Strong sequential equilibria and the existence of others

There are three classes of equilibria of this game.

The first class is sequential:
\begin{equation}
(s_1,s_2)=(y,r)
\end{equation}
and the beliefs are
\begin{equation}
\mu_1(a)=\mu_1(b)=\mu_2(a\mid y)=\mu_2(b\mid y)=\frac12.
\end{equation}

The second class is not sequential, but weak perfect Bayesian:
\begin{equation}
(s_1,s_2)=(x,l)
\end{equation}
and the beliefs are
\begin{equation}
\mu_1(a)=\mu_1(b)=\frac12,\quad\text{but }\mu_2(a\mid y)=1-\mu_2(b\mid y)=p>\frac58.
\end{equation}

The third class is actually a boundary case of the second class, which permits P2 to use mixed strategies. This equilibrium requires P2's off equilibrium belief to be
\begin{equation}
\mu_2(a\mid y)=1-\mu_2(b\mid y)=\frac58,
\end{equation}
so that P2 can play a mixed strategy that puts sufficiently high probability on $l$ ($\sigma_2(l)\ge\frac35$). It is also necessary that P1 does not play $y$ with positive probability, because if they do, Bayes rule would kick in, requiring that P2's belief be $(\frac12,\frac12)$ instead of $(\frac58,\frac38)$ which rationalizes P2's mixed strategy.


First-price symmetric auction with discrete value

Suppose you are one of the bidders and fix the strategy of the other one to be the one supposed.
Your payoff will be $\mbox{Pr(winning the auction, i.e., b>b')}(v-b)$, with b being your bid and b' being the other's bid and v being your valuation (or type).

Now, imagine you are the type-5 player.

If you bid an integer less than 4, you lose for sure because the other player bids more regardless of their type. Hence, your expected payoff is 0.

If you bid 4, (we assumed that) you win with probabilty 0.5 if the other one is of type 5 (and hence bids 4) and lose in every other case.
Hence, your expected payoff is $0.2*0.5*(5-4) = 0.1$.

If you bid 5, your payoff from winning is zero such that your expected payoff will be zero in any case.

If you bid more than 5, you make a loss when winning, making the expected payoff negative.

Hence, bidding 4 when being type 5 is a best response (with this restricted bid space!).

Similarly, imagine being a type-7 bidder.
Bidding less than 3 makes you lose the auction, and bidding 7 or more does not seem to be a good idea (see above). Now calculate the winning probabilites for bidding 4 (same as above, but higher winning payoff), bidding 5 (winning against type 5 for sure, but losing against other types), bidding 6 (winning against type 5 for sure, but against type 7 only with probability 0.5). Next, you repeat the exercise for type 10.
If you don't find a profitable deviation, you have got an equilibrium.


Ruling out boundary solutions in Utility Maximization

No, it isn't.

A simple counterexample is
\begin{align*}
\max_x \hskip 10pt & x_1 + x_2 \\
\\
\text{s.t.} \hskip 10pt & x_1 + 2x_2 = 10.
\end{align*}
The utility function is monotonic (strictly monotonic even), but the solution is a corner solution at $(x_1,x_2) = (10,0)$.

The above answers the question, but it is worthwhile to note that defining a sufficient condition (that is not too strong) seems to be difficult.

Not even convexity is sufficient. (The utility function above is convex, and could be made strictly convex with an extremely slight bend.)

Seems like you would have to compare the level curve gradients at all the possible corner solutions to the gradient of the budget set, but that does not really help you in higher dimensions.


Information Partition and Common Knowledge

The situation is not described by $K_A(K_B(3))$. $A$ does not know that the draw was 3, and hence does not know that $B$ thinks it is in the set $K_B(3)$. What $A$ does know is that the draw
is in set $K_A(3)$, hence $A$ thinks $B$ should have narrowed it down to $K_B(K_A(3))$.


Depreciation and second-hand markets for bicycles

Interesting observation. Perhaps a part of the answer lies in Akerlof's lemons' problem?

Admittedly judging the quality of a 2nd hand bike is easier than that of a car, but from what you're describing in the US there doesn't seem to be a well functioning market for 2nd hand bikes in the US. Moreover, bikes are becoming more and more like cars in that respect with all the gadgets and technology that one cannot fix in their own shed or backyard.

Perhaps it also plays a role that US citizens are not used to bikes and therefore have a harder time judging their quality (just guessing here, and perhaps stereotyping too much). Here in the Netherlands we seem to have a well-functioning market for them that serve all segments (including, I might add, a high theft rate), but everyone rides a bike here, so we should be better at judging their quality.


Why does falling global bond yields signal coming deflation?

Bond yields falling from their current near-zero position will place them in negative yield territory. Negative bond yields are deflationary by definition.
Paragraph 3, sentence 5 of the article says:

With Bank Rate already close to the floor, and some UK bond yields now in negative territory... [emphasis added]

To understand why negative bond yields are deflation markers, consider the example of negative Asian and European central bank rates described in the same article:

This is because a negative rate is effectively just a tax on the banks, forcing them to pay for the privilege of holding reserves with the central bank.
Someone has to shoulder the burden of this tax; either borrowers pay more, or depositors get less, or profits go down, and if it is the latter, then it damages the banking sector’s ability to rebuild capital, crimping credit availability accordingly. Whichever it is, the end result is a monetary tightening, rather than the loosening intended.

So you can think of the effect of negative central bank rates (and, similarly, negative bond yields) as having the inverse impact as quantitative easing which creates an inflationary effect.
This page explains the relationship between prices and money supply as follows:
$$P=\frac{MV}{Y}$$
where $P$ represents prices, $M$ is the money supply, $V$ is the velocity of money and $Y$ is the real GDP.
Therefore,
$$\Delta P = \Delta M \cdot \frac{V}{Y}$$
and for constant $V$ and $Y$,
$$\frac{\partial P}{\partial M} = \frac{V}{Y} > 0$$
So an effective restricting of the money supply caused by negative rates/yields via the mechanism above described will tend to put downward pressure on prices (a/k/a deflation).
Edit:
In the comments to this answer, OP asked about the relationship between negative interest rates and negative bond yields.
Whereas central banks set rates by policy, long term bond yields are determined by the market (i.e., supply-demand equilibrium). An inverted yield curve signals recession and implies rates are set too high relative to market expectations. If central banks set rates in positive territory when long term bond yields are negative, they will de facto force the yield curve to invert. Although this sometimes happens, it is usually counterproductive to the policy goals of the central banks (in most environments) and, therefore, typically avoided.


Is there unnecessary duplication for borrowing in this circular flow of income diagram?

No because the financial markets do not provide firms with capital they provide them with funds to buy capital which is a difference. You can call those funds financial capital but it’s not capital in economic sense.

In common usage in English capital is often used as a synonym for money. For example in Oxford dictionary under the word capital the second entry is simply money (see here).

However, in economics capital, especially when we are talking about factor markets, is used in the following definition:

“The equipment and structures used to produce goods and services.”

The above definition is taken from Mankiw and Taylor (2014), Economics the 3rd edition. In different textbooks the exact wording may change but to my best knowledge no conventional textbook would include money as it’s definition.

Also generally money is not considered a factor of production in itself hence it’s a tautology that factor markets should provide only factors of production.

Moreover, generally financial markets are often considered quite distinct from other economic activity. In fact often you can hear economists speak of ‘real’ economy and financial markets separately. This is due to the classical dichotomy which makes sharp distinctions between real and monetary (nominal) variables. Factor markets are considered to be part of real economy, where real goods, services, and factors circulate, while financial markets are part of monetary sector which covers the circulation of money and other financial goods and services.

In comment @user161005 asks additional question of why the arrow going to factor markets includes interest, since answer to this might be of interest (no pun intended) to other users who skip comments I put the answer also here:

In economics the income derived from labor is called wages, the income from land rent, and income derived from capital interest.

The last often confuses students because many textbook models call the interest earn from capital $r$ instead of $i$ but this is because $i$ is already used widely in other macroeconomic models.

The profit is what is left after the wages are paid to the labor, rent to landowners and interest to capital owners. Profit is considered to be the reward for entrepreneurship which can be considered as another factor of production.

A good entry level explanation of this is provided in this FED St. Louis podcast that you can listen to (or read transcript to) here.


Vocabulary/Name for Utility of a Set of Choices

What you seem to be interested is the subfield of decision theory that goes by the name of "menu choice." The starting point of this literature is the paper

Kreps, David M. "A representation theorem for preference for flexibility." Econometrica: Journal of the Econometric Society (1979): 565-577.

This paper might be directly relevant to the problem you want to address.


Strong monotonicity and weak monotonicity

Let $x,y\in X$. Suppose $y \gg x$ (so that in particular, $y\geq x$ and $y\neq x$). By assumption, $\succsim$ satisfies strong monotonicity; and so, $y\succ x$.

(We've just shown that for any $x,y\in X$ such that $y \gg x$, we have $y\succ x$. Hence, $\succsim$ satisfies weak monotonicity.)

The argument you've given above is correct and roughly the same as the proof I've just given. The only problem is that your argument is somewhat indirect, convoluted, and unclear.


You are allowed to use logs in 2sls model. The reason why that article does not use logs is that the article assumes linear supply and demand.
At the beginning the author states:

Consider the following supply and demand functions
Supply:$Q=\beta_1 P + \epsilon_s$
Demand: $Q=\alpha_1 P+ \alpha_2 I+ \epsilon_d$

If the author would instead assumed supply and demand to have following form:
$Q=P^{\beta_1}$ and $Q=P^{\alpha_1}I^{\alpha_2}$ then taking logs would make sense but if the supply and demand are already assumed to be linear taking logs is unnecessary as you already have the right functional form and you can just directly apply linear model without log-linearizing it.
For example, Soderbery (2015) who estimates import supply and demand elasticities uses logarithms and models his supply-demand system using 2sls model with log-log form.


How to estimate if goods are complementary or substitutes when one good is always offered for free?

One answer is to try to find variation in the non-pecuniary price of the free good. Here is a famous example from estimating demand for public parks:

In 1949, Harold Hotelling wrote a letter to the National Park Service,
outlining a method where the park visitor’s round trip distance could
be used to proxy the recreation trip price, so that consumer’s surplus
estimates might be recovered. The idea was to obtain the net benefit
for outdoor recreation for a particular geographic area. The CVM
method was suggested in the same year in an article by
Ciriacy-Wantrup, although the first actual application of the method
(to deer hunting in Maine) was by Robert Davis in his Harvard PhD
dissertation, in 1963.

The new economics of outdoor recreation
Hanley, N. ;  Shaw, W. S ;  Wright, R. E. (2003)
Chapter free here
It seems like something very similar is possible here. For example, say you were asking about complementaries between the radio (which is free) and cars (which are not). You could look at hard of hearing people, and how demand changed for the radio as more and more expensive equipment is required to listen to the radio.
Or consider evaluating if public schools and Disney World are complements or substitutes. Using distance from school as a proxy for the variation in the cost of "consuming" school, you could use that variation to trace out the cross-price elasticity of Disney with respect to school.


What is the difference between a transitory and a permanent shock?

"Permanent shock" is a shock whose effects on current values of a variable never die out in absolute terms. "Transitory shock" is a shock whose effects gradually die out. Consider the simplest one-lag model

$$y_t = \beta y_{t-1} + u_t, \;\;\; y_0 = 0$$

By direct substitution one gets that

$$y_T = \sum_{t=0}^{T-1}\beta^t u_{T-t} = u_T + \beta u_{T-1} + \beta^2u_{T-2} +...+\beta^{T-1}u_1$$

If $|\beta| <1$ the absolute effect of say shock $u_1$ on $y_T$ diminishes as time passes, $T \uparrow$. While mathematically it doesn't die off completely except at infinity, its absolute contribution decays exponentially.

On the other hand, if $\beta =1$ (a "unit root" model) then

$$y_T = \sum_{t=0}^{T-1}u_{T-t}=u_T + u_{T-1} + u_{T-2} +...+u_1$$

and no matter how far we move into the future the shock at period $1$ will fully contribute to the value of $y_T$.

Of course, if autocorrelation is strong (high $\beta$) then a "transitory" shock will influence visibly many periods in the future, so it may be "ignored" when examining "long-term" outcomes, but not when one deals with mid-term effects and policy.


Let $G$ be the cumulative distribution function for the buyers' willingness to pay. If there is no cost, expected profit is simply expected revenue. The revenue is quantity times price. If the unit of the good is actually sold, this is just the price. Now the probability that the good is sold at price $p$ is the probability that the willingness to pay is not smaller than $p$, which is $\big(1-G(p)\big)$. So in total, expected profit is simply

$$\big(1-G(p)\big)p.$$

Two things are implicit in the problem already: First, the willingness to pay already includes all relevant aspects of the buyer's income. Moreover, unless we have the willingness to pay for each amount of the commodity, it is likely that the expected quantitty is a unit quantity of $1$.


Prove that $u$ is a utility function for $\succsim$

You're asked to prove that $u(x)\ge u(y)\;\Leftrightarrow\;x\succsim y$ for any $x,y\in X$, where $u(x)=|\{z\in X:z\prec x\}|$, i.e. the utility of $x$ is measured by the number of other alternatives that rank strictly below it.
Since $X$ is finite, let's suppose without loss of generality that $X=\{1,2,\dots,N\}$ where $N$ is some finite number.

I'll prove the case in which there is no indifference among the alternatives, say, $1\succ2\succ\cdots\succ N$. I'll let you finish the proof by establishing the case where there are indifferences among subsets of alternatives.

Step 1. Establishing $u(x)>u(y)\;\Rightarrow\;x\succ y$.

Suppose $u(x)>u(y)$. By the definition of $u$, the number of alternatives strictly worse than $x$ is larger than the number of alternatives strictly worse than $y$. If $y\succsim x$, this would simply contradict the previous statement. Hence, we must have $x\succ y$.

Step 2. Establishing $x\succ y\;\Rightarrow\;u(x)>u(y)$.

Suppose $x\succ y$. Since we assume no indifference among alternatives, the set of strictly worse alternatives to $x$, $\{z\in X:x\succ z\}$, must contain more elements than the set of strictly worse alternatives to $y$, $\{z\in X:y\succ z\}$. In other words, $|\{z\in X:x\succ z\}|>|\{z\in X:y\succ z\}|$. Therefore, we obtain $u(x)>u(y)$ as a result.

Taken together, steps 1 and 2 demonstrate that $x\succ y\;\Leftrightarrow\;u(x)>u(y)$ for any arbitrary $x,y\in X$.


Interacting covariates with the instrument in the first stage

Short answer: No.
Your model is $Y=\alpha + \beta X + \varepsilon$. Even when $X$ is exogenous, if you regress $Y$ on $X$, $W_1$ and $W_2$, then the OLS estimator is inconsistent (for $\beta$) unless $W_1$ and $W_2$ do not affect $Y$ on average (after controlling for $X$) or $X$ is uncorrelated with $W_1$ and $W_2$. When $X$ is endogenous, there is no reason why your strategy should work.
Now suppose that your model is $Y=\alpha + \delta X + \gamma_1 W_1 + \gamma_2 W_2 + \varepsilon$, instead. (This is not your model; $\delta \ne \beta$.) You are saying $X$, $W_1$ and $W_2$ are endogenous, while $Z$ is exogenous. You need instruments for $X$, $W_1$ and $W_2$, but you don't. The parameters are not identified.


Limit of utility function in Ramsey-Cass-Koopmans model

The answer of @tdm shows you how it is done using De L'Hospital's rule. If you want to avoid this rule you can write $u(c)=\frac{c^{1-\theta}-1}{1-\theta}$ and note that for $\theta\ne 1$ this implies $u(1)=0$. Differentiating with respect to $c$ then gives you $u'(c)=c^{-\theta}$ and therefore $\lim_{\theta \to 1}u'(c)=c^{-1}$. Then integrate again to get $\lim_{\theta \to 1}u(c)=\ln(c)+K$ and comparing for $c=1$ gives you $K=0$, so $\lim_{\theta \to 1}u(c)=\ln(c)$.


In this lecture it is stated at chapter 2.3 that "The Markov switching model and its variants discussed in the preceding sections are
only suitable for stationary data". Perhaps you'd like to read into that more thoroughly.
Possible solution:
Since differentiation is allowed, you should also be able to produce an Error-Correcting Model -and its variants- through cointegration if stationarity is not achieved.


Why Chinese GDP does not correspond to Chinese GDP growth rate?

Google's figures for China's GDP are simply taken from the World Bank's "GDP (current US\$)" (NY.GDP.MKTP.CD):

2016: 11190992550229.5 USD -- Google rounds this off to US\$11.19 trillion.
2017: 12237700479375 USD -- Google rounds this off to US\$12.24 trillion.

(As you point out, the growth rate from the above figures should be ≈9.4%.)

Google's figure of 6.9% for China's 2017 GDP growth is simply taken from the World Bank's "GDP growth (annual %)" (NY.GDP.MKTP.KD.ZG). Which, as you point out, is a discrepancy with our computed 9.4%.

But look at the "Details" on the last linked page, which explains:

  Annual percentage growth rate of GDP at market prices based on constant local currency.

That is, the growth rate is calculated from the World Bank's "GDP (constant LCU)" (NY.GDP.MKTP.KN), for which we have these China figures:

2016: 73521858296478.3 CNY
2017: 78594866518935.4 CNY

And now we do indeed get the figure of 6.9%.*

The lesson here is that when making GDP comparisons across time or space, there are some adjustments that one can choose to make. What adjustments one makes can lead to different results.

*A little strangely, the 6.9% is exactly precise. Which suggests that instead of calculating the growth rate using estimated GDP figures, the World Bank's GDP figures for China are calculated using the estimated 6.9% growth rate.

In general, with national income and product accounts (NIPA), there is a great deal of crude estimation and sausage making. You shouldn't take too much faith in any numbers past the first one or two significant figures. This is especially the case for China where, in addition to measurement difficulties faced everywhere, economic data are manipulated for political purposes.


Short cuts to solve Cobb Douglas Utility function (minimization)

A shortcut to solve $$\min_{x,y} x^{\alpha} y^{\beta}$$ s.t. $$xp_x + yp_x \leq m$$ and $$0 \leq x,y$$ is to set $x = 0$ and/or $y = 0$, as this results in $U(x,y) = 0$, and the function does not map to negative values.


The effects of Quantative Easing (QE) on FX rates?

You've identified the first-order effect: the supply of local currency has increased, so its price (and from the perspective of other currencies, that's its exchange rate) decreases. That's assuming that the move wasn't already (fully) priced-in by the market.

There are lots of other effects.

There will be a change in domestic interest rates. The particular rates affected, depend on which debt has been bought. And there will be knock-on effects in those markets. This may affect inward investment in those markets.

QE in itself sends a couple of signals to the market. Firstly, that the Bank sees deflationary pressures a year or two ahead (assuming its mandate is an inflation-rate target); and secondly, that it intends to apply inflationary pressure to deal with it. The market will react to those signals, depending on how far they diverge from what the market had already anticipated. Those reactions might include increased optimism or pessimism about future domestic demand, and / or about the health of domestic industries, and that can have an affect on exchange rates.

In short, the central bank is pressing levers that touch on much of the macro-economic picture, so there are lots of second-order effects. The nature and direction of these effects will depend on expectations before and after the announcement, as well as what's actually going on in the economy.


The Savage sure thing principle and Subjective utility representation

In Kreps' (1988) book "Notes on the Theory of Choice", the issue is dealt with in chapter 9 "Savage's Theory of Choice Under Uncertainty", after discussing subjective probability in chapter 8. As usual, Kreps' style helps: he has the ability to seamlessly inject his -always formal- approach with very down-to-earth comments and examples that are strong in intuition (and he does it better than Savage, I might add). But also, here "formal" does not translate into "complete exposition": he explicitly refrains from formally proving parts of the whole apparatus, mentioning that "this is a two-page proof", and "this is another two-page proof", and "if you want to prove this, good luck". For these parts he falls back on Fishburn's (1970) "Utility Theory for Decision Making" book, chapter 14 "Savage's Expected Utility Theory". And Fishburn is formal alright (more symbols than words in a page).

My impression is that combining these two sources can be beneficial.


How does printing money cause inflation?

WHY and HOW monetary issue causes inflation.

When a central bank emits money (or decrease key interest rates), commercial banks will be able to refinance themselves at a lower cost. Which means that at the end, these commercial banks will also be more, say, flexible, and thus will stimulate demands (here and there) in the national economy by lending money (here and there). If supplies stay constant, markets will reach a new equilibrium via prices rising due to these stimulated demands $\implies$ inflation.


How does using HP filter allows us to estimate long-run values for variables such as output and unemployement?

I've found the answer to my question. By using the HP filter, we are able to extract the long-term component of a time-series, namely it's trend component. Using that, we can then take the difference between the series and its trend component at every observation. This leaves us with a cyclical component; if we are interested in estimating gaps between observed values and potential values or natural rates, this is what we are looking for.
Of course, this is the english explanation of what happens; for a mathematical/statistical approach, I recommend reading the original paper.


Finland plans to give every citizen 800 euros a month and scrap benefits. Will it become successful?

It will be successful in increasing economic growth. Removing benefits and giving everyone a fixed sum of money per month will disincentivize voluntary unemployment, remove moral hazard from benefit schemes, and encourage poor people to seek ways out of poverty. Also there should be less pressure on public infrastructure, including healthcare.
It will be successful in reducing the number of migrants seeking refuge in Finland.
It will NOT be successful in making life easier for the most poor and unfortunate.

From economic perspective, it should certainly be successful. It is hard to judge it from ethical point of view though.


Existence of Symmetric Pure Strategy Equilibrium

There is no guarantee that symmetric games have symmetric equilibria.
See this paper for concrete examples.
There is also no guarantee that symmetric games have pure-strategy equilibria. For example,  the following game is symmetric and has no pure-strategy equilibria
$$
\begin{array}{cccc}
& \mathrm{a} & \mathrm{b} & \mathrm{c}\\
\mathrm{a} & 1,1 & 0,2 & 5,0\\
\mathrm{b} & 2,0 & 3,3 & 0,4\\
\mathrm{c} & 0,5 & 4,0 & 1,1\\
\end{array}
$$
There are two special kinds of games that are known to always have pure strategy equilibria: supermodular games and potential games.


Raise/Lower Seller Fees on Platform - Expected Seller Behaviour?

I sell digital books, and will discuss their particular situation. I do not know whether what I describe applies elsewhere.
I and have read a fair amount of discussions about how to price books. To what extent academic research would be referred to, it’s wherever marketing research ends up (business journals?), and not straight economic models.
The first thing to keep in mind is that for electronic books, there is “arbitrage” across platforms, and there is pressure by some stores to keep your price in line with other platforms. As such, if one platform changes their royalty payout, you have to either eat the lower revenue, drop the platform, or raise price across all platforms. If the platform is small, you might be able to just raise the price on that one platform, but that might anger customers. As such, the decision is complex. E.g., is it possible to stop using a platform? Is the lower revenue per unit on one platform enough to justify a global price hike?
After that, pricing is somewhat of a black art. Each market has its own conventions for pricing. For example, dollar prices often end in .99 (this convention is in fact enforced on one platform). Therefore, you cannot make small percentage adjustments to price (unless the price is large, which is rare for e-books).
After that pricing is relative to competitors, any paperback edition (where printing cost partly determines price), and how you want to position your book. For example, you might keep the price high if you are emphasizing the quality of your books. Once again, pricing is a marketing decision, since the sales price is far above the marginal production price.


Can't government experiment on price in the case of a monopoly?

If I understand your question correctly, you are proposing that the government sets prices and observes the behavior of market actors to elicit the demand and supply functions.

I see two possible concerns:
Unless the monopoly is obligated to serve all customers, the market may not be in equilibrium, full demand may not be observed.
Similarly the government would have to buy any unsold goods, otherwise
a too large $p$ could result supply being larger then demand, resulting in a loss for the monopoly.

The other concern is that the monopoly's behavior significantly affects the government's observations. It could potentially benefit by behaving suboptimally now and earning more in the regulated future. One way it could do that is to make the government believe it has higher costs by supplying lower quantities given any price $p$.
-While I was writing this answer, @Dan has posted the same idea in a comment.


There is nothing more to it than the equation

$$Y_t - (c_y+v)Y_{t-1} = C^a + I^a$$

This is a linear non-homogeneous first-order difference equation, and it is non-homogeneous because there is a non-zero constant, $C^a + I^a \neq 0$.

This is not some "habit in economics", but rather standard mathematical terminology.


How can autonomous private spending change budget surplus?

In a very rudimentary sense, the question can be answered in the following way :
We see that
$BS = tY - G - TR $

As you put in the quote, Dornbusch is arguing that budget deficit/ surplus is not a good measure to gauge government policy.

EXPLANATION:
For example, a deficit doesn't always mean that $G$ is large or that the government is spending to increase income. An increase in budget deficit can occur if there is a reduction in private spending (which is $I$) - since that leads to a reduction in $Y$, which reduces $tY$ (tax collections) that increasing deficits.
When he says that private spending can affect budget surplus, he means that private spending $I$ can affect the level of income $Y$ which can affect $tY$ (tax collections) which can impact Budget Deficit/Surplus.
Hope this helps!


Multivariate linear regression: how to test for whether the slopes are the same?

Estimate the model
$$W_i=a_1+a_2E_i+a_3G_i+a_4E_iG_i+\epsilon_i,$$
where $W,E,G$ denote wage, education, and an indicator for gender (e.g. 1 if male, 0 if female), respectively. The term $EG$ is an interaction term obtained by multiplying the gender dummy by education. A simple t-test of the hypothesis $H_0: a_4=0$ will tell you if the effects of additional education on wage are significantly different for different genders.


Economists studying the eldery and intergenerational transfers

As far as intergenerational transfers are concerned, I would suggest you start with Wojciech Kopczuk:

Kopczuk, Wojciech. "Taxation of Intergenerational Transfers and Wealth." Handbook of Public Economics 5 (2013): 329.


Since in this case, both utility function and the budget line are perfect substitutes and thus you can't equate the Marginal Rate of Substitution to the price ratio.

This problem can be solved this way,

$$\min_{x,y} \quad  3x+3y\\
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{s.t.} \quad  2x+5y=10\\$$

or

$$\min_{x,y} \quad  x+y\\
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \textrm{s.t.} \quad  2x+5y=10\\$$

Here,
     $$MRS=\frac{MU_{x}}{MU_{y}}=\frac{2}{5}<1=\frac{P_{x}}{P_{y}}$$
or,  $$\frac{MU_{x}}{P_{x}}=2 < 5 = \frac{MU_{y}}{P_{y}}$$

Therefore the expenditure is minimum when the consumer spends his entire income on good y, that is $(x,y)=(0,2)$ is the bundle that minimizes the expenditure given the utility $u(x,y)=2x+5y=10$. This is depicted in the figure below:


Price discriminating monopolist question

MC = 2Q1 + 2Q2 for both demand functions. Set MR1 = 2Q1 + 2Q2, and MR2 = 2Q1 + 2Q1 and then you have a system of 2 unknowns and 2 equations. The solution ends up being Q1 = 15, P1 = 72.5, Q2 = 17.5, P2 = 82.5. In this case Profits under PD > Profits Under single price. See you in 468 tomorrow.


How are working hours statistics computed?

Referring to USA, the place to go and learn a lot (if not everything) about the matter is The Bureau of Labor Staitstics

Indicatively, for one of BLS's many outputs, they report that:

  Payroll survey — methods and measurement issues

  The Current Employment Statistics (CES) program, also known as the
  payroll survey or the establishment survey, is a monthly survey of
  approximately 143,000 businesses and government agencies representing
  approximately 588,000 worksites throughout the United States. From the
  sample, CES produces and publishes employment, hours, and earnings
  estimates for the nation, states, and metropolitan areas at detailed
  industry levels.

  The CES employment series are estimates of nonfarm wage and salary
  jobs, not an estimate of employed persons; an individual with two jobs is counted twice by the payroll survey. The CES employment series
  excludes employees in agriculture, private households, and the
  self-employed.

  ...The entire sample is redrawn annually, and a supplemental sample of
  new business births is selected midway through the year. About
  one-fourth of the sample is rotated out each year and replaced with
  newly selected businesses.

  All new sample is solicited by computer-assisted telephone interview
  (CATI), and data are collected for the first 5 months via this mode.
  After the initiation period, many sample units are transferred to one
  of several less costly reporting methods that are self-initiated by
  the respondent...

  Estimation Methodology.
  CES monthly employment estimates are made using a two-part estimator.
  The sample reports are used to estimate month-to-month employment
  change from continuing businesses and a birth/death model is used to
  account for new firm births that otherwise would not be sampled in a
  timely fashion.

  Estimate review.
  CES uses automated edit and screening techniques to identify potentially erroneous sample data; respondents are
  re-contacted as needed to validate or correct their reported
  information. After the microdata edit process is complete, monthly
  estimates are calculated. Automated edits of the estimates are
  supplemented by analysts who look for errors and outliers and provide
  final validation of the series before publication.

  Seasonal Adjustment.
  The seasonal adjustment process removes from the series the effects of
  normal variation from recurring events within a year, such as holidays
  and weather changes, and helps reveal underlying economic trends. CES
  uses a concurrent seasonal adjustment methodology, meaning that it
  incorporates estimates up through and including the current month’s
  data to achieve the best possible series.

  Monthly Revisions.
   CES first preliminary estimates of employment, hours, and earnings are published each month approximately 3 weeks
  after the reference period. Estimates are then revised twice before
  being held constant until the annual benchmarking process. Second
  preliminary estimates for a given month are published the month
  following the initial release, and final sample-based estimates are
  published 2 months after the initial release.


The optimal level of inflation is very debated with unclear answers. There are many reasons, and a great answer would be very long. It should also distinguish between expected inflation and surprises.

I'm not going to do any of this, but giving you three reasons for a desirable positive level of inflation. This list is of course incomplete, also there are many reasons against too high inflation.

Downward-rigidity of wages

It is really unclear to economists why it is happening, but nominal wages seem to be downward rigid. It appears to be a behavioral thing (and might not be true at all, see Barro (1977)), but it appears that in crises, once presented with the choice of more firings or reduction of wages, most firms/workers decide to not cut wages but rather respond to the slump with increased separations.

To the extent that we believe this lack of reduction in wages is suboptimal, the central bank can enforce a reduction in real wages through increased inflation rates, and thus preventing separations to some extent.

Redistribution

This argument is based on Keynesian theory. Keynes claimed that the marginal propensity to consume out of income is smaller for rich households (and indeed, we do find this in the data  to some extent). Unexpected inflation is similar to redistribution from debtors to borrowers, as long as debt contracts are not indexed to inflation.

To the extent that poor people consume more out of this unexpected wealth shock and than rich people decrease their consumption, this redistribution will lead to an increase in aggregate consumption.

Room to cutback interest rates

This argument was most prominently brought forward by Krugman around 2008. In recessions, you want to be able to decrease nominal interest rates in order to punish households and firms for "holding cash" and incentivize them to spend it instead. If you start with low (say, 2-3%) nominal interest rates during normal times, you don't have a lot of space to cut back nominal interest rates during the crises.

If, instead, you would have higher nominal interest rates (Krugman argued for around 8%) and relatedly higher inflation during normal times, you could easier cut back nominal interest rates during busts and stimulate the economy.


In what sense are "new-Keynesian" models "new" and in what sense are they "Keynesian"?

In essence New-Keynesians adapt micro to macro theory. This is in contrast to new classicals which adapt macro theory to orthodox neoclassical market-clearing microfoundations. New-Keynesians adapt the rational expectations hypothesis but accept that market may fails due to wage and price stickiness and Friedman's natural rate hypothesis. The New-Keynesians, the RBC school and New Classicals focus on issues relating to aggregate supply and have been the dominating schools since the 1970's, especially the new-Keynesians have dominated the last one to two decades. On the other hand "old"-Keynesians and the orthodox monetarists mainly focused on issues relating to aggregate demand and these dominated economic thinking pre-1970's.

As an example of the differences between New-Keynesian New Classicals and the RBC school consider an increase in money: This increase will have real effect in a New-Keynesian model due to these market imperfections. In a new classical model on the other hand money will only have real effects if they are unanticipated (Lucas' Island model) while in a RBC model the increase will only feed into higher prices due to perfect clearing markets and rational expectations.

Note that it has been argued by N. Gregory Mankiw that the New-Keynesian school could just as well have been called New-Monetarist.

For an excellent reference on the different schools, what they represent and their differences see Snowdon and Vane: "Modern Macroeconomics: It's Origins, Development and Current State"


price elasticity: Linear regression low r square

I would raise two issues about the dataset to which the regression line was fitted:

The range of prices in the dataset.  If the datapoints all lie within a very narrow range of prices, then even small variation (whether real or due to measurement error) in the associated quantities can lead to a low coefficient of determination $R^2$.  In terms of the formula:

$$R^2 = 1 - \frac{\text{Residual sum of squares}}{\text{Total sum of squares}}$$

a narrow range of prices tends to result in little of the total sum of squares being explained and most of it being residual, resulting in a low $R^2$.  Note that it is assumed here that the regression takes price as the independent variable and quantity as the dependent variable.

The number of datapoints.  If the number is small, then, even within its range of prices, the dataset may happen to be unrepresentative of the distribution of points within the population of interest. This can result in the squared residuals calculated from the dataset being on average either much smaller or much larger than is representative of the population.  Thus it is possible that $R^2$ has been correctly calculated from data which is not representative of the population.


Under what condition would the law of one price hold?

Converting my comments to an answer. The law of one price (LoP) is an economic concept which posits that "a good must sell for the same price in all locations". This law is derived from the assumption of the inevitable elimination of all arbitrage. Source wikipedia.

Arbitrage opportunities imply the law of one price but the reverse is not always true. In logic, LoP is a necessary condition for arbitrage but not a sufficient one. In common language, we cannot out rule out the fact that the law of one price may hold without full arbitrage. If the price of an asset is not different for different people, the law of one price holds without assuming full arbitrage.


What does it mean if a country's exchange rate is said to be “overvalued”?

This is hard to define because in general in economics the definition of overvaluation of currency is very vague. In economics overvaluation or undervaluation would be a deviation from a person’s expectation of purchasing power parity (PPP) should be (see the textbook Money, Banking and Financial markets from Schoenholtz).

So technically if you think that 1 dollar should purchase 1 euro in long run, but actually the real-life exchange rate is such that it can purchase 1,1 euro you would say the dollar is overvalued.

This being said in actual research we don’t just try to guess what the proper exchange rate should be, but we estimate some model and if exchange rate deviates from model predictions we would say it’s over or undervalued. For example, now outdated but still popular model for exchange rate determination is the monetary model of exchange rates. In this model exchange rate depends on ‘fundamentals’ such as price levels, money supply, velocity of money, real output and interest rate.

In such model we say exchange rate is overvalued if the actual observed exchange rate has a higher value than we would expect given the values of country’s fundamentals (price level, output, interest rates etc.).


Thesis-advisor/student pairs that won the nobel prize

This seemed like a fun exercise, so using just the info on Wikipedia, I was able to compile this:

Honourable mentions, who I classify as economists who have at least two students who have won the Nobel include:

Wilson (Roth and Holmstrom)
Lovell (Prescott and Mortensen)
Tucker (Nash and Shapley)
Cassel (Myrdal and Ohlin)

Another interesting fact worth mentioning is that, while Krugman’s advisor, Dornbusch, did not win the Nobel, Dornbusch’s supervisor, Mundell, did. A similar comment applies so Lovell, who was Leontief's student, and had two students who won the Prize.

More interesting observations: predictably, for the vast majority of these pairs, the advisor won the prize before the student. Exceptions are Leontief and Samuelson; Fama and Scholes; and Hurwicz and McFadden.

Other triplets:

Miller-Fama-Scholes
Leontief-Samuelson-Merton
Leontief-Solow-Akerlof/Diamond/Stiglitz
Leontief-Schelling-Spence

We even have a quadruplet: Kuznets-Friedman-Markowitz-Sharpe.


features of Generalised Method of Moments estimation

It is a fancy, confusing and indirect way to say that GMM, being a Method-of-Moments estimator, does not require distributional assumptions in order to estimate regression coefficients, just like Ordinary Least Squares, and in contrast to maximum likelihood that needs to make distributional assumptions.

The wording is confusing, because "robust to distributional assumptions" means that we get consistent estimates even if we have made the wrong distributional assumptions.


What effect would destroying large amounts of physical currency have on government finances

In most modern economies, the central bank certainly has the authority to print money, when it does so to implement its legally defined objective. Let's consider what the implications of this 'elimination' of cash are.

The Balance Sheet

From the asset management side of the central bank, currencies are always a liability - because, amongst other things, they can be deposited in an account with the central bank, which will usually accumulate interest payments. Historically, many central banks used to allow currency to be exchanged for gold.

Central banks must always be concerned with their balance sheet. The reason why they cannot simply issue and distribute it to the masses (known as helicopter money) comes down to this management of liabilities - there is a real constraint here for these institutions. Issue too much currency, and commercial banks might not think you will be able to pay interest on their deposits. Or more aptly, they may believe the only way you can pay this interest is by printing money, decreasing the value of money and  potentially leading to inflation or a runaway zimbabwe situation

So the destruction of currency reduces the financial liabilities of a central bank, benefitting its balance sheet.

Impact on the Economy

Now we can look at the implications for the economy. Shown below is the fisher equation, a simple but valuable representation for money interacts with the economy.

$MV = PY$

This indicates that real output (the right hand side) is equal to the amount of currency in circulation, multiplied by how quickly that currency is circulated around. So if a central bank wanted to keep prices and output constant, it would want to eliminate any changes to the money supply. Otherwise, as you have intuited, there may be a decrease in output or the price level.

But we have to consider if this is really currency in circulation. Are people really using these large denomination notes in actual transactions, or just as a store of wealth? If this currency was unused to begin with then maybe there won't be much of a real impact.

We must also consider that central bank's typically interact with commercial banks only, and are typically most immediately concerned with maintaining their target interest rate with those commercial banks.

How might this play out in practice?

First, these protesters having burned some of their money and so may replace it by withdrawal from their accounts with commercial banks. These commercial banks distribute this physical currency out to consumers, and have less remaining to loan to between themselves. Since for them, loanable funds are more scarce, there will be upward pressure on interest rates.

To maintain the target interest rate, we will then see the central bank response, purchasing bonds from the commercial banks in exchange for cash. Thus the balance of money is restored.

But perhaps this destruction of money was great enough that it has made consumers considerably poorer. Now their bank balance is reduced, they may not want to spend as much. Over the short term, this would be expected to drive down prices and output.

If the central bank sees this play out, or is worried enough to anticipate it, it would likely reduce its interest rate target. With a lower cost to borrowing, we have increased spending in the economy, inflation of asset prices that make people feel richer and hopefully, a restoration of balance.

This is a hypothetical, and relatively simple policy analysis. A less conventional central bank could throw replacement money out of a helicopter, but this seems unlikely to happen in practice.

How does this effect the government?

It will likely benefit them. As I discussed, currency is a liability for central banks. So now some of it is gone, they can print more without having to demand any kind of asset in return. The article you have edited into your question argues that the central bank may print and give money to the government. Normally, because money is a liability, they would have to demand government debt in return. But because the balance sheet already incorporates this currency (which is just replacing what has been destroyed) it need not demand government bonds --- so the government can spend without incurring any debt.


The "identically - independently distributed" assumption (or any restriction thereof) on a sample, refers to the relation between random variables in different observations, not between the variables forming the same observation.

So in a multidimensional sample "of size $n$"

$$\{(y_1, x_{11}, x_{21}),(y_2, x_{12}, x_{22}),...,(y_n, x_{1n}, x_{2n})\}$$

the assumption applies between the variables of observation $i$ and all variables in any other observation. So $y_1$ is assumed independent from, say $y_2$, but it is also assumed indepedent from $x_{12}$. Also $x_{11}$ is assumed independent from $x_{12}$ but also from $x_{22}$, etc.

Certainly, $y_i$ is assumed stochastically dependent with $x_{1i}, x_{2i}$, while also $x_{1i}$ and $x_{2i}$ can be stochastically dependent for the same $i$, i.e. inside each observation.


Self-teaching 1st year PhD Microeconomic Theory

My prof is always telling us, if we want to pursue PhD level Econ in the future, we should master the full content of the following book:

Microeconomic Theory. Andreu Mas—Colell Michael D. Whinston and. Jerry R. Green. New York Oxford OXFORD UNIVERSITY PRESS 1995.

He also mentioned that there's main difference in viewpoint between graduate-level and top-level PhD Microeconomics. While in Grad School students calculate and theorize in the vector space, PhD students do mathematical operations using set theory almost all the time.


Oaxaca decomposition - Interpretation Interaction

In principle the idea is that there are three sources of the 29.5 change:

Lets assume we are taking period 1 as the 'baseline'. The first source of the 29.5 change is that the independent variables changed. To see the importance of this effect, you estimate education's effect on time for period 1. Then you apply the estimated model to the period 2 education and you get a predicted increase in time because the period 2 were more educated on average and the model predicts that higher education leads to more time. Suppose that estimate amounts to 22.78 minutes in this case.

The second source of difference is found by estimating the model with period 2 data. Then you look at the different coefficients on education from the period 1 and period 2 models and apply the difference to the period 1 data. In other words, you see how much the function changed, as measured for the period 1 education levels. Suppose that in the second period, the effect of education seems to be larger. This difference in the coefficients explains 50.38 minutes of the difference between 1 and 2.

Finally to interpret the interaction term, you have to imagine that you apply the difference in the models to the difference in the data. While the period 1 education using the period 2 model implies a high time and the period 1 model applied to period 2 education also implies a high time, it turns out that the period 2 model applied to the period 2 education yields a small time. This idea that when you interact the period 2 data with the period 2 model you get a different result than if you had just summed up the education change by the period 1 model and the model change by the period 1 data is what is at stake. In this case it turns out that the combination in yields a lower value than what is predicted by the sum of the differences. The interaction term is then difference between what you expected from the two individual differences and the result. In this case its a negative number, -43.66.

Another way to see this is to imagine we are decomposing the difference between f(x1,x2) and f(x2,y2). As you go from x1,y1 to x2,y2, you will advance over the x axis. You expect the function to change Dx=(x2-x1)*df/dx(x1,y1).AS you go from x1,1 to x1,y2 you will advance over the y axis. You expect the function to change Dy=(y2-y1) *df/dy(x1,x2). Lastly you compare your results and realize that there's an error, xi, so that f(x2,y2)=f(x1,y1)+ Dx+Dy +xi. Dx here corresponds to the endowments effect, Dy to the coefficients effect, and xi to the interaction effect.


If central banks become shareholders, where can investors go?

What would be left for investors to invest in?

Stocks and bonds, as always. Just imagine the CB as another institutional investor. Institutions invest in the capital markets all day every day. Another large player will put upward pressure on prices due to increased demand. Upward price pressure can either be good or bad depending on what side of the trade you sit. From the POV of other investors entering the market it could be bad. But for investors seeking to exit and the corporate management and employees who either own company stock or their compensation is impacted by their company's stock price, it could be good.

  ...would the ECB becoming a shareholder / bondholder amount to the same thing?

It depends. The scenario you describe simply amounts to a shift in the risk pool. The bond risk pool is shifted from sovereign debt to corporate debt. Which is inherently riskier but, again, it depends on which corporations. Risk quantification would require a thorough analysis in order to be useful.

  Could it soften or break the opposition against taking hair cuts?

Whose to say? Anything is possible. Further comment would be speculative.

  Who would be paying ultimately?

As with any bank or corporation, all profit and loss of the ECB is borne by its shareholders. In this case, these are the central banks of the EU's member states. Primarily, Germany, France and (soon-to-be formerly, post-Brexit) UK. Closely followed by Italy and Spain. See this page for more details.


Where can I find data on income and social mobility over time? How far back is data available?

Maybe check Clark's work. He allegedly uses long time series.
http://jasoncollins.org/2014/09/30/the-genetic-basis-of-social-mobility/


Economics of Health Textbook Recommendation

The two main textbooks in any Health Economics class are: Michael F. Drummond - Methods for The Economic Evaluation of Health Care Programmes and Andrew Briggs - Decision Modelling for Health Economic Evaluation. These are must read books. Michael Drummond is a legend in the area of Health Economics. He is the author of two major textbooks and more than 600 scientific papers, has acted as a consultant to the World Health Organization and was Project Leader of a European Union Project on the Methodology of Economic Appraisal of Health Technology. He has also served on the Boards of Directors of the International Society of Technology Assessment in Health Care and the International Society for Pharmacoeconomics and Outcomes Research.


How does inflation impact the welfare of the economy?

First, there are direct negative effects from inflation, known as the shoe leather cost and menu costs of inflation. These are direct costs which come from price changes: People will carry less cash, need to update their knowledge about prices, firms need to update their prices and wages, and similar. These are usually small, but typically higher given higher inflation rates.

Second, there are direct redistributional effects, which come from the devaluation of money and everything denoted in nominal terms, including most importantly nominal bonds and debt (credits). Usually, we imagine borrowers to be less wealthy than lenders (1). Here (unexpected) inflation will redistribute wealth from the rich to the poor. Under most standard welfare functions, this will be a first-order welfare improvement.

Third, we typically think that poorer people have higher marginal propensity to consume (MPC). Assuming that we are in a situation with inefficiently high savings rates (for example in Keynesian traps), the aforementioned redistribution will improve allocations, but this is typically a second-order effect (but also this will depend on what you believe is true about the Keynesian multipliers).

Wage contracts belong into the second argument, but deserve a special mentioning due to their extensive treatment in the literature.

(1): This is sufficient, but not necessary. In order to get a higher MPC from borrowers than from lenders, it is sufficient that the borrowers are not optimally smoothing consumption over time (for example, because they are up against a borrowing constraint) - we call these types of consumers Hand-to-mouth, because they will typically consume all their disposable income. As long as the distance between the (aggregated) borrowers consumption and what they would like to consume in a frictionless environment is larger than the distance between the (aggregated) lenders' consumption and what they would like to consume in a frictionless environment (2), devaluing debt will increase consumption.

(2): It may sound surprising, but there is a significant mass of wealthy hand-to-mouth consumers. See Kaplan et al, 2014


What is the actual numerical value of the velocity of money?

(It appears that the answer was supplied in the original question. The third link was to the time series for the M2 velocity for the United States. This answer is just taking text out of the comments.)

The linked chart link to FRED website is of the ratio: (nominal GDP at an annual rate) divided by M2 (for the United States).

The units for annualised GDP are \$ per year, and the units for M2 are \$; the \$ cancels and we end up with 1/years as units. (This is tracking units the way that is done in the physical sciences.) If we took non-annualised quarterly GDP, the ratio would be 1/4 as large.

Some economics charts can be confusing from the perspective of physical sciences, because the inverse time units for flows are suppressed. This is a convention that has held for decades; you just have to know what is being done. It probably causes problems because people often look at mixes of stock and flow variables, such as the infamous debt-to-GDP ratio.


The European Commission produces reports on tourism as part of its Eurobarometer surveys, which you can find here: http://ec.europa.eu/public_opinion/index_en.htm. If you do a bit of searching you might find some useful data in these reports. For example, a report of European's attitudes towards tourism can be found here: http://ec.europa.eu/public_opinion/flash/fl_370_en.pdf. I know this is not exactly what you asked for, but there is usually reference to cultural tourism in these kinds of reports and from there you may get further breakdown in terms of music tourism. This, I cannot guaranteed, however.

A second place to look would be on the websites of national tourism offices (they usually have a publications section with reports and statistics) and also on websites of lobbying groups; for example, UK Music has a research section and there they cite numbers on economic impact of music tourism. See here: http://www.ukmusic.org/research/measuring-music-2015/.

Generally speaking, it's quite hard to come by good data for arts and tourism - except maybe in France. Hope my suggestions are helpful and best of luck.

(I made this an answer because it was too long for a comment).


The reason why you find it mainly for large companies is that the market capitalization is by definition the market value of company’s equity.

If the company is not public it might not be possible to get the market value as for that you need to observe the price at which the company trades its equity (like shares on stock market for example). Most of the time only big companies go public because it involves a ton of extra regulatory requirements that small firm might not be able to even cope with.

Hence, small - medium size partnership or private limited companies do not generally sell equity publicly. The best you could do there would be some estimate based on some expert’s judgement of market valuation but I imagine that will be very hard to find and inherently very inaccurate. Sometimes even big company might decide to stay private to avoid greater regulatory burden but for bigger companies it’s usually easier to find some expert valuations or estimates of the firm value based on some other observable characteristics.


Basic question on trade based on comparative advantage

You did not factor in the 4 Darwinian goals of global trade:

Find foreign markets to absorb excess production, that is, where excess production can be dumped.
Extract foreign resources at low prices.
Deny geopolitical rivals access to these resources.
Open foreign markets to domestic capital and credit so domestic capital can buy up all the productive assets and resources.


The difference is that in buying the bond, the central bank now owns a bond, and a fiscal deficit has not been directly monetised.

That means that:

the government will have to keep paying interest on the bond, to the central bank, and will have to redeem it if it's not a perpetual. Remember, some governments are completely distinct entities from central banks, with no guaranteed way to move money between them in a consequence-free manner: the Eurozone governments are a good example.
The central bank can sell that bond when it wants to, giving it an extra means of contracting the money supply.
Currency speculators are less likely to start circulating around the currency like vultures that have spotted dying prey.


Mathematical analysis of an economic model - stuck in calculations

There exist implicit assumptions in the set up that exclude various cross-price effects on the demand side, related to business behavior as well as to consumer behavior. Accepting the set up (and assuming downward-sloping demand and upward-sloping supply), the old-fashioned verbal treatment goes as follows:

An increase in $G$ will move the demand schedule $Q_1^d$ outwards, in the $\{Q_1, P_1\}$ space. This will tend to bring a new equilibrium for good 1, at a higher price $P_1$, and a higher quantity $Q_1$.

A higher $P_1$ increases costs for the supply of good $2$, and so it will shift upwards the supply curve for good 2 in the $\{Q_2, P_2\}$ space, while leaving the demand schedule for good $2$ unaffected. This will produce an equilibrium with higher $P_2$ and lower $Q_2$. A higher $P_2$ in turn will tend to shift upwards the supply curve for good 1 in the $\{Q_1, P_1\}$ space (again, because due to higher costs, lower quantity is supplied at any own price than previously).

This will tend to increase equilibrium $P_1$ even more, but it will tend to offset the increase in the new equilibrium $Q_1$.

So we have learned:
In market $2$, an increase in $G$ for good $1$ will lead to an equilibrium with higher $P_2$ and lower $Q_2$.
In market $1$ an increase in $G$ will lead to an equilibrium with higher $P_1$, but ambiguous effect on the new equilibrium $Q_1$.

I'll leave to the OP the task of mathematizing these results and see also whether the ambiguity can be resolved.


The Myth of Exploitation According to the Labor Theory of Value

The whole point of the (specific) Labor Theory of Value that you apparently refer to, is that, what Labor "injects" as value into goods produced, is more than what Labor needs to consume in order to survive. This is a fundamental assumption/argument, in order to answer the question "how profits come about?".

Then, since the theory argues that profits spring solely from Labor, it follows that they should not be appropriated by the holders of capital (the holders of capital should only get the value of capital etc that goes into the goods produced, a value that does not "inject more" than what it already is). Simply put, after paying third parties for purchases etc out of Sales, and calculating depreciation (which is what the holders of capital should receive), the rest, that would appear as profits, should be distributed to the workers (because it is the workers that created this surplus, not the capital) and not retained by the capital holders. But because they are not thus distributed, the theory of exploitation emerges.

Even more simply, I need to eat one Kgr of bread to produce two Kgr of bread, in which production other goods, equipment etc have participated, whose value amounts to only half-Kgr of bread. I take as wage the one Kgr of bread, and all other factors of production take their half-Kgr of bread. Another half-Kgr of bread remains. It is taken by the owner of the bread business as "profits". It should be given to me, because it materialized solely because of the participation of my labor in the production process -or so the theory argues.

Therefore the phrase in the question

  "We might then go further and assume that the labor put into it was as
  valuable as the goods used to fuel that labor (and so on and so on)"

negates the fundamental premise of this theory, and leads to some other theory, one apparently incorporating a general principle of "preservation of value" (which by the way, is what the Accounting discipline is based on).


Literature Request: how does the house quality distribution change endogenously over time?

A paper of interest but it is mostly about trend changes in quality and not business cycle fluctuations:

  This paper employs a new national data set to estimate constant
  quality dwelling prices for four regions of the country. The data set,
  collected by the National Association of Realtors, provides sale
  prices, locations, and characteristics of 5581 dwellings sold during
  the years 1986–1992 inclusive. The paper provides the first national
  hedonic analysis based upon transaction prices of sample dwellings.
  The estimated equation implies that more than half of dwelling price
  increases during the period resulted from quality improvements in
  dwellings.

New Hedonic Estimates of Regional Constant Quality House Prices (Millsa and Simenauerb (1996))

Because housing is a long lived asset and it is somewhat unusual to significantly change the size of a house after it is built we could explore your question with the quantity and quality of new housing, assuming existing housing is largely unchanged. The US Census's New Residential Construction Survey has rich information about new housing. For example, on Characteristics of New Single-Family Houses Sold one can see many attributes of new houses including size, bedrooms, bathrooms, number sold, and much more. The Median and Average Square Feet of Floor Area in New Single-Family Houses Sold by Location table goes back annually to 1978. So you could compare it with RGDP per capita to get an idea of how this varies with wages.


Why stochastic dominance is "stochastic"?

In the below figure, CDF $F(\cdot)$ is first-order stochastically dominated by $G(\cdot)$. But $X_1$ and $X_2$ fall within the support of both distributions. So it would be possible to draw $X_1$ from $F$ and $X_2$ from $G$, or to draw $X_2$ from $F$ and $X_1$ from $G$.

More generally, if $X_G$ is a draw from $G$ and $X_F$ is a draw from $F$ then $X_F-X_G$ will sometimes be positive and sometimes negative. In this sense, the dominance is only stochastic: $G$ produces larger draws than $F$ on average, but not all of the time.


What does it mean when a bank is "fully loaned up"?

The reserve requirement on deposits ensures that (for a given $reserve\ requirement(\%)$ and $deposits(\$)$): $$loans(\$) \leq deposits(\$)  \cdot (1 - reserve\ requirement(\%))$$
Fully loaned up means that this holds with equality:
$$ loans(\$) =  deposits(\$)  \cdot (1 - reserve\ requirement(\%)) $$


Can I get a more detailed explanation of the Weak Axiom of Profit Maximization?

You don't need more detail, you need less:

Suppose a firm could make production choice A, or production choice B.  Suppose the firm chooses A, and is maximizing profits.  Then, this implies that the firm is making as much or more profits as it would make if it had chosen B.


How does the federal funds rate compare to the federal discount rate?

You are right the two series closely follow each other for the reasons you mention. During quite some time the discount rate actually used to be a ceiling for a funds rate. This is precisely, because if the federal funds rate was below
the discount rate, most banks adjusted their reserve positions in the federal funds market and when the federal funds rate rose to the level of the discount rate it was cheaper to borrow at discount rate so banks simply do that.

However, they are not always as close as you would expect, especially the discount rate does not anymore acts as a ceiling. There are several reasons for that in literature, primarily because borrowing at both of those rates has a different strings attached to it.

For example, Meulendyke (1998) argues that:

  Since the mid-1960s, the basic discount rate frequently has
  been below the prevailing Federal funds rate. The Federal Reserve
  has relied on administrative procedures to limit access to the window by restricting the frequency and amounts of borrowing.
   ... The Federal Reserve’s administrative
  restrictions used to be the primary factor that discouraged borrowing, but in the last decade banks themselves have been responsible
  for much of the limitation. Heavy borrowing in the 1980s by a few
  banks with financial difficulties caused others to avoid the window
  for fear depositors might conclude that they were also in trouble.
  Reluctance to borrow contributes to a seemingly contradictory
  result—namely, that increases in the amount of reserves in the banking system, when provided through the discount window, make
  reserve availability more restrictive on the margin because such
  increases put banks under pressure to find other sources of reserves
  to repay the loans.

and she further argues that:

  Until the mid-1960s, the Federal funds rate did
  not trade above the discount rate. During “tight money periods,” when the
  Trading Desk was fostering significant net borrowed reserve positions, funds
  generally traded at the discount rate, and the funds rate was not considered a
  useful indicator of money market conditions. When free reserves were high,
  funds often traded below the discount rate and showed some day-to-day
  variation. At such times, the funds rate received greater attention as an
  indicator of reserve availability.
  There was considerable surprise when the funds rate first rose above
  the discount rate, briefly in October 1964 and more persistently in 1965. As
  large banks became more active managers of the liability side of their balance sheets, they borrowed funds in the market in a sustained way. Banks
  had introduced large negotiable certificates of deposit (CDs) in 1961. But CD
  borrowings were (until 1991) subject to reserve requirements and (until
  1970) to interest rate ceilings under Regulation Q. Borrowings from other
  banks through the Federal funds market were free of reserve requirements
  and interest rate ceilings. Furthermore, they were not subject to the restrictions on prolonged use that were applied to the Federal Reserve’s discount
  window. The changes in liability management techniques meant that
  individual banks could expand credit even when they did not have free
  reserves if they were willing to bid aggressively for wholesale funding from
  other banks.

These and possibly also some other reasons will lead the two to diverge in some situations as borrowing at those rate might be subject to different conditions and have different implications. You can see on the graph below which plots of the federal funds rate in red and the discount rate in blue (based on FRED data) that they can sometimes differ quite substantially but indeed tend to follow each other closely.


How to find the corresponding APR to a given APY?

The length of the CD is not relevant. The point of APR and APY numbers is that they annualize the rate you pay over a time frame that may be smaller or larger than a year. What matters is the compounding frequency. CDs generally compound on either a daily or monthly basis, so n will be 12 or 365, depending on the CD.
Is this a real-life problem or an homework/practice question? If it's the latter, you should probably assume n=12 because teachers seldom want you to mess with daily compounding without also telling you how many days you should assume there are in a year for compounding purposes.


Why is the Chilean economy so much better than the rest of the countries in South America?

This is honestly a pretty complicated question - it's very difficult to pinpoint any exact factors that have led to Chile being so successful relative to its Latin American counterparts. I'm not particularly knowledgeable on the political factors, so I'll focus on some economic ones.

As a developing country, Chile has several factors that have really helped with exports.

Copper

Chile is by far the world's largest producer of copper. Copper is essential for electronics - meaning that Chinese factories have been buying massively - and Chile has been able to use its natural resources to boost its own economy. Copper is also far less volatile in price than oil; prices have only begun to go down because of slowing growth in China.

Unique position in Southern hemisphere

Since the seasons are swapped between the northern and southern hemispheres, Chilean farms can produce fruits and vegetables to sell to the Americans in the middle of winter. If you live in the States, go to a local supermarket in the middle of winter and look at the fruits. Some of them (for me, I see grapes) are from Chile.

Chile also exports fish. It has the longest coastline in the world, and has had some success in fish farming; both of these factors help Chile sell a lot of fish.

This is just part of why Chile has been successful (I haven't even mentioned government policy!) but hopefully it'll give you some background on why Chile has been more of a success story than other Latin American nations.

Source: http://atlas.media.mit.edu/en/profile/country/chl/#Exports


Why are marginal rates of substitution are identical at equilibrium consumption levels?

One explanation I could think of: MRS (which is equal to $MU_x/MU_y$) is equal to the price ratio $P_x/P_y$. In equilibrium, all consumers will consume where MRS = price ratio, and hence MRS will be "identical" across consumers.

I agree that this might not be a terribly good way of putting it.


Fisher Effect vs Quantity Theory of Money and how an increase in the money supply lowers interest rates?

ceteris paribus - other things staying the same

What happens with a changing money supply depends on what other things stay the same in your thought experiment. Possibilities might include:

If real interest rates stay the same but you manage to increase the money supply and inflation, then nominal interest rates must rise by definition.  Historically, nominal interest rates have been higher in inflationary periods than in periods of price stability or deflation
If nominal interest rates stay the same but you manage to increase the money supply and inflation, then real interest rates must fall by definition.  This is likely to lead to an inflationary spiral as people borrow cheaply in real terms, increasing demand and the money supply further and so pushing prices higher
If you force inflation to stay the same but you manage to increase the money supply excessively compared with that inflation, you may need other controls such as price regulation.  This could lead to shortages of goods or an asset price bubble, at which point things other than money become important while interest rates could become less relevant


All correct up to 6), although you may want to be careful with "$D$ to $A$ is correct" kind of answer, "$D$ to $A$ is consistent" seems a better phrasing - one has no idea how hard it comes out to quantify those shifts, because they impact each other.

Point 6) changes interest rate, which means that money supply, demand and interest rate do not coincide: the equilibrium is lost on the money market. As it is lost with higher interest rate than the equilibrium rate, it means that people want less money than is available. Like every good, this causes money to lose value, which in turns means either inflation on the open market, or a quick reaction via a decrease in supply from the central bank. As the question explicitly ask for the effect on the market, you can suppose the central bank doesn't make a move, and let the market move into inflation. This does happen in cases when the central bank doesn't want to or cannot cut money supply, usually because the central bank is controlled by the government, that needs the money, and led to some disastrous effects in the past. For this reason, usually central banks are relatively independent from governments, and hopefully the situation is addressed by central banks rather than by the market.

Here is an example of a government-dependent central bank having decided not to cut money. Wheelbarrows were commonly used on the way to buy loaves of bread in 1920s Germany...


All-Pay Auction Mixed Strategy Equilibrium

In response to our policy on homework questions, I will work on the following abstract version of your question:

A single prize is to be allocated by an all-pay auction. Assume the value of the prize is 1 to both bidders, and this is common knowledge. Both bidders have a budget constraint $m$. If there is a tie, each bidder receives the prize with $\frac{1}{2}$ probability. Find a Nash Equilibrium in this game.

We shall focus on the symmetric equilibrium in this game, that is, both bidders are using the same equilibrium strategy. Before we get started, let's make two observations:

Claim 1: Whenever there is a mass point in the equilibrium strategy (i.e. there is positive probability to submit one bid), this mass should be put on the upper bound $m$. For if not, one bidder can always deviate to a slightly higher bid and obtain higher surplus.

Claim 2: Whenever a bidder randomizes, the bid should be uniformly drawn from $(0,t)$, and the probability that this happens should be $t$. This is because only in this case, a bidder is indifferent between any bid $b\in(0,t)$:
$$\text{Expected payoffs}=t\cdot \frac{b}{t}\cdot 1-b=0.$$

Therefore, we can guess that the equilibrium strategy takes the form
$$t\cdot \text{Uniform}(0,t)+(1-t)\cdot\delta_m.$$
That is, with probability $t$, a bidder submits a bid uniformly drawn from $(0,t)$, and with complementary probability she submits $m$. The only thing left is to pin down $t$. Note that if this is an equilibrium strategy, a bidder should be indifferent between any bid $b\in(0,t)\cup\{m\}$. Therefore, the payoffs from submitting $m$ should also be 0, i.e.
$$t\cdot 1+(1-t)\cdot\frac{1}{2}-m=0,$$
which implies $t=2m-1$. (Notice that for the existence of such an equilibrium, we must have $2m>1$.)

Therefore, a Nash Equilibrium in this game is that both bidders play the strategy
$$(2m-1)\cdot \text{Uniform}(0,2m-1)+(2-2m)\cdot\delta_m.$$

How does it map back to the original problem? I believe the OP can take it from here.


If you read Fed's balance sheet, you'll see a corresponding increase in its liabilities under various deposits. They simply credit the accounts of institutions who sold them the assets.


The question is quite straightforward, and you do not need the first
step that you have. You have (for some reason) a different multiplier
here for each time period. That is not the case- you simply have a
no-ponzi scheme condition, aka a transversality constraint. The problem
is expressed as $max\,\sum\beta^{t}U(c_{t})$ st $c_{t}+k_{t+1}\leq f(k_{t})$.
The stream of utility to the agent is $U=\sum\beta^{t}U(c_{t})=U(c_{0})+....\beta^{t}U(c_{t})+\beta^{t+1}U(c_{t+1})+...+\beta^{T}U(c_{T}).$

Substitute in $f(k_{t})-k_{t+1}$ for $c_{t}$. Then, we derive wrt
$k_{t+1}$ and set equal to 0:$\frac{\partial U}{\partial k_{t+1}}=\beta^{t}U'(c_{t})-\beta^{t+1}U'(c_{t+1})f'(k_{t+1})=0\Rightarrow U'(c_{t})=\beta U'(c_{t+1})f'(k_{t+1}).$

You do not need the other equation you referred to.


Pure exchange economy: Given an initial endowment are multiple equilibria possible?

Yes. The Debreu version of the Sonnenschein-Mantel-Debreu theorem guarantees that excess demand has to satisfy very little restrictions if there are as many consumers as commodities.

An explicit example of multiple equilibria in a $2\times 2$-exchange economy can be found in

  Shapley, L. S., and M. Shubik. “An Example of a Trading Economy with
  Three Competitive Equilibria.” Journal of Political Economy, vol.
  85, no. 4, 1977, pp. 873–875.

In the example, both consumers have even quasi-linear preferences, which are (necessarily for nonuniqueness) linear in different commodities.


Utility function used to indicate bliss point

An answer that meets all the current demands of the question:

Let $(x_b,y_b)$ be the blisspoint. Let
$$
U(x,y) = \left\{
\begin{array}{cc}
1 & \mbox{ if } (x,y) = (x_b,y_b) \\
0 & \mbox{ if } (x,y) \neq (x_b,y_b).
\end{array}\right.
$$
The demand function for $x$ in this case is
$$
D_x(I,p_x,p_y) = \left\{
\begin{array}{cc}
x_b & \mbox{ if } p_x \cdot x_b + p_y \cdot y_b \leq I \\
\left[0,\frac{I}{p_x}\right] & \mbox{ if } p_x \cdot x_b + p_y \cdot y_b > I.
\end{array}\right.
$$


Identification of switching costs from price shocks

This is not a complete argument but I'm going to give a brief sketch. Assume that producers set prices instead of quantity (which seems to be the model above). Assume that cost shocks do not affect demand, in other words $q(P,C,S) = q(P,S)$, and that they are not correlated with some kind of variation in switching costs. For simplicity, imagine also that this is a constant cost model. Then:

$$\frac{d q^\star}{d C}= \frac{\delta q}{\delta P}\frac{d P^\star}{d C}$$

Assuming switching costs are such that $\frac{\delta q}{\delta P}$ is low for high switching costs, this could give you an idea. If you enough form for $q$ you should be able to pin down $S$ using just-identified GMM (1 moment, 1 parameter).

Let's take a more concrete example, using a more concrete problem. Assume a two period model. In the first period, prices are announced, in the second period there is a surprise shock to the costs of $B$ (WLOG they get cheaper) which $A$ does not experience. This causes a switch from $P_B$ to $P_B^\prime$. Suppose a continuum of consumers, $i$, each with a time invariant valuation of each good, $j$, $V_i^j$, jointly distributed according to $F$. Then people who switch to $B$ from $A$ this period are those individuals for whom:

$$U(V^B_i - P_B^\prime -S)> U(V^A_i - P_A)$$
but:
$$U(V^B_i - P_A - S)< U(V_i^A-P_A)$$
This means:
$$q^\prime - q = \int_V1_{U(V^B - P_B^\prime - S)> U(V^A-P_A) >U(V^B - P_A - S)}dF(V^A,V^B)$$
Note that this is basically analogous to the argument above using derivatives. With form on $V$ and $U$ estimation is straightforward.

More complicated setups, (for instance, consumers with infinite horizons) will require more complicated arguments, but the basic identification will be the same. In this setting what you really need are forms for demand and not costs. As far as demand goes, with only this data you are restricted in what you can estimate. With linear utility, and one of the goods valuations normalized to 0, you could probably pin down the mean of $V_i^j$, and maybe a bit of its cross-sectional distribution. More detail would be possible with consumer level data.


I think you're getting things backwards. Marginal cost (at a certain quantity $q_0$) really only makes sense as the derivative of the total cost function with respect to quantity at point $q_0$:
$$
MC(q_0)=\frac{dTC(q_0)}{dq}.
$$
The interpretation of marginal cost as "the cost of producing the last unit of output or the cost of producing the next unit of output" is based on the (linear) approximation
$$
\widehat{MC}^+(q_0)\approx\frac{TC(q_0+1)-TC(q_0)}{(q_0+1)-q_0}\quad\text{or}\quad
\widehat{MC}^-(q_0)\approx\frac{TC(q_0)-TC(q_0-1)}{q_0-(q_0-1)}
$$
This interpretation is usually introduced in undergraduate textbooks because of its intuitive appeal, not for its mathematical rigor. Imagine trying to explain the concept of marginal cost as the cost associated with an infinitesimal amount of change in quantity to a freshman who probably has no idea what "infinitesimal" means, let alone the concept of derivative.

A first problem related to interpreting marginal cost as the cost associated with the previous or the next unit of output is that producing the two units (the previous and the next) may imply different costs. This is illustrated by the formulas $\widehat{MC}^+$ and $\widehat{MC}^-$. Suppose total cost is quadratic, i.e. $TC(q)=q^2$. Then it can be easily verified that $\widehat{MC}^+(q_0)\ne \widehat{MC}^-(q_0)$, for any admissible $q_0$. This is problematic because we want the marginal cost (at a particular $q_0$) to be unique.

Second, and this relates to your question at the end, the concept of "unit" is somewhat arbitrary. And this arbitrariness makes measures such as $\widehat{MC}^+$ and $\widehat{MC}^-$ unstable as well. To illustrate, continue with the quadratic total cost example. When the unit of quantity is kg, the "marginal cost" at $1$kg--as you would refer to the cost associated with producing one extra unit--is $$\widehat{MC}^+(1)=2^2-1^2=3.$$
But what if we suddenly want to change the unit of quantity to gram (g)? Then the "marginal cost" at $1$kg, or $1000$g, would change as well:
$$
\widehat{MC}^+(1)=(1.001)^2-1^2=0.002001.
$$
To avoid such instability caused by changes in units of measurement, marginal cost is defined as a derivative, not as a difference.

If you want to measure the change in total cost due to change in some arbitrary quantity, simply apply the first formula in your question. But keep in mind that this cost is not marginal cost as an economist would understand it.


"omission bias" vs "action bias" -- how can they be reconciled?

This is a great question, you're not the only one confused. We just completed a working paper on this very topic: "What is action, what is inaction? A review of action-inaction biases and recommendations for term use and typology" and we cover these two biases and many other action-inaction biases.

See page 5:

  Building on the action-effect is the omission bias. The omission bias
  has been defined in many different ways, and is sometimes regarded to
  be the same as the action-effect. We view the two biases as different,
  with the omission bias extending the action-effect to focus on the
  choices made between action and inaction or about moral attributions
  to action and inaction, rather than the experience of negative
  emotions or regret as a consequence of action-inaction. In its
  broadest definition, the omission bias is the phenomenon that people
  prefer omissions to commission when there is the possibility of a
  negative outcome (Spranca, Minsk, & Baron, 1991; Ritov & Baron, 1990).
  Using this definition, if we consider the classic action-effect
  scenario, the two investors facing a risky situation and presented
  with the options of taking action and switching versus not taking
  action and sticking to the initial investment would prefer inaction to
  action. Linking with the action-effect, the bias could partly be
  explained as a strategy to minimize anticipated-regret (Anderson,
  2003; Wong & Kwong, 2007; Zhang & Fishbach, 2005), possibly further
  heightened by related biases such as loss aversion (Tversky &
  Kahneman, 1991) and risk aversion (Kahneman & Tversky, 1979; Tversky &
  Kahneman, 1992).

Briefly, the "action effect" is the tendency to regret actions more than inaction (negative outcomes) leading to a preference for omissions. In Bar-Eli's paper the action-effect is reversed to an inaction-effect (Zeelenberg etal. 2002) due to normality, the expectations for goal keepers to act. Therefore, the reconciliation is by using normality from norm theory (Kahneman & Miller, 1986).

Hope that helps, we'd appreciate feedback if things aren't clear.


Difference type of crowding out effect in economics

A vertical LM curve means that any change in government expenditure will result in a change of the interest rate which exactly offsets the initial change of public demand (private demand increases/decreases).

A vertical AS curve indicates the level of long-run output. If AD shifts, all that this will create is a change in prices in the long-run. This is also called the Classical Dichotomy: real variables are independent of monetary variables.

So, both diagrams tell the same story about crowding out: changes of private expenditure offset changes of public expenditure. The diagrams just highlight different parts of the same mechanism.


How should I regress income inequality on economic growth?

If your are interested in the statistic relationship between income inequality and economic growth, you should use GDP per capita growth since the others indicate level rather than growth.
More generally, the choice of the variable in your (and any other) regression depends on your economics model, otherwise you are just doing some linear projections.
A good reference you may start with this topic is "Banerjee, A. V., & Duflo, E. (2003). Inequality and growth: What can the data say?. Journal of economic growth, 8(3), 267-299."


How to find data about earning per share of S&P 500

A Google search turns up sites like

http://www.wsj.com/mdc/public/page/2_3021-peyield.html
http://www.multpl.com/table?f=m
https://www.macrotrends.net/2577/sp-500-pe-ratio-price-to-earnings-chart
https://ycharts.com/indicators/sp_500_pe_ratio

which slightly disagree about whether the current trailing ratio is just above or just below 20


Osborne, Nash equilibria and the correctness of beliefs

Introducing the language of beliefs here is slightly strange, given that beliefs do have a very specific meaning in other parts of game theory.

Indeed, Osborne's description is reminiscent of a Bayes Nash Equilibrium. We could introduce the notion of beliefs into the normal form of a complete information game as follows: suppose that with probability $a_i$ each player, $i$, is a "strategic" type who will play according to (Nash) equilibrium, and with probability $1-a_i$ he will select some strategy uniformly at random (because, say, he is indifferent across all actions). We thus have a Bayesian game where thinking about beliefs is more natural.

The Bayes Nash solution concept then says that $i$'s strategy must be optimal given the expected play induced by the other players' strategies and the beliefs over their types implied by $\{a_j\}_{j\neq i}$. If we look at the limit as $a_i\rightarrow 1$ for all $i$ then the Bayes Nash equilibrium of this game will coincide with the solution concept described by Osborne.

I guess the reason Osborne wrote it like this is a pedagogical one, given that this is an introductory text. When we introduce students to static games, we tell them that player $i$ best responds to the actions of the other players. Students naturally want to know "how can they respond to a strategy chosen simultaneously without knowing what that strategy will be?" This is, in many senses, a philosophical question. Common answers are

If the game is one that is played often then (putting aside issues of
other outcomes that can be sustained in repeated games) we can think
of Nash as being an equilibrium in the sense that if we converge
there we can develop a norm whereby people continue to play that
equilibrium indefinitely (and expect others to do the same).
If the game is really one-shot then we usually invoke the idea that players are going to try to predict what others will do—and our equilibrium notion embeds the idea that these predictions must be correct.

It seems that the predictions in the second point correspond to the "beliefs" invoked by Osborne. However, it is important to stress that these predictions/"beliefs", are merely an informal/intuitive tool for helping us to conceptualise what is going on in an equilibrium and are not part of the definition of such an equilibrium. The concept of Nash equilibrium itself is completely agnostic on the notion of beliefs (as you note in a comment, it is defined only over actions), which is why, when Osborne goes on to formally define Nash equilibrium, he does so without invoking the idea of beliefs at all.


This depends on the kinds of beliefs players can have about their opponents’ play. If opponents' strategies can be correlated, then the strategies surviving IESDS are exactly the rationalizable ones. But if you only allow opponents to choose independently (this is the standard approach in most textbooks), then these two sets are identical for 2-player games but need not be identical for games with more than 2 players.

Here is an example: Player 3's strategy Z is undominated, but not rationalizable (assuming independent mixtures of players 1 and 2).


What are the logistic implications of using bottle caps as currency?

How would inflation work in such an economic environment?

Supposedly, whenever a wastelander opens a bottle of Nuka-Cola, "new money" is injected into the world, so that would theoretically drive up inflation, but very minimally. Also, information asymmetry would certainly cause prices to fluctuate differently in different regions of the game. I think we need to assume that banks do not communicate with each other, at least not regularly.

  How can people buy expensive objects like houses or maybe vehicles, (they cost at least 10000)?

While caps are definitely accepted as a form of currency in the Fallout world, many would opt to convert their caps to a larger capital asset that is easier to carry around. Then, the purchaser can use that to barter. It's not ideal to carry tens of thousands of caps, as you're probably implying.

  How would a bank know if another has launched a new version of purple coin that is worth more than any other single bottle cap?

The bank would probably find out when someone tries to convert the new currency for purified water or whatever the good was. My guess is that the new currency would also be commodity-backed based on something else (stimpacks, perhaps?).

  If a new bottle cap of unique color appears, how to corroborate that a bank in another village supports its value? (that is, that is not false).

Not sure how this would work. Maybe show the bank the bottle which the cap was found on? Show that it was indeed Nuka-Cola branded. Then, the bank can take-it-or-leave-it. I'm only aware of regular Nuka-Cola and Nuka-Cola quantum, and I think they have the same cap style.


Why do many occupations show a gender bias?

I should say pre-emptively that I'm not satisfied with ideas that merely may explain the effect, as yesterday's question produced a proliferation of rationalisations and claims that lacked rigour.

I'd prefer to see an explanation that attempts to be comprehensive, or supports contentious claims (such as those about women's productivity) with research.

Your question and the yesterday's one are contentious even in the literature, which means there is no consensus among the economists and no one can give an answer that satisfies everybody. Given the complexity of these questions, they can involve and interact with many different issues, all of which matter to some extent. Even in the literature, the common way is to attack them from one specific perspective rather than to offer a "comprehensive" answer. Also a convincing explanation usually requires careful data analysis, qualified modelling and many robustness checks, which suits more to an academic paper rather than to an answer in this forum.
However, I agree with you that we should avoid rationalisations and claims that lack rigour and the support of academic references. In the comments of this question someone has already mentioned that some recent papers argue that commuting requirement can be one potential reason. I thus list some recent papers that suggest other deep-in-root causes below.
Adda, Dustmann, Stevens 2017 finds that different occupations can diverge in the amenity "child raising value" and in the loss of accumulated skills when interrupting work careers. Given the fertility consideration, women would thus more likely to choose the occupations that have lower wage growth.
Occupation choice is also related with education choice. Women's underrepresentation in the math-intensive fields (i.e. STEM fileds) translates into their underrepresentation not only in high-paying occupations but also in occupations where the gender wage gap within the occupation is particularly small. A survey paper by McNally 2020 reviews many arguments behind this gender differences in tertiary education.
Hsieh, Hurst, Jones, Klenow 2019 studies the roles that the discrimination in the labor market, the barriers to forming human capital, and the differences in preferences or social norms play in individuals' occupation choice. They find that the improvement in the discrimination in forming human capital is the most important driver behind the increased participation of women in high-wage occupations in recent decades.
In additional to typical economics reasoning, Bertrand 2020 suggests that endogenous preferences and self-fulfilling prophecies caused by gender stereotypes can be important for the low share of women in STEM careers and occupations. Also, Grosjean & Khattar 2019 shows that cultural persistency can affect womens' occupational choices.


Did the idea of “making money” originate in the United States?

No, the idea of wealth as something to be created did not originate in the United States.  It was part of the mercantilist approach to national economic policy that was widely adopted in Europe in the 16th to 18th centuries.

Mercantilism involved a range of policies, many of which were designed to increase the wealth of one country at the expense of others and probably reduced wealth overall, eg restrictions on international trade, and warfare to gain control over sources of goods.  However, it also involved policies that were not at the expense of other countries, including promoting industry via research or subsidies, maximising the use of domestic resources, and (see under France in the link above) decreasing internal barriers to trade, reducing internal tariffs, and improving infrastructure.


Is the mode of wage distribution a meaningful economic indicator?

Yes, it is a very useful indicator.  You are correct that it has greater noise, but, depending upon how it was calculated it can be a very useful measure.

Statistical efficiency is a nice thing, but it implies a very specific loss function.  Let us imagine you are a government.  For the US government, the mean wage includes individuals like Bill Gates.  The implicit loss function is quadratic loss.  However, is this a politically useful measure?

You want to get over 50% of the votes, and this is a highly skewed distribution.  That does not attest to the well-being of the middle of the body politic.  For that, you want to look at the median, which minimizes absolute linear loss.  It is less efficient. It tells you the experience of the median voter, provided votes are roughly distributed randomly across the income spectrum. However, since voting is an all-or-nothing proposition, you don't just want to know the health of the middle.

This brings us to the all-or-nothing loss function.  It is the populist loss function.  It is the most common experience of your people.  Hillary Clinton and David Cameron managed median voter theory perfectly; they both lost.  This is a problem that, in politics, used to be called the "farm-block," problem.  The median is not representative if any one large body can move the vote.  The body most likely to do this are the individuals at the mode.  If they act together, they move an otherwise tied election, and almost all elections are close to being tied if competition is real.  Voting is an all-or-nothing proposition.  The mode minimizes the all-or-nothing loss function. To ignore the mode is to watch a government collapse.

Do not think of this as an attempt to find the center of location.  Think of it as a gamble that no government wants to lose.

Don't use bins, find the supremum of a kernel density estimate.