost certainly increase with the "scale of production" (credit expansion) as the bank expands to less trustworthy debtors. So no "decreasing average costs here" - the banking sector is not a natural monopoly.

A case of negative externalities can be made, that moreover cannot be reversed: the choices a bank makes as to where it will lend its capital, and for what purposes, may not satisfy the needs of the economy as a whole: the economy may need transportation infrastructure -but the banks may lend capital for building office space, which are less needed. But once capital is sunk into buildings, it cannot be transformed into roads (certainly not directly, while indirectly, it won't be timely and it will have dead burden costs). But externalities are regulated in other ways, not by imposing a state monopoly.

What about the various imperfections that inadvertently characterize all actual markets, banking sector included? Incomplete and asymmetric information, rigidities, straightforward inefficiency, etc? Here, the matter becomes quantitative, exactly in view of the criticality of the banking sector: a given level of imperfection and inefficiency would matter less for society if observed in another market, compared to the banking sector. So we have to measure it -and then compare this with the level of inefficiency and imperfection that we would observe if we had a public monopoly instead.

This is what I had to offer as food for thought. I close with some responses to the OP's specific projections:

(In case of a state monopoly, the businesses would have)
A lower interest rate
Why? The central bank would lend public money. To private businesses. It would also have to bear administrative costs. Also, it would be accountable for public money. This would introduce conservatism into the system, to avoid accusations for favoritism for example. Either higher interest rates, or much more stringent criteria to grant a loan. Would that be desirable, as there will be no-one else (officially) lending capital in the economy?) What about the possible increase in loan-shark activity?

Control on their interest rate
I do not understand what that means.

Theoretically no limit
No way. Either the OP has in mind that the central bank will have vastly larger amounts of capital to lend (it will, but will it lend it "easily"? -see previous remark), or he has in mind creation of capital out of thin air based on the "state's authority", which of course would have solved all humanity's materialistic issues, if it could happen.


How do I run a regression with the restriction that one set of parameters are proportional to another?

Just to make things simple I'll show how to do it for a simpler case.

Let's say you have a regression $Y = \beta_1 X_1 + \beta_2 X_2 + \varepsilon$ and you want to impose $\beta_1 = 2\beta_2$.

This is equivalent to saying $Y = 2 \beta_2 X_1 + \beta_2 X_2 + \varepsilon = \beta_2 (2 X_1 + X_2) + \varepsilon$. That is, you can generate a new variable $X_3 = 2X_1 + X_2$ and regress $Y$ on $X_3$.


A company makes decisions that have positive NPV. No list will ever be completely exhaustive since obligations are not necessarily zero-sum.

How do you increase profits for the shareholders? Mostly by taking care of the customers. How do you win customers? One way is to portray your company as environmentally friendly. This, in turn, helps the (wo)mankind and the nation. Do you see how it all pretty much ties together?

Of course, for a given company, one thing may be more important than the other.


Suppose there are two goods $X$ and $Y$. A consumer has lots of $Y$ and very little $X$, and as a result his marginal utility of $Y$ is a lot lower than the marginal utility of $X$ (it helps here to think about a convex utility function such as $U=XY$). We could make him better off if we could convert some of those $Y$'s into $X$s (again, this follows from the convexity of utility, a standard assumption. Intuitively, people want more of everything but still prefer balanced consumption bundles to extremes). The question then becomes, "how many $X$s can we buy for each unit of $Y$ we give up?" The answer is that each $Y$ can buy $\frac{P_X}{P_Y}$ units of $X$.Thus, we convert $X$ into $Y$ until the slopes of the marginal utilities equals the price ratio, at which point additional conversion would result in a larger loss from less $Y$ than gain from more $X$.

A more general way of looking at this is through calculus. For a constrained function that is continuously differentiable, the slopes at any interior maxima and minima must be parallel to the slope of the constraint. In this case, the function is the utility function, $\frac{P_X}{P_Y}$ is the slope of the constraint.


Code finding all stable matchings in one-to-one problem

Patrick Prosser has some great java code at http://www.dcs.gla.ac.uk/~pat/roommates/distribution/ which, among other things, can compute all the stable matchings in roommate problems.

The code is for roommates problems, but Patrick's code allows preferences over roommates to include unacceptable roommates. To implement a two-sided market, just make sure any roommates on one side of the market views any other roommate on the same side of the market as unacceptable, and you're good to go.

If (like myself) you are not used to java, you might struggle a little to get the code working. Here is a little tutorial for Mac OS, which worked for me as of today.


Notice that you only  consume out of your disposable income, that is, adding transfers and removing taxes. Thus, from that formula of the multiplier, yes, it should be there, in the $(1-t)$.

As (assuming) $C=cY^D=c(1-t)Y$ and thus, $MPC=\partial C/\partial Y = c (1-t)$, then $Y-Y^D=tY=\text{Taxes}-\text{Transfers}$, with $t$ "condensing" all the information.


Hint:
You can convert the game into the following normal form:
\begin{equation}
\begin{array}{|c|c|c|}
\hline
& Steal & Split \\\hline
Steal & 100\beta, 0& 100(1-\alpha),0 \\\hline
Split & 50(\alpha+\beta),50+50(1-\alpha-\beta) & 50,50 \\\hline
\end{array}
\end{equation}
Note that Steal is a weakly dominant strategy for player 2, so (Split, Split) can never be an equilibrium.


WARP implies completeness, transitivity and thus rationalizability. What is wrong with the statement?

In Notes on the Theory of Choice (assuming this is what you refer to by "Kreps (1988)"), Kreps does not appear to mention WARP as such. But he does refer to "Houthakker's axiom" (Houthakker 1950, Economica), which is actually SARP, not WARP.

Your conditions 1 and 2 are equivalent if there are only two goods. If there are more than two goods, WARP in condition 1 must be replaced by SARP to preserve the equivalence. Jehle and Reny (2011) have an intuitive explanation for this:

  [W]ith two goods, the pairwise ranking of bundles implied through revealed preference turns out to have no intransitive cycles. [...] And when this is so, there will be a utility representation generating the choice function.

SARP basically posits conditions that explicitly rule out intransitive cycles in revealed preferences with a general $n$-good environment.

WARP/SARP differ from GARP in that the former presumes the availability of infinitely many choice data to build a choice function with, whereas GARP only requires a finite set of choice data.


What are the widescale economic ramifications of people holding onto their money instead of spending it?

The amount that people save depend on the marginal propensity to consume (MPC), i.e., part of the additional income that a person consume, hence, $1-MPC = savingrate$. Keynes assumed that with increase in income MPC would increase, while neo-Keynesian assumed it to be constant in their analysis. But real data shows that MPC increases with elevation of income but decreases with large income gains. Hence, in a way, rich people save more relatively to others. But that is not the reason why they are rich, more saving is the effect, not the cause. Mostly, the reasons behind getting rich are the apt investment decisions.

When an economy hold more money, consumption reduces leading to reduced outputs. This in return decreases real income. Since, in equilibrium savings equals investment ($I = S$), so actually there is no increase in savings and we are left poorer with lower income levels. This is called the paradox of thrift, suggested by Keynes.


I'm answering from my memory of Reddit comments like blitzkrieg9
 and AdmiralAkbar1. Don't hesitate to edit this post to link to them, but I can't remember where exactly I saw them. And I haven't just copied and pasted – I've rewritten out typos.

  Alternatively, I would like to get paid for receiving a barrel of oil (I can keep it in my back garden) and then sell it in the future.

Unprocessed crude oil is full of toxic and nasty stuff, and if not properly contained, many dangerous gasses like benzene or hydrogen sulfide will just evaporate out of it. You definitely don't want to be anywhere near an open swimming pool full of crude oil.

You can't really do anything with it at home, and if you dump it, you go to prison. You get nasty byproducts that you'll have trouble selling in the tiny quantities you produce, but which are illegal to dump. But hey, you might get a gallon or two of usable gas out of it!

  But, why don't they just hoard it until the coronavirus crisis is off?

Crucially, this is the price for crude oil which still has to be refined into usable products, and cannot easily be disposed of. The negative price reflects the costs associated with storing the stuff indefinitely.

And you can't turn off producing wells like a tap. It takes time to shut them down without ruining them, so that they can be turned on again. Look how long it took to get Kuwait up and running again after the first Gulf War, when the retreating Iraqis destroyed wells.

It's simple to shut in natural gas by turning the valve, as natural gas has a natural pressure. Oil does not have natural pressure. Once oil's shut in, it's not easily "turned back on". It's part of what causes wild price swings as supply can't move as fast as demand.

The leases for oil and gas in America have "shut-in clauses" where after being shut in for a certain period of time (usually one year or so, but they vary) the lease holder has to go back to the mineral owner, to re-lease. Again with gas, operations can go and turn on a gas well for a month, then turn it back off just to get enough production time to avoid the shut-in clauses. You can't do this with oil.  Even old stripper wells need about $18-20 to pay to keeping them operating.

Even if you can stop extracting oil, the oil producing countries are often in a sort of Mexican stand-off. If you produce too much, oil drops and you lose money. If you produce less, the other countries may produce more and you lose money.

  Why don't speculators jump in and bring up the demand again to positive?

A futures contract is simply a contract to buy something in the future, like next month or next year. Some futures contracts "settle to cash". This means we determine the price when I bought the contract and the actual real-world price on expiration date, then one of us pays the other person the price difference.

But oil and most commodity contracts settle for the actual product! If you don't close out your soybean contract, your exchange calls you asking what you'd like to do with your soybeans that are waiting for you in Kansas City! Same with oil. Tankers are showing up in Houston with millions of gallons of oil and somebody needs to take delivery. Nobody can, because all the storage facilities are already full. So people are literally paying you over $30 a barrel for you to possess oil!

You're referring to a specific contract for specific delivery to an oil pipeline in Oklahoma on Apr 21 2020. That is the crux of the issue. The supply vs demand kerfuffle is very localized at a major crude oil pipeline junction in middle America. This is unprecedented. But it is a LOCALIZED phenomenon.

What's happening now is what happened in Trading Places when the real report comes out: people realized that they overpaid for their futures and are selling to try and recoup some of their losses. There's less demand for oil right now, there's less hope of more demand for oil in May, and there's little storage space left. Low demand + high supply = low prices. A few days ago, the settlement price for a barrel of oil for futures contracts with May settlement dates was locked at ~\$18 a barrel. People expect it to be way lower than \$18 next month. So they're trying to sell off their futures contracts. It's going negative because they're literally paying you to take it off their hands if you have room for it. Although they know it'll be a loss, they figure it's less of a loss than holding onto worthless oil that they can't store.


Finally, I believe it is easy to answer it. Well, in case where some player will not cooperate and the triger strategy is enabled, we have the following soloutions:
Soppuse that p2 does not cooperate in the first round so, he will gain a payoof of 1, but in the second round p1 plays $a_2$ and he won't cooperate with p2 in perpetual. Thus, we have that p2 will play $a_1$, because by playing $a_2$ he increases his looses and the value function is:
$$V_2^{NC}=1-1\cdot\delta-1\cdot\delta^2-1\cdot\delta^3+...=1-\delta\sum_{j=1}^{+\infty}\delta^{j}=1-\dfrac{\delta}{1-\delta}$$
Hence, they will cooperate if:
$$0>1-\dfrac{\delta}{1-\delta}\Rightarrow \delta>\dfrac{1}{2}$$
Differently, if both players choose trigger strategies, so p2 also chooses $a_2$, then both will make looses of -2 and the value function is:
$$V_2^{NC}=1-2\cdot\delta-2\cdot\delta^2-2\cdot\delta^3+...=1-2\delta\sum_{j=1}^{+\infty}\delta^{j}=1-\dfrac{2\delta}{1-\delta}$$
and hence they will cooperate if:
$$0>1-\dfrac{2\delta}{1-\delta}\Rightarrow \delta>\dfrac{1}{3}$$


Why would using real-time settlements have helped combat some of the chaos that occurred during the January 2021 GameStop saga?

It would not prevent speculation driven bubble like GME but it would arguably prevent the situations such as Robin Hood being forced to restrict trading.  This is because when you buy stock your transaction is actually not processed immediately but takes few days to clear (usually 2 days see here).
This creates a problem because by law brokers like Robin Hood are required to keep portion of their own capital until transaction clears (e.g. SEC net capital obligations and clearinghouse deposits). This can cause a problem for a firm if people make too many transactions as it simply might not have enough capital to set aside to fulfil the regulatory requirement. In fact brokerage firms often raise additional capital even if they themselves not need it just because they anticipate that their customers could increase in trading activity but the volume of trading is not easily predictable.
This problem could be alleviated if stock trades were settled immediately instead of waiting days before actually carrying out the transaction. This is because if the stock trades are cleared immediately there is no need for brokerage firms to have capital buffers forcing them to restrict trading as the rationale for the regulatory buffer is to have some funds set aside if something happens in those 2 days.


Has any country attempted to inflation adjust capital gains before taxation?

Mexico has a 3% per annum adjustment to basis on real property when calculating capital gains, though they still have a 5% floor on capital gains tax.

I cannot weigh in on the "success" of the program given everything else they're juggling.

Source


Production Possibility Frontier and Opportunity cost

Your answers in the first and second questions are correct, as another post noted. However, the last question cannot be answered: we would have to know what would be the number of Toyota produced when production of Lexus is zero, in order to answer that.

By the way the Labels in your graph are wrong - it should read "Lexus" in the vertical and "Toyota" in the horizontal axis.


What the paragraph actually said is

  "Fervent demand for renminbi from international investors has driven down rates in Hong Kong and thereby created incentives for companies considering using the renminbi for trade or financing," explains Financial Times correspondent Robert Cookson.

This means that international investors were buying offshore Renminbi (CNH) in Hong Kong and using that to buy offshore bonds denominated in Renminbi. That pushed the price of those bonds up, corresponding to reduction in their effective interest rates ("driving down rates").


Does immigration help to sustain welfare in rich European countries?

Here is a chart from the OECD, which shows the net fiscal impact of migrants on their recipient country (i.e. by how much to they contribute or withdraw from the welfare state, albeit excluding in-kind benefits such as healthcare). For most Western countries, Denmark included, migrants are modest net positive contributors ("they pay in more than they take out").

This is because migrants tend to be disproportionately working age, whereas the native population contains many retired or young people. Thus, migrants tend to pay a lot of income tax relative to the average native. Luxembourg and Switzerland stand out because both are small economies that attract many high-wage knowledge workers.

On the vertical axis of this chart is the net contribution to the fiscal budget. So a value of 1% means $(\$\text{tax paid}-\$\text{welfare received})/\$\text{GDP}=0.01$. Note that this is different to (and less than) migrants' overall contribution to GDP because, for example, migrants keep much of their contribution to GDP in the form of after-tax wages.

I took the chart from Oxford's migration observatory, which has more information specific to the case of the UK.


Central bank loss function (I did a solution, but it doesn’t totally make sense I guess)

You need to differentiate with respect to Y, as you said! You substitute the phillips curve as you did (because it is a constraint for the central bank's optimization problem), then differentiate w.r.t $\bar{Y}$ (which I assume here stands for gap) and set this to 0 to obtain the optimality condition, which in turn will give you the monetary rule. Recall that once the optimal output-inflation combination is determined using the monetary rule, the central bank sets the interest rate to implement its choice.


In this scenario, which coin is more valuable?

The word 'valuable' is not well defined in this question.

If I am willing to buy an apple from you for \$1 and but I am only willing to sell an apple to you for \$4 which is more valuable: An apple or \$2? You cannot tell because neither option is sufficient to get the other option by trade.

You are faced with the same dilemma in your problem. There are different prices at which to sell and buy so neither one of the options is clearly more valuable.

EDIT:
As @Adam Bailey points out in his excellent answer in this situation even though the exact value ratio is ill defined, it is clear that coin B is more valuable than coin A, because you can exchange one coin B for five coin A, thus you can get more by chosing coin B.


Can marketing efforts be considered to be a factor of production?

Marketing is obviously relevant. We can observe that firms spend money on marketing. If it had no effect, they wouldn't do it. Whether it is a "factor of production" depends on your definition.

In my book, a factor of production is any input that directly contributes to the production (creation) of the good or service. That is, it enters some production function $f$ that maps "factors of production" into "output", a quantity of the produced good. In that sense, marketing is not a factor of production, because it has no direct effect on how many goods are produced. In my eyes, marketing is supposed to influence the demanded quantity, not the produced quantity. Of course the two are related, but not directly.

Belleflamme and Peitz have a nice subsection on adverstising (6.1), where they discuss three distinct aspects of how advertising (which I use synonomously with marketing) affects demand: it can be persuasive, informative or complimentary. Such a distinction matters, because the welfare effects differ. For instance, informative ads may increase welfare as consumers can make better-informed choices, whereas persuasive ads may decrease welfare as it distorts consumers' real preferences.


The figure provided by that author is widely inaccurate according to the official statistics.
According to the OECD the general government debt to GDP ratio ratio in 2008 was just $102\%$. Moreover, here is comparison to other countries:

Moreover, general government is by OECD defined thusly:

General government consists of central, state and local governments and the social security funds controlled by these units. General government accounts presents data on fiscal balance, debt, revenues, expenditure, costs and reserves of governments. Central government consists of the institutional units making up the central government plus those non-profit institutions that are controlled and mainly financed by central government. The political authority of central government extends over the entire economy. Central government can impose taxes on all residents and non-resident units engaged in economic activities within the country.

So it does include central, state and local government and even non-profits that are mainly financed by public funds.
Hence either there are some caveats that you are not mentioning in your question (perhaps it does not include just government debt but also some private debts or present value of unfunded future liabilities) or the author of the book made a mistake.


In one dimension if the functions $f_i$ are bounded, e.g. defined on a compact interval, then yes. Take $g_1$ such that it makes $f_1$ almost constant, e.g. $g_1(x)=\frac{x}{N(\max f_1-\min f_1)}$ where $N>0$ is large. Take $g_2(x)=x$ Then $g_1\circ f_1+g_2\circ f_2\approx f_2$, which is strictly quasiconcave by assumption.

In one dimension a strictly quasiconcave function is either strictly increasing, strictly decreasing or single-peaked.


For Quantitative Easing to happen, a central bank needs to expand its balance sheet by, more or less, "printing money" from scratch. Quantitative Easing is a difficult policy in that you want to encourage inflation to force investment, but at the same time you need to control that inflation. For this reason QE should be used only when central banks believe economy is growing slower than it could/should be, which does not happen a lot in Africa.

For QE to be efficient, you need a strong central bank/currency, ie printing new dollars will create more dollars, but printing new 1920s Deutschmarks will only bring down the value of an individual Deutschmark.

To my knowledge, the only African country that wandered in Quantitative Easing is Zimbabwe. The success of this policy may by itself discourage any other comparable country to engage in such a policy.

QE is a last resort policy for stalling economies. Japan, Western Europe and the USA are countries where QE has been widely discussed because it is sometimes believed investment is delayed in hope of better days - QE should prevent this, but this is not a typically African problem, to my knowledge.

TL;DR: Even if African central banks were strong enough to engage in QE, African economies don't need such a policy to date.

Note that as mentionned in a comment, Quantitative Easing is a set of policies that has no standardized definition and that some examples above are anachronic with regard to the relatively recent introduction of the notion. While the term dates back to the early 2000s, what we know describe as QE was never practised as strongly as in the 1930s and 1940s against the Great Depression.


I apologise for the poor formatting and such, this is one of my first posts so please let me know if anything needs improvement (including, possibly, the correctness of my answer!).

For part a.), I'm assuming that a per unit tax of $2q$ is levied on the consumer in this market. The inverse demand function and inverse supply function are given by $p = \frac{393}{2} - \frac{q}{2}$ and $p = 4q + 48$ respectively. Thus, a per unit tax of $T = 2q$ will result in the inverse demand function being $p + 2q = \frac{393}{2} - \frac{q}{2} \iff p = \frac{393}{2} - \frac{5q}{2}$. Equating supply and demand, we find that the new equilibrium quantity is $q = 22.846$, the price that consumers pay is $185.077$ and the price producers receive is $139.385$.

For part b.), I'm assuming that a tax of $20$% of revenue is levied on the producer in this market. The inverse demand function and inverse supply function are given by $p = \frac{393}{2} - \frac{q}{2}$ and $p = 4q + 48$ respectively. Thus, a tax of $20$% of revenue will result in the inverse supply function being $p - 0.2p = 4q + 48 \iff p = 5q + 60$. Equating supply and demand, we find that the new equilibrium quantity is $q = 18.2$, the price that consumers pay is $187.4$ and the price producers receive is $120.8$.

I hope this makes sense (and is correct)! Feedback would be great!


Explain some All countries excluding residents in BIS Debt Securities Data

Your link is not particularly helpful - perhaps you mean something like https://stats.bis.org/?m=6%257C346%257C644#ppq=CBS_IB_INT_C_BY_SEC_TYP_IB;pv=11~10,9~0,0,0~name

"All countries excluding residents" is a phrase BIS use for international debt securities, so represents debt or similar instruments issued by somebody in one country and held by somebody else in a different country, i.e. excluding residents (human or corporate) of the same county

Note that residents of the same country may have a different nationality or be owned by non-residents.  So for example if an American-owned subsidiary incorporated in London issues a debt security to a German-owned subsidiary incorporated in London denominated in Japanese Yen, this would be excluded from these statistics since the two banks are both resident in the UK. But if the American-owned bank in London holds a debt security issued by an American bank in New York, this should not be excluded, and should count as a US liability to the UK

BIS provides a glossary, which is sometimes helpful and sometimes less so


What are the economic arguments against (a) variolation (analogue) as a solution for Covid-19?

Although some models would surely be better than mere qualitative [counter]arguments, Tyler Cowen, who teaches economics at George Mason University, put forward some counter-arguments, although his arguments are against a more specific proposal of Robin Hanson (another GMU economist) who proposed creating "Hero Hotels" where [young] people would get "variolated" with Covid-19. Some of Cowen's [counter]arguments (I'm skipping some):

Qualified medical personnel are remarkably scarce right now.  I do not see how it is possible to oversee the variolation of more than a small number of individuals.  [...]

The NBA has an amazing amount of money, on-staff doctors, the ability to afford tests, and so on.  And with hundreds of millions or billions of dollars at stake they still won’t restart a crowdless, TV-only season.  They could indeed run a “Heroes Hotel” for players who got infected from training and play, and yet they won’t.  “Stadium and locker room as Heroes Hotel” is failing the market test.  Similarly, colleges and universities have a lot at stake, but they are not rushing to volunteer their dorms for this purpose, even if it might boost their tuition revenue if it went as planned (which is not my prediction, to be clear).

Does small/marginal amounts of variolation do much good compared to simply a weaker lockdown enforcement for activities that involve the young disproportionately?  Just tell the local police not to crack down on those soccer games out in the park (NB: I am not recommending this, rather it is the more practical version of [Hero Hotels]; both in my view are bad ideas.)

Society can only absorb a small number of very blunt messages from its leaders.  You can’t have the President saying “this is terrible and you all must hide” and “we’re going to expose our young” and expect any kind of coherent response.  People are already confused enough from mixed messages from leaders such as presidents and governors.

(Some of Cowen's later arguments are more into the realm of behavioral economics.)
Cowen is actually somewhat sympathetic to the idea of sorting who is (most) exposed to the risk, but he concludes by saying that

I don’t think Heroes Hotel, and accelerating the rate of deliberate, publicly-intended infection, is the way to a better solution.

It would be more interesting (of course) if these competing ideas/intuitions (Hanson's, Cowen's) were more formally modelled.
I suppose I could add that China did run fairly large centres when they quarantined everyone they thought were infected (though not infected on purpose), the so-called "Fangcang hospitals". There were up to 16 such facilities opened at one point in Wuhan, with about 10,000 people in them at the peak, but reportedly these have all be closed now.
Hanson doesn't seem to make any quantitative estimates how many "hero hotel" rooms would be needed (e.g. per capita) to achieve some concrete effect in reducing the economic downturn, although he does advance a scenario in relative population risk terms as:

For example, in a simulation where old and sick people are selected for isolation, while only the young and healthy are eligible for variolation, there are 40% fewer life years lost, compared to no variolation and random selection for isolation. Each variolation volunteer suffers only an additional 0.20% chance of death to save a random other person from a 6.5% chance.

Alas, he doesn't seem to (even) quantify the (relative) effects of variolation "on top" of non-random isolation of the old (which is a more plausible alternative than random isolation, which Hanson uses as the yardstick.)
I see Hanson has (also) written a reply/rebuttal to Cowen's points. I won't get too much into that here because Hanson e.g. says

If variolation cuts death rates by a factor of 5 (median estimate in this poll) [...]

Where the "poll" is a Twitter poll Hanson ran rather than some medical expert poll (at least). So the arguments seem based on rather dubious/uncertain parameters... which is what I suppose also precludes more serious research/publications on this proposal.


In quantitative tightening do banks have no choice in refusing to buy bonds from Central Banks (mainly Federal Reserve)

The question as it stands now contains points that are debatable. So I will just attempt to answer a similar question; the question could be revised in response.

There are a number of other questions on QE that offer some background.

Difference between QE and regular monetary operations
Limits to QE

My revised version of the question is: “In quantitative tightening, the Federal Reserve sells bonds on its balance sheet. Can banks refuse to buy these bonds?

The answer to that question is yes, but it does not matter. Note that my description here is highly simplified, technical details are being skated over.

Let’s assume that Betty is an investment dealer, but not a bank. She has a bank account at Acme Bank. The Federal Reserve has an auction to sell bonds. Acme Bank refuses to participate for some reason or other, but Betty buys \$1 million in bonds from the Federal Reserve.

Betty transfers the money on deposit at Acme Bank to the Federal Reserve. What then happens.

Acme Bank reduces Betty’s deposit balance (at Acme) by \$1 million.
Acme Bank transfers the funds to the Federal Reserve by losing \$1 million of reserves it had on deposit at the Federal Reserve. This eliminates \$1 million of excess reserves in the system.

These operations are neutral for Acme itself. It loses an asset (the reserves), but also loses a liability (Betty’s deposit).

That is, bank reserves are just deposits by banks at the Federal Reserve. A central bank is called a central bank for a reason - most banks have deposits there. It’s just that in American banking jargon, these deposits are called “reserves”. Other countries (like Canada) have abolished resserve requirements, and these deposits are just called “settlement balances.”

Reserves are destroyed by anyone transferring money to the Federal Reserve. The only way a bank could stop it is by not transferring the money, which would be a breach of some law or another.

The only way that quantitative tightening could be stopped is if nobody was willing to buy Treasury bonds at any price. Unless the Treasury was on the verge of default, we normally do not see such behaviour.


$V=E+D=300+100=400$

So

$\frac{E}{V}=\frac{3}{4}$ and

$\frac{D}{V}=\frac{1}{4}$

You probably overlooked one zero and thought equity was 30, but in your question you yourself state that $E=300$.


Validity condition for Instrumental Variables

Essentially, $Cov(u,W) = 0$ is implied by $E(u|W) = E(u)$ by the Law of iterated expectations


Can we use hotelling's lemma as a rule of thumb when wage rate can vary?

I don't quite know what it means to apply a mathematical result to a situation in which the assumptions are violated, but the same idea as in the classical version of Hotelling's lemma can be used to understand the marginal effect of wage changes and you can combine the two to understand what happens on the margin if you change both prices and wages.

The underlying logic of Hotelling's lemma was that for a profit-maximizing firm, the marginal effect of adjustment outputs (and therefore inputs) is zero. This tells us that the marginal effect of wage changes on profits should be proportional to inputs used, that is
$$\frac{\partial \pi(p,w)}{\partial w}=-x(p,w),$$
with $x(p,w)$ the optimal input bundle. This is indeed true and the proof is exactly the same as for the marginal effect of $p$, just replace the derivative with respect to $p$ by the derivative with respect to $w$. If $f$ is the production function, the profit function is given by
$$\pi(p,w)=pf\big(x(p,w)\big)-wx(p,w).$$
Taking the derivative with respect to $w$ now gives us
$$\frac{\partial \pi(p,w)}{\partial w}=pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial w}- x(p,w)-w\frac{\partial x(p,w)}{\partial w}.$$
The first order condition for $x(p,w)$ being a maximizer of $pf(x)-wx$ implies
$$pf'\big(x(p,w)\big)- x(p,w)-w=0$$ and therefore
$$pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial w}-w\frac{\partial x(p,w)}{\partial w}=0.$$
Substituting this in, we get
$$\frac{\partial \pi(p,w)}{\partial w}=-x(p,w).$$

The envelope theorem generalizes the underlying idea that marginal benefits of adjustments for optimizing agents are zero for more general parameterized optimization problems.