Are there any states that don't have debt?

The classic answer here would be Libya and Brunei, but I think Libya now has debt.

Brunei is a strange case in that it uses a joint currency with Singapore dollar, controlled by the monetary authority of Singapore, so in effect you can use Singapore debt as a substitute for Brunei dollar investment.

Not having any debt, and having a free currency is generally a bad idea, since you cannot control interest rates. Furthermore, if you can afford not to have debt, then people will be begging to lend to you at such favorable rates it would be foolish not to borrow a bit.

On the other hand there are many counties with no net debt. Singapore and Norway don't need to borrow but do so to provide bonds for the financial markets. They then invest the money they borrow. Norway can borrow so cheaply they can buy US treasuries and make a profit.


Why does profit maximization implies that both of the following are equal?

The production side is modeled as follows:
There are $m=1,...,j$ identical firms (the number of firms is not necessarily equal to the number of workers of course), that operate in a perfectly competitive environment. This means that firms are price takers, both in the goods market and in the production input market, i.e. they take prices as given when they seek to attain their objective. It also means that markets "clear": in particular, prices adjust without frictions/delay so that all capital and all labor are employed. The firms solve a static (not intertemporal) problem: maximize profits period-by-period separately. Labor is provided totally inelastically, no labor-leisure choice here from the part of workers. Moreover each firm has a constant-returns to scale production function in capital and labor, i.e. the function is homogeneous of degree one.

Omitting the time subscript, the typical firm's production function is

$$F(K_j, L_j),\;\; j=1,...,m$$

and the objective of the firms is to maximize profits which are defined as the surplus of production over payments to labor $wL_j$, net payments to rented capital $rK_j$, and depreciation $\delta K_j$:

$$\max_{K_j, L_j} \pi = F(K_j, L_j)-wL_j-rK_j-\delta K_j$$

Note that these are "real" magnitudes in the economics sense of the word, i.e. that we have divided throughout by the price of output (we do not show it usually, we just say, "expressed in real terms").

Denote $\kappa_j \equiv K_j/L_j$, the capital-labor ratio at firm level. Due to the homogeneity of degree one we can re-write the maximization problem of the firm as

$$\max_{\kappa_j} \pi = L_j\cdot \big[F(\kappa_j, 1)-w-r\kappa_j-\delta \kappa_j\big]$$

Note that $L_j$ has become a multiplicative factor, so we can maximize only the term in brackets, and so only with respect to the capital-labor ratio. We also set $F(\kappa_j, 1) \equiv f(\kappa_j)$ to arrive at

$$\max_{\kappa_j} \pi = \max_{\kappa_j} \big[f(\kappa_j)-w-r\kappa_j-\delta \kappa_j\big]$$

The first order condition for a maximum is for the first derivative to be set equal to zero so

$$f'(\kappa_j) - r - \delta = 0 \implies f'(\kappa_j) - \delta = r$$

This is not yet the funky equation $(6.32)$,although it looks a lot like it, because the latter is expressed in terms of "per capita" capital $k\equiv K/N_1$,  i.e. at the level of individuals/consumers/workers, not at firm level.

How do we arrive at $(6.32)$? Well, since we have assumed that all firms are identical, that labor is provided inellastically, and also that the markets for production inputs clear, we have that

$$mK_j = K \implies K_j = K/m,\;\;\; mL_j = N_1 \implies L_j = N_1/m $$

So

$$\kappa_j = \frac {K_j}{L_j} = \frac {K/m}{N_1/m} = K/N_1 \equiv k$$

and now we have obtained $(6.32)$.

Note how all the assumptions made have been used in order to arrive at this result.


Dummy variable regressor OLS coefficient formula

To start, let's derive $\hat{\beta}$. As you note:

$$\hat{\beta} = \frac{\sum(D_i - \bar{D})(y_i - \bar{y})}{\sum (D_i - \bar{D})^2} $$

The key to evaluating this is to recognise that $D_i$ is a binary variable that equals $0$ or $1$. Thus, we can use standard results on Bernoulli random variables.

To evaluate the numerator, note that if $D$ is Bernoulli distributed:

$$ \mathrm{Cov}[D, y] = \mathbb{E}[Dy] - \mathbb{E}[D]\mathbb{E}[y] = \mathrm{P(D=1)}\mathbb{E}[y|D=1] - \mathrm{P(D=1)}\mathbb{E}[y] $$

By the sample analogue of this:

$$ \frac{\sum(D_i - \bar{D})(y_i - \bar{y})}{n} = \bar{D}(\bar{y}_{D_i=1}-\bar{y})$$

To evaluate the denominator, note that if $D$ is a Bernouilli random variable:

$$\mathrm{Var[D]} = \mathrm{P(D=1)}[1-\mathrm{P(D=1)]}$$

Sample analogue:

$$\frac{\sum (D_i - \bar{D})^2}{n} = \bar{D}(1 - \bar{D})$$

We therefore see that:

$$\hat{\beta}=\frac{\bar{D}(\bar{y}_{D_i=1}-\bar{y})}{\bar{D}(1 - \bar{D})}
= \frac{\bar{y}_{D_i=1}-\bar{y}}{1 - \bar{D}}
\\ = \frac{\bar{y}_{D_i=1}-(\bar{y}_{D_i=1}\bar{D}+\bar{y}_{D_i=0}(1-\bar{D}))}{1 - \bar{D}}
\\ = \bar{y}_{D_i=1} - \bar{y}_{D_i=0}$$

which is what we wanted to show.

It is then trivial to derive:

$$\hat{\alpha} = \bar{y} - \bar{D}\hat{\beta} =\bar{y}_{D_i=1}\bar{D}+\bar{y}_{D_i=0}(1-\bar{D}) - \bar{D}(\bar{y}_{D_i=1} - \bar{y}_{D_i=0}) = \bar{y}_{D_i=0}$$


Is it fair to compare 1950 US economy to 2016 US economy?

Yes. It is a fair comparison.

Human nature, market behavior and principles of finance and economics have not changed much in the recorded history of mankind. For example, the principles set forth by Adam Smith in his opus The Wealth of Nations are just as relevant today as they were in 1776 when the book was published.


After how many years will loan be paid with constant payment

Some of your equations are wrong and you need to account for the time value of money correctly in some places. It might be useful to write down a timeline and make sure to bring all payments to the same point in time (let's take time 0 as our reference)

Therefore, what you receive should have the same present value to what you pay (I'll use American conventions for punctuations):

$$100,000=\frac{5,000}{0.05}\left[1-\frac1{(1.05)^5}\right]+\frac1{(1.05)^5}\left(\frac{10,000}{0.05}\right)\left[1-\frac{1}{(1.05)^n}\right]$$ and solve for $n$.

So the first calculation was correct (though  your formula is not), since doing the first operation you get:

$$78,352.61665=\frac1{(1.05)^5}\left(\frac{10,000}{0.05}\right)\left[1-\frac{1}{(1.05)^n}\right]$$

However, hopefully you can see that you failed to account for the fact that the $10,000$ payments start coming at period 6 (so the annuity formula will  bring the payoffs to period 5 and you have to further discount them 5 more periods), further, you also did something funny with the signs when you solved for n. Instead I get that:

$$n=-\frac{\ln\left(1-\frac{(78,352.62)(0.05)(1.05)^6}{10,000}\right)}{\ln(1.05)}=14.2067$$

That means that they will finish paying the debt in period "$19.2067$", i.e. in period 20. That is after 5 years of paying $5,000$, 14 periods of paying $10,000$ and paying the remainder (less than 10,000) on period 20.


In a game with alternating moves and complete information, the Nash equilibrium cannot be a non-trivial mixed equilibrium?

This statement is wrong. Consider Alternating Matching Pennies with imperfect information (the follower doesn't observe the leader's move). The strategic form of this game is just the classical (simultaneous-move) Matching Pennies Game and the unique NE has both players mixing.


Is there a change over the business cycle in how much of economic activity is market-based?

Yes, the two figures are different because that is how the economy works in a circular flow. To calculate the size of economy (GDP), we will find the total expenditure, total income or total production. In theory, all three calculations should converge at the same figure. However, in economies they may give different figures. It is a fault of whole concept of GDP that it does not take into account any transaction for which there is not a market price available. So, no matter how much gardening you do at your home, you are not adding up anything in the economy. So, whatever mutual understanding you have with you neighbor and whatever transactions you make, it won't be considered.
In the second part of your example, you are spending 10 dollars and your employer is spending 50 dollars so that add up to 60 dollars. This happens because of circular flow, your income is somebody else's expenditure.

Production for which there is no price will not be counted.


When treating a relative, normalized utility function as a pmf, what is the interpretation of Shannon entropy or Shannon information?

After the exchange with the OP in my other answer, let's work a bit with his approach.

We have a discrete random variable $X$ with finite support, $X = \{x_1,...,x_k\}$, and probability mass function (PMF), $\Pr(X=x_i)=p_i, i=1,...,k$

The values in the support of $X$ are also inputs in a real-valued  cardinal utility function, $u(x_i) > 0\; \forall i$. We then consider the normalized utility function

$$w(X): w(x_i) = \frac {u(x_i)}{\sum_{i=1}^ku(x_i)},\;\;i=1,...,k \tag{1}$$

and we are told that

$$w(x_i) = p_i \tag{2}$$

Note that we do not just make the observation that a normalized non-negative discrete function of finite domain, satisfies the properties of a probability mass function in general -we specifically assume that $w(x_i)$ has the functional form of the PMF of the random variable whose values $w(x_i)$ takes as inputs.

Since $w(x_i)$ is a measurable function of a random variable, it, too, is a random variable. So we can meaningfully consider things like its expected value. Using the Law of the Unconscious Statistician we have

$$E[w(X)] = \sum_{i=1}^kp_iw(x_i) = \sum_{i=1}^kp_i^2 \tag{3}$$

This is a convex function, and if we try to extremize it over the $p_i$'s under the constraint $\sum_{i=1}^kp_i=1$ we easily obtain

$$\text{argmin} E[w(X)] = \mathbf p^*: p_1=p_2=...=p_k=1/k \tag {4}$$

and we have obtained a general result:

  The normalized utility function as defined above has minimum expected
  value iff the distribution of $X$ is Uniform.

Obviously in such a case $w(X)$ will be a constant function, a degenerate random variable with $E[w(X)]=1/k$ and zero variance.

Let's turn to Shannon's Entropy which is the focus of the OP. To be calculated, Shannon's Entropy needs the probability mass function of the random variable... so we should find the PMF of the random variable $w(X)$...

But it is my impression that this is not what the OP has in mind. Rather, it views Shannon's Entropy as a metric that has some desirable algebraic properties and perhaps can measure compactly in a meaningful way something of interest.

This has been done before in Economics, specifically in Industrial Organization, were Indices of Market Concentration ("degree of competition/monopolistic structure of a market") have been constructed. I note two that look particularly relevant here.

A) The Herfindahl Index, has as its arguments the market shares of the $n$ companies operating in a market, $s_i$, so they sum to unity by construction. Its unscaled version is

$$H = \sum_{i=1}^n s_i^2$$

which is an expression that has the exact same structure with the expected value of $w(X)$ derived above.

B) The Entropy Index
$$R_e = -\sum_{i=1}^n s_i\ln s_i$$
which has the exact mathematical form with Shannon's Entropy.

Encaoua, D., & Jacquemin, A. (1980). Degree of monopoly, indices of concentration and threat of entry. International Economic Review, 87-105., provide an axiomatic derivation of "allowable" concentration indices, i.e. they define the properties that such an index must possess. Since their approach is abstract, I believe it may be useful to what the OP wishes to explore and attach meaning to.


Speed of convergence of output on the balanced growth path

Okay so, in the book they looked at the first order taylor series of $\dot{k}$, so as @denesp said, we need the time here too!  So rather here we should look at the taylor series around $\dot{y}$

$$ \displaystyle \dot{y}(y) \simeq \left[ \frac{\partial \dot{y}(y)}{\partial y(k)} \bigg|_{y=y^*} \right] (y(k) - y(k^*)) $$

$$ \frac{\partial \dot{y}(y)}{\partial y(k)} \bigg|_{y=y^*} = \left( \frac{\partial \dot{y}(y)}{\partial k(t)} \bigg|_{y=y^*} \right) \left( \frac{\partial k(t)}{\partial y(k)} \bigg|_{k=k^*} \right) $$

First we look at:

$$ y=f(k) $$

$$\Rightarrow \dot{y} = \frac{d}{dt} f(k) = \frac{d f}{d k} \frac{dk}{dt} = f'(k) \dot{k} $$

We know the key equation of the Solow model is:

$$ \dot{k}(t) = s f(k(t)) - (n+g+\delta)k(t) $$

$$ \Rightarrow \dot{y} = f'(k) \left[ s f(k(t)) - (n+g+\delta)k(t) \right] $$

We take the derivative of this with respect to capital:

$$ \frac{\partial \dot{y}}{\partial k} = f''(k)\left[ s f(k(t)) - (n+g+\delta)k(t) \right] + f'(k) \left[ s f'(k) - (n+g+\delta) \right]$$

The value $k^*$ is the golden-rule level of the capital stock so:

$$ s f(k^*) = (n+g+\delta)k^* $$

And hence

$$ \left( \frac{\partial \dot{y}}{\partial k} \bigg|_{y=y^*} \right) = f''(k^*)*(0) + f'(k^*)\left[ s f'(k^*) - (n+g+\delta) \right] $$

Next we use the hint (not much of a hint, and rather misleading to begin with in my opinion)

$$ \left( \frac{\partial k(y)}{\partial y(t)} \bigg|_{k=k^*} \right) = \left( \frac{\partial y(k)}{\partial k(t)} \bigg|_{y=y^*} \right)^{-1} = f'(k^*)^{-1} = g'(y^*) $$

Plugging both of these in to our first order partial derivative:

$$ \frac{\partial \dot{y}(y)}{\partial y(k)} \bigg|_{y=y^*} = \left( f'(k^*)\left[ s f'(k^*) - (n+g+\delta) \right] \right) * \left( f'(k^*)^{-1} \right) $$

$$ = s f'(k^*) - (n+g+\delta) $$

since around the balanced growth path $s = (n+g+\delta)k^*/f(k^*)$ and putting

$$ - \frac{\partial \dot{y}(y)}{\partial y(k)} \bigg|_{y=y^*} = \lambda $$

$$ \lambda = (n+g+\delta) - \frac{(n+g+\delta)(k^*)*f'(k^*)}{f(k^*)} $$

With $\alpha_k = \frac{k*f'(k)}{f(k)} $ being the elasticity of output with respect to capital we obtain:

$$ \lambda = (n+g+\delta)(1-\alpha_k(k^*)) $$

So $y$ converges to its balanced growth-path value rate $\lambda$ the same as $k$ converges to $k^*$


Is my solution to finding the Hicksian demand correct? Maximize $x_1^{1/2} + x_2^{1/2}$ subejct to the budget constraint

Here is what I am thinking (and i could have made a mistake since CES algebra is a bit tedious)
First you have the MRS condition from Lagrange which boils down to
$$p_1 x_2^{\rho-1} = p_2 x_1^{\rho-1}$$
from here I always aim to reconstruct the utility function (because it is a constant equal to the max utility or in this instance you know it will satisfy constraint and be u). So to get the utility function to appear I do this
$$p_1^{\frac{\rho}{\rho-1}} x_2^{\rho} = p_2^{\frac{\rho}{\rho-1}} x_1^{\rho}$$
I then add $p_1^{\frac{\rho}{\rho-1}} x_1^{\rho}$ to both sides
$$p_1^{\frac{\rho}{\rho-1}} x_2^{\rho} + p_1^{\frac{\rho}{\rho-1}} x_1^{\rho}= p_1^{\frac{\rho}{\rho-1}} x_1^{\rho}+p_2^{\frac{\rho}{\rho-1}} x_1^{\rho}$$
and isolate
$$p_1^{\frac{\rho}{\rho-1}}( x_2^{\rho} +  x_1^{\rho})= (p_1^{\frac{\rho}{\rho-1}}+p_2^{\frac{\rho}{\rho-1}} )x_1^{\rho}$$
utility function appears on LHS and I impose constraint
$$p_1^{\frac{\rho}{\rho-1}}u= (p_1^{\frac{\rho}{\rho-1}}+p_2^{\frac{\rho}{\rho-1}} )x_1^{\rho}$$
finding
$$\frac{p_1^{\frac{\rho}{\rho-1}}}{(p_1^{\frac{\rho}{\rho-1}}+p_2^{\frac{\rho}{\rho-1}} )}u=x_1^{\rho}$$
which reduces to
$$\frac{p_1^{\frac{1}{\rho-1}}}{(p_1^{\frac{\rho}{\rho-1}}+p_2^{\frac{\rho}{\rho-1}} )^\frac{1}{\rho}}u^\frac{1}{\rho}=x_1$$
Inserting value of $\rho = 1/2$ I get
$$\frac{p_1^{-2}}{(p_1^{-1}+p_2^{-1} )^2}u^2=x_1$$


Speculation: Why is it referenced and how do we account for it?

It is problematic to use as a word.  Some of the better authors in the last one hundred years have defined it, but there is no consistent definition.  I provide an operational definition for a narrow purpose in a paper as those possessing risk-loving preferences.  That definition, of course, would exclude many traders.  I use it with reference to implications for prudential regulation, but I didn't need to measure it for what I was doing as their presence or absence didn't impact the empirical component.

The problem of talking about "speculators," aside from the definition problem, is that it is not a behavior.  Indeed, it's not meaningful.  Imagine seeing a newspaper article about a plane crash, and "gravity" was blamed.  Excluding planes flying into mountains, how useful would it be to discuss the effects of gravity on planes instead of mechanical failure causing gravity to suddenly become more meaningful?

I can think of possible ways to mathematically segregate speculative behavior if you can clearly state what you mean by the word, but I would really want to ground that in a model so it would very clear what the predictions might be.  However, I can also imagine definitions where central banks could fall into that categorization in some cases so rigor will be both difficult and require a clear statement that it is an operational definition and not a canonical definition.

In the housing market, I doubt it could be readily measured because the measures in the housing market are so easily manipulated and there are plenty of incentives to manipulate.  That may not be true in the Canadian system, but in the American system, particularly with REITs, there is just a ton of ways to play games.  I think you would want a tamer market, such as equities, to build test runs of a methodology.


Prices set by firms are the same in a Salop circle. But why?

tl;dr: Prices are identical because firms are identical.

Profits of firm $i$ depend on own price $p_i$ and on neighbors' prices $p_{i-1}$ and $p_{i+1}$. Prices are set simultaneously, therefore competition in this market is described by a (normal form) game. A strategy is a price, and your formula for $p_i$ is the best response function, showing firm $i$'s profit maximizing price, given its neighbors' prices. Since consumers are distributed uniformly along the circle and the firms are equidistant and have the same marginal costs, the game is symmetric. It is therefore natural to look for a symmetric Nash equilibrium as a solution of the game. To do this, you assume that your own price and your neighbors' prices are symmetric, i.e., identical, while at the same time your price is the best response to their prices. This gives you the common price of all $N$ firms in symmetric Nash equilibrium.


Why traditional banks tend not to provide finance to the poor?

This is actually heavily studied question in development economics. Here are some common answers:

Lack of secure property rights in developing countries. More substantial loans require collateral, in many developing countries especially in rural areas people don’t have deeds of ownership for their houses or farms. Even when they have access to some deed the property rights might be not secure.
Lack of identification - this is related to the previous answer. In many developing countries it’s hard to prove that you are who you are. Again especially in rural areas people might not even have birth certificates. Hence, a bank who does not have local knowledge like loan shark might not get repaid because person just disappears. Hence the costs of private monitoring for classic bank are prohibitively expensive.
In some places it might still be physically very hard to reach the people or you can put it other way around people in many places can’t physically access banking services - due to lack of transport or internet.

Also I got these from the book Poor Economics by Banerjee and Duflo (recent Nobel prize winners) they go in their book into much greater detail of this issue, so I would recommend it and the papers cited therein for more info. Although the book deals with multiple issues so it’s not only about this - just mentioning it so you know what to expect if you plan to get it.


Is there a name for this generalization of the stable marriage problem?

Yes, this has been studied. Look up "matching with contracts" and the earlier papers by Crawford & Knoer (1981), and Kelso & Crawford (1982) in Econometrica.


For a "stable" base, you can use any average of previous years that "makes sense", that is an average with minimal variance and give an explicit explanation
If the last year doesn't deviate from that average then it is unnecessary

To demonstrate the momentum effect you can use a moving average, in that case there is no meaning for the base and edge cases are normalized


You can imagine a demand curve continuing through the x-axis as monetary price becomes negative. A negative price can be interpreted as payment to pick up shirts, a kind of subsidy.* That might tempt those who are still unwilling at price of zero because there's still a opportunity cost of picking up the shirt, or because they only wear collared shirts, or because disposal is not free (the unwanted shirt would need to recycled).

In the simple model, the demand curve that represents willingness to pay is holding these other factors fixed. In a more complicated model, the price drop might induce congestion at the student union, so the demand curve would also shift down or up, depending on your desire to spend time in a crowd of college students.

*Obviously, you have to impose some sort of limit per student to keep quantity demanded from shooting off to infinity.


In Indonesia, reducing fuel subsidy strongly impacts the price of everyday goods. Why?

Consumers rarely have to pay the full price of petrol / gasoline. That's true in most of the world, and the US is no exception. There are explicit subsidies and favourable tax regimes in many extraction countries; and of course there are the implicit subsidies in the unpriced pollution - oil's externalities.

Those explicit and implicit subsidies have been there for decades: and consequently, oil products are hugely over-used, compared to the alternatives: that's a direct consequence of oil being a product with demand that's not perfectly inelastic - indeed, estimates of long-term price elasticity of demand for petrol are of a figure around -0.6 or so. That means that oil is embedded in large quantities in the supply chains for very many products.

The larger the historic subsidies, the bigger the impact of changes in subsidy levels to the price of other goods and services. And the poorer the population, the bigger the impact. That's why you see a bigger impact in Indonesia than in the US.

Now, although in the long run removing some of the subsidies will correct some of those imbalances, that will require capital reallocation and some industrial restructuring, all of which takes time.

And although in the long run, rising prices of some products, in the absence of rising wages, will choke off demand for other products, leaving inflation at no net change, the short-term effect of rising oil prices is to raise the costs in a significant part of the supply chain for a lot of goods and services, given an instant boost to inflation.


How does an UHN individual leverage their wealth?

Leveraging is done with debt. Most high net worth individuals are debt free and even if they take out loans, it is to reduce taxes.

However, high net worth individuals normally have their wealth in a form of equities or stocks. For example Bill Gates' wealth is mostly in Microsoft stock, Warren Buffet's is in Berkshire Hathaway's, etc. The corporations that these rich people own can take on a lot of debt and therefore become leveraged. Debt to equity ratio of corporations like Microsoft and Berkshire Hathaway is not very high though, in fact it is below 50% for those two. You can say that Microsoft's use of debt, or leverage, amplifies its impact on the economy by 1.4x, which is not a bad thing.

A hedge fund or some investment institution, on the other hand, can easily have several times more debt on their balance sheet than equity. On top of that, hedge funds can buy/sell leveraged derivatives like futures, for examples, which have a 10x leverage of their own. As a result, owners of such investment institutions CAN have a considerably impact on the economy as a result of the use of leverage.


Relation between linear utility function and U=max{x,y}

The optimal choice set for a max function and a perfect substitutes function with equal relative prices share some solutions [i.e, boundary solutions], but in general, the indifference curves, and hence non-boundary solutions, are different.
Main Idea
For both a max(x1 x2) and perfect_sub(x1 x2) utility function, the point, say, m/p1 (or m/p2) would maximize utility. So both utility functions share boundary solutions. But think about the IC of m/p1 for a perf_subs consumer and think about the IC of m/p1 for a max consumer. You see the points that the two consumers think are 'just as good' as that m/p1 point are very different.
That is, the same boundary bundle might be a solution to both utility functions, but the other solutions will differ.
Logic
The reason is that U = perfect_subs is a (not-strictly) convex utility function, whereas U = max isn't. That is: the consumer is either indifferent between, or actually prefers, less extreme combinations to more extreme combinations for the former. As for the latter? Well, they just care about the max; they like extreme combinations (e.g, [C = (m/p1, 0)])
That's why for U = perfect_subs with relative prices the same, both A = (100,0) and B = (ß(x1), (1-ß)x2) have the same utility. That is,B lies on a straight line IC connecting (100, 0) to (0, 100). Whereas for U = max(x1 x2), B would not be on the same IC as A (unless ß is 0 or 1, in which case, we're back to talking about boundary solutions!)
The word, max is confusing here. Essentially, the perfect_subs (x1 x2) consumer wants to maximize total goods. The max(x1 x2) consumer just cares about maximizing the larger quantity in his {x1, x2} bundle.
Tl;DR
To be more concrete: If A = (100,0) and B = (75, 25), a perf_subs consumer is indifferent between A and B; a max consumer is not.


Travel Cost Method - how to treat people who live at a site

Short answer.  It is perfectly correct, and probably simplest, to exclude from the analysis the members of the fishing society, provided  that you present your result as “the tourist value of the harbour” or “the value of the harbour to tourists” (and not as the total value of the harbour).

Long answer. The individual travel cost method involves two stages.  In the first stage, the data collected is used to estimate a trip-generating function, something like:

$$V_i  =  f(TC_i,\pmb{A}_i)$$

where $V_i$ is the number of visits by individual $i$, $TC_i$ is the round-trip travel cost from $i$’s location of origin, and $\pmb{A}_i$ is a vector of characteristics judged to be relevant, eg age, income.  In the second stage, a demand curve for visits to the site is estimated on the assumption (call this the key assumption) that individuals would respond to different hypothetical prices added to their actual travel costs in the same way that, according to the trip-generating function, they respond to different  travel costs.  The site value is then taken to be the consumer surplus, which if access to the site is free is the whole area under the demand curve.

The fact that travel cost for some individuals is zero is not in itself a reason for excluding them from the analysis.  One reason is that their data is relevant in estimating the trip-generating function: a data point with $TC_i = 0$ is still a data point, and to exclude such points would shift the line of best fit.  Also, the second stage methodology means that, other things being equal, individuals with zero travel costs make a disproportionately large contribution to the total consumer surplus (because the sum of a hypothetical price and zero is less than the sum of that price and a positive travel cost, so that the number of visits associated with the former will be greater).

However, there are other reasons for excluding members of the fishing society.  The origin locations of the tourists, it can probably be assumed, are to a large extent randomly spaced.  It seems unlikely however that this is so for the fishing society.  Very likely they live at the harbour because, given their dependence on fishing, they have chosen to do so.  If so, one of the assumptions of the travel cost method is undermined.  Freeman, Herriges & Kling (2014) list as one of the assumptions of the method (for the case of recreation but the point is more general):

  … it is assumed that the individual’s choice of where to live (which is one determinant of the cost of a trip to a recreation site) is independent of preferences for recreation visits.  If people choose residential locations so as to be near preferred recreation sites, then the price of a visit is endogenous.  (p 273)

As a consequence, including the fishing society would tend to result in a biased valuation.

A second reason for excluding members of the fishing society (which would apply even if their travel costs were positive) relates to the key assumption above.  The assumption may be reasonable for a fairly homogeneous group such as tourists, or indeed for the fishing society considered on its own.  But if one puts together two such very different groups, the assumption becomes much more dubious.  Suppose, for example, that the incomes of the tourists are typically much larger than those of members of the fishing society.  In that case the former  might respond to a certain price by visiting less frequently, while for the latter the price might be so large in relation to their incomes that they could not afford even one trip.  Thus a ‘hybrid’ demand curve, based on data for the two groups combined, would be unlikely to give a valid consumer surplus estimate.


What is the impact on the money supply of a reduction in the required reserve ratio?

Theoretically, reducing the required reserve ratio will increase the money supply by allowing banks to produce more loans and keep less in reserve.  Theoretically your simple (ie basic homework) calculation would show that the money supply could increase by that amount.  In practice there are of course many other factors, such as the interest rate, velocity and reliablility of the people taking out the loans.


Preferences where wealth effect dominates

In open macroeconomic models some cases of logarithmic preferences cause the wealth effect to dominate the intertemporal substitution effect.

If I recall correctly, at least Vegh's book "Open Economy Macroeconomics in Developing Countries" (2013, MIT Press) presents this case.


Abolishing all tariffs without reciprocity

You get cheaper access to inputs. For a large advanced economy that is relatively diversified, it is possible that unilateral tariff reductions across many or all sectors can be good for the economy by freeing up resources presently occupied in low-value activities with few prospects to contribute to technological progress. The additional ease of doing business in that country may also attract additional financial investments and human resources talent from abroad, which would tend to increase technological progress and productivity at the same time.

For a small economy that specializes in some specific sector (e.g., tourism, finance), a unilateral tariff reduction may improve the competitiveness of the sector they specialize in, and therefore be good for the economy and/or technological progress over both short and long time frames.

These are somewhat special conditions, and most often countries do not see benefit in unilateral tariff reductions.

As for what politicians do, this is a somewhat different question and should be considered separately from the economics argumentation about unilateral tariff reduction. It is very straightforward to argue that some concentrated interest(s) in just one or two industries could be major campaign contributors in a winning campaign, leading the politician to try to impose tariffs which benefit specific industries while causing overall negative impacts across the entire economy.

Theoretically, it would be possible to be in a situation where both a) unilateral tariff reductions could benefit the economy and b) at the same time concentrated interests could legally and/or illegally obtain political influence which resulted in an increase in their own profits at a cost to the overall economy.


To continue the example of the street performer:

Suppose there are two people offering the performer money, but in exchange they want her to perform different songs. Person A prefers song A while person B prefers song B. Whoever offers the higher sum gets their wish. In this case there is rivalry between the consumers. Yet there is also non-excludability: Even if someone loses the bidding war, they can listen to the song chosen by the other person.

Note that the rivalry results in higher profits than if both people preferred song A played, because then they would have had no incentive to bid against each other due to non-excludability.


Do perfect complements have to be normal goods? If so, why?

A good is normal if its demand is increasing in income. So let $p_x$ and $p_y$ be the price of the goods with quantities $x$ and $y$ and let $m$ be income.

Suppose $ax>by$. Then $\min\{ax,by\}=by$. By slightly reducing $x$ by and spending the saved money on $y$, one gets a better bundle. For an optimal bundle, this cannot be.

Similarly, it cannot be optimal that $by>ax$. So in the optimal consumption bundle, it must be the case that $ax=by$. It is also not that hard to see that the consumer will spend all her income. So rewrite the condition as
$$y=\frac{a}{b}x$$ and plug it into the budget equation $$p_x x+p_y y=m$$ to get
$$p_x x+ p_y\frac{a}{b}x=m=x\Big(p_x+p_y\frac{a}{b}\Big).$$
Therefore, we get the demand function given by $$x(p_x,p_y,m)=\frac{m}{p_x+p_y\frac{a}{b}},$$ which is clearly increasing in $m$. Similarly, one shows that the other good is normal too.

Pedantic remark: A differentiable function can be increasing at every point without the derivative being strictly positve everywhere. The function given by $x\mapsto x^3$ has derivative $0$ at $0$ but is everywhere increasing.


What is an estimate of the value of the world's assets?

One place that estimates the world's wealth (measured as net value of assets) is the Credit Suisse Global Wealth Report.
According to the report their estimation of worlds net assets is 361 trillion of USD in 2019. Their site has also statistics for other years. They also offer some country level statistics in their reports (not sure if that's granular enough).


How could monetary policy be affected if cash "gifts" to consumers were more frequently used?

This is only inflationary if the monetary authority monetizes the borrowing. If the Treasury borrows $1000 on the open market by issuing a bond and sends the money to a consumer, no money has been created— it has simply been borrowed from one person and given to another. However, if the monetary authority purchases that bond (swapping base money for the bond), then it is indeed inflationary, as the amount of money now in circulation has increased.

As far as its role in monetary policy— the decision of whether to monetize fiscal authority borrowing is part of standard monetary policy. The phrase you'll often hear is, "the monetary authority moves last," which reflects the fact that the monetary authority can either accommodate spending by monetizing it, increasing inflation, or do nothing, which has the effect of increasing interest rates on government borrowing.


Dimensional analysis for the qdf/quantile function corresponding to the pdf/CDF for the size distribution of income

Your random variable is $X$ = "Yearly Income per capita measured in dollars". So $X$ is a real-valued function. Its domain is usually left unspecified, while its range is measured in "dollars per person-year", as the OP writes it.

The density and the distribution (pdf/cdf) functions of $X$ ,say $f_X,F_X$ have as their domain the range of $X$. So the domain of $f_X,F_X$ is a set that carries the unit of measurement "dollars per person-year".

Being strict in dimensional analysis, indeed we could say that the density must then be measured in "person-years per dollar" (irrespective of whether this is meaningful or not).

So the expected value that is the integral $\int_0^{\infty} x f_X(x) dx$ keeps having "dollars" as a measurement unit (I just assumed $X$ is non-negative), applying the rule the OP states (the one unit is the reciprocal of the other, and we are left with the third which is again "dollars per person-year").

Moreover the integral that is the cdf $F_x(x) = \int_0^x f_X(s) ds$ has a range that comes out "dimensionless", since probability is a relative measure.

(Digression: Note that by definition, the distribution function of a random variable measures the probability of the random variable being smaller or equal than a threshold, $F_X(x): = Prob(X \leq x)$. Additional assumptions are required to "tranlsate" this probability as representing also "income shares", meaning I guess "percentage of population with per capita income below or equal to $x$")

Turning to the Quantile function, it is defined as the inverse of the distribution function (generalized inverse to cover all cases, but let's keep it simple). Being the inverse, it uses as its domain the range of $F_X(x)$, so the domain of $Q(p)$ carries "probability" as unit of measurement, while its range is the domain of $F_X$. And the domain of $F_x$ is the range of $X$: the Quantile function has the same range as $X$ and so it is measured in "dollars per person-year" too (i.e. its range caries the "dollars per person-year" unit of measurement).

it follows that the derivative of $Q(p)$", $q(p) = dQ(p)/dp$ has the same units of measurement as $Q(p)$, while the integral $\int_0^1 pq(p)dp$ is measured in "dollars per person-year" also.


Interpretation: Elasticitity of Substitution

The ratio of marginal products $\frac{\partial P}{\partial x_1}/\frac{\partial P}{\partial x_2}$ (the marginal rate of tech. substitution) is equal to $1/x_1+r$ where $r\equiv x_2/x_1$.

The derivative of the MRS wrt to $r$ ($\frac{d MRS}{d r}$) is equal to $1-\frac{g}{MRS-r}$, where $g\equiv \frac{dln(x_1(r))}{dr}$.

Note how $\frac{dln(MRS)}{dr}$ which is equivalent to $\frac{d MRS/d r}{MRS}$ is equal to $\frac{MRS-r-g}{MRS(MRS-r)}$. Also note how $\frac{dln(r)}{dr}=1/r$.

Rearranging the last two relations in order to acquire expressions for $dln(MRS)$ and $dln(r)$ and substituting into the definition of the elasticity of substitution yields $-\frac{MRS(MRS-r)}{(MRS-g-r)r}$.

Finally, note how $r=0$ when $x_2=0$. Also, $MRS=1/x_1$ when $x_2=0$. The behavior of σ depends of the value of $g\equiv \frac{dln(x_1(r))}{dr}$ at $x_2=0$. If the value of $g$ is defined and is finite and constant then $σ$ approaches $- \infty$.

Intuitively, the behavior of $g$ near $x_2=0$ depends on the shape of the isocurves. $g$ denotes the (proportionate) rate of change in $x_1$ as a result of an infinitesimal change in $r$. The following graph shows that for values of  $x_2$ near zero (notice the values on the vertical axis), the isocurves are really steep.

Very informally, in this setting, (really) small changes in $r$-which is equal to the ratio of $x_2/x_1$ (which is equivalent to the tangent of angle $θ$)- have almost no effect on $x_1$. For $Δθ$ approaching zero $Δx_1$ is also small and probably close to zero too. Equivalently and very loosely speaking, in this region $g$ should be close to zero too.


How encouragement design works practically?

The fact that you can only encourage but not force the teachers to participate puts you squarely in the world of Local Average Treatment Effect (LATE) and Intention to Treat analysis.

Essentially if you use the voucher as a instrumental variable, you will get an estimate of the effect of the voucher on "the sort of person who you can encourage to take a class by giving them a voucher".

Read the Instrumental Variables section in Mostly Harmless Economics, to get a good easy to understand explanation of exactly the situation you are interested in.


Quantitative Marxian/Marxist micro and macro economic models?

There is a very interesting paper by John Roemer, published in Econometrica in 1980, that presents a mathematical general equilibrium Marxian model. I am not sure someone have ever estimated it though. Find it here (subscription required).
General equilibrium models (oftentimes idyosincratic) were also developed in the USSR. Check here for an account of them.
Models also informed (or attempted to inform) policy in USSR. The 1985 book "Macromodels of the National Economy of the USSR" gives an account of such models and policy applications.


why do we analysis the housing market from 'short run' and 'long run'?

Housing markets are complex and the following are just some basic points.

Key features of houses as economic goods are: 1) their very long life; 2) their differentiation by type and location. In the short run housing supply (the total existing stock, plus any new houses completed) is price-inelastic because the number of new houses can be only a tiny proportion of the existing stock.  Hence a short-run analysis of house price changes will focus primarily on short-term changes affecting demand (eg changes in interest rates or in housing-related taxes).  In a long run analysis, however, consideration also needs to be given to changes in supply, and to long-term changes affecting demand (eg population growth, rising incomes).

Cyclical factors are those that tend to reverse over a period of a few years. The cause of a cycle in a housing market could be internal to that market, eg a self-fulfilling expectation at one time that prices will rise, leading to a price bubble and eventual collapse, as described here.  The cause could also lie in the wider economy, eg house prices tending to rise when the economy is booming and fall in a recession.  Structural factors are permanent or at least long-lived, eg land availability and planning constraints, housing-related policies that are considered unlikely to change, and the distribution of the existing stock between types and locations.


How do I verify Philips curve empirically for recent years?

Simple correlation is not a good approach to evaluate the Phillips curve, because you are mixing many periods together. To get a better grasp of what this relationship might have been in the past, use a graphical approach.

I suggest you to do something like this:

So basically, you plot the points, but add a line that connects them temporally. This might give you a better idea of some dynamics. It is clear from that graph that a simple correlation cannot capture the complex dynamics of inflation and unemployment.

For example, this is an animation of the evolution of the Phillip curve for the US:

The author identifies three or four PC over 40 years, reflecting different political/economic periods. A simple correlation analysis however would show none.

Source of images: here.


Who determines the exchange rate and how?

Technically you are changing the BDT/USD exchange rate, or at least you're changing one of the reasons. If supply/demand changes, the value of a particular currency is going to change as well. But there are other factors - interest rates and political climate also affects the rate. To give you an example, Venezuela is now going through some rough times, perhaps not many people are willing to buy their local currency (VEF) today because tomorrow it may be not be worth the original investment, as as consequence the exchange rate BDT/VEF will increase: you can buy a lot of Venezuelan Bolivar with a few Taka. Each country then has its own policy to set exchange rates - basically there are two of them.

  Floating exchange rate

In this system the rate is fixed by supply/demand. Roughly speaking: a currency is worth whatever buyers are willing to pay for it. So, for example if (for some reason) many people in US are buying Euro (EU) then the demand increases, and so will the price, the rate USD/EU will drop and you will get fewer Euro for each US Dollar. Conversely, for Europeans the Dollar will depreciate as more USD were supplied to the market.

Usually countries with very mature/estable currencies have these systems (e.g. Canada, Switzerland, United States), allowing the exchange rate to adjust itself to local conditions (inflation, supply, demand, $\dots$). Although this may sound great, it actually has its drawbacks, for instance, lack of certainty may refrain some investors. One may argue that the fact that the economy is stable in these countries, then we're indeed certain of what's going to happen before an investment, but I guess that's a whole other story.

Below there's a graph of the evolution of the exchange rate for the Swiss Franc to USD:

  Fixed exchange rate

In contrast with the previous system, the exchange is artificially maintained fixed by the government. The way they do it is pegging the local currency to a foreign currency (say USD) and compensate the supply/demand in order to keep the exchange fixed. A good example of this system was the Chinese Yuan, when the local demand of USD increased, the government had to release US Dollar into the market in order to mitigate the fluctuation.

In the curve bellow I show the exchange CHY/USD, you can immediately see that in the period 1998-2005 (for instance) the Yuan was pegged to the Dollar:

After 2005 the model is changed, and they decide the exchange rate from a basket of external currencies.

  Managed exchange rate

In practice all governments make local decisions to modify the exchange rates, for instance to help import/exports they can slightly provide supply/demand so that the rate is modified.


Game Theory: Continuity in equilibrium profits?

Let me first try to answer Statement 2:

The agent's utility profiles are continuous in x

Let's try to flesh out the problem. Let $S_1$ and $S_2$ be compact, convex, strategy sets of the players (say subsets of $\mathbb{R}^n$) (for simplicity) . Let $u_1(s_1, s_2, x)$ and $u_2(s_1, s_2,x)$ be the continuous utility functions of the two players.
Let's solve for the SPNE. At stage 2, player 2 maximizes her utility given the action $s_1$ of player 1. This gives:
$$
v_2(s_1, x) = \max_{s_2 \in S_2} u_2(s_1, s_2, x)\\
s_2(s_1, x) = \arg\max_{s_2 \in S_2} u_2(s_1, s_2, x).
$$

The problem is well defined due to the theorem of the maximum.
The function $v_2(s_1, x)$ is continuous (Berge's maximisation theorem).
The best response $s_2(s_1, x)$ is only guaranteed to be an upper hemi-continuous, non-empty correspondence.
If $s_2(s_1, x)$ is single valued, then it is continuous. However single valuedness will depend on the functional form of $u_2(s_1, s_2, x)$ (e.g. strict quasi-concavity) and therefore does not follow from the assumptions so far.

Let $\tilde s_2(s_1, x)$ be a selection of $s_2(s_1, x)$. In stage 1, Player 1's optimisation problem is:
$$
v_1^\ast(x) = \max_{s_1 \in S_1} u_1(s_1, \tilde s_2(s_1, x), x).\\
s_1^\ast(x) = \arg\max_{s_1 \in S_1} u_1(s_1, \tilde s_2(s_1, x), x).
$$

If the selection $\tilde s_2(s_1, x)$ of Player 2 is the optimal choice from $s_2(s_1, x)$ for Player 1, then the objective function appears to be upper semi-continuous, (but not, in general, continuous). Due to upper semi-continuity of the objective function, the maximisation problem is well defined.
Although $v_1^\ast$ will be (I think) upper semi-continuous, it is not necessarily continuous. Anyway, the best response correspondence $s_1^\ast(x)$ does not need to be continuous.
If $s_2(s_1, x)$ was single valued, then it is continuous, which means that $v_1^\ast(x)$ is also continuous. If, in addition $s_1^\ast(x)$ is also single valued, then it will also be continuous.

Let $\tilde s_1(x)$ be a selection from the best response $s_1^\ast(x)$ from player 1. Then, we can define:
$$
v_2^\ast(x) = v_2(\tilde s_1(x), x),\\
s_2^\ast(x) = \tilde s_2(\tilde s_1(x), x).
$$

In general, neither $v_2^\ast(x)$ nor $s_2^\ast(x)$  need to be continuous.
If both $s_1^\ast(x)$ and $s_2(s_1, x)$ are single valued, then $v_2^\ast(x)$ and $s_2^\ast(x)$ will both be continuous.

It thus appears that Statement 2 holds if both $s_2(s_1, x)$ and $s_1^\ast(x)$ are single valued.

Single valuedness of $s_2(s_1, x)$ holds if (for example) $u_2(s_1, s_2, x)$ is strictly quasi-concave in $s_2$
In turn, single valuedness of $s_1^\ast(x)$ holds if $u_1(s_1, \tilde s_2(s_1, x), s_1)$ is strictly quasi-concave in $s_1$. Notice that strict quasi-concavity of $u_1(s_1, s_2(s_1, x), x)$ is a strong assumption as it puts shape restrictions on the best response $s_2(s_1, x)$ (and not only on the function $u_1$ itself).

I hope this answers Statement 2.
For statement 1:

This game has at least one SPNE, but in general multiple. In case of multiple SPNE these will have the same utility profiles, that is, yield each agent the same utility.

As shown above, the existence of a SPNE can be shown using only continuity of $u_1$ and $u_2$ and compactness of the strategy sets, and under the assumption that $\tilde s_2(s_1, x)$ selects the best option for Player 1.
Different SPNE will in general not give the same utility. Assume that at stage 2, Player 2 is indifferent between more than one optimal strategy, so $s_2(s_1, x)$ is multivalued (for example, assume that $u_2(s_1, s_2, x)$ is the constant function, so Player 2 is indifferent between every strategy in $S_2$). If different selections $\tilde s_2(s_1, x)$ from $s_2(s_1, x)$, provide different utilities for player 1, then it is possible that $v_1^\ast$ takes different values depending on the choice of player 2 in stage 2 of the game. So different SPNE may give different utilities.


Does 0% interest rate mean the minimum value of a currency?

Can the inflation rate rise even if the interest rates are below 0%?

In short, yes. The inflation rate rises when more money chases goods and services than before. This can happen because there is

more money circulating
less goods and services output
money is circulating at a faster pace (increased money velocity)
money is chasing a narrower set of goods and services

Note the last point decreases the inflation rate for the goods and services that are no longer chased.

In the short term, when interest rates lower, people borrow more money to spend as borrowing is cheaper. Assuming no change in the goods and services output, the inflation rate should rise.

As you mention, you can raise government employee salaries. And you don't have to necessarily limit the salary raise to government employees: an earned income tax credit would have the effect of raising most people's salary. Or more generally, you can expand on fiscal spending: build bridges, fix roads, launch rockets, etc. Assuming this fiscal spending is financed through government debt and there is a budget deficit, there is now more money chasing existing goods and services in the short term, and thus the inflation rate should rise in the short term. Over the long term, productivity gains from fiscal spending should increase the output of goods and services and offset the inflation spike.

You can also just print money and dump it off a helicopter. This will again cause more money to circulate and raise the inflation rate in the short term. What happens over the long term depends on how effectively people spend helicopter money.


Empirical research into sectoral comovement

A relevant paper could be:

Veldkamp, Laura & Wolfers, Justin, 2007. "Aggregate shocks or aggregate information? Costly information and business cycle comovement," Journal of Monetary Economics, Elsevier, vol. 54(Supplemen), pages 37-55, September.

NBER Working Paper version: https://www.nber.org/papers/w12557

Published version: https://www.sciencedirect.com/science/article/abs/pii/S0304393207000529


How is employment related to aggregate demand and aggregate supply?

In an economic model, we normally have a production function that increases with the amount of labour. Since inherited capital is fixed, in the short run,the only way to increase out (supply) is via increasing the amount of labour employed. The real world is somewhat fuzzier; some service industries (like software) can have greater sales without increasing the number of employees. However, such behaviour is limited, and greater output is normally associated with more people working.
The relationship between aggregate supply and aggegate demand somewhat depends on prior economic views. To summarise the Keynesian view, if aggregate demand rises (or is expected to rise), firms will increase output (hiring more labour) to meet expected increased demand.


Iet's say we have n identical firms and an infinite horizon of time.

The n firms sustaining the collusion, will find optimal to fix the same price $p_m$ where $p_m$ is the price of the monopoly level and we define $\frac{\Pi^m}{n}$  as the profits each firm is obtaining by sustaining the collusion in each moment t.

Now, of course each firm can betray the others by fixing a price lower than $p_m$, namely $p_m-ε$, where ε is small, and by doing so, the firm will capture the entire demand because in this market the firms are doing the Bertrand competition. In other words, the firm by betraying the others, will get almost π_m at the time T=t. We will also assume that in all t > T no firms will make any profits, because they will punish the firm, by fixing the price in Bertrand competition.

The firm will defect if:

$π_m/n + δπ_m/n + δ^2π_m/n.... < π_m+0+0....$

Where δ is the discount factor.

This can rewritten as:

$(\frac{π_m}{n})(\frac{1}{(1-δ)}) < π_m$

We can now see that if n, the number of firms, increases then the profits by sustaining the collusion will decrease, so the above inequality will be more likely to be true. This means that a firm has less incentives to sustain a collusion when there are too many participants, because the profits will be divided among too many firms and the punishment will be seen as less heavy.


It is because supply of saving is equal to the cost of saving which is forgone consumption as the graph notes. To explain this further the vertical distance from $x$-axis to the supply curve gives you the value of forgone consumption for some infinitesimal increase in the supply (along the supply curve). This is because value of the forgone consumption needs to be at least equal to the ‘quantity’ of output you decided to save rather than consume times the return you earn on those savings (otherwise you would not supply those savings to the market to begin with).
So the total cost of saving will be the sum of all costs of savings for every small increase in the number of saving (i.e. every difference between the point on supply curve and the $x$-axis). The sum of these adds up to the area under supply curve up to the point where actual return intersects the supply of savings.


Formula for splitting profit and loss for every person

Either I totally misunderstand the question, or this is just adding up numbers, there are no economic or mathematical principles involved. (In which case this is not the right SE for it.)

Denoting the "balances" by $b_1,b_2,...$ and the "supplies" by $s_1,s_2,...$ total profits are
$$
\Pi_{total} = \sum_i (b_i-s_i)
$$
You want everyone to end up with a proportional share of this, so that in the end they have
$$
\frac{s_i}{\sum_i s_i} \Pi_{total}.
$$
To compensate for their investment, each person should get
$$
x_i = s_i + \frac{s_i}{\sum_i s_i} \Pi_{total}.
$$
One can verify that
$$
\begin{align*}
\sum_i x_i & = \sum_i s_i + \sum_i\frac{s_i}{\sum_i s_i} \Pi_{total} \\
& = \sum_i s_i + \Pi_{total} = \sum_i b_i
\end{align*}
$$
so we made no mistakes, you can indeed give people their values $x_i$.


If there were such a spectrum, it would not be one dimensional. For instance, we could compare the competitiveness of markets with homogeneous products in terms of the number of firms in that market, and have the following spectrum:

monopoly: one firm, no competition
(Cournot) oligopoly: $n$ firms ($n<\infty$), imperfect competition
perfect competition: infinitely many firms, perfect competition.

However, this spectrum only works homogeneous products. When you talk about monopolistic competition, i.e. firms competing with differentiated products, then a new dimension opens up: non-price differences in products. An example would be cell phones with different operating systems (e.g. iOS and Android). A small increase in the number Android-powered phone brands is not necessarily going to reduce the market power of iPhones.

So the spectrum of market competitiveness is not as straightforward as you envisioned. While the two extreme cases, perfect competition and monopoly, are pretty clear, things in between are much less so. For example, a 2-firm Cournot market could be more competitive than a 2-firm monopolistically competitive market, if you carefully choose the demand functions for the latter. So at least in the comparison between oligopoly and monopolistic competition things are not as deterministic as you thought.


What is the economic interpretation of this utility function?

This looks like constant relative risk aversion (CRRA) utility . Usually CRRA is written like $U = \frac{C^{1-\rho}-1}{1-\rho}  $ (I omitted second part for brevity) in your case $a=1-\rho$. $\rho$ is the relative risk aversion. By extension $a$ is the function of $\rho$ so as $a$ increases (due to smaller $\rho$) the person should become less risk averse.


How to close the gap between non-Econ background and Macroeconomics

Sorry if this sounds harsh, but:

You need to start asking more. You're not stupid for asking questions - you're stupid for not asking questions. Be upfront about your different background and ask away.

Of course you should try to research things by yourself, but a lot of assumptions are next to impossible to know or find out in a reasonable amount of time. I've got a 5 years of university economics - primarily in macro and I still struggle with assumptions. So, if you can't find it - ask! And ask your professor about some relevant books / articles.


Have the stocks and flows of hunter-gatherer economies been formally analysed?

Anthropology is concerned with understanding a society in qualitative terms rather than quantitative terms. To try to understand it in these terms is to misconceive how those societies work and what anthropology is.


$x\sim y$ implies $x+a\sim y+a$ for any $a\geq0$ and $x,y\in\mathbb R^n$, then the preference is linear?

It is not true. Let us consider $\mathbb{R}^2$ so bundles are $x = (x_1,x_2)$.

Consider the preference:

(i) If $x_1 \leq 0$, preferences are lexicographic, i.e.
$$
x \succ y \Leftrightarrow
\begin{cases}
x_1 > y_1 \\
\text{ or } \\
x_1 = y_1 \text{ and } x_2 > y_2
\end{cases}
$$
(ii) If $x_1 \geq 0$, $u(x_1,x_2)=x_1+x_2$.

Notice that no indifference occurs on $\mathbb{R}_{<0} \times \mathbb{R}$, and the required condition holds on $\mathbb{R}_{\geq 0}\times \mathbb{R}$ since it is linear there.

However, $(0,4) \sim (2,2)$, choose $a = (-1,-1)$, we get
$$(2-a,2-a) = (1,1) \sim (0,2) \succ (-1,3) = (0-a,4-a)
$$
hence the second condition fails.


Explaining fluctuations in inflation rate

Given the expectations-augmented Phillips curve, national output which is less than the natural level of output (the level of GDP which is associated with full employment) causes deflationary pressure at an accelerating rate (assuming people's expectations of inflation is last period's inflation rate).

The financial crisis of 2008 was caused by a credit-driven asset price bubble. This was caused by financial institutions who invested in assets that were at the time priced at a value greater than their fundamental value. By holding these assets that were rising in price, they were used as collateral and/or used as capital buffers in order to make leveraged investments on these same overvalued assets. Causing a feedback loop, as these leveraged investments created a further increase in demand for these assets (raising their price, once more) which caused financial institutions to make further lending on these assets, which drives the asset price higher, and so on.

When the asset-price bubble "burst" and financial institutions subsequently made substantial losses (as they did not have enough capital to subsidise these losses) they had to cut down on their asset holdings to at least mitigate these losses, which caused a significant contraction in lending to the economy, as these financial institutions were more focused on collecting a large reserve base. A decrease in lending causes a decrease in investment in the economy, and subsequently decreases consumption, hence why aggregate demand decreased and the global economy was facing a period of mass deflation.


What did Adam Smith mean by "putting [the pins] into the paper"?

Alecos is correct that Smith is talking about packaging, but I don't believe that "paper" refers to a paper bag.

Straight pins used to be sold on sheets of paper (usually as sets of a dozen), through which they passed twice, like so:

So by "put them into the paper" Smith means exactly what he said: the pins are put into paper before being sold.


Can $u(x) = \sqrt{x_1 x_2} + \sqrt{x_3 x_4}$ be solved by Kuhn–Tucker conditions?

Why do you want to guess that some $x$ is zero? What is the problem of having all four goods strictly positive at the solution?

In fact, due to the presence of the square root in the first order conditions, the solution looks straightforward, since the inequality cannot hold (and assuming exhaustion of the budget).

ADDENDUM
Given the f.o.c's provided by the OP, all $x$'s have to be strictly positive at the solution, since otherwise we have division by zero and/or the undetermined form $0/0$.

This in turn implies that $\partial L/\partial x_i = 0$ at the solution, which now becomes easy to compute.


Does the H-1B visa program have the effect of reducing wages in the U.S

The program would increase the supply of labor to those specific sectors. As a result more foreign people would get into the sectors bidding the price down (lowering the wage). In addition, some Americans would not be able to get hired, that otherwise would have.

In addition, companies would become more profitable so more companies would be started and the reduced labor costs would be visible in other sectors, increasing labor demand elsewhere.

In the very long run free movement of labor is at least in theory more efficient than restricted movement, as people will more efficiently be able to navigate to the most efficient job locations available.


Why is the Euro a bad idea compared to other currencies?

One of the key requirements for a currency area to be successful is that there must be a sustainable political mandate to support fiscal transfers between regions, to provide the balancing mechanism that would otherwise have been performed by fluctuating currency rates between those regions.

Another is that labour can move freely between regions.

The Eurozone, it turns out, met neither of those requirements:

Language barriers have impaired freedom of movement.

And although Northern Europe was happy to reap the benefits of currency union during the good times, it has proven to be unwilling to make the fiscal transfers to the South during the bad times.

Exacerbating this, Greek admission to the Euro zone was created on the back of some fraudulent accounting courtesy of Goldman Sachs. Based on the real numbers, Greece would never have been admitted.


You can easily change the informal explanation to a "quantity-based" approach, using a dynamic approach. Assume that firms in the market produce this period $S_1$ but the consumers that show up buy only $D_1<S_1$. What would you do if you were a producer, for period $2$, given the information you have about where demand stands?

Assume the good is perishable. Then the unsold quantity from period $1$ is destroyed. In period $2$ are you going to produce the same amount, or something less? I would say less, to reduce somehow the quantity that will go to waste and your loss from that fact.

Assume the good is not perishable, so you have inventories left over from the first period. Now, the tendency is to produce even less than when the good is perishable, because you can sell during period $2$ the inventories from period $1$.

So we see, that starting from a quantity supplied above quantity demanded, it is reasonable to expect lower quantity supplied next period.
One can easily make the analogous argument starting with quantity demanded higher than quantity supplied. This is a gradual adjustment towards "market equilibrium", where quantity supplied will equal quantity demanded, or approximately so.

...and I didn't mention prices anywhere. You can think that this is a local market where the producers are forced to sell at a specific price, "imposed" on them by other similar markets and the expectations of consumers. In other words, what I described above can roll out while the price remains fixed from period to period.

The above was just an informal description. The benchmark approach to the dynamic adjustment towards market equilibrium is the Cobweb Theorem, see Ezekiel 1938.


Definiton of information sets in rational expecations models

Two notes.
A. "Conditioning on information" has always been applied in economics without much attention to probability theory-rigor, because it has (indeed) such a strong intuitive sense: "based on the information I have (where "information" here means data, processing algorithms, psychological makeup, almost anything) I somehow form through a black-box process an expectation for the value of some variable".
B. Conditioning is always done with respect to a sigma-algebra. But in economics it is customary to write just the generator of the sigma-algebra, in our case $I_{t−1,i}$ and expect to be understood as $\sigma\left(I_{t−1,i}\right)$. So all you have to do is imagine $I_{t−1,i}$ as a set that can generate a sigma algebra. In that case, each element of the set can be anything really.
Note that the $\Omega$ in the probability space and the sigma-algebras that can come from it can literally be anything. We restrict $v$ to be a random variable, namely a function whose range is some numeric set, like the Reals or the Naturals, but the domain of $v$ can be, again, literally anything, and as multi-dimensional and as non-numeric as we like.


Does a perfectly price discriminating natural monopoly cause externalities or any type of market failure?

The question seems to be in between theoretical model and reality. In one type of theoretical model one may argue for the monopolist, in another against him. In reality it is worthwhile to mention that no monopoly can achieve first degree price discrimination. It is barely theoretically possible and not technically feasible. Other forms of price discrimination do not necessarily yield Pareto-optimal outcomes. Real world economists usually argue against monopolies in general, not against theoretical first degree price discriminating monopolies.

Even if there were such a monopoly there is an answer to "Why would these things be advocated if pareto efficiency is achieved?"

There may exist a multitude of Pareto-efficient allocations. A rather usual consideration for any social planner is some sort of fairness (or envy-freeness, or inequity-aversion, etc.)

In a setting corresponding to a monopoly with first price discrimination as you say all the surplus is captured by the monopolist. This is such a distribution of surplus/goods/utility that it will probably not meet any of the usual fairness criteria.

So while the monopolist might create an efficient allocation of goods, the allocation might not, and usually will not, meet secondary criteria deemed desirable by society.


Utility function for introductory microeconomics

One that was not mentioned in question is the quadratic utility (aka preference for extremes):
$$U(x,y) = x^2+ y^2$$
This one is less common but still used in micro courses.


The decline in the number of domestic servants

I think that what Stigler called 'domestic servant' would be nowadays considered 'domestic worker'. For example, according to the Luebker, Oelz, & Simonovsky (2013). Domestic workers across the world: Global and regional statistics and the extent of legal protection (a technically semi-academic source as it is a technical report but based on comments I believe this would be acceptable to you), implicitly equates domestic workers with servants (and provides statistics on domestic work). According to the report:

With respect to North America, national sources also show a relatively low share
of domestic workers in total employment – 0.5 per cent in the United States (2010)
... For the United States, the 2010 Current Population
Survey put the number of domestic workers at 667,000 men and women. This represented a substantial decrease, of roughly 140,000 workers, following the onset of the
economic crisis; the same survey counted more than 800,000 in the pre-crisis years
2005 to 2008. Job losses among domestic workers were therefore quite substantial –
an aspect that is not commonly highlighted in discussions of the employment impact
of the global economic and financial crisis. ...
Historical sources indicate that the number of domestic workers was substantially
higher at the time of industrialization in the United States. The censuses of 1870 and
1900 counted 975,000 and 1,455,000 “servants”, respectively (see Rubinow, 1906). [emphasis mine]

Note that the number reported above by Rubinow roughly match statistics reported by Stigler.
According to the report indeed the number of domestic worker ('servants') declined in the second half of 20th century as in 2005-2008 the number of these workers was roughly half of what it was in 1900.
The data sources for US are listed as:

U.S. Bureau of Labor Statistics, Current Population Survey 2010 (Labor Force Statistics: tables 17 and 18 at www.bls.gov).

Additionally they frequently reference data from published papers/reports/books like the Rubinow in the excerpt above.
As to the reason what is the explanation of this decline - the authors do not mention specifically reasons for the historic decline in the U.S. (although for U.K. they cite changes to immigration policy which in past allowed servants to get settled status after 5 years), but the authors and also sources they cite argue that historically most domestic work was done by; a) women, b)  immigrants.

There is a peculiar historical continuity with respect to the overrepresentation
of international migrants among domestic workers. Based on United States Census
data from the late nineteenth century, Rubinow observed (1906, p. 508) that “[a]s
most means of employment are closed to the foreign-born (even English- speaking)
women, almost every second woman is forced to become a servant”. At the time, the
largest numbers of migrant domestic workers in the United States came from Ireland
(195,000), Germany (161,000) and Sweden (45,800).

In the report they also argue that most of domestic work is still performed by migrants or people who face barriers into entry to other occupations. Hence, my first guess would be that the number of domestic servants declined because nowadays especially women but migrants as well face less restrictions when entering into other occupations (although to be clear this is my conjecture).


Why are "transfer payments" not part of GDP?

Transfer payments aren't included in GDP to prevent double-counting.

The reason the author's question is troubling you is because the answer externalizes everything that happens after the payment has been made (which is when the money from that payment DOES get factored into GDP, see below). I assume it does this to reinforce that transfer payments aren't counted in GDP, but you are right to question what's going on here.

That \$100 million is transferred from the government to consumers, who will presumably spend a good portion (say, \$98 million) of those funds on goods and services, at which point those funds would be added to GDP. If we had included the transfer funds in GDP, the change reflected would be \$198 million instead of the more accurate \$98 million.

You can also have an example on a much smaller scale:

You give your child \$5 to buy a burger; none of that is added to GDP.
Your child spends \$4.75 on the burger; \$4.75 is added to GDP.


I will denote the demand function by $Q(p)$ and the inverse demand function by $P(q)$. Then
$$
\forall q: Q(P(q)) = q
$$
so for any $h > 0$ and $q$ we have
$$
\begin{align*}
p & := P(q) \\
p_h & := P(q+h) \\
q & = Q(p) \\
q_h & := Q(p_h) = q+h
\end{align*}
$$
From the definition of derivatives
$$
\frac{\text{d} P(q)}{\text{d} q} := \lim_{h \to 0} \frac{p_h - p}{q_h - q}.
$$
If this is a non zero real number then indeed
$$
\frac{1}{\lim_{h \to 0} \frac{p_h - p}{q_h - q}} = \lim_{h \to 0} \frac{q_h - q}{p_h - p} = \frac{\text{d} Q(p)}{\text{d} p}.
$$

There are some technical considerations. We did not prove that
$$
\lim_{h \to 0} p_h - p = 0.
$$
This is true if $P$ is continuous.
We also assumed that at $q$ and $q+h$ both functions are single valued. If there is a price $p$ where $Q(p) = 0$ then $\frac{\text{d} P(q)}{\text{d} q}$ does not exist.


Incentive differences payroll and income tax

Just to add a basic point not mentioned in other answers.  Payroll taxes relate specifically to circumstances involving employers and employees.  Income taxes apply to income from employment, but also to income from other sources which (depending on a country's tax rules) may include self-employment, pensions, savings and investments, property rents, etc.

This makes a significant difference to the incentive effects of the two types of tax. In particular, a payroll tax can create an incentive for workers to become self-employed and/or for their 'employers' to encourage them to do so. As a consequence it makes it important for the authorities to define clearly the criteria under which a worker is deemed to be an employee or self-employed (in the UK for example the criteria are as outlined here).


Integral solution (or a simpler) to consumer surplus - What is wrong?

Your calculation is correct. We can doublecheck your work with a graphical approach.
As shown in the figure below, $CS$ at some arbitrary and not necessarily equilibrium price $p_0$ is the gray-shaded area. If we take the non-integral approach, we get
\begin{equation}
CS=\frac12\left(\frac{A}a-p_0\right)D(p_0)=\frac12\left(\frac{A}a-p_0\right)(A-ap_0)=\frac{(A-ap_0)^2}{2a}.
\end{equation}
Differentiating with respect to $p_0$, we get
\begin{equation}
CS'=-A+ap_0.
\end{equation}


There is no mistake. The solution: it is assumed that there is only one price at the domestic wheat market. Hence domestic producers will not sell at price $P_w$ and price $P_{Quota}$ as well. This makes sense: Suppose you are a domestic producer and you are aware that the price that results from the quota system is $P_{Quota}$. Knowing this you would not sell your goods for the lower $P_w$.


Calculating marginal tax rates when near a cliff

The marginal tax rate is supposed to reflect incentives for the individual. In order to derive a meaningful metric, the calculation should hence try to reflect the actual decision an individual is facing. An increase in \$1 pay is probably not something people are considering. If, however, someone earning \$10 per hour considers to increase his/her labor supply from 35 to 37 hours per week, you will end up with a pay increase of sth like \$1,000 per year. So I'm rather sympathetic to the \$1,000 step than to the $1 step.

The OECD also applies a change in hours worked to obtain the change in gross income used for calculating marginal tax rates.


Portfolio choice problem of a CARA investor with n risky assets

The problem is equivalent to maximizing

$$
\max_\phi \left( \phi ' \mu - \frac{1}{2} \alpha \phi ' \Sigma \phi \right)
$$

subject to

$$
\mathbf{1}' \phi = w_0.
$$

(boldface 1 is column vector of ones, ' stands for transposition).

The lagrangian is

$$
L(\phi, \lambda) = \phi ' \mu - \frac{1}{2} \alpha \phi ' \Sigma \phi - \lambda \left( \mathbf{1}' \phi - w_0 \right),
$$

and its jacobian wrt. $\phi$ is

$$
\mathrm{D}_\phi L(\phi, \lambda) = \mu' - \alpha \phi ' \Sigma - \lambda \mathbf{1}'.
$$

First order conditions require that the jacobian is equal to zero (in each element), plus the original budget constraint, which together (after transposing the jacobian) yields a system of linear equations in $(\phi,\lambda)$:

$$
\begin{split}
\Sigma \phi + \mathbf{1} \lambda &= \frac{1}{\alpha}\mu \\
\mathbf{1}' \phi &= w_0
\end{split}
$$

We can solve for $\phi$ as a function of multiplier $\lambda$:

$$
\phi = \frac{1}{\alpha} \Sigma^{-1} \mu - \lambda \Sigma^{-1} \mathbf{1}
$$

Plugging this into the budget constraint and some algebra yields an expression for $\lambda$:

$$
\lambda = \frac{\frac{1}{\alpha}\mathbf{1}' \Sigma^{-1} \mu - w_0}{\mathbf{1}' \Sigma^{-1} \mathbf{1}},
$$

and substituting this into the expression for $\phi$ should yield the result as given in the question.


How to test if the effect of one regressor entirely comes from other regressors?

For Bivariate Regression

Your model would be :

IQ = B0 + B1(Your Education)

For Multiple Regression:

You would use a correlation matrix/plot to do this. Basically it would give you a matrix with correlation values of all variables to each other 0 being none and 1 being perfect. This will also allow you to see the multicollinearity problem which is when independent variables are related to each other.

You can do this in R or Stata. I can post the R script of a similar regression I have done if you would like.


What's the relationship between Output elasticity and Returns to scale?

Define the production function $Q=f(x_1,..,x_k)$, where $x_i$ denotes the ith input.
Next recall that the total differential of output can be written as
 $\Delta Q = \sum_{i=1}^{\ k}\frac{\partial Q}{\partial x_i}\Delta x_i$
 Now we are interested in returns to scale, which is the change in output due to the multiplication of every input with a constant. We can write: $x_{1i}=ax_i$, for every x where a is a constant and the subscript 1 means the new x due to the multiplication. It follows that:
 $\Delta x =x_{1i}-x_i =ax_i-x_i=(a-1)x_i$
 Substitute this last result into the total differential. We obtain:
   $\Delta Q = \sum_{i=1}^{\ k}\frac{\partial Q}{\partial x_i}\ (a-1)x_i$
  which can be rewritten as
$\Delta Q = (a-1)\sum_{i=1}^{\ k}\frac{\partial Q}{\partial x_i}\ x_i $
 Now recall that the output elasticity with respect to an input i can be written as
$\epsilon\ = \frac{\partial Q}{\partial x_i}\ \frac{x_i}{\ Q}$
rewrite this to the expression
 $\epsilon\ Q = \frac{\partial Q}{\partial x_i}\ x_i$

Substitute this result into the last version of the  total differential which becomes:
$\Delta Q = (a-1)\sum_{i=1}^{\ k}\ \epsilon\ Q = (a-1)Q\sum_{i=1}^{\ k}\ \epsilon$
Rewriting this we obtain the multiplicator of the output associated with multiplication of every input with $a$:
 $\frac{Q_1}{Q} = (a-1) \sum_{i=1}^{\ k} \epsilon +1$
Where $Q_1$ is output ex post.

From this last expression it can be immediatly seen that constant returns to scale ($ax \implies aQ$) follow from the sum of elasticities being equal to 1, increasing returns to scale from it being greater than 1 and decreasing when it's smaller than 1.

The graph is plotted for a=2. In red all points which correspond to decreasing returns to scale, in green for increasing returns to scale. Further, it can be seen that
 $\sum_{i=1}^{\ k} \epsilon = 1 \implies \frac{Q_1}{Q} = a$


Is it proven that prestige gained from sports victories helps the country economically?

There is research into this topic but it's not proven.

There's too much variability to consider.

Take the '04 Detroit Pistons (it's a City, I know, but the premise scales). In 2004 the Detroit Pistons were underdogs to the Los Angeles Lakers. Economically, LA was in a much better place than Detroit but Detroit scrapped it's way to a victory (and a return to the finals but a loss the next year to the San Antonio Spurs (Go Spurs Go)). Detroit's victory may have led to sales growth in player's jerseys (Richard Hamilton and Ben Wallace for instance) but the sports victory did not save the city from it's demise and subsequent bankruptcy.

I believe the economic benefits of professional sports victories are more cosmetic than practical. A real estate agent may have a few more selling points in an economic transaction but the gains are usually temporary as it's hard to have back-to-back victories in professional sports...

Controversially, the theory MAY ACTUALLY apply to collegiate sports and the economic prowess of a university. Ever been to University of Georgia or Alabama? Those walls aren't lined with money from their journalism schools, I can tell you that much.

It'd be hard to prove that professional sports wins tie to economic success given how many other exogenous factors there are to economic success but there's data available to analyze this question (specifically, GDP growth and World Cup/Olympic Cup/World Title Events).

I don't know if governments are tossing up grant money for this research though.


Why is the Bank of Japan buying all the government bonds?

Central banks usually buy back government bonds to combat deflation and facilitate economic growth. This policy is commonly called "quantitative easing" or QE. After the arguable success of QE during 2008-2009 recession in the United States, many other countries choose to adapt this monetary policy.

It is very unlikely that the Bank of Japan will keep this policy until 2026 because at some point inflation will pick up and they will have to stop buying bonds and perhaps even raise interest rates. In the US, UK, and EU the bond-buying quantitative easing programs haven't lasted longer than 5 years and they are usually closely monitored and can be stopped at any time.

When the central bank buys government bonds, they stay on its balance sheet until maturity. Normally, central banks are happy to keep them as a convenient monetary tool. They can sell the bonds back to reduce inflation. Whether the government debt on the central bank's balance sheet can "cancel out" depends on the countries laws.


How does the Federal Reserve unwind the asset purchasing scheme announced in response to the 2020 Coronavirus Pandemic?

The unwind depends upon what the central bank did.

If the central bank did a repurchase agreement (“repo”) or lent against assets for a fixed time, the agreement automatically unwinds at the term of the deal (which is short). They would need to enter into new deals to keep their balance sheet size unchanged.
If they bought the asset outright, it will either mature, or the central bank can sell ahead of maturity.

There are no “charging a premium” involved; the cost of the operation is embedded in the lending cost (or purchase/sales price of an asset).

There was a very large debate about unwinding central bank balance sheets in the 2010-2016 (?) period. The Fed was eventually able to slowly reduce the size of its balance sheet in the last cycle, and there is no reason to believe that this time is different. There is no sign that an “unwind” was necessary, nor are the effects of unwinding a deep concern.


What were the factors that caused the Philips curve to stop working during the 1970s?

The general way people figured the Phillips curve worked was that a shock in aggregate demand or fiscal stimulus would cause labor demand to increase as government spending generated growth, which makes labor more scarce and causes firms to compete for workers by raising nominal wages. Wage costs then rise, and the firm passes on some of the cost to the consumer by raising the prices of their products.

So in this way, unemployment and inflation become tradeoffs. The problem is then this can be easily exploited, but also easily noticed. Lucas's critique states that people's expectations of inflation--and future inflations--will affect price levels as well. So if a policy maker tried to create inflation all the time to lower unemployment, what would happen is that the Phillips curve would actually shift around, so the tradeoff would hold, but you would be getting worse tradeoffs as people's expectations shifted.

By raising inflation constantly by the central bank and by politicians, you'd fall into a liquidity trap (when raising inflation fails to decrease interest rates and stimulate and economy). The reason why is because if everyone expected the rise in inflation, there'd be no reason to hire more workers, since real demand is staying the same. So instead of a short run rise in output, firms just went straight to raising prices instead of hiring more people.

That said, today, Lucas's critique has its limitations. Even transparent policies that are fully known to increase inflation by firms can still boost output in the short run. This is probably because after the 1970s in America, the new head of the Federal Reserve, Paul Volcker, decided that the central bank should commit more heavily to a set inflation rate in the long run, so that their policies would be more credible.

We can set up a game-theoretic model for optimal monetary policy, as such:

$$\max_{u, \pi} V(u, \pi) = -(u^2 + \pi^2)$$

The central bank wants to minimize a combination of unemployment and inflation. Their constraint is a simple Phillips curve:

$$u = u^* - k(\pi - \pi^e)$$

Where $k > 0$ is some weight for the degree of the tradeoff ($-k$ is the slope of the Phillips curve). $u^*$ is the natural rate of unemployment and fixed, while $\pi^e$ is the expected inflation.

Comparing the Ramsey and Nash equilibria of this game, where consumers and firms set expectations of inflation and the central bank tries to game that, we'll find that if firms trust the central bank to be honest about what inflation they are targeting, the central bank will always have the incentive to lie, if only acting for one period.

Substitute the Phillips curve into the central bank's maximization problem:

$$\begin{align}
& V = [u^* - k(\pi_t - \pi^e)]^2 + \pi^2 \\
& \frac{\partial V}{\partial\pi} = -2[u^* - k(\pi_t - \pi^e)]k + 2\pi = 0\\
& \implies \pi_{\text{opt}} = \frac{k}{1+k^2}(u^* + k\pi^e)\\
& u_{\text{opt}} = u^* - k(\pi - \pi^e) \\
& \implies u_{\text{opt}} = \frac{1}{1+k^2}u^* +  \frac{k}{1+k^2}\pi^e \\
\end{align}$$

Now assume the private sector knows the Fed's optimal monetary policy problem, and set $\pi^e = \pi_{\text{opt}}$. Say the Fed is credible.

$$\pi_{\text{opt}} = \pi^e = \frac{k}{1+k^2}(u^* + k\pi^e)$$
$$\boxed{\pi^e = ku^*}$$

$$u_{\text{opt}} = \frac{1}{1+k^2}u^* +  \frac{k}{1+k^2}\pi^e$$
$$= \frac{1}{1+k^2}u^* +  \frac{k}{1+k^2}ku^*$$
$$\boxed{u = u^*}$$

These are the best responses for the Fed...if they are worried about telling the truth. Investigating Ramsey's time inconsistency problem, the Fed can technically do better. Suppose it announces $\pi = 0$ as a target.

$$\pi_{\text{opt}} = \frac{k}{1+k^2}u^*$$
$$u_{\text{opt}} = \frac{1}{1+k^2}u^*$$

You can verify that the Fed has an incentive to deviate in a one-period game.


Under what condition is a cost function strictly concave in prices?

No function that is homogeneous of degree one, is at the same time strictly concave in its arguments. If the function is differentiable (or non-differentiable at a finite number of points), then the Hessian of a linear homogeneous function is singular. So if you want to end up with a unit cost function that is strictly concave, you have to drop at the same time the linear homogeneity in $w$.


In the standard theory of games, simultaneous and sequential games are distinguished by the means of something called "Information Sets". Alluding to the origins of Game Theory (von Neumann and Morgenstern 1944), simultaneous move games can be thought of as special cases of sequential move games. In their landmark work, vN-M give an explicit method to construct simultaneous move version a sequential game. For them, every game is essentially a sequential move game. But that's just history.
From a theoretical standpoint, all that matters is what does a player "know" about the previously played moves while making a decision in a game. This notion of the "information that a player has before making a move" is characterized in sequential move games (and their game trees) via Information sets.
Consider, for example, prisoner's dilemma. But suppose that it is played sequentially, in two different ways:

Player 1 moves. Player 2 learns Player 1's move. Player 2 moves.
Player 1 moves. Player 2 does not know about Player 1's move, but knows that Player 1 has moved. Player 2 moves.
In most cases, 2. is essentially a simultaneous move game.

Here's how we conventionally represent a simultaneous move game in extensive form. In the above image, the dotted line joining the two decision nodes of Player 2 signify that while making her decision, Player 2 does not know at which node she actually is. If there were no dotted line, it would mean that Player 2 knows at which node she is, and hence has learned about Player 1's choice. In the pictured version, all that Player 2 knows that if she plays C, she would get either 3 or 1, but that's it. Note that this is exactly similar to the situation we have in a simultaneous move game. A repeated version of this one shot simultaneous move game is then easy to imagine.
In GT jargon, we say that these two nodes which are joined are contained in the same Information set. So, an Information set is the collection of nodes among which a player cannot differentiate.


Why is the production possibility set convex?

Perhaps you are confusing two things.

If $Q_1$ and $Q_2$ denote the production of goods 1 and 2 in a single country and you are in the space defined by them given $L_1 + L_2 = L$ then it is indeed the inferior set (the production possiblity set defined by the feasible $(Q_1,Q_2)$ pairs) that is convex. Formally this means that for all feasible $L_1,L_2,L_1',L_2'$ labor allocations you have
\begin{eqnarray*}
\mu \cdot Q_1(L_1) + (1-\mu) \cdot Q_1(L_1') & \leq &
Q_1(\mu \cdot L_1 + (1- \mu) \cdot L_1') \\
\\
\mu \cdot Q_2(L_2) + (1-\mu) \cdot Q_2(L_2') & \leq &
Q_2(\mu \cdot L_2 + (1- \mu) \cdot L_2').
\end{eqnarray*}
As this is equivalent to saying $Q_1$ and $Q_2$ are concave this is true given the condition that you describe. In this situation the function $Q_1 + Q_2$ has no meaning at all. What are we adding up? Ducks and cars? What will be the unit of measurement here?

But if $Q_1$ and $Q_2$ represent the production of a single good in countries 1 and 2 given the  amount of labor allocated and the space is $L_1, L_2$, the curves being levels of $Q_w$, then it is the upper level curves that are convex. You can see this straight away because the marginal products of $L_1$ and $L_2$ are decreasing hence their rate of teachnical substitution is 'decreasing' as well. Formally:

By definition the function $Q_w(L_1,L_2)$ is concave if for all $L_1,L_2,L_1',L_2'$ and for all $\mu \in [0,1]:$
\begin{eqnarray*}
\mu \cdot Q_w(L_1,L_2) + (1-\mu) \cdot Q_w(L_1',L_2') & \leq &
Q_w(\mu \cdot L_1 + (1- \mu) \cdot L_1',\mu \cdot L_2 + (1- \mu) \cdot L_2').
\end{eqnarray*}
We know that the functions $Q_1$ and $Q_2$ are concave, meaning
\begin{eqnarray*}
\mu \cdot Q_1(L_1) + (1-\mu) \cdot Q_1(L_1') & \leq &
Q_1(\mu \cdot L_1 + (1- \mu) \cdot L_1') \\
\\
\mu \cdot Q_2(L_2) + (1-\mu) \cdot Q_2(L_2') & \leq &
Q_2(\mu \cdot L_2 + (1- \mu) \cdot L_2').
\end{eqnarray*}
Adding these up we get
\begin{eqnarray*}
\mu \cdot \left(Q_1(L_1) + Q_2(L_2)\right) + (1-\mu) \cdot \left(Q_1(L_1') + Q_2(L_2')\right) \leq \\
\leq
Q_1(\mu \cdot L_1 + (1- \mu) \cdot L_1') + Q_2(\mu \cdot L_2 + (1- \mu) \cdot L_2').
\end{eqnarray*}
And since
$$
Q_w(L_1,L_2) = Q_1(L_1) + Q_2(L_2)
$$
we have
\begin{eqnarray*}
\mu \cdot \left(Q_1(L_1) + Q_2(L_2)\right) + (1-\mu) \cdot \left(Q_1(L_1') + Q_2(L_2')\right) \leq \\
\leq
Q_1(\mu \cdot L_1 + (1- \mu) \cdot L_1') + Q_2(\mu \cdot L_2 + (1- \mu) \cdot L_2') \\
\\
\mu \cdot \left(Q_w(L_1,L_2)\right) + (1-\mu) \cdot \left(Q_w(L_1',L_2')\right) \leq \\
\leq
Q_w(\mu \cdot L_1 + (1- \mu) \cdot L_1',\mu \cdot L_2 + (1- \mu) \cdot L_2').
\end{eqnarray*}
which is what we set out to prove.


Clarification of iterated prisoners dilemma

The payoff depends entirely on how you set the game up. Here's one example of the case where $2R \leq T + S$:

                                  Player B
                           Cooperate  |    Defect
Player     Cooperate        (1, 1)    |  (-2, 100)
A          Defect          (100, -2)  |  (-1, -1)

Here, $R = 1$, $T = 100$, and $S = -1$, and $2R < T + S$.

Edit: To add a bit more value to the answer... any game could be played repeatedly (and thus is "iterable" in that sense.) However, my understanding is that the Wikipedia page writer means that for (Cooperate, Cooperate) to be sustained, you at least need this condition to be true.


If prices stay the same, how's lending money the same as lending real goods?

Lending money is on fundamental level equivalent to lending goods because you can use the money to buy some goods and services. If a truck cost \$100,000 and you finance that truck purchase via getting bank loan, then on fundamental level that is equivalent of the bank just lending you the truck.
Next, in economics 'real' essentially means inflation/deflation adjusted. Opposite of 'real' variable (e.g. real consumption, real investment, real spending, real output etc) is not imaginary or unreal variable but nominal variable (that is variable that does not take into account change in price level e.g. nominal consumption, nominal investment, nominal spending, nominal output etc.).
If prices do not change from period to period (i.e. there is no inflation or deflation), then any interest rate on a loan will represent real interest rate because by Fisher equation (see Mankiw Macroeconomics pp 110) the real interest rate is given by:
$$r \approx i - \pi$$
where $r$ is real interest rate, $i$ is nominal interest rate and $\pi$ is inflation. If $\pi=0$ then any nominal interest rate you pay at the bank is virtually equivalent to real interest rate.
Moreover, since prices do not change borrowing money from bank is on fundamental level equivalent to borrowing real goods from the bank since if you borrow \$100  you can finance \$100 worth of real consumer spending.


Computing an equilibrium is not needed for implementing it. This was the mistake of Lange and co. and was decisively rebutted by Hayek.
If you want a mathematical formulation, simply take a tatonement process. Given prices, each individual needs to compute her net demand correspondence. Under different tatonement mechanisms, which are an idealization of the dynamic process of price adjustment in the real-world economy, prices should converge to approximate equilibria. In both experimental settings and in financial markets, there is ample evidence for convergence.


Are there big waves of entrepreneurship over time?

I think there are waves indeed. When you look at historical evidence you see a swinging between 'more free market' and 'more central planning (thus less free market)'. The more free a market is (less regulation, less taxes), the more entrepreneurship you will see.
In the Soviet-Union for example entrepreneurship was dead because everything was centrally planned by government. It made no sense taking risk or sticking your neck out for something.

For statistics, if I were you I would try to look for historical data about 'number of people self employed', 'number of starting businesses', ... Maybe you can start by looking over here: http://www.oecd.org/std/business-stats/ or here http://ec.europa.eu/eurostat/web/structural-business-statistics/entrepreneurship/indicators

hope this helps a little


Redding and Weinstein in their paper "A Unified Approach to Estimating Demand and Welfare" develop a unified price index that nests all existing major price indexes. Existing price indexes can be thought of as arising from the imposition of parameter restrictions on their unified index.

Their examination with barcode data suggests that standard methods of measuring welfare overstate cost of living increases by ignoring new products and demand shifts. However, contrary to the Boskin report, they do not put an overall number on the bias.


Should unemployment caused by the government closing non-essential businesses during quarantine be counted as structrural unemployment?

There are lots of different people unemployed, and the answer will be different depending on the person. In your prostitute example, if you left your medical assistant job because prostitution paid more, then it's frictional, because there are lots of medical assistant jobs, but it might take you time to go through the interview process and be onboarded. If you have no marketable skills other than sex work, then it's structural unemployment. If there is another job that you're qualified, but you don't get the job because there are so many other people competing for it, then it's cyclical (which is a bit of a misleading term, since there it's not like there's a "quarantine/no quarantine" cycle, at least not yet, fingers crossed).


If house prices appreciate, why do developers sell them?

They are different businesses.

Developers make more money developing than landlording. Some do both. But they are different businesses entirely. Also, there is a limited supply of capital and carrying that capital has a cost.

For example. Let's say Bob the Builder borrows $10M to develop a project. The bank lends on the development. Not the completed project. That requires a different type of loan structure, depending on the use: residential vs. commercial, owner occupied residential vs. residential rental, etc.

After the project is completed, Bob will usually look to sell. If he holds the property as a rental or a commercial, he then switches hats and becomes a landlord. And the asset will require refinancing, as above described. Some builders choose this route. But most do not. As their skills, expertise, equipment, personnel, interests, etc. are all used best and most profitably building and developing.


How to show that a homothetic utility function has demand functions which are linear in income

I think what you need is that if $U(x,y)$ is homothetic then
$$
\forall \alpha \in \mathbb{R}_{++}, \forall (x,y) : \hskip 6pt
\frac{\frac{\partial U(x,y)}{\partial x}}{\frac{\partial U(x,y)}{\partial y}} =
\frac{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial x}}{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial y}}
$$
and love.