Supply and demand curves are a function of price and quantity.  If anything else changes other than P or Q that is relevant to the curve, the curve shifts.

For supply, these shifters generally fall into three  categories:

Technology
Number of producers
Price of inputs

For demand:

Number of buyers
Price of complements or substitutes
Customer tastes and preferences
Consumer income

If you come up with something that didn't fit into these categories, but is not P or Q, the result is still a shift! You probably just need your imagination to squeeze it into one of these 7 formal categories.  For example, if the supply curve was P=2Q+3, and there was a decrease in the cost of inputs, the demand curve could shift to P=2Q+2.  Note how the price levels are lower at every level of Q.

Changes in supply and demand that are not "shifts" are called "slides along the curve". They are any direct change in P or Q. This is easiest to see with a linear, mathematical example.

Let us say the government wants to set the price of a product. If P=2Q+3 is the supply, then consider that if you set the price (by law) to be 7, then Q is now 2. The curve remains steady, but we slide along it to get to the new P=7, Q=2 position. From wherever P and Q started (P=12 in the example), we now end at point P=7,Q=2, and the curve remains unmoved along the entire line P=2Q+3.


For a certain good, if there is a one-time fixed cost for a consumer to switch to a different firm, then this good is?

The good has a switching cost / is subject to a switching cost. Rarely used: it is a Diamond good. (Referring to the Diamond 1971 paper.)


How is the growth rate form of the exchange equation derived?

Taking logs of the equation $MV=PY$ gives: $ln(M) + ln(V) = ln(P) + ln(Y)$.

Next, differentiating with respect to time yields $\frac{dln(M)}{dt} + \frac{dln(V)}{dt} = \frac{dln(P)}{dt} + \frac{dln(Y)}{dt}$.

Or, using $\frac{dln(X)}{dt} = \frac{\dot{X}}{X}$, with $\dot{X} = \frac{dX}{dt}$, the derivative of a variable $X$ with respect to time, you can immediately derive $\frac{\dot{M}}{M} + \frac{\dot{V}}{V} = \frac{\dot{P}}{P} + \frac{\dot{Y}}{Y}$, which equals $\%\Delta M + \%\Delta V = \%\Delta P + \%\Delta Y$.


Original Work for Pareto Refinement of NE under Strategic Form

I think you're referring to the criterion of payoff dominance in equilibrium selection. This idea can be attributed to Harsanyi and Selten (1988):

John C. Harsanyi and Reinhard Selten. 1988. A General Theory of Equilibrium Selection in Games, MIT Press.


Derive demand function from utility function with constant elasticity

You stopped a bit too early.

Let me re-write the FOC for good $i$ as follows:

$$ \lambda=\frac{\theta \left(\sum^n_{i=1}q_i^\beta\right)^{\theta-1}\beta q_i^{\beta-1}}{p_i}
$$

As this is identical across goods, equalise for good $i$ and good $j$, where, after simplification, you get:

$$ \frac{q_i^{\beta-1}}{p_i} = \frac{q_j^{\beta-1}}{p_j} $$

This relation hold for any pair of goods. This means you can re-write the demand for any good in terms of $q_i$ and relative prices. Then, you can use the budget constraint to obtain the final demand for good $i$, in terms of income $I$ and all the prices.

Hopefully this information is enough for you to move forward and solve the problem.


ARCH Model - Expectation of Absolute Value

I don't believe $w>0$ is necessary or sufficient to get a positive standard deviation everywhere. I think you need $\sigma_0 \geq 0$ and $w\geq 0$ with one of the two inequalities strict.

If $\gamma  > 1$ then the series $\sigma$ will grow explosively if $w + \gamma \cdot \sigma_{t-1}  >0$ or behave explosively harmonic if $w + \gamma \cdot \sigma_{t-1} <0 $. Neither of those seems particularly desirable most for economic situations.


Problem with a Hal Varian Question from chapter 5

I think the answer to your question is that it would be a matter of Ralph's preferences. Perhaps he is indifferent to eating at 11:00 am or 1:00 pm, as you suggest. But he might have non-parabolic preferences (for reasons unspecified), and both this situation and the one you describe are consistent with his eating at 11:00 am, so the answer provided is a correct solution to the question asked. Perhaps the particular indifference curves drawn are a way to avoid multiple solutions (the author might have thought it simpler).

I'm going exclusively off the information in your post though, so I'm assuming that we do not know if Ralph has any preferences over time independent of how it implicitly affects the price he pays.


I have tried to grab the article to read it, but my library apparently doesn't subscribe to JME as full text.  However, when I have seen the word singular used in economics it is usually in the framework of a singular matrix.  As this article appears to be a criticism of another model, it would appear they are arguing the other model would not give rise to an invertible matrix.  In the above set of comments, it is implying that the stochastic component is vanishing and that the model is deterministic, despite the fact, there are no actual deterministic models in economics if you have to measure things.

My guess is that they are criticizing the other model by showing it can't work.


How does an increase in the minimum reserve requirement affect the monetary base?

In a fractional reserve banking system, the maximum portion of the money supply represented by bank-issued loans relates to the reserve requirement as $M_{loans} = \frac{deposits}{r} - deposits$ where $r$ is the reserve requirement expressed as a decimal.

For example, a bank holding \$1000 in deposit liabilities facing a reserve requirement of 1% is free to issue $\$1000/.01 - \$1000 = \$99,000$ in new money in the form of loans.  Doubling the reserve requirement to 2% would reduce this number to $\$49,000$.

Reserve requirements in the US depend on the total amount held by a given bank and range from 0% to 10%, with larger banks facing the larger requirement.  https://www.federalreserve.gov/monetarypolicy/reservereq.htm.  Some countries, like Canada, do not have a reserve requirement.  Banks determine their reserves according to their internal risk models, including assumptions on inflation risk.

Of course, banks don't have to lend money out until they hit the reserve requirement, and as noted in comments above, the money supply in the US is largely decoupled from banking operations thanks to quantitative easing measures.


What does irrationality mean under the notion of revealed preferences? An example with terrorism.

I think I have worked out some sort of solution to this question.

In Handbook of Defense Economics Chapter 9: Terrorism Theory and Applications, Enders and Sandler give the example of metal detectors in airports to explain the notion of rationality under terrorism.

They show empirically that the inclusion of metal detectors in airports not only decreased the number of skyjackings, but that terrorists substituted towards other means such as more conventional hostage taking. They also give the example of when America fortified their embassies, terrorists attacked America's allies' embassies more. These substitution effects are an example of rationality as either cost benefit analysis is being applied or there optimization with constraints is taking place.

However, my question related to "What if skyjackings didn't decrease after metal detectors were put in", would this imply terrorists aren't rational and we can't use these models to understand their behavior?

The answer to this is that we don't know and that two competing solutions exist:

(1) Terrorists are irrational and act arbitrarily to express anger.

(2) Terrorists simply don't have full information or their preference for skyjacking is extremely high. In this case they could have done a cost benefit analysis but may have calculated incorrectly or not been privy to full information.

I lean towards the later, but it is impossible to prove either.


Just how much calculus is involved in the study of economics?

Calculus is one of the most widely-used branches of mathematics in economics. Here are two (non-exhaustive) examples of important ways in which economists use calculus:

to optimize functions. As a simple example, suppose we are thinking about a firm that must choose its price in order to maximize profit. Provided the profit function satisfies a few regularity properties (i.e. is continuous, concave, and smooth), locating the profit maximizing price amounts to identifying the price for which the profit function has zero slope (see figure below). Since we can calculate the slope by differentiating, calculus gives us the means to identify the optimal price.

to perform 'comparative statics' analysis. In the simple problem above, the firm must choose its price to maximize profit. But often the result of this exercise will depend on a number of other factors. For example, the optimal price will probably depend upon competitive the market is, or how closely substitutable rival products are. A relevant question for economics is therefore "if I make the market more competitive, what will happen to the price?" The typical way of answering this question would be to calculate the optimal price, $p$, as a function of market competitiveness, $h$, and then compute the derivative: $p'(h)$.* Indeed, $p'(h)$ is the answer to the question "how much would $p$ change if I change $h$ a little bit?" Most types of policy involve going into a market and changing something, so the ability to predict what the effects of such a change are likely to be is very useful.

In your principles courses, you have probably seen problems that could be solved in this way, but were instead solved through some other method. almost certainly, the courses were organized like this in order to avoid making calculus a requirement. But, as the problems you tackle become more sophisticated, there comes a point at which the effort needed to learn calculus is much less than the difficulty of trying to figure out a way to do economics without calculus.

To answer your explicit question, there is lots of calculus in economics. But, to answer the implicit question in the background, I don't think you should be too worried by this because:

Economics involves a lot of fairly easy calculus rather than a little very hard calculus. Primarily, this means calculation of simple derivatives and the occasional bit of integration.
Doing economics is a great way to become good at calculus! You will get lots of exposure to simple calculus problems. That will give you lots of practice. Moreover, the problems will usually have a fairly concrete application and hopefully make calculus more interesting than trying to learn it from an abstract math book.
From long personal experience as a student and teacher: it is easier to become good at things you love than it is to learn to love things you are good at.

* If you are wondering why I use $h$ for competitiveness, take a look at the Herfindahl index.


How successful will OPEC's predatory dumping strategy be?

This is the cost curve of oil production (North American shale at $65):

Source

And this is the cost curve of major American projects:

Source

While "Citi's Ed Morse highlighted this chart, showing that for most US shale plays, costs are below $80 a barrel."

And clearly, many of these new projects would lead to losses with oil prices at $60:

Source

The problem with predatory pricing (if it's purposeful) is that OPEC itself depends on high oil prices to ensure balanced budgets in their (mostly dictatorial) regimes. Their sovereign funds are large enough to survive a brief (2-3 years) period of low oil prices, but in the long run, dumping endangers their rulers' positions:

Source

You can't kill American oil production this way—it would return after these two-three years anyway.

The \$60-80 band leaves space for most American projects, while oil below \$60 kills too many OPEC members. So, it does not seem to be a good dumping strategy.


Consider two firms with production functions $A_1 F(k,l)$ and $A_2 F(k,l)$. Both have the same curvature (and are CRS), but one is more productive. Here we can solve the social planner's problem (why?) instead of looking at a competitive equilibrium. He will want to equalize marginal returns across the firms. As one is more efficient, it will get more inputs.

And what is the maximum? Give all of the inputs to the more productive firm. This is because of CRS (which indeed means that as you scale up one production function, it's marginal product will always be larger than the other one's).

That is, define $G(l, k)$ as the aggregate production function. It is given by

$$ G(l,k) = \max_{(l_i, k_i)_{i=1,2}} A_1F(k_1,l_1) + A_2F(k_2,l_2) \text{ s.t. resource constraint}$$

The solution here is $G(l,k) = A_1F(k, l) $ if $A_1 > A_2$, and vice-versa.

Adding Up Makes No Sense

If you think that adding them up makes sense, you're not thinking things through:

If one cook can bake one cake with each egg, and the other cook can also bake one cake with each egg, then the two cooks together can bake two cakes with each egg?

Decreasing returns to scale

Now, it won't be the case that one firm is more productive than the other one always. Consider $F_1(k_1) = A_1 k_1^\alpha$ and $F_2(k_2) = A_2 k_2^\alpha$. You will want to split up capital between the two firms such that marginal returns equalize: $k_1, k_2 : F_1'(k_1) = F_2'(k_2)$. Due to DRS, this necessarily means that both $k_1, k_2 > 0$. I leave the actual derivation of the aggregate production function as an exercise.


The New Palgrave Dictionary of Economics Article on Adjustment Costs gives a nice overview of the use of adjustment costs in economic models.

What is the purpose of scaling investment costs with K here?

This function form implies that adjustment costs of size X% of capital $I/k$ cost the same fraction of capital ($c(I/k)$) regardless of the amount of capital($k$) involved. It is a constant returns to scale form of adjustment cost.

It seems that this gives the property that marginal cost of investment decreases in existing capital stock.

I think you have the marginal cost of adjustment wrong here.

$$ \frac{\partial}{\partial K} K \cdot c\left(\frac{I}{K}\right)$$
$$ = c(\frac{I}{K})  + K \cdot c'(\frac{I}{K})$$

The first term is decreasing in $K$ for fixed $I$ but the second need not be.


How to approximate stochastic volatility model with finite-state Markov chain?

You could do a binomial-tree approximation to the process for $z_t$ and then have a different process control the number of steps you take on the tree. This preserves recombining property and it is essentially the method explored in On the Computation of  Continuous  Time Option Prices Using Discrete Approximations (Amin (1991))

  We  develop  a class of  discrete, path-independent  models to
  compute prices  of American options  within the Black-Scholes  (1973)
  framework,  including  models in which state variables have
  time-varying volatility  functions and models with  multiple  state
  variables. Time-varying volatility  functions are illustrated with
  applications  to term structure models developed by  Vasicek  (1977)
  and  Heath, Jarrow,  and Morton  (1988), (1990).  Distinct from
  previous  work in the  literature,  the multivariate models suggested
  in this  paper are consistent with  arbitrarily large, though
  constant,  covariance functions. Finally,  we compare  and contrast
  the numerical  accuracy  of a  large  number of models with simulation
  results.

You could then use a binomial AR(1) process for the number of steps.


Computing pure strategy Nash equilibria in finite games

If $10^{10}$ pure strategies is too large for Gambit, it'll likely be too large for any other software as well. In the comments, I suggested that you could first compute the expected payoffs for each player in a separate program such as Matlab/R/Excel, and then export the resulting matrix to Gambit where the BNEs can be calculated. This way, you may be able to save some resources for Gambit on converting the game into its Bayesian normal form.

If you're okay with significantly reducing the dimension of your problem, then you can rely solely on Gambit (or, Game Theory Explorer (GTE), which is now a part of the Gambit project). In the following example done in GTE, I assumed that bidder $i$'s valuation and strategy spaces are binary: $V_i=S_i=\{0,1\}$ for $i=1,2$. The values are assumed to be uniformly distributed.

First, you enter the game in its extensive form. In the figure below, the four branches at the top correspond to Nature's choices that determine the valuations. From left to right, the branches represent the valuation profiles $(v_1,v_2)=(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$ respectively, each occurring with probability $1/4$.

After you finish entering the extensive form game, press the "Matrix Layout" tab to see the (automatically calculated) Bayesian normal form. There's a gear button at the top that will compute the BNE of the game.

The result looks like the following. In this case, the result was returned in seconds (in part thanks to the relatively small dimension of the problem). The unique BNE, in which each player bids zero regardless of their value, was found.

You can of course try to increase the size of the value/strategy space. This method would also allow you to specify asymmetric distributions of bidder values. However, GTE only solves two-player games. But Gambit seems to be able to solve games with more than two players.


How did the Price/Value of any goods measure in money determined in the first place

No prices exist even without money when you barter. Early money was just some sought-after commodity that was originally bartered (like grain/livestock/metals), but people just figured that such commodity can be used in all transactions to avoid double coincidence of wants.  Simple forms of money precede countries, as we understand them today.
As the first ancient states developed this just got better organized. Coins developed from state stamping pieces of metal to certify its quality. A more complex form of money such as Mesopotamian clay tablets that were exchangeable for barley at harvest time developed from the government starting to enforce contracts. At that point, the value of money started to be also underpinned by the fact that the government required taxes to be paid in its money which created a demand for it.
So at the point when money was created, all goods and services already had prices in terms of money. Relative prices exist irrespective of whether money exists, and early money was just some commodity that already had all relative prices to other commodities. Once it was selected as money just other values started to be expressed per unit of the selected commodity as well. You can have look at The Ascent of Money by Ferguson, it is a nice overview of the history of money from ancient times to the present day.


Does neoclassical economics explain or assume private property rights?

It is true that mainstream economics (which includes neoclasicall economics) is empirical but note that any explanation can  only arise from theory (whether theory is explicitly formulated or only implicit in the background), and any theory is based upon assumptions. Empiricism does not mean you don't build theoretical models with assumptions, it means you go and test those theoretical models rather than just rely on deductive reasoning alone.
For example, theory of relativity is explanation of gravitational attraction deduced from some assumed premises. This theory can be tested by observing whether light bends around massive object. However, note it does not work the other way around. Observing that light bends around massive objects does not provide any explanation for gravitational attraction - on its own it is just a empirical curiosity. This is not to say that one might not first observe something and then develop theory, but point is that explanations themselves can't be empirical.
That out of the way these are common explanations for property rights in mainstream economics with references that also contain empirical examples:

Private Property Solves the Tragedy of Commons: The tragedy of the commons is a situation that arises when goods/resources are commonly owned where individuals, acting independently behave contrary to the 'common good' by depleting the shared resource through their collective action. This happens because if everyone owns let's say fish in the lake, then everyone has incentive to extract those fish as soon as possible otherwise someone else will do so. This is an example of prisoner dilemma type situation. Hence, excessive pollution, overfishing, deforestation and other such examples are observed virtually  every time there is a lack of well defined property rights. Property rights (where possible to apply) solve this issue because if one person has right over some resource they have an incentive to exploit it sustainably  (see Mankiw Principles of Economics pp 218-223).  This is why ivory trade makes elephants almost extinct (as they are 'commons' in many places), while caws are not endangered even if they are much more intensely used for food or skin or why rainforests are disappearing while privately owned forests are managed sustainably.
Private Property Encourages Investment: This was already correctly mentioned in your answer. Some altruistic preferences/utility aside people will invest less in something when they will not reap all the benefits. This is why it is very uncommon to see people making large investment in leased car or property. If you rent a house why would you renovate it just so later on when your lease expires someone else is going to enjoy the investment.
This is even more important in context of investing in factors of
production like capital goods. This is crucial for economic development. Economic growth arises from investment (see Barro & Sala-i-Martin: Economic Growth 2nd ed). Empirically the above two are very important as there are virtually no societies that would achieve sustained economic development without property rights and other inclusive institutions, while a common factor among societies that remain  poor lack secure property rights (see Acemoglu & Robinson: Why Nations Fail? and sources cited therein).
Private Property Rights Encourage Productivity: Private property rights provide incentive for production and encourage productivity. For example, if you are a farmer but there are no property rights so anyone can just walk on your farm and take any produce they want this destroys any incentive for people to produce anything beyond immediate consumption (again see Acemoglu & Robinson: Why Nations Fail? and sources cited therein).
Private Property Rights are Prerequisites for Well Functioning Markets: This is connected to the two point above, but I list it as a separate reason as it goes little bit further. Lack of well defined and secured property rights is actually considered market failure and in fact government providing well defined and secure property right is considered to be necessary intervention to make markets work. As argued by Stiglitz (Economics of the Public Sector pp28):

An important activity of the government, but one that accounts for very little expenditure, is the establishment of the legal framework within which firms and individuals can engage in economic interactions. ... Without laws defining property rights, only the exercise of force would stop one individual from stealing from another. Washout the ability to protect property, individuals would have little incentive to accumulate assets. Needless to say, economic activities would be severely restricted.

Later on pp 77:

But even for markets to work, there needs to be a government to define property rights and enforce contract.

In fact Mueller in Public Choice III pp 10 argues enforcement of property rights is one of the reasons for people developing 'constitutional contracts' and moving beyond state of Hobbesian anarchy toward having some sort of organized state.

Private Property Reduces Transaction Costs: This was again already mentioned in your question and it is also one of the reasons you see in the literature. Often property rights can be used to provide better incentives for people to behave in certain way. For example, a firm my require employee to own certain asset (e.g. car) rather than provide it to ensure that its taken good care, which might be cheaper than having some monitoring in place  (see  Williamson The Economics of Transaction Costs and sources cited therein).

There might be other explanations,  that I have missed and hopefully someone other users can fill those gaps, but I think I listed all the major ones.


Reliable US yearly population and netmigration estimates by state from 1970 up to today

Census State to State Migration Flows are available from 2004 - 2016. Census's State Intercensal Tables: 2000-2010 has annual state population data over the 2000-2010 period. Combining this with Census's State Population Totals and Components of Change: 2010-2017 gives population series for back to 2004.


Why wouldn't competition prevent "usurious" payday loan rates?

The posted quote is economic nonsense.

  If a lender chooses to innovate and reduce cost to borrowers in order to secure a larger share of the market, the competing lenders will instantly do the same, negating the effect.

This applies to any industry without intellectual property protection -- it is hardly unique to the payday loan industry. By this logic, we'd expect to encounter massive price gouging across dozens of industries.

Besides, if payday loan innovation is in the form of software that better predicts default (the most likely path), it will be protected under copyright law and potentially software patents. And while business model innovations are not patentable, there's still a first mover advantage.

The payday loan business is not highly profitable.

Profit margins of payday-loan corporations are publicly available, and lower than most other industries. One study found that "despite the common belief, payday lending firms do not always make extraordinary profits. In fact, when compared to many other well-known lending institutions, payday lenders may fall far short in terms of profitability."

This is not surprsing, since the payday loan market is highly saturated, which suggests substantial competition.

"Usurious" APR rates are misleading.

A typical payday loan charges \$17 for a two-week \$100 loan.
Expressed as an annualized rate, this is an "outrageous" 390% APR.
But the loan's short-term nature means transaction costs will likely prevent a large profit.
(The source of this information is potentially biased. Be sure to read critically.)


Partial differential equations in economics

Here are a couple of suggestions. There are some PDEs in some recent continuous-time models, for instance in:

A Continuous-Time Version of the Principal-Agent Problem, by Sannikov, Review of Economic Studies (2008)
Persistent private information, by Williams, Econometrica (2011)

I would not be surprised if Yuliy Sannikov used PDEs in some of his other papers (although I have nothing precise in mind).

You can refer to the review by Achdou et al. (Partial differential equation models in macroeconomics, Philosophical Transactions of the Royal Society, 2014) for other references.


What is the mechanism that triggers a stock price change?

I won’t discuss the fundamental reasons why stock prices change (discussed in another answer), but the mechanics (roughly) work like this. (Real world is more complex, since there are multiple exchanges, and high frequency trading.)
An exchange matches orders from buyers and sellers. The sensible way of making an order is to put a limit price on it. So you either make a bid up to a maximum price, or sell at a minimum.

If your order cannot be matched to an existing order, it is added to the queue of orders. There is a list of bids (buy orders) and offers (sell orders), which are ordered by price. E.g., if the highest bid is to buy at \$90, a bid at \$100 is better and is added to the front of the queue. (If the order price matches an existing price, the orders are processed first-in, first-out.) No transaction has happened, so there is no recorded stock price change. (Exchanges report the best bid/offer, which might change.)
If your order can be matched (either you pay as much as someone is willing to sell at, or sell at a price people are willing to pay), you buy/sell at the prices specified by the existing orders. E.g., if there were orders to buy 100 shares each at \$100 and \$90, and you are willing to sell 200 at \$90, you first sell at 100, then 90. The pricing history will note the transactions, and the price drops.

Other orders are “market orders,” where you buy/sell at the best offer/bid. In the era of high frequency trading - where the prices move extremely fast - this is surprisingly risky. A market order can be considered to be a bid with a limit of infinity (!) or a sell at 0 (which explains the risk, if orders can jump extremely rapidly). In a market where most orders are market orders, they will account for most of the transactions - limits are set not to trigger a transaction, rather they wait for a market order.
One thing that is often not appreciated is that professional traders will continuously monitor their open orders. They will remove them and add them back at new prices in response to news. This means that the prices can jump without any buying or selling: people can adjust prices without there being any transactions. This effect means that it is safest to think about prices as being set based on traders’ views, and not some mechanical supply and demand effect based on buying and selling flow numbers.


Cobb Douglas relation with uncompensated law of demand

Cobb-Douglas
Short answer
The Cobb-Douglass utility function is additively separable (take logs). Additively separable utility functions have normal goods (i.e. demand increases with income). Normal goods have negative own price effects, so they satisfy the law of demand.
Long answer
Consider the Cobb-Douglas utility function $u(x_1, \ldots x_n) = \prod_{i = 1}^n (x_i)^{\alpha_i}$ with $\alpha_i \ge 0$ and assume wlog that $\sum_i \alpha_i = 1$.
Then demand for good $i$ is given by:
$$
x_i = \frac{\alpha_i y}{p_i}
$$
Take two price vectors $p_1 = \begin{bmatrix} p_{1,i}\\ \vdots \\ p_{1,n} \end{bmatrix}$ and $p_2 = \begin{bmatrix} p_{2,i} \\ \vdots \\ p_{2,n}\end{bmatrix}$ with demands $x_1$ and $x_2$. We want to show that $(p_1 - p_2)' (x_1 - x_2) \le 0$.
We have:
$$
\begin{align*}
(p_1 - p_2)'(x_1 - x_2) &= p_1' x_1 - p_1' x_2 - p_2' x_1 + p_2' x_2,\\
&= y - \sum\frac{p_{1,i}}{p_{2,i}}\alpha_i y - \sum \frac{p_{2,i}}{p_{1,i}} \alpha_i y + y,\\
&= 2 y - y \sum_i \alpha_i\left(\frac{p_{1,i}}{p_{2,i}} + \frac{p_{2,i}}{p_{1,i}}\right),\\
&= 2 y - y \sum_i \alpha_i \frac{(p_{1,i})^2 + (p_{2,i})^2}{p_{1,i}p_{2,i}},\\
&= 2 y - y \sum_i \alpha_i \frac{(p_{1,i} - p_{2,i})^2 + 2 p_{1,i}p_{2,i}}{p_{1,i}p_{2,i}},\\
&= 2 y - y \sum_i \alpha_i \left(\frac{(p_{1,i} - p_{2,i})^2}{p_{1,i}p_{2,i}} + 2\right),\\
&= 2 y - 2 y - y \sum_i \alpha_i \frac{(p_{1,i} - p_{2,i})^2}{p_{1,i}p_{2,i}} \le 0
\end{align*}
$$
Homothetic preferences
Demand functions from homothetic preferences are linear in income:
$$
x(p,m) = \alpha(p) m.
$$
where $\alpha(p)$ is the unit income demand. i.e.:
$$
x(p,1) = \alpha(p) \ge 0.
$$
Then:
$$
\frac{\partial x(p,m)}{\partial m} = \alpha(p) \ge 0,
$$
so homothetic preferences generate normal demands. This means that their won price effect is negative, so they also satisfy the law of demand.


Το take this one out of the Unanswered queue:

The completeness property of the preference relation, implies that for all non-negative $t$ we will be able to form and declare the preference relation. Hence

$$A \cup B = \mathbb{R_+}$$

By monotonicity we have $B=[0, \overline{t}],\;\; A=[\underline{t}, \infty)$.

Ad absurdum, assume that $\overline{t} < \underline{t}$. Then there exists an open interval $(\overline{t},\underline{t})$ that does not belong to the union of $A$ and $B$. But then we arrive at $A \cup B \neq \mathbb{R_+}$ which contradicts the implications of the completeness property.

So we conclude that $\overline{t} \geq \underline{t}$, which implies that the intersection $A \cap B$ is non-empty (even if it may contain a single element, when $\overline{t} = \underline{t}$).


Faulty reasoning in two person card game problem

Suppose Mike plays a randomised strategy of playing "0" with probability $\dfrac58$ and playing "1" with probability $\dfrac38$

Then @Herr K.'s table becomes (in expectation)
$$\begin{array}{|c|c|c|}\hline&\text{'0'}&\text{'1'}\\\hline\text{Mike's }\frac58,\frac38 \text{ combo}&\frac18,-\frac18&\frac18,-\frac18\\\hline\end{array}$$

and there is nothing Cheng can do to get a better result than an expected loss of $\frac18$.  So in that sense the game is advantageous to Mike if Mike takes this approach

Incidentally, here Cheng can restrict expected losses to this amount by playing the same mixed strategy, and both players doing so is a Nash equilibrium


Rampini elaborates on your idea (and also the Pratchett quote) in his AER article "Financing Durable Assets". See the abstract:

  This  paper  studies  how  the  durability  of  assets  affects    financing.  We  show  that  more  durable  assets  require  larger  down  payments  making them harder to finance, because durability affects the price of assets and hence the overall financing need more than their  collateral value.  Durability  affects  technology  adoption,  the  choice  between  new and used capital, and the rent versus buy decision. Constrained firms invest in less durable assets and buy used assets. More  durable assets  are  more  likely  to  be  rented.  Economies  with  weak  legal  enforcement invest more in less durable, otherwise dominated assets and are net importers of used assets.

Similarly, Orhun and Palazzolo study this argument in "Frugality Is Hard to Afford" in the context of bulk buying large packets at low prices vs. only being able to buy small packets.

  Intertemporal savings strategies, such as bulk buying or accelerating purchase timing to take advantage of a good deal, provide long-term savings in exchange for an increase in immediate spending. Although households with limited financial resources stand to benefit the most from these strategies, they are less likely to make use of them. The authors provide causal evidence that liquidity constraints impede low-income households’ ability to use these strategies, above and beyond the impact of other constraints. Exploiting recurring variation in household liquidity, this study shows that when low-income households have more liquidity, they partially catch up to higher-income households’ ability to use intertemporal savings strategies. The findings provide guidance to marketing managers and researchers regarding targeted promotional design and measurement of deal-proneness. For policy makers, they suggest a new path for decreasing the higher prices low-income households have been documented to pay for everyday goods. Policies have traditionally focused on increasing financial literacy or access to supermarkets. Our work suggests that providing greater liquidity can help low-income households make better use of savings opportunities already available to them.


Clarification on the definition of Bayes Correlated Equilibrium

I think the reason you're having difficulty is that your definition is actually not equivalent to Bergemann and Morris' definition of BCE, except under specific assumptions which include the ones you state regarding independence. However, in general, we do not wish to make these assumptions. (For example, assuming that $t_{-i}$ is independent of $a_i$ conditional on $\theta$ and $t_i$ means that the mediator cannot reveal any additional information about other players' types once they have revealed the state to a player. This seems unnecessarily restrictive.)

In the paper, a BCE induces a probability distribution $\Pr$ that satisfies

$$ \sum_{a_{-i},t_{-i},\theta}\Pr(a_i,a_{-i},t_i,t_{-i},\theta) u_i(a_i,a_{-i},\theta) \ge \sum_{a_{-i},t_{-i},\theta} \Pr(a_i,a_{-i},t_i,t_{-i},\theta) u_i(a^\prime_i,a_{-i},\theta) $$

for all $a_i^\prime \in A_i$. If we wish to express this inequality in terms of player $i$'s expected utility given their information, then we divide both sides by $\Pr(a_i,t_i)$. Similarly, from the inequality

$$ \sum_{a_{-i},t_{-i},\theta}\Pr(a_{-i},t_{-i},\theta \vert a_i,t_i) u_i(a_i,a_{-i},\theta) \ge \sum_{a_{-i},t_{-i},\theta} \Pr(a_{-i},t_{-i},\theta \vert a_i, t_i) u_i(a^\prime_i,a_{-i},\theta), $$

we can multiply both sides by $\Pr(a_i,t_i)$ to recover the first expression above. To see how this squares with your calculation, notice that

$$ \begin{align*} \Pr(a_{-i}\vert a_i,t_i,t_{-i},\theta) \Pr(t_{-i}\vert a_i,t_i,\theta)\Pr(\theta\vert a_i,t_i) \Pr(a_i,t_i) &= \Pr(a_{-i},t_{-i},\theta \vert a_i, t_i) \Pr(a_i,t_i) \\
&= \Pr(a_i,a_{-i},t_i,t_{-i},\theta) \\
&= \Pr(a_i,a_{-i}\vert t_i,t_{-i},\theta) \Pr(t_i,t_{-i} \vert \theta) \Pr(\theta)\\
&= \sigma(a_i,a_{-i}\vert t_i,t_{-i},\theta) \pi(t_i,t_{-i} \vert \theta) \psi(\theta).
\end{align*} $$

In particular, we have that $$ \Pr(a_{-i}\vert a_i,t_i,t_{-i},\theta) \Pr(t_{-i}\vert a_i,t_i,\theta)\Pr(\theta\vert a_i,t_i) = \frac{\sigma(a_i,a_{-i}\vert t_i,t_{-i},\theta) \pi(t_i,t_{-i} \vert \theta) \psi(\theta)}{\Pr(a_i,t_i)}. $$ This makes it difficult to get the decomposition you want without making needlessly strong independence assumptions.

When $N=1$, there is only one player, so the equilibrium is (in some sense) degenerate. However, the difference between one-player BCE and one-player BNE is that the BCE set is larger, and includes any BNE. (Indeed, this is true even when $N>1$.)

For your second question:

  a BCE is still intended as mapping from state and signal to a probability distribution over actions. From reading at p.26 of the linked paper, the authors then say "[...] In that case, the set of BCE correspond to joint distribution of actions and states [...]". I'm confused.

Formally, a BCE is a probability distribution of actions for each realisation of the state and signals. This probability distribution induces a joint distribution over actions and states. Just integrate out $(t_i,t_{-i})$ in $\Pr(a_i,a_{-i},t_i,t_{-i},\theta)$ to get that joint distribution.

Finally, this equilibrium concept is called Bayes Correlated Equilibirum because it extends our definition of correlated equilibrium (defined by Aumann for complete information games) to games of incomplete information, in the same way that Bayes-Nash equilibrium extends Nash equilibrium.


Is Buy Nothing Day a classist protest of Black Friday?

in relation to things people would be buying anyway, doesn't boycotting Black Friday just amount to paying full price (or finding other sales) at some other time of the year versus taking advantage of these sales?

Yes.  Regarding necessities or scheduled expenditures, participating in BND requires a time-shift in consumption that may carry a higher price for the same goods or services, in addition to other potential non-pecuniary opportunity costs.  I suppose in the extreme this could lead to hoarding of perishables, fuel, etc. that might be measurable.

  And thus wouldn't that be something that people of lower economic status would be excluded from participating in, as they wouldn't necessarily have the luxury of choosing to pay more just to make a statement?

Yes.  If BND were enshrined in policy, it would be considered regressive in the sense that its negative welfare impacts are likely to be higher for those of lesser means.  It may also constitute a redistribution of wealth from BND adherents to business owners (although this would be complicated to show empirically).


Is there a book or notes that explain all about inflation?

If you are looking something for student level the best place to start are textbooks. You will not generally find these just for single topic such as inflation (for that you would have to look at research papers but you explicitly state you are not looking for that) but they offer rigorous treatment with references to the literature.
Mankiw Macroeconomics Chapter 5 gives nice undergraduate introduction to  inflation, for more full undergraduate treatment you can have look at Lewis and  Mizen Monetary Economics.
Walsch Monetary Theory and Policy offers graduate level treatment of many models including inflation. In fact virtually all models there involve inflation in one way or another.
You could supplement Walsch also with Romer Advanced Macroeconomics or with Woodford Interest and Prices.


How to estimate "necessary" economic output using Leontief input-output impact analysis? In particular: how to handle imports

If you want to model exports and imports you need  interregional (or multiregional) Leontief input–output model instead of standard one.
The way how these models work is that you will have separate foreign sector in your input output model, along what you can see in the table below (taken from Hewings, G. J., & Jensen, R. C., 1987):

This is in spirit similar to assuming that foreign are just some other production sector - but the issue is that spending and productivity might not necessary be the same as at home. There is also more nuance to it. I would recommend going over the above mentioned Hewings, & Jensen (1987) which has all the details on how input-output analysis is different for open economy.


$P=\frac{\epsilon}{\epsilon +1} MC$

This is the markup rule for price-setting firms.

Now, it is clear that a firm with monopoly power will never set $P=MC$. However, we can assume for now that the term involving epsilon is fixed for our firm with monopoly power.

If this firm adopts technology that reduces costs, what do you think will happen to the marginal cost curve (in what direction does the marginal cost term move in the above equation)?

Given the expected change in marginal cost, in what direction should price change?


Question about skill-biased technical change, Acemoglu

I won't replicate the model -interested readers should read the link the OP provides.

Acemoglu's argument as to when a high relative supply of skilled labor will lead to skill-biased technological change is laid out in page 13 (below eq. $(18)$), it includes the all-important condition eq. $(19)$ and ends in the middle of page 14. Here the author explains what must hold in order to observe this phenomenon.

Then at the the bottom of page 18 the author writes

  "Moreover, recall that when condition ($19$) is satisfied, equation $(18)$ traces an upward-sloping long-run relative demand curve for
  skills. Therefore, the induced skill bias of new technologies can be
  sufficiently pronounced that the skill premium may increase in
  response to a large increase in the supply of skills."

So here, the author simply invokes the necessary theoretical condition obtained previously.
So what is this condition? It reads

$$\sigma > 2- \delta \tag{19}$$

where $\delta$ is an index raging in $[0,1]$ (and serving some purpose) and  $\sigma$ is the elasticity of substitution between skilled and unskilled labor.

So we see that in order to obtain skill-biased technological change together with high relative supply of skilled labor in the model (as appears to be the historical experience), namely, in order for condition $(19)$ to hold, we must have, as a necessary but not sufficient condition $\sigma >1$ according to this model.

Now, at the bottom of page 9, we see

$$\sigma \equiv \varepsilon - (\varepsilon-1)(1-\beta), \;\;\; \beta \in (0,1)$$

and where $\varepsilon$ is the elasticity of substitution of the two goods that are combined in private consumption (though not directly in the utility function as the author writes somewhere -see eq. $(6)$).

So for our necessary condition $\sigma > 1$ we must have

$$\sigma > 1 \implies \varepsilon - (\varepsilon-1)(1-\beta) > 1$$

$$\implies (\varepsilon-1) > (\varepsilon-1)(1-\beta)$$

which will hold iff $\varepsilon >1$.

So the real-world rationalization is shifted

from "is there high substitutability between skilled and unskilled labor"

to "is there high substitutability in final product/consumption between intermediate products made using skilled labor and intermediate products made using unskilled labor"?

...because if there is such high substitutability at this "second" stage, then indirectly, skilled and unskilled labor acquire a high degree of subsitutability, even though looking directly at a single production function, it would appear reasonable that they should not be highly substitutable.
In other words, here "substitutability" emerges as an even more economic, rather than mainly an engineering/technical concept.

I will leave to the OP the task of contemplating further.

ADDENDUM

Responding to a comment by the OP, in the specific model, "intermediate goods" are combined at the consumption level. In real life of course, in almost all products, skilled and unskilled labor are combined to reach the ready-for-consumption good. For example consumers consume services of a building as a whole, not design services and/or brick-laying services separately.

One way to rationalize this is by thinking in quality terms: degree of substitution between buildings made by mediocre craftsmen alone ("unskilled" labor), against buildings made using experienced builders and engineers. Since the substitution degree is a preference parameter, not taking into account prices, one could reasonably argue that the degree of substitution between similar products of different quality can be high (again, as a tendency coming from preferences, the actual consumption decisions will also depend on prices of course).


They are related but they are not same.

An interest rate is broad term for what you pay for the use of assets as a percentage of the value of the asset or principal. See for example definition here.

A discount rate can be the interest rate that Fed charges at its discount window, or it is often in theory used more broadly as the rate at which people discount future cash flows. The latter is also in a sense an interest rate but not interest rate charged by any institution but the one that person would expect to receive to  be compensated for lending/investing their resources and postponed consumption.


Pure-Strategy Bayesian Nash equilibrium with general common prior

Matrix looks correct. To final all pure strategy BNEs, you'll have to discuss cases based on the value of $p$.
For example, if $p\in(0,1)$, then $FT$ is player 2's unique best response to $F$. Thus, to have a BNE, you'd want $F$ to be player 1's best response to $FT$ as well, meaning that you'd require $3p>1-p$, or $p>\frac14$. Hence, $(F,FT)$ is a BNE if $p\in(\frac14,1)$.
You should be able to find other BNEs following a similar line of reasoning. (Hint: don't forget the edge cases where $p=0$ and $p=1$.)


Its because the student did not produce any value added in the production of the book. She used the book and then basically transferred it to the bookstore. On the other hand, the bookstore may repair the book and sell it for a \$10 profit.  (the bookstore may be advertising, displaying, and other services). The \$10 reflects this additional value added in the economy in the current year, and is thus added to the current GDP.


Is real income the same as present value of income?

Real income refers to the income of an individual or group after taking into consideration the effects of inflation on purchasing power. For example, if you receive a 2% salary increase over the previous year and inflation for the year is 1%, then your real income only increases by 1%. Conversely, if you receive a 2% raise in salary and inflation is at 3%, then your real income shrinks by 1%.

Net present value is the value today of a stream of income over time - it is influenced primarily by interest rates.

There is some relationship between the terms - because they both discuss the value of money in relationship to to time, but real income discusses the difference of buying power over time due to inflation and NPV focusses on the theoretical equivalence of money today versus a stream of money over a period of time - largely influenced by the risk of receiving that income and the available alternatives of investment during the time period.

Investopedia definition of Real Income


Forward Guidance - What happens when the FED unloads its balance sheet

The direct effect of changes in the money supply on the economy (and other variables) is not clear. The Fed purchases of assets created excess reserves in the banking system (which the banks cannot offset by themselves). The presence of excess reserves does not greatly affect the ability of banks to lend. Hence, there is limited evidence of direct effects on the real economy. (I am unaware of any such studies, but that does not mean that they do not exist.)

The usual argument is that the effect of the large scale asset purchase program was that it reduced bond and mortgage yields. This note by Bonis, Ihrig, and Wei (link to note) gives a recent estimate of its effect. The effectiveness of the program to lower bond yields is questioned. (As a disclaimer, I see little evidence that bond yields are measurably lower as a result of the asset purchases.)

If we grant that the large scale asset purchases lowered bond (and mortgage) yields, the effects on the economy are indirect. The precise impact of lower bond yields on the economy is uncertain (it depends upon which econometric studies on the effects of bond yields on economic activity you believe). Presumably, the effect of a gradual unwind of the Fed's purchases would reverse those effects.

Returning to your question, the linkages to the areas you ask about (the value of the U.S. dollar, bubbles, etc.) would be even more controversial. For example, in my opinion, the Canadian real estate bubble was entirely driven by Canadian policy changes; it is very hard to see how the size of the Fed's balance sheet had a measurable effect. I am certainly unaware of any studies finding such a linkage, and there is no reliable rule for relating changes to American bond yields to Canadian bond yields (and mortgage rates). (If such a rule existed, it would imply that it was easy to make money on cross-currency spread trades.)

Obviously, very rapid sales of securities by the Federal Reserve would be disruptive for the bond market, which presumably would cause contagion in other financial asset markets.


Are there multiple equilibria in the second price auction?

Sure. An example: if both valuations are drawn from the $[0,1]$ interval then the strategies
$$
b_1(v_1) = v_1
$$
and
$$
b_2(v_2) =
\left\{\begin{array}{cc}
v_2 & \text{ if } v_2 < 1 \\
5 & \text{ if } v_2 = 1.
\end{array}\right.
$$
Another, slightly more annoying equilibrium for $v_1,v_2 \in [0,1]$:
$$
\begin{align*}
b_1(v_1) & = 0 \\
b_2(v_2) & = 2.
\end{align*}
$$

EDIT: The truth-telling bidding strategies are weakly dominant strategies. If we only allow trembling hand perfect equilibria, players assign positive probabilities (in this case densities) to all truncated strategy profiles. In such a setting, the weakly dominant strategies become strictly dominant. (Because of sequential rationality even zero measure deviations such as my first example are eliminated.) In a game where all players have strictly dominant strategies, there is only one equilibrium.


Statistical tests/models to use for Macroeconomic and Financial Research

One of the reasons why there are so many different models is that "having impact" is not that easy to define. It is not to test whether a variable has "any" effect on another variable, so researchers first have to come up with a model (based on theoretical considerations) that they can estimate and test. And of course the tests are different depending on the model they assume.

Basic model

One of the most basic (but powerful and widely used) concepts for these type of hypotheses is Granger-causality. Simply put, variable $A_t$ is said to Granger-cause variable $B_t$, if $A_t$ is useful in forecasting $B_t$. So you might try to find out which macro variables Granger-cause your indices. However to do formal tests you need to restrict the previous definition a bit.

Let us say, that variable $A_t$ Granger-causes variable $B_t$ if
$$E[B_t|B_{t-1}, B_{t-2}, ...] \neq [B_t|A_{t-1}, B_{t-1}, A_{t-2}, B_{t-2}, ...]$$
or in words, knowing the past values of $A_t$ helps us get a better prediction of the current value of $B_t$. The nice thing is, that the above hypothesis is easy to test: you just have to fit a linear model.

So if your index variable is $Y_i$, and your explanatory variables are $\{X_{1t}, \dots X_{mt}\}$, then you just have to estimate the simple linear model
$$ Y_t = \alpha + \sum_{s=1}^S \gamma_s Y_{t-s} \sum_{k=1}^m \beta_{ks} X_{k, t-s} + \epsilon_t $$
by OLS. Then $X_k$ Granger-causes $Y$ (in the expectation sense), if $\beta_{k1}, \dots \beta_{kS}$ are jointly significant.

Practically to test which explanatory variables Granger-cause $Y_t$ you have to do the following:

Run a linear regression with $Y_t$ as the dependent variable and some lags of $Y_t$ and the explanatory variables at the right hand side
Test whether all the lags of a given explanatory variable are jointly significant
Repeat it for each of your indices

By estimating these models you also get an answer to your second question (you just have to look at the coefficients).

Another way to go would be to estimate a VAR model where you include all of your indices and explanatory variables, and then see whether the lags of a given explanatory variable for a given index are jointly significant. This will have a slightly different interpretation: you would test whether your explanatory variables Granger-cause your indices even if you condition on all the indices. In your case I would be careful with it due to the relatively low number of observations.

Practical concerns

Deciding how many lags to include in the regression might be tricky. It depends on the amount of data you have and the model you have in mind. For quarterly data, you should probably include at least 4 lags.
As have panel data you could be inclined to include a country dummy to get fixed effects estimates. However because you have lagged dependent variables on the right hand side, your estimates would be biased if you did this. There are ways around it, but they are quite advanced.

If you have questions how to implement it in practice feel free to ask for details.


If strict convexity of indifference curves isn't assumed, does MRS have to be negative?

The marginal rate of substitution is the slope of the indifference curve multiplied by -1. Since the indifference curve is negatively sloped (as soon as the person prefers to consume the goods), the MRS is always positive. This has nothing to do with the convexity of the preferences.

If the indifference curve is convex without being strictly convex, then it is linear, i.e. it has constant slope. In that case, the marginal rate of substitution is constant (but still positive!).


The bank runs were not caused by the crash; they were both caused by the same phenomena.  In the run-up to the crash banks around the United States sent requests to city banks and the reserve banks for gold.  In part due to the poor information collection system at the time, there was no way for the Fed to distinguish between a heating up economy and a true shortage of gold.

The Fed interpreted the requests for gold as a heating up economy and so contracted the money supply by 3%.  This exacerbated the shortage of gold around the United States.  It quickly propagated through the world financial system because of the shared gold standard.  The market crashed because banks were issuing margin loans at up to 90% of value.  The older system in use, before the Fed, was to create a banking suspension where banks would collect gold, but not disburse gold.  The Fed made that functionally impossible forcing runs.

The thing to remember about stocks is that they are purchased with your spare money.  Given a choice of eating food or buying shares in Apple, people will choose food.  When the money supply contracted, funds didn't exist for businesses to continue operations, to pay employees, so people sold stocks.  The sales forced margin calls, which triggered more sales, which triggered more margin calls and so forth.

The NY Fed did provide liquidity at the beginning of the crisis, but the Board of Governors were still convinced that the contraction was necessary and forced the end of the NY Fed's liquidity provision.

The stock market was the thermometer, not the cause.  The system had become sick.

To be fair to the Fed, it was itself part of a much larger set of global events without an information base to act on.  It was also true for the Bank of England or the main European banks.  The gold standard propagated economic issues in ways that were not understood at the time.

When World War II ended the nations of the world created a variety of treaty organizations such as the United Nations and the OECD to prevent such a sequence of events happening again.  The problem is that restricting those events from happening means that some large business owners miss out on opportunities and so there is pressure to dissolve or weaken the treaty organizations.  In addition, the major nations sometimes endanger the system.  You can see this very directly through the actions of the Trump administration.

The postwar era institutions really do need reform as the threats have changed, but the political will to protect the world doesn't play well in local political processes.


Factor-augmenting technologies (Acemoglu and Restrepo 2018c)

Chain rule! To make it simple call $\tilde{L} = A_L L$ and $\tilde{K} = A_K K$, so that

$$
Y = F(\tilde{K}, \tilde{L})
$$

so that

$$
\frac{\partial Y}{\partial L} = \frac{\partial \tilde{L}}{\partial L}\frac{\partial F}{\partial \tilde{L}} = A_L F_L(A_K K, A_L L)
$$

the trick here is that you should read $F_L$ as the derivative of $F$ with respect to its second argument.


Can an overall improvement in technology make the owners of one factor worse off?

1x2 model

Consider a mode where production of a single good is given by a constant returns to scale CES production function:

$$Y=A(\alpha L^\rho +(1-\alpha)K^\rho)^{\frac{1}{\rho}}$$

where the elasticity of substitution between the two factors is

$$ \sigma = \frac{1}{1-\rho} $$

It can be shown that the marginal product of labour (equal to the real wage in a competitive economy) is

$$ MP_{L} = A\alpha \left( \frac{Y}{L}\right)^{1-\rho} $$

To understand the consequences of biased technical change on real wages, we need to look at the three components separately.

For simplicity, let's assume a fixed supply of labour, $\bar{L}$. Now, a technical change that improves productivity of both factors, but more that of capital is such that $A$ increases (the neutral component), and $\alpha$ falls (the bias). On top of this, any increase in productivity (regardless of its bias) will increase $Y$. Yet, labour is fixed. Then, you have that the three determinants of the real wage change. In particular, two are increasing and one is falling. The change in the real wage - and the nature of growth (enriching, immiserizing, neutral) depend on how these play-out. In particular, the greater the bias, the smaller the neutral technical change, and the higher the elasticity of substitution (higher $\rho$), the more likely you will have immiserizing growth. If $\rho = 1$, the third term disappears altogether.

2x2 model

You can get similar results by having a 2x2 model (2 goods and 2 inputs). For instance, say that capital productivity increases more than labour productivity. Furthermore, say that the elasticity of consumption among the two goods ($Y_{a}$ and $Y_{b}$) is very high. For example,  individuals are willing to consume a lot of the former and little of the latter. If the production of $Y_{a}$ is very intensive on capital, and the technology has a high degree of substitution between capital and labour (so firms could produce by expanding capital and not much expanding labour), then you will end up with a reduction in aggregate demand for labour. This will lead to a fall in the real wage, and an absolute loss of income for workers (i.e. immiserizing growth).

Thus, in the end you can get any type of behaviour depending on:

technology used in either good, and in particular, the degree of substitution between factors
preferences of consumers, and in particular, the degree of substitution between goods

Think of robots increasing the productivity of capital by much more than that of workers. If consumers are happy to rebalance their consumption bundles toward more robot-produced goods and away from labour-intensive goods, then the wage rate will fall, affecting workers.


Can a progressive marginal taxation be regressive in terms of average tax rate?

$$
\frac{\partial\frac{T(Y)}{Y}}{\partial Y}= \frac{T'(Y)}{Y} - \frac{T(Y)}{Y^2}
$$

This can only be smaller than 0 if

$$ T'(Y) < \frac{T(Y)}{Y} $$

In other words, the marginal tax rate needs to be smaller than the average tax rate. This can't happen because you start with an average tax rate of zero and a positive marginal tax rate. From then on, in a progressive scheme, the tax due increases faster than taxable income. Progressive taxation is actually characterized by the fact that the marginal tax rate exceeds the average one. See here, p.27.


Did Bitcoin increase the money supply of the world?

The concept of “money supply of the world” is vague. I think there’s at least two ways of looking at it.

Economics is normally concerned with the formal definition of “money”. Money is associated with a currency. One key property of money that its value is (nearly) fixed (some broad money definitions allow short-term debt instruments that have a fixed value in the near future). Bitcoin does not have a fixed value in any country’s currency, and will not be part of the money supply. If we somehow aggregate all countries’ money supplies (not obvious how to do), it would not change.
The obvious response to the above is that Bitcoin is a new currency. One could argue that it should be added (somehow) to all the other money supplies. The problem with this is that the domestic money supply figure has some theoretical relationship to transactions in a country. There is no obvious relationship between the supply of Bitcoin to any country’s domestic transactions, nor is there any reason to believe that a “global money supply” figure is meaningful.


The weights used in the U.S. CPI (from 2016) are available here: link to BLS weights. Note that these weights are based on consumption data, so that they are meant to align with actual spending patterns.
If you look at them, you will see that the majority of the weight is either processed food or food away from home. Note that some categories explicitly state “processed,” but even things like salad dressing are put together in a factory.
Factories have a comparative advantage for processing food, but not for delivering hot food. Meanwhile, food is normally shipped in a form that can be stored. A box of pasta can be held in a store a lot longer than pasta that has been boiled and put into a container. Households do the final heating steps. As such, the premise of the question is somewhat flawed - processing is being done largely in an “efficient” fashion (although there are debates about food quality).
Restaurants are required to pay workers and rent and expect to make a profit, whereas food prepared (or just heated up) requires no payment. Cooking is labour intensive, and so it is hard to compete against \$0/hour if we look at the cost of the meal. Individuals might value their time at more than \$0/hour, but how they value their time does not matter if they cannot get more paid work and face a strict monetary budget constraint. Meanwhile, travelling to get a meal takes time. Finally, being reliant on food away from home poses risks, as seen in the pandemic.


A weaker definition of local non-satiation can also imply indifference "curve"

(2) does not imply (1). Consider a utility function with "circular indifference curves", e.g. $u(x,y)=-(x-1)^2-(y-1)^2$. At the bliss point $(1,1)$, the function satisfies (2) but violates (1).

(2) does imply (3), and the proof should be similar to the one showing that (1) implies (3).


How does the money supply grow in the long term?

Central banks often do not create cash.  For example, in the United States, it is the Treasury department that prints cash.

What a central bank does most often that has the effect of creating money is to engage in open market operations.  In an open market operation, it does one of two things:

Sell an asset that it is holding, usually a government bond.
Buy an asset, again usually a government bond.

If it sells, it takes money out of circulation.  Obviously it does this less often, as overall the money supply is growing.  More often, it buys.  This transfers money from its account to someone else's account.  Most of the time, this is a purely electronic transaction.  While the central bank may track its account balance, there's nothing keeping it from being negative.  As it grows more negative, we call this money creation.  But again, this is an electronic transaction.  No actual cash needs to change hands.

Now, someone may go to the bank and ask for the money.  The bank hands that person cash, which it has to replenish.  The bank may ask the central bank to ship it some actual cash.  If the central bank had none, it would have to go to whomever prints the money to get some.

Because of the way that fractional reserve banking works, most money is actually created by the banks.  The way this works is that someone deposits money in the bank.  Let's say it's \$10 and the reserve fraction is 10% (as in the US).  The bank keeps \$1 or deposits it with the central bank as a reserve.  The other \$9 it can loan out to someone else.  That entity either spends it or banks it.  If it spends, then someone else faces the same decision.  Eventually all of that \$9 is deposited.

In the new deposit, the bank (which may be a different bank or the same one) has to reserve \$.90 but can loan out the remaining \$8.10.  But now we have two deposits of the same money.  So there's \$19 of deposits with only \$10 of money.  \$9 was created by the banks.  The process repeats until there is \$10 of reserve and \$90 more (10% reserve fraction).  So for every \$10 created by the central bank, regular banks can create up to \$90 more.

Most of this money will never be exchanged as cash.  People will write checks or use credit cards.  The money will never exist physically.  It only exists as accounting transactions.

It's not even necessary to involve banks.  Let's say I sell you something for \$10.  But you tell me that you have no cash.  I loan you the money implicitly by allowing you to take the item.  I've implicitly created \$10 of money.  When you pay off the loan, you are implicitly destroying that \$10 by closing the loan.

In summary, central banks often don't print any cash.  But regardless of who prints cash, most money in the money supply is never printed.  Most money only ever exists as accounting transactions, and most money is generally created by regular banks rather than the central bank.


You cannot show this because it is not always true. Consider the following situation:

Two consumers (Romeo (R) and Juliet (J))
Two goods (Apples ($A$) and Chocolates ($C$))
Romeo's endowment is $(a = 1, c = 0)$ and Juliet's endowment is $(a = 0, c = 1)$
Preferences are given by $u_R(a_R, c_R, a_J, c_J) = a_R + 2a_J $ and $u_J(a_R, c_R, a_J, c_J) = c_J + 2c_R $

Observe that Romeo only cares about apple consumption and Juliet only cares for chocolate consumption. Even though Romeo likes apples himself but he loves to see Juliet consume his favorite fruit twice as much. Similarly, Juliet prefers to watch Romeo consume the chocolate than consuming it by herself. The only Pareto efficient allocation in this economy occurs when Romeo offers the apple to Juliet and Juliet gives her chocolate to Romeo i.e. $(a_R, c_R, a_J, c_J) = (0, 1, 1, 0)$.

The competitive equilibrium in this economy is defined as $(p_A^*, p_C^*)$ and an allocation $(a_R^*, c_R^*, a_J^*, c_J^*)$ satisfying the property that both consumers choose their best bundle taking prices and other individual's choice as given. Since both can only choose for themselves and not for the other person they'll both end up buying what they like i.e. Romeo will buy only apples and Juliet will buy only chocolates. So endowment allocation $(a_R^*, c_R^*, a_J^*, c_J^*) = (1, 0, 0, 1)$ is the only competitive equilibrium and is supported by any price vector $(p_A^*, p_C^*)$ satisfying $p_A^* > 0, p_C^*> 0$. Clearly it is not efficient.


Refinements in signaling games with three sender types

Both of these criteria refine equilibria where there is an unused message or action. Notice that if there is one action that is not used by any of the types, it can be either because they are all pooling on some other action, or because they are all separating, but there are more actions than types, or anything in between (for example some types are pooling and some other are separating, etc.).

Now, unless the unused action is dominated by all types, for most games you will see that no type is using that action because if they do, they anticipate that the other player(s) will react and choose an outcome that no type would rather have.

The idea behind these two criteria is to realize that perhaps the only reason why the other player(s) react in such a bad way is that they have an (off-equilibrium path) belief about the type of the first player that is un-reasonable.

Because no type is actually choosing that action, the second player's beliefs are not restricted in a perfect Bayesian equilibrium. The intuitive criterion, however, imposes the restriction that the second player should not base her strategy after that unused action based on a belief that she is facing a type that would not possibly benefit from deviating to said action.

In the typical signaling game, this means that an employer should correctly believe that if a worker has sufficiently more years of education than what expected in a pooling equilibrium, it is not the low type worker since he has no incentives to signal their type by acquiring more education.

Similarly, the criterion of Divinity imposes stricter restrictions on off-path beliefs. Not only the second player should not believe she is playing against a type that cannot benefit from deviating to this action, but also it should assign a higher probability in her beliefs to be playing against a type who has a higher incentive to deviate.

Due to these extra restrictions, it should not be surprising that while many separating equilibria survive the first criterion (included the pareto-efficient outcome), with stricter restrictions, only the Riley outcome survives. Of course, the result that the efficient outcome is the one that survives is not obvious at all.

I hope this helps you form an intuition about whether the pooling equilibria you are considering survives these criteria or not. I am not familiar with the model you reference, so I can't help much with that. But I'm happy to give further thoughts if you introduce the model here, or expand on what you have been able to figure out.


There seem to be (at least) two ways of interpreting the concept of a 'data generating process':

The data generating process is a correct description of the causal process
that generates the values of the dependent variable $Y$.
The data generating process specifies the expected value of the dependent variable $Y$ given the values taken the independent variables.

Now the answer to your question seems to depend on which interpretation we take:

If we follow interpretation 1, then these cannot be both data generating processes: either variable $Z$ causally determines $Y$ or it does not.
If we follow interpretation 2, then these can both be valid data generating processes. As you point out, there is no reason to suppose that $\mathbb{E}[Y|X = x]$ is the same as $\mathbb{E}[Y|X = x, Z = z]$.


The Russian economy has been in an recession since 2014, when oil prices plumited. The Russian debt to GDP ratio is still very low though:  below 20%, compared to over a 100 for the United States. This is because Russia had a huge stash of cash before the recession, which has shrunk from over 91 billion in 2014 to 32 billion. If this trend continues at some point they will have to start borrowing money to pay for expenses. But I don't see them going bankrupt anytime soon, because their debt to GDP ratio is so low so investors will lend them money against low interest rates.


Formal definition of perfect information

A simultaneous-move game is not a game of perfect information. It is a game of imperfect information.

Let me quote Gibbons (Chapter 2, p.58).

  The key features of a dynamic game of complete and perfect information are
  that
   (i) the moves occur in sequence,
   (ii) all previous moves are observed before the next move is chosen, and
   (iii) the players' payoffs from each feasible combination of moves are common knowledge.

Picking up the discussion in the comments, let me again quote Gibbons, p.122, Footnote 19:

  This characterization of perfect and imperfect information in terms of singleton
  and nonsingleton information sets is restricted to games of complete information
  because, as we will see in Chapter 4, the extensive-form representation of
  a game with perfect but incomplete information has a nonsingleton information
  set. In this chapter, however, we restrict attention to complete information.


Take a country like Uganda, a net importer. Let's say that the government borrows foreign capital to finance a dam construction. Using the funds, it pays for construction materials, most of which are probably imported; it pays for the design of the dam, the architects and specialists are foreigners; construction companies purchase fixed capital stock (trucks, cranes, etc), most of which are imported, and builders and workers receive wages and they spend most of their income on consumption of imported goods.
Hope this helps.


The answer is in the the first line and title of the articles. Not all debt is government debt. One calculation also includes household and corporate debt.


How to write budget constraint if the agent sell the goods that buy

Agent is a buyer of gasoline in the black market if his consumption of gasoline exceeds 20 i.e. $G > 20$, and he buys $(G-20)$ units from the black market in this case. So, his budget constraint is $R + 20 + 10(G-20)=100$

Likewise, agent is a seller of gasoline in the black market if his consumption of gasoline is below 20 i.e. $G \leq 20$. In this case, he sells $(20-G)$ units in the black market. So, his budget constraint is $R + 20=100 + 10(20-G)$

To conclude, his budget constraint (in all cases) is $R + 20 + 10(G-20)=100$.


Trying to apply in practice the theory of finding the optimal price in a cournot competition

Suppose you are willing to believe that the demand curve has a constant elasticity $\epsilon$. Then the optimal price $P^*$ is given by
$$ P^* = \frac{|\epsilon| \hspace{0.1em}c}{|\epsilon|-1}$$
where $c$ is the marginal cost of producing gold rings (assumed constant). So if your friend can estimate the elasticity $\epsilon$ (e.g. by randomly varying the price and seeing what happens to the number of rings sold), they should be able to estimate the optimal price.
To illustrate, suppose that your friend thinks that increasing the price by 1% will decrease demand by 2%. Then $\epsilon \approx -2$, i.e. $| \epsilon | \approx 2$. The formula then proposes charging $2c/(2 - 1) = 2c$; that is, a 100% markup on the constant marginal cost.
As an aside, note that this formula holds independently of the market structure. So you don't need to worry about whether the market is described, say, by the Cournot model (except insofar as this bears on your estimate of $ \epsilon$).


Math basics for Macro&Micro economics and econometrics

I recommended familiarity with the following topics:

partial differentiation and optimization of multivariate functions
study fixed point theorems. Kakutani and Brower are good ideas.
Set theory is very important
analysis (especially sequences, sub-sequences, convergence of sequences, etc.)
Topology (basics is good enough. For example, understanding the open ball for local non-satiation)
Contraction mapping theorem (For Bellman equations in Macro)
Value functions / Bellman equations
Envelope Theorem
Be very comfortable with calculus (so...do some review if you aren't)
Familiarize yourself with the simplex
basic ideas of function properties like concavity, convexity, quasi-concavity, quasi-convexity, continuous, differentiable, discontinuity etc.
Take a look at point-to-set mappings

Edit:

I'll add for point to set mapping that upper, lower-hemicontinuity and hemicontinuity are necessary as well
Taylor expansions around a point c for multivariate and log functions
glance over methods of log-linearization
This isn't too hard but very important: binary relations over finite and infinite sets. To be a bit more specific - completeness, transitivity etc.
I also recommend being familiar with cyclicity etc.
The separating and supporting hyperplane theorems

This may sound silly, but also make sure you are ridiculously comfortable with algebra. Sometimes I feel like the unending onslaught of algebra is the thing that slows me down the most during exams etc.

Edit: I'd like to recommend a book for math prep that covers almost everything I've listed: The Foundations of Mathematical Economics by Michael Carter

It's a solid book with good exercises and readily available solutions to confirm your answers.

Alright - I tried to give you some specific things rather than provide the broad strokes. I figure it is easy enough to know that you should study a subject and so I've tried to highlight what I remember being the important little bits of math that we use most often.

Good luck with graduate school.


How to calculate the quantity of an incumbent that sets the profit of a challenger equal to 0?

Some steps are a bit unclear to me, but I would say that they are wrong. For example step 1 is wrong because the meaning of this equality is: I will set the q1 that makes the challenger do zero profits once (after) it enters, but the point of this problem is that we have to block it before it enters.

so here my version.

The first thing to do is to calculate the challenger best response function, which you obtain by taking the derivative to respect to q2 (challenger)with given q1(incumbent). You will then obtain the optimal q2 which you will substitute in the challenger profit function, and only now you choose the q1 by setting the challenger profit equal to zero.

intuition: once the challenger enters, he will face a price and a quantity given by the incumbent, by doing the derivatives, he will also choose a quantity that best help him maximize his profit. So we substitute the optimal q2 in his profit function

The incumbent knowing this, and knowing the challenger best response function, will choose a q1 that set the opponent profit function to zero.

Now you may wonder what is the difference with step 1 of your solution.
Notice that in your solution the incumbent doesn't know what quantity the challenger will set ex-ante, but only ex-post, while in my analysis the incumbent knows that the challenger is rational and will set an optimal q2 which depends on the quantity q1, so in this case the incumbent knows ex-ante.


Is there a difference between deadweight loss and welfare loss?

I would say all deadweight loss is welfare loss but not all welfare loss is deadweight loss.

For example an unregulated polluter causing a negative externaly results in a welfare loss compared to the social optimum. There is no 'missing production' and hence no deadweight loss in this case.

Edit: According to Wikipedia, the loss created by externalities is also classified as deadweight loss.


What is a what is meant by a "workhorse" macroeconomic model

It is meant in a way that these models are highly influential and many papers are based on them.

From my own experience it is impossible to attend macroeconomic conference without having couple of papers that are based on models of overlapping generations or on neoclassical growth model.

The reason for that is that the models are quite useful and easily to adapt in lot of different settings. You can use overlapping generations model to examine anything from simple household savings decision to topics that are even outside the realm of macroeconomics such as optimal wealth taxation or environmental policy.

When it comes to the Neoclasical growth theory, for better or worse most of the research in the literature on economic growth in past few decades heavily focused on Solow-Swan model. In mainstream the only competition to Solow-Swan is the endogenous growth model (for which Romer got his Nobel Prize), but due to difficulties in verifying it empirically Solow-Swan is still dominant growth model.

Both of the above mentioned models can be adapted to so many settings that it would not be completely incorrect to think of them more of as a mathematical tools like for example Lagrangian optimization and hence many call them workhorse models since they do most of the work in the mainstream macroeconomics. Essentially, with a little bit of demagoguery you could say that for a decent academic career in macro it is enough to learn how to solve and play with one of the above mentioned models.

For an illustration consider this: In quite popular macro handbook - Advanced Macroeconomics from David Romer, first two chapters are about Solow-Swan and overlapping generations model. By word count almost 1/6 of the handbook is devoted to them, and beside these two models, only other models that have separate chapters devoted to them are the endogenous growth, real business cycle and dynamic stochastic general equilibrium model, other chapters usually squeeze multiple different models in themselves showcasing how important the above mentioned are.


Yes there absolutely is. It is generally agreed upon that utility is (at least) ordinal. That means I can compare utility levels for a single person (not necessarily across people) and the numbers have meaning in this sense. So for a single person:

$U(down, right) = 4 > 1 = U(Up, Left)$.

This means (Down,Right) is strictly better for each agent than (Up, Left). You will notice this satisfies the definition of a pareto improvement. So (Down,Right) pareto dominates (Up, Left).

It is also generally agreed that a good way to compare outcomes in economics is pareto efficiency (no person can be made better off without someone being made worse off).

Hence we can safely say that the equilibrium with (Down,Right) each is comparable and better than (Up, Left). This is generally accepted, not controversial and doesn't require any assumptions we dislike.

Note that it is not necessary to assume here that utility is cardinal. That would mean we can compare utility levels across agents. E.g. for the statement: if U(agent 1)=4, U(agent 2)= 1, then U(agent 1)>(agent 2). Another example: saying the sum of utilities for everyone is larger under policy A than under policy B and hence A is better than B requires cardinal utility. This is a wrong assumption, but sometimes used (for example in public and welfare economics) to make comparisons. It is however controversial to use, since it is actually wrong. Public economists see it as a necessary evil. What everyone accepts is that utility is ordinal and that is all we need to compare the two equilibria.


Under what BLS line do "people who want a job but haven’t searched in the last 12 months" show up?

That's reported in A-38 under "not in the labor force" but "want a job" and "did not search for work in the previous year". The line/figure doesn't have a U-style indicator.


Why does BOJ unchanged rate lead to Nikkei and Topix declines?

It doesn't in general.

It did in this specific case because the Nikkei and Topix had earlier priced in the possibility that the Bank of Japan would do something other than leaving rates unchanged this time. Now that that possibility has gone, the market no longer needs to price in the possibility, and so has corrected.

That's not just specific to this case: that's true of all market responses when an unknown event becomes known.


Predicted recessions that never happened

One example is this UK Government analysis:

  "Britain’s economy would be tipped into a year-long recession, with at least 500,000 jobs lost and GDP around 3.6% lower, following a vote to leave the EU, new Treasury analysis launched today by the Prime Minister and Chancellor shows…Speaking at B&Q in Eastleigh, Hampshire, the Prime Minister and Chancellor set out the Treasury’s analysis of the impact on the nation’s economy over the immediate period of two years following a vote to leave. This analysis shows that such a decision would cause an immediate and profound economic shock across the country"

But growth has been consistently positive in the almost three-year period since the vote.

I'm not sure this a very good example, because there are clear political incentives surrounding this forecast, which might mean it was less than sincere.


Monopoly second degree discrimination - two part tariff

I'm inclined to say "yes" to $T_1>T_2$ and no to $P_1<P_2$.

For simplicity, suppose the monopolist's marginal cost is constant. Then consumer surplus is the area under the (inverse) demand curve and above the marginal cost. In a two-part tariff scheme, the monopolist would charge entrance fee equal to consumer surplus and unit price equal to marginal cost.

If, in your language, consumers with "high demand" are understood as those who are willing to pay more for each unit of the good than those consumers with "low demand", then high demand consumers generate higher consumer surplus than the low demand ones.

Suppose monopolist can distinguish between the two types and offer different two-part tariff bundles to each (e.g. a theme park company operating in US and India charging different prices in the two countries). Then it will be the case that the monopolist charges a higher entrance fee to the high type (because they generate higher consumer surplus) but the same unit price for both types (because this is related to the marginal cost of production, which is the same regardless of the consumer types).


Is there objective evidence that the US economy is "booming"?

A fairly standard measure to use would be Gross Domestic Product (GDP). Over the past 30 years, US GDP has been growing at approximately 2.5%.

According to trading economics

  The US economy expanded an annualized 2.3 percent on quarter in the first quarter of 2018, below 2.9 percent in the previous period but beating market expectations of 2 percent

Using this measure, calling it a boom is a bit much.

Another way to cut the data is by comparing it to the ten-year trend of about 1.5%. Growing 80bps above trend looks a bit better.

So, to answer your question, depending on how you cut the data, it could be considered booming but I wouldn't.


I will expand this into an answer since I've had a chance to look at a bit more of the literature on price salience, which I consider to the primary consideration here. That being said, I think there is something to say about the fact that the company might do away with a pricing policy that some subset of potential customers feels is deceptive. This will obviously lead to a more positive perception of the company (from these customer types). Whether or not this positive increase can offset the impact of increased price salience is an open question.

My original comment:

One must also consider the antithetical effect that price salience has on purchase decisions. It might be the case that people are more likely to tolerate a product's price ex-post than they are ex-ante. There is a robust literature on the effect of salience on elasticities and prices (for example, Finkelstien's tax salience paper). This definitely doesn't settle the matter but it does inform whatever answer you ultimately accept.

Her paper: economics.mit.edu/files/7878

Other papers to read:

Bordalo and colleagues:
http://scholar.harvard.edu/files/shleifer/files/pdf_1.pdf
This paper looks at consumer choice and characteristic salience of goods. This is a good paper to help build an intuition of what I mean by salience and what it means for a good or a good's characteristics to be salient.

Dias et al:
https://ideas.repec.org/a/taf/jgsmks/v20y2010i2p149-163.html
This is a paper about how price salience impacts consumer perceptions and purchase intentions. I added this paper because I think it hits the nail on the head here. Also, it gives a good review of the literature that is most related to your question.

Hope this helps.


Suitable terminology for the inverse elasticity?

Economists generally call it the "inverse elasticity". But note
$$\frac{1}{\eta}=\left(\frac{\partial Q}{\partial P}\frac{P}{Q}\right)^{-1}=\frac{1}{\left(\frac{\text{% change in quantity}}{\text{% change in price}}\right)}=\frac{\text{% change in price}}{\text{% change in quantity}}$$

Thus, whilst the price elasticity of demand can be interpreted as (roughly)

  The percentage change in quantity sold that would follow a one percentage point increase in the price.

the inverse elasticity (the "demand elasticity of price") could be explained as

  The percentage change in price necessary to achieve a one percentage point increase in quantity.


Consumer demand: a function of present endowment?

Although it's often left unstated, demand for a good is measured as quantity per unit of time.  Broadly, the longer the time period over which demand is measured, the smaller the (proportional) effect on demand of the initial endowment.  Whether you have a soda in your fridge one morning may make the difference between your demand for soda that day being $0$ or $1$. But over the next year, if you often consume sodas, it may make the difference between your demand being $n-1$ or $n$, only a very small proportional difference if $n$ is large.  For this reason, even at the individual level it can often be appropriate to ignore endowments as a determinant of demand for frequently consumed goods.

At market level, a further consideration is that the effect of endowments may average out over many individuals. At a certain time, for example, A has a recently purchased washing machine, B's washing machine is in mid-life, and C's is old and needs replacing soon.  In the absence of changes in income or major improvements in washing machines, it may be that in any year approximately the same proportion of people have a demand for a new washing machine, so that both the aggregate endowment of washing machines and the market demand for new washing machines are approximately stable.  Again, therefore, it could be appropriate to ignore endowments.

But there are also many situations where neither of the above apply and it is important to consider endowments as a factor affecting demand.  These include: in consumer markets, take-up of goods embodying new technologies; in housing markets, the size and condition of the existing housing stock; and in commodity markets, stocks of for example wheat or copper held by commodity traders.


Is measurement error in self-reported health a problem in this scenario...?

Check the correlation between health status variables and employment status. Start with simple pairwise correlations and then perform auxiliary regressions where you regress a health indicator on the employment status variables.

If there is visible explanatory power in employment status for health indicators, you may want to treat health indicators as endogenous variables and employment status as instruments. Depending on the number of variables, this may be the Two-stage Least Squares estimator.


Rawlsian SWF and Arrow Impossibilty Theorem

Independence of irrelevant alternatives prevents you from using the information needed to implement a Rawlsian SWF; the information who is society's worst-off cannot be used.
Indeed, the relevant information is not even specified in a profile of preferences.


Textbook/Reading Suggestions for Urban Economics

Masahisa Fujita is considered as one of the most famous theorist in urban economics. He is one of the pioneers of the New Economic Geography. His book
Urban Economic Theory: Land Use and City Size, 1989, Cambridge is a master piece, laying down the theoretical foundations of urban economics. Also, you may read

Fujita, Masahisa; Krugman, Paul; Venables, Anthony J. (1999). The
Spatial Economy: Cities, Regions, and International Trade. Cambridge,
Massachusetts: MIT Press.
Fujita, Masahisa; Thisse, Jacques-François (2013). Economics of
Agglomeration: Cities, Industrial Location, and Regional Growth (2nd
ed.). Cambridge University Press.

Esteban Rossi-Hansberg from Princeton is a rising star in urban economics. You may check out his syllabus, which is pretty useful. He does not recommend any textbook, and provides a more modern approach.


Calculating the elasticity of substitution between factors of production

First of all, I think that 'linear and homogeneous' is a typo of 'linearly homogeneous.' Indeed, it can be shown that if the production function $V$ is linearly homogeneous, Allen Elasticity of Substitution between $K$ and $L$ can be expressed as
$$
\sigma = \frac{V_K V_L}{V \cdot V_{KL}}
$$
by using the fact that $V_K$ and $V_L$ are homogeneous of degree 0, resulting in
$$
V_{LL}L + V_{LK}K=0
\\
V_{KL}L + V_{KK}K=0
$$
.

Moreover, by the definitions in the paper (Lu, 1967), note that
$$
X \equiv \frac{K}{L}
\\
Y \equiv \frac{V}{L} = F(\frac{K}{L}, 1) \equiv f(X)
\\
$$

Now, by substituting the results given by the author, we have
$$
\sigma = \frac{\bigg[\frac{1}{K}(V-V_LL)\bigg]  \bigg[\alpha X^{-\frac{c}{b}}Y^{\frac{1}{b}}\bigg]} {V \cdot \frac{\alpha}{bL} X^{-\frac{c}{b}-1}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )}
$$
, or
$$
\sigma = \frac{b}{V} \cdot \frac{L}{K} \cdot \frac{(V-V_LL)XY}{X\frac{dY}{dX}-cY}
\\
= \frac{b}{V} \cdot \frac{(V-V_LL)Y}{X\frac{dY}{dX}-cY}
\\
= \frac{b}{V} \cdot \frac{V-V_LL}{\frac{X}{Y} \frac{dY}{dX}-c}
$$

Note that, since $V$ is linearly homogeneous,
$$
V-V_LL = V_KK = \frac{dY}{dX}K
$$
where the second equality holds for (3.18) in the paper.

Therefore,
$$
\sigma = \frac{b}{V} \cdot \frac{\frac{dY}{dX}K}{\frac{X}{Y} \frac{dY}{dX}-c}
\\
= b \cdot \frac{\frac{X}{Y} \frac{dY}{dX}}{\frac{X}{Y} \frac{dY}{dX}-c}
\\
= \frac{b}{1-c(\frac{X}{Y} \frac{dY}{dX})^{-1}}
$$

Finally, note that
$$
\frac{X}{Y} \frac{dY}{dX} = \frac{X}{f(X)} \frac{dY}{dX} = \frac{X}{f(X)}f'(X)
$$
, and thus we have
$$
\sigma = \frac{b}{1-\frac{cf}{Xf'}}
$$

(You have a typo on this expression)


Can an economy grow without its population growing?

Yes, there can be economic growth without population growth.

As a reminder: gross domestic product (GDP) is the total value of all of the goods and services produced within an economy over a period of time (usually a year). GDP growth is simply the tendency for the GDP to increase over time.

Roughly speaking, there are two ways for GDP to increase. The first is for each worker to keep producing the same stuff, but to increase the number of workers (i.e. population growth). But the second is to keep the population the same and increase the amount produced by each worker. This increase in productivity might happen because

workers become intrinsically more productive (i.e. through education)
workers have better tools to work with (e.g. through technological progress).

If you live in a subsistence society in which people tend the fields by hand and produce 100 bushels of grain per year and then someone invents a combine harvester that allows them to produce 500 bushels per year then the output of the economy will increase even if the population stays the same.


It's highly likely that the coal will be pretty much worthless in 20 years from now.

That's because we've got far more fossil fuels than we can burn safely: the negative externalities far exceed the consumer's private benefit.

The world is on the edge of rejecting coal forever. Several countries have already done so, and that number is growing steadily.


I suppose you have in mind a game that looks like this:

  she knows her type when she is called upon to move, so why do we calculate an expected payoff for player 2 when converting the game into an normal-form?

The fact that "she knows her type" and therefore has no need to calculate an expected payoff presupposes that the play has reach a particular information set for player 2. Formally, let $\theta\in\{R,B\}$ be P2's type. Her utility  conditional on her type being $\theta$ is
\begin{equation}
u_2(\sigma_1,\beta_2(\color{red}{\theta})\vert\color{red}{\theta}).
\end{equation}
where $\beta_2(\theta)$ is P2's behavior strategy at the information set associated with $\theta$ (i.e. the left decision node for P2 when $\theta=R$ and the right decision node for P2 if $\theta=R$). Here, no expected value is calculated since the uncertainty is resolved for P2.
In contrast, in the normal form representation, P2's utility is an unconditional one:
\begin{equation}
u_2(\sigma_1,\sigma_2)=\Pr(R)u_2(\sigma_1,\beta_2(R)\vert R)+\Pr(B)u_2(\sigma_1,\beta_2(B)\vert B)
\end{equation}
where expectation is to be taken.

The reason is that by converting an extensive form game into the normal form, the temporal aspect of the game is lost and everything is seen from the ex ante perspective.