If I fill a vending machine with \$0.50 sodas and sell them for \$1, am I creating 50 cents of wealth for the economy?

You have to be very careful with your definitions here. Following the basic definitions (e.g. see use of terms in Mankiw Principles of Economics):
Value: Value depends on your marginal utility and typically it is the amount of money you are willing to give up for something.
Wealth: Is by the value of net assets.
Income: Is the net return to some activity.
You are right that economy is not necessary zero-sum game (although there are economic interactions which can be zero-sum), but your use of the term wealth is improper.
If someone pays you to fill the soda machine, you have costs \$0.5 but get paid \$1 your income will be \$0.5, you created at least \$0.5 of value because the person who paid you clearly valued the filled vending machine for at least \$1 otherwise they would not exchange that 1 dollar for your work, but at the same time you had 0.5 costs, so you are creating at least \$0.5 extra value.
The wealth is accumulated by saving (whether you save by buying house or putting money on your account value of your net assets increases). Hence if you have \$0.5 income and you save it all you also create \$0.5 wealth. If you consume half you only create \$0.25 wealth and so on.

So in the above example with the vending machine, does the federal reserve have to create an extra 50 cents of the currency in order to avoid deflation?

Here the answer is maybe. Inflation/deflation does not depend just on the value of output that you create.
Inflation/deflation is just positive/negative change in the price level which is in turn determined by the money market equilibrium. The money market equilibrium, in its simplest form is given by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY$$
Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ output.
Solving for price level and log-linearizing (so % changes in right hand side variables give us the % change in $P$) we get:
$$\ln P=\ln M + \ln V − \ln Y$$.
If you perform those extra services you increase real output by \$0.5, ceteris paribus, you are correct that Fed would have to create additional \$0.5 dollars to prevent inflation. But the, ceteris paribus, assumption might not hold in real life because velocity of money can change as well (plus in more complex models of money market equilibrium expectations play role and so on). Consequently, the correct answer here would be maybe. Under ceteris paribus assumption yes, but it is not guaranteed.

PS: Note sometimes even economists in casual speech equate wealth creation with value creation, so you might have heard some pundit or economists use similar example to argue wealth was created, but that is not how the two terms are rigorously defined in the literature.


Testing the long run phillips curve / Contradicting empirical results?

You are assuming the PC has not shifted over the period. For example, you can see at the top right some sort of PC. To see this better, you need first to analyse the data in evidence for regime shifts. The first suggestion is to connect the points over time. US example below:

This will let you see possible structural changes in the PC, together with movements between two PCs. This gif shows this more clearly:

Knowing the monetary and economic history of Canada should certainly help you understand the movement and changes of the PC. For example, you cannot expect the PC to hold over the oil crisis period, where the economy was basically in stagflation.

Once you identify sub-periods, estimate the PC for each subperiod.

A paper from the Central Bank of Canada estimating the PC and its regime shifts can be found here. Notice that standard analysis of PC are much more elaborated than a simply bivariate correlation. Strictly speaking, you are assuming there is no other variable involved in the correlation. The example of structural changes might for example be captured with changes in a variable you are omitting.

PS: I gave an answer to a related question here.


Why is it a problem if China starts "dumping" US Treasuries? What does it mean to be dumping?

It would increase the total supply of US treasuries available at any point in time, which would require a higher interest rate for the market to clear. Higher interest rates would increase the cost of capital in the US economy, which in most conditions will reduce investment.

However, higher supply would also decrease the exchange rate of the USD, which could be beneficial for exports (which needs to be considered in a context of higher cost of consumer imports and higher cost of production inputs).

"Dumping" would be fine as a descriptive word for many informal contexts, but "dumping" has specific meaning in an international legal context, such as in the WTO, and so regarding the specific question of what it would mean to unload large volumes of US treasuries at the same time, e.g. for the purpose of negatively impacting the US economy, "dumping" would not be the correct term.

It should also be mentioned that China would immediately lose significant financial assets if the 1.2 trillion in treasuries were to lose 100 billion or 200 billion dollars in equivalent value. Considering the relatively large volume of credit extended in the Chinese economy relative to financial assets, this implies a relatively major economic cost to China if it were to make such a move. On the matter of strategic reasons to hold large volumes of US treasuries, the face value argument of the Chinese government is legitimate, if not the complete story: What else should they be holding if not US treasuries?


Widening the acceptable band for the overnight rate would reduce the effectiveness of monetary policy to a certain extent. If the actual market rate is near the middle of the band, then it would be possible for the policy rate band to be moved by 50 basis points, and the actual rate could remain unchanged. This implies a potential need for greater movements in the policy rate band to achieve the same overnight rate trajectory.

Furthermore, allowing more variation in the overnight rate creates greater uncertainty about the average over time, even with the policy stance unchanged. This creates greater risk for fixed income instruments of maturities greater than overnight. Under the assumption that investors are risk averse, this would raise term premia.


What's the actual relationship between deflation, spending and saving?

Think about the permanent income hypothesis. People's spending is dependent not just on present income, but also the future stream of income. A deflation usually indicates a recession. The central bank is trying to optimize monetary policy subject to a short run Phillip's curve.

$$\min_{u, \pi} V(u, \pi) = -(u^2 + \pi^2)$$
such that
$$u = u^* + k(\pi^e - \pi)$$

where $u$ is unemployment, $u^*$ is the natural rate of unemployment, $k$ is the sensitivity of the tradeoff, and $\pi^e$ is the market expectation of inflation.

If you find a Nash equilibrium for this game with credibility in the model, then the Federal Reserve will want to target a constant steady rate of inflation. If the economy falls into recession, they will in return try to raise inflation, because in the short run it serves as a tradeoff with unemployment, so though the Fed is causing more inflation, hopefully they will stop too many people from losing jobs (note the loss function for the Fed in this case is convex). If the Fed tries out monetary policy and still sees deflation, then you can tell the recession is pretty rough, if that makes sense.

So even though the price of goods falls, wages on average will fall, and production will fall. So demand in the economy will fall. Those who do still have a good job are doing better on average than those who become unemployed, and they might dissave in periods where expected income in this recessionary period might be lower than future periods where the economy is doing well, but in reality, most people will be liquidity constrained and will be saving what little they can while drastically cutting back on spending. Usually. There are exceptions (think of payday loans).


What is the difference between the discount window and Fed repos?

Economically, they are both loans against financial instruments. The difference is the collateral posted - repos by the Fed are generally against Treasury securities. This is not a small difference: it makes it much easier for a bank to access liquidity, since not all of its assets are Treasury securities.

There’s a mechanical difference: a discount window operation is borrowing, while a repurchase agreement is matched pair of buy/sell orders.

Historically, there’s a big difference: banks only went to the discount window at the Fed if they faced liquidity problems. I.e., the Fed is the lender of last resort. This was greatly discouraged, as it was a signal that the markets did not consider the bank to be viable, and the regulators would want to know why. Since the Financial Crisis, there have been efforts to remove that stigma.


I went over your calculations and they seem to be right. In principle, this is a valid answer, but I understand that the numbers seem suspicious. Are you sure this is the right problem?

Update: everything checks. I think your answer is correct, although it looks "weird" for a textbook problem.


Consider the following strategies. In any period a player makes an offer, they offer a share of $0.33$ to the other player, keeping $0.67$ for themselves. In any period a player accepts or rejects an offer, they accept any offer that gives them at least a share of $0.33$ and rejects anything worse.

Clearly, the outcome of these strategies will lead to the immediate acceptance of the offer. All that remains is to check that these strategies form an equilibrium. We do so by appealing to the one-shot deviation principle.

Consider the perspective of a player deciding to accept or reject. They will not want to deviate from the prescribed strategies only if

$$ 0.33 \ge 0.67 \delta \iff \delta \le \frac{33}{67} $$

However, this violates the assumption that $\delta = \frac{1}{2}$. Thus, the given strategies cannot form an equilibrium.

For characterising other equilibria, see van Damme, Selten and Winter (1990).


What would happen in a modern economy if growth plateaus?

Caveat
 Your main question is simply too broad to cover in SE format so this answer will focus only on the 3 further sub-questions.
tl;dr
The premise of the question is actually not generally valid. Economic growth is not driven primarily by population growth, especially not per capita economic growth (which is what matters for living standards). However, suppose that there will not be any economic growth for some other reason. The short answers to your 3 questions would be:
A1: This would put stress on the pension system. However, It would not make the pension system unworkable as pensions can work on the same principle as an insurance and that does not require economic growth for it to be viable. Nonetheless, at the same time it would make pension schemes more expensive which would mean people would have to either; a) work longer; b) have lower pensions; c) payments into these pension plans (or taxes in case of government) would have to increase.
A2: I can't see any economic reason why this would affect fractional reserve monetary systems in any way whatsoever. In fact this won't be further addressed in full answer because there is not much more to say about it. Fractional reserve system simply works equally in a falling, stagnant or growing economy.
A3: This would not have any effect on inflation in itself ceteris paribus, but there could be some effects caused by the way the government chooses to respond to the issue of no growth.
Full Answer
Addressing the Premise
Before addressing the questions let me address problems with your premise. You argue that:

If the growth rate for the population is low, it would take considerable effort to maintain existing growth rates in the economy, since the number of people available to produce capital would be growing at a lower rate or not growing at all.

This is simply not a generally accepted view in growth literature.
First, what matters for the economy is growth per capita (per person) not just economic growth in general. For example, if GDP doubles from 100 to 200 but also the number of people increases from 1 to 2, GDP itself grows but per capita GDP in both scenarios is just 100. It is growth per capita that matters for material welfare of people  (see discussion of living standards in Mankiw Macroeconomics or Blanchard et al. Macroeconomics: a European Perspective).
Second, most economists don't believe that per capita growth has much to do with population growth. In economics there are two main strands of literature when it comes to economic growth. There are exogenous theories of economic growth and endogenous theories of economic growth. Both of the above mentioned strands of literature agree that per capita economic growth is caused by increase in technology but they disagree on what can cause increase in technology. In the exogenous growth literature (which is actually more dominant) population growth has no effect on GDP per capita whatsoever. In the literature on endogenous growth it might have an effect on GDP per capita through R&D (more people means more scientists) but even most endogenous growth models what matters more than population is investment in R&D and share of population devoted to producing R&D. Even in the small subset of models zero population growth would not lead to zero growth per capita - only to lower growth than otherwise (see discussion in Ch 1 and 3 in Romer Advanced Macroeconomics and Barro & Sala-i-Martin Economic Growth). This being said for the rest of the answer we can just assume zero economic growth as a thought experiment.
Answer to question 1 on pension systems
Zero economic growth and population growth would definitely put a stress on pension systems but they do not require economic growth or population growth to operate.
Generally speaking pension schemes work on a similar principle as insurance. In fact you can consider it an insurance for reaching a certain age. Such a scheme does not require growth in population or economic growth per se. However, economic and population growth make these schemes cheaper for the government (respectively people in case of private pension schemes).
Population growth makes  pension schemes cheaper because it improves the dependency ratio (i.e. how many working age people support one retiree - see further explanation at Eurostat). However, dependency ratio is affected by large number of things that are completely in control of government (company offering pension). For example, retirement age directly affects dependency ratio because people at some point die from old age. Thus, in principle if you set retirement age high enough you can get as favorable dependency ratio as you want.
Next the economic growth helps in the case of government pay as you go schemes because it increases peoples income and thus indirectly taxes that government collects (20% tax from €1000 is higher than 20% tax from €100). Hence as government budget grows over time these pensions can become more generous. In absence of any economic growth government would just have to face tighter constraint on its spending. In the case of private insurance where retirement money are invested an additional problem would be that if there is no growth investment opportunities will be low to none. However, this just means that money people save for retirement do not accrue interest. But pensions can still exist they will just not be such a good deal for people anymore as before.
If there would be no population and economic growth governments (people) could in principle adjust to the situation by:

Increasing the retirement age. This automatically improves the dependency ratio as if there is less people who are eligible
Stop increasing pensions over time.
Increase taxes (payments with private schemes) to raise more resources for such schemes.
Make pensions less generous.

The above might be serious political issue but from economic perspective it is completely feasible. Political problems are for political scientists to worry about.
Answer to question 3 about effect on inflation
There does not need to be any necessarily.  Inflation is determined by changes in price level which in turn depends on what the money market equilibrium is. This is given by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY \implies
\ln P=\ln M+ \ln V− \ln Y $$
Where M is the money supply, V velocity of money, P price level and Y output. In the log-linearized version the effects of right hand side variables can be interpreted as % changes. If there is  zero economic growth that  means $\Delta Y = 0$, which holding everything else constant would have no effect on price level.
When it comes to the other variables $M$ is under control of the government. Hence, it can stay put in principle. One could argue that due to lack of growth government might face higher pressures on monetary financing (due to the problems discussed in previous answer for example). But this would not be caused by the lack of growth itself.
Lastly, $V$ is historically very stable, although it does depend on interest rate. To be more specific generally when interest rate decreases velocity also decreases. To the extend that zero growth would cause drop in interest rate it would lead to deflation equal to the rate at which velocity dropped. However, velocity cannot drop forever (if velocity is 0 that means money are not used in economy at all). So there would be temporary deflation, which could be offset by government  increasing money supply proportionally as to keep prices constant.


"If $\lambda$ is greater than than 1, the system explodes." Why does the system explode?

Eq. (2.64) can be written as (at first order)

$$
k_{t+1} - k^* = \lambda (k_t - k^*) \tag{1a}
$$

Define the quantity $\kappa_t$ as

$$
\kappa_t \stackrel{\rm def}{=} k_t - k^* \tag{2}
$$

So that

$$
\kappa_{t+1} = \lambda \kappa_t \tag{1b}
$$

This can be solved very easy by realizing that

\begin{eqnarray}
\kappa_1 &=&\lambda \kappa_0 \\
\kappa_2 &=&\lambda \kappa_1 = \lambda(\lambda\kappa_0) = \lambda^2\kappa_0 \\
&\vdots& \\
\kappa_t &=& \lambda^t \kappa_0 \tag{3}
\end{eqnarray}

So the dynamics is $\kappa$ is driven by the value $\lambda^t$. You can see that if $\lambda = 1$ then $\lambda^t=\lambda$ for each $t$ and $\kappa$ remains constant. However, if $\lambda > 1$ the values $k^t$ will become increasingly large (explodes) as $t$ increases, that is

$$
\lim_{t\to \infty} \kappa_t \stackrel{\lambda > 1}{=} \infty
$$

Similarly

$$
\lim_{t\to \infty} \kappa_t \stackrel{0< \lambda < 1}{=} 0
$$


Taylor Series Approximation around steady state in Solow

The steady-state value $k^*$ must be a fixed point :

$(1+g)(1+n)k^* = s(k^*)^{\alpha} +(1 - \delta) k^*$

Taking the difference between this equation and the dynamic one :

$(1+g)(1+n)(k_{t+1} - k^*) = s((k_{t+1})^{\alpha} - (k^*)^{\alpha}) +(1 - \delta) (k_t - k^*)$

Now if you denote by $d_t = k_t - k^*$ the distance to steady-state, this gives :

$(1+g)(1+n)d_{t+1} = s((k_{t+1})^{\alpha} - (k^*)^{\alpha}) +(1 - \delta) d_t$

But :

$(k_{t+1})^{\alpha} = (k^* + d_{t+1})^\alpha = (k^*)^\alpha + \alpha d_{t+1} (k^*)^{\alpha - 1} + o (d_{t+1})$

(Here was the Taylor development.) So that finally, this equality holds approximately around steady state ($d_t$ small):

$(1+g)(1+n)d_{t+1} = \alpha s (k^*)^{\alpha - 1} d_{t+1} +(1 - \delta) d_t$

From the steady state condition you can also compute (assuming that $ng << n+g$):

$(k^*)^{\alpha -1} = \frac{n+g+\delta}{s}$


Can an In the Money Put Option's price $>$ its Strike Price?

If you own a put, you have the right but not the obligation to sell at the strike price. For most financial assets generally including stocks and bonds(although some commodities have been known to take negative prices), the minimum price they could possibly sell for is zero, because of limited liability. The worst they can be worth is zero, meaning that they generate no cash flows. If an asset has a price ($P$) of zero and you can sell it for the strike price ($K$), then you take a profit of $K-P = K-0 = K$. This is the maximum in the money value of the put because the asset price can't be below zero).
As you can see in the figure below (a call option, but it is the same idea), the time value of an option falls as the underlying price gets farther from the strike price. Therefore, when the ITM value is maximized you are as far as possible as you can get in one direction from the strike price, and we should expect the time value to be small. In fact, if you have a American option where you can exercise early, you will want to exercise this put option early if the price of the underlying ever reaches 0, because you can't make any extra money by waiting, which implies the time value is not just small but actually zero. NB: I believe this is only true when interest rates are positive. With negative rates it pays to wait just to avoid getting negative interest.

I believe it is not generally possible for the total value (ITM + time value) of an option to be above the strike price, which includes when it is out of the money, again as long as interest rates are positive. At exercise, the most the option can be worth is $K$. The expectation of a random variable that takes only weekly positive values less than $K$ must be less than or equal to $K$ ($0 <  E[X / (1+r)^t| 0 \leq x \leq K]  <  K/(1+r)^t < K $ where $t$ is the remaining option life.
Let me say this another way. The most you can ever get at this option at maturity is $K$. What's this worth to you? The absolute maximum this should be worth to you is if you knew for certain that it would be $K$. Then you would have a certain payoff of $K$ and the price would be $K/(1+r)^t$, just like any risk free cash flow. If anything else happens and the price of the underlying is anything higher than zero, this option is worth less than this. An option is out of the money if the price of the underlying is above the strike $P>K$. But the higher $P$ is, the lower the probability of the stock hitting zero is. This means that the total value of the option (ITM + time value) should be falling as the underlying price increases. This convinces me that that it cannot be that the total value of the put option is ever higher than strike price.


Gapminder's Dollar Street and the role of self-supply

On Gapminder's website they write

  If you have a question, maybe it's already answered in our Q&A.
   Q&A

The underlined text links to a Google document, which answers several questions. Under one of these answers

  The households’ door numbers represent the consumption values (US dollars) that each adult in the household has per month. This figure comes from a combination of the household’s self-reported consumption and income levels. We then checked official cost of living data in each country to adjust the values for purchasing power parity (PPP) and converted the value into US dollars.

there is a link to a further Google document titled Detailed income calculations for Dollar Street. This second one seems to answer your question, but it is several pages long, so I will not copy paste its full body of text here.


I'm not sure what the "fuss" is.  Is TRPF really so heavily disputed?  In a sense, profits are just a kind of arbitrage, and so the reasoning behind TRPF is similar to the reasoning around zero-arbitrage assumptions (profit-maxing agents under perfect competition with free entry/exit and full information will drive supply up and prices down until the profit gap closes).

Obviously, these assumptions make for boring problems; the real world is far from boring, so profits happen and can be sustained for a really long time in some cases.  I think the diversity of viewpoints you're seeing are more explanations for the varying degrees of failure of TRPF in real markets, than they are explanations of empirical evidence of TRPF.


Are Price and Quantity Conjugate Variables?

Lauwrence Lau (1969) has pointed out that "Samuelson has pointed out the basic duality that exists between the direct and indirect utility functions: They are connected by Legendre’s duaI transformation."
Lau (1973, chapter 1.3, section 1.4) makes intensive use of the Legendre transformation for deriving dual relationships between production and profit functions. This chapter is available here.
Most duality theory between direct and indirect utility functions, and between production, cost and profit functions rely on these relationships (See also Blackorby, Primont and Russell, 1978).
Blackorby, Charles, Daniel Primont, R. Robert Russell, 1978, Duality, Separability, and Functional Structure: Theory and Economic Applications, North-Holland.
Lau, L., 1973, Applications of Profit Functions, in Production Economics: A Dual Approach to Theory and Applications, edited by Melvyn Fuss and Daniel L. McFadden, North-Holland.
Lau, L., 1969, Duality and the structure of utility functions, Journal of Economic Theory, 1, 374-396.


What is the point of all the models in an economics degree?

Models are more than just math. Model is a simplification of a reality that allows you to study the underlying mechanisms. Models do not need to be mathematical. Many people actually create models without even realizing it. For example, if someone says "minimum wage will not lead to unemployment" that person is actually having a model even if they never present any mathematics. Such person definitely does not take into consideration full reality - every choice of every single individual, movement of every atom and so on.

Hence, whether you use mathematics or not you are using some model anytime you are describing any mechanism, such as x causes y, or there is such and such relationship between x and y because of this or that. Thinking otherwise is just self-deception as no human mind can take into account full reality when examining any issue. You have to simplify whether you do it with math or without. However, I get that at the heart of your question is why economists use mathematical models, but I wanted to highlight that there is no way of escaping models.

Mathematical modelling has advantages over plain text.

Dani Rodrick put it best when he said:

  We need the math to make sure that we think straight--to ensure that our conclusions follow from our premises and that we haven't left loose ends hanging in our argument. In other words, we use math not because we are smart, but because we are not smart enough.

  We are just smart enough to recognize that we are not smart enough. And this recognition, I tell our students, will set them apart from a lot of people out there with very strong opinions about what to do about poverty and underdevelopment.

Hence the reason why economists use mathematical models is that they are tremendously helpful in disciplining your thinking and by forcing you to say exactly what you mean. Written text is imprecise, words can have double meaning, people can easily cover up unsound logic by good use of literary devices or appealing to your emotions all of which can cloud your judgement.

This makes text  harder to actually analyze analytically. For example, even in moral philosophy - the least mathematical field I can think of, to fully analyze argument made in text and determine whether it makes sense you have to often resort into translating it from plain English into symbolic logic - which is form of math. Mathematics itself is just a language based on logic, that's also why in mathematics we often talk of equations as sentences or statements.

When you reduce model to set of mathematical expression you are forcing yourself to make your own thinking bare. You cannot easily conceal faulty logic anymore. You can argue that math can be used to 'fool policy-makers' as you put it but you cant easily fool other scholars with math while with text you can easily fool both policy makers and scholars.  My argument here is not that the math is bullet proof, as argued by Romer some scholars can try to mislead with using "mathiness", but lying with math is much much more harder than lying with text.

Models actually do help predict what will happen. First, do not confuse prediction with forecasting. For example New Keynesian model predicts that if there will be sudden shock to demand wage rigidity will lead to higher than natural level of unemployment - that is a prediction but its different from forecasting as the prediction is based on the ceteris paribus conditions while the word is constantly changing.

Predictions that model make can be useful in itself for several reasons. Predictions actually allow us to test whether one theory is superior than another. In fact the only way of testing a theory is to test its prediction against empirical observations. If someone has elaborate theory written in text that says that minimum wage does not affect employment and other person will make equally compelling case in text against that how will you decide which person is right?

One way how to do that is to turn the first person text into a mathematical model. The person says that minimum wage does not affect unemployment? Okay that means he postulates the following relationship:

$$ U(w_{min}) = a + bw_{min}$$ where $b=0$

Now converting the thinking of that person into mathematical model actually allows us to test it - we can based on the model above construct regression of unemployment on minimum wage:

$$U = \beta_0 + \beta_1 w_{min} +\epsilon $$

and test whether the first person was correct by testing the null hypothesis of $\beta_1=0$ against alternative hypothesis $\beta_1 \neq 0$.

Predictions also allow us to do counterfactual analysis. We can ask ourselves what would happen if in the presence of wage rigidities government increases spending during the recession. New Keynesian models provide you with framework to think analytically about such questions and fully explore all their logical conclusions.

Does this mean that math is the only way of studying economics? No. Of course, every individual is different. Some people are visual learners, some people are auditory learners.  Some students respond better to narration than to math. Moreover, as you yourself admitted math helped you to understand some problems so it worked for you well at least a couple of times. While you might have learned better most concepts by reading about them  other student might have completely opposite experience.

Moreover, depending on your career choices you might end up not using most of the math in real life. However, a master degree especially if its master of science and not just master of arts is often a stepping stone (at least in the Europe) towards PhD or more academic career in policy institutes or government institutions where you will need the math if not directly then at least indirectly to understand new advances presented in academic journals. For example, even routine psychologists must have some statistical understanding if they want to keep up to date with new techniques to help their patients as in order to see if some technique is better than other some testing has to be done. Otherwise you will be just at mercy of trusting what authors say in conclusions, or some intermediary like academicians who like to blog for wide audience etc.

Furthermore, you should not interpret anything I said above as saying math is the only way of doing economics or that narrative analysis is useless. Sometimes narrative analysis can be more nuanced, and with some problems we might not even yet discovered ways how to model them mathematically in satisfactory way. However, models and especially mathematical models are incredibly important tool in any economist's toolkit.


What does "capital used as structure" means?

This is not really an economic terminology. Many words can have multiple meaning in common English. According to the Oxford dictionary:

structure /ˈstrʌktʃə  /
▸ noun
1 the arrangement of and relations between the parts or elements of something complex.
...
2 a building or other object constructed from several parts.

In case of that article the first meaning of the word is used instead of second one.
To be more specific, the authors Hemous and Olsen (2018)  use that sentence when they talk about production process and production function of the company so what they mean is the that the capital can be used in the meaning 1 that is used in the arrangement of and relations between the parts of the production.


Does bond yield always equal the prevailing bank interest rates?

Each fixed income instrument has an interest rate implied by a transaction price, and these prices can move in different directions.
The FRED website has a list of some of the more important interest rates in the United States:
link to FRED. However, these are mainly “generic” yields, which are based on fitting a curve through traded bond quotes. For example, the “10-year Treasury bond” yield shown on the website is going to be close to the the market yields of Treasury bonds/notes that are near a 10-year maturity, but there are spreads between them.


Is this a realistic solution to world poverty?

I am afraid the solution to world poverty will not be two pages long. Some flaws:

You cannot simply give poor people in other countries money, their government controls their financial system. Even if the government agreed and you transferred millions of dollars it would not make more housing and consumer goods available at their location, they would just have credit. They could use these dollars to import the desired things but this may cause a lot of additional costs and again the government might interfere with large scale imports by controlling the distribution.

On hiring soldiers in Syria: This was an idea early on, but how can you be sure that they are doing what you pay them to do? Also there might not be a sufficient number of soldiers for hire at the desired location.

On giving poor people in the US unlimited access to basic goods. I like this idea, but to my knowledge no country in the world has implemented such a scheme. Also access to some goods does not eliminate crime, look at all the white collar crime that wealthy people commit or consider that they also struggle with drug problems.

Humans are complex creatures who will always compete with their environment.


The put option allows you to decide to sell at the exercise price $K$ rather than the market price $S$.

So if you exercise the put option, in effect it is worth $K-S$ at that point as you could have sold at the market price.

You will only do that if it is positive. Hence the $\max\left\{K-S,0\right \}$ expression.

If you could only exercise at a particular future point in time, the option would now be worth the expected future benefit $\mathbb{E}[\max\left\{K-S,0\right \}]$

But with a so-called American put option you can exercise at any time up to a limit, making the value the highest of the future expectations, leading to an option value now of $$\sup_\tau \mathbb{E}[\max\left\{K-S,0\right \}]$$

Typically the option value now is higher than the payoff if exercised now of $K-S$.  This difference is called the time value of the option and reflects the benefit from the future payoff being asymmetrical since it cannot be worse than $0$


A decrease in the cost of borrowing (interest rate) will case

I believe this is a trick question testing to see if the introductory economics student has overcome the usual confusion over "demand vs quantity demanded" and "supply vs quantity supplied".

When the price of an apple falls, this may cause quantity demanded or quantity supplied (of apples) to change, but cannot cause demand or supply to change.

Similarly, when the interest rate (or the price of loanable funds) changes, this may cause quantity demanded or quantity supplied (of loanable funds) to change, but cannot cause demand or supply to change. Hence, the answer is E.


Why would China and Russia selling the US' debt ruin the economy?

It wouldn't "ruin the US economy". The US (the country as a whole, not the government) bought goods and services from rest of the world by selling them IOUs. So far, the world has been content to let the US roll over these obligations. If the world decided that they wanted to trade these IOUs for goods and services, the US would have to start running current account surpluses and shipping goods abroad to pay for its past current account deficits. These surpluses have to come from somewhere, and they would result in less consumption and less investment in the US economy.

Of course, an economy can't borrow its way in global capital markets forever - the US does eventually have to ship some stuff over to China in exchange for all of the stuff that China shipped over to the US in exchange for (electronic) green pieces of paper. If China decides to unilaterally impose capital controls and stop lending goods to the US, then this hurts both China and the US - the reason capital in China was flowing to the US is that it could get higher returns there. (The functioning of domestic capital markets in China has become a lot better over the past decade, which is why China can now allocate its savings to productive uses domestically instead of lending large amounts of it to foreign countries.)

What some people are specifically afraid about ties into the USD's status as the international reserve currency - the Chinese government and central bank own plenty of US government debt as part of their foreign exchange reserves, and if they want to redeem this debt in exchange for goods and not roll it over, the scenario I mentioned above comes to pass. This seems to me like a grossly exaggerated concern at a time when global real interest rates are quite low. If China wanted to sell its US government bonds and the US had no intention of stopping the rollover process, returns would rise until the market for US external debt cleared, and there's no reason to think that the return response would be large given the low level of global real interest rates. The US (as a country) would have to pay more interest on its debt outstanding, which isn't great, but it's not a disaster of a scale that would "ruin the US economy".


So why did we need a central bank? Well, lets look at what the co-author of what has gone down in history as the Aldrich Plan, named after Senator Nelson Aldrich. His co-author on the bill was Edward Vreeland who wrote in August 25th, 1910, in the Independent, a newspaper owned by Nelson Aldrich that, "Under the proposed monetary plan of Senator Aldrich, monopolies will disappear, because monopolies cannot continue at such a low rate. Also, this will mark the disappearance of the Government from the banking business."
An interesting piece of trivia is that when our modern day "Federal Reserve System" was being planned, Paul Warburg who was a European from Germany that worked for the banking house Kuhn, Loeb and Company informed his other colleagues that were also planning for this new system that we have today, to avoid using "central bank" in the name. That's how we got this name of "Federal Reserve System", but it's basically a central bank with a new label, new to the early Twentieth Century.
Another piece of trivia is that the argument for a central bank in 1812 was presented as being that financial distress brought about by the war (War of 1812) required financial relief in the form of a new national bank.
This was at least the argument made for the second central bank that was charted in the history of the United States, the House and Senate passed a bill creating the Second Bank of the United States. I am speeding through history here.
One last piece of trivia that speaks to your statement of, "surely the government can print currency and regulate banks by conducting audits through finance ministry and so on".
At the outset the opinion of Thomas Jefferson made the powerful argument regarding the proposal for the first central bank of the United States:
I consider the foundation of the Constitution as laid on this ground; That "all powers not delegated to the United States by the Constitution nor prohibited by it to the states, are reserved to the states, or to the people."
To take a single step beyond the boundaries thus specifically drawn around the powers of Congress is to take possession of a boundless field of power no longer susceptible of any definition.
The Bill delivers us up bound to the National Bank, who are free to refuse all arrangements, but on their own terms, and the public not free, on such refusal, to employ any other bank.
The bill that Thomas Jefferson referred to was one put before him by proponents of central banks, namely Alexander Hamilton.
Citation:

Charles A. Lindbergh, Sr., Banking Currency and the Money Trust
The Writings of Jefferson, vol. 7 (Autobiography, Correspondence Reports, Messages, Addresses and other Writings)(Committee of Congress: Washington, D.C., 1861) page 685.


How is labor disutility modeled in Arrow-Debreu model?

In the Arrow-Debreu model, households are endowed with some commodities, producers can use feasible production plans to transform commodities, and households ultimately consume commodities - which they choose optimally given their preferences subject to budget constraints.

These commodities can include (potentially time and state-contingent) labor services provided by households. In this framework, provision of labor can simply be viewed as a form of negative consumption, which decreases utility but provides earnings that can be spent elsewhere. Indeed, the original Arrow-Debreu paper mentions this interpretation several times. For instance, on page 268 when they describe households in the model (which they call "consumption units"):


The preference relation is not transitive. As a counterexample take $x=(0,0), y=(0,1), z=(1,1)$.
For monotonicity let $x=(x_1,x_2)=(y_1+1, y_2)$ so that the $x$ bundle contains more of some commodity but no less of any, than $y=(y_1,y_2)$. Then
$$ (y_1+1+y_2-y_1-y_2) = 1 \leq 1 $$
so that the preference relation is not monotonic.
so that the preference relation is strictly monotonic.
The preference relation is continuous since
$$ \{x| x\succeq y\} = \{x_1, x_2 | x_1 + x_2 - y_1 - y_2 \geq -1\} $$
is closed and
$$ \{x| x\preceq y\} = \{x_1, x_2 | x_1 + x_2 - y_1 - y_2 \leq 1\} $$
is also closed
Finally, it is not locally non satiated. Take any $\epsilon\leq 1$ as an example.
EDIT: It is not strictly monotonic, see the comment below


Where did Robert Lucas say "market is always in equilibrium"?

I think this is a poor oversimplication of Robert Lucas's maxim. It's one thing that Robert Lucas models markets as if "always in equilibrium", it's quite another Robert Lucas claims it's an empirical truth that  "market is always in equilibrium".

Lucas(1978) explained briefly why he chose such methodology which doesn't faithfully reflects reality:

  As always, there are innumerable ways for the economy to be out of equilibrium, so we must expect any treatment of out-of-equilibrium behavior to have considerable arbitrariness, not resolvable by economic reasoning.


Question on significance of different ways of measuring Price Elasticity of Demand

Ideally the points $(x_1,y_1)$ and $(x_2,y_2)$ are very close. The assumption is that that they are so close that the elasticity of the demand function on the curve between them is nearly constant. This is what elasticity is trying to measure, local changes, because on the far side of the range of the demand function consumer behavior may be unobserved and totally different. E.g. to my knowledge no one has tried to sell a bottle of Coke for \$40,000 so we don't know what value the demand function would take. If you would choose a very distant point $(x_3,y_3)$ instead of the local $(x_2,y_2)$ you would get a very different value of elasticity.

So what does the book mean by initial value? Ideally the elasticity measured in points $(x_1,y_1)$ and $(x_2,y_2)$ would be constant because they are very close. Of course in practice this will not always be the case, meaning
$$
\frac{(x_1-x_2)/x_1}{(y_1-y_2)/y_1} \neq \frac{(x_1-x_2)/x_2}{(y_1-y_2)/y_2}.
$$
Instead of taking one of these values as the elasticy of the curve we can improve our estamite by taking the arc elasticity which usually yields a value between these two.


Quasi-concave utility functions represent convex preferences. But that preferences are convex means that the weakly-better sets are convex, it is not about the convexity of any function.


Clarifying the meaning of market capitalization

The "shares outstanding" variable used in the calculation of market capitalisation is defined as all shares that exist (and not was issued in public offerings). The market capitalisation is the share price times the outstanding. Therefore, the numbers in question do not apply.

There is no need for shares to be transferred in a public offering; an initial investor in a firm before it was listed could sell to another investor privately. This means that there is no good way of tracking how many shares are held by "the public," and that number is not used in calculations.


Ln differenced price level times 400, why 400?

You already accounted for multiplication by 4. Further multiplication by 100 to make the units of measurement percentage points? I.e. inflation is usually referred to as 3%, not as 0.03.


Can government just print a little of money to purchase some foreign resources?

We know that government shouldn't just print money

I partly disagree. The government (or, well, the central bank) should print the amount of money that is needed to keep the inflation stable. Anything above that shouldn't be done.

Bad what if to do it in a small scale, just a little

What if you stole something small from a nearby shop, just a little?

What if you took advantage of insider information, just a little?

What if you committed a small accounting fraud, just a little?

Printing money increases the monetary supply. That leads to inflation, which leads to increased inflation expectations. The increased inflation expectations raise the nominal interest rates.

Those who have net debt will benefit (if it's fixed rate), those who have financed the debt of others by depositing money into bank will suffer (if they are fixed rate deposits). The suffering will be greater than the benefit, because of the unpredictability of inflation (if the government is free to print money, you never know how large the inflation is going to be).

Generally, governments are in debt (and they should be because it offers investors the ability to invest into government bonds, which are a core part of a stable investment portfolio) and the debt is fixed rate. Thus, the people having the government bonds will suffer. Essentially, printing money steals from the government bond holders.

Those who have real assets like real estate, forest or non-financial stocks will not see any effect (expect perhaps by taxation because capital gains are typically fully taxed even if they occur in times of increased inflation).

Note the government will eventually suffer as well. The unpredictable inflation means government bonds have a higher yield, meaning if the government is going to finance most of its operations by obtaining debt from the public (even though printing money for the rest of its operations), the interest rates will rise. So, printing money is a bad strategy, even in a small scale. Soon there will be no other way of funding the government operations apart from printing even more money, which leads to hyperinflation.


Negative correlation conditional variance and return

This is simply saying that the falling returns are usually correlated with high variance (i.e. sharp movement in prices). This is not very surprising... you can see from the graph of S&P vs VIX below. Whenever the stock market plunges, volatility shoots up.

What @heh said was right. You need to compare assets that are more or less substitutable. For example, for what you said to hold true, it must be that investors in the US could say, returns in the US relative to the volatility is pretty low, so I'll move my money to Germany (where returns relative to volatility is higher) and invest there instead. In reality, such investors would have to face with interest rate differentials, FX risks, etc.

The "high volatility -> high return" applies to assets that are in the same "portfolio" (for me, this means in the same country at least). For example, you could say that a stock of a small company offers very high return but is very risky. You then have another asset, of a large company, that offers low return but is more stable. You also have the "risk-free" treasury bill that offers the lowest return and no risk at all.


Welcome to SE.

The two goods that the consumer consumes in this model are leisure and consumption. If the agent works more, he can consume more, but he has to give up leisure. Similarly, if he undertakes more leisure, he must consume less. All this assumes the wage rate is unchanged.

However, consider an increase in the wage rate. The opportunity cost of lying in the sun (leisure) has increased, as the agent could have worked instead, earned a higher wage as compared to before, and consumed more. Thus, the relative price of leisure has gone up, and the agent substitutes towards consumption, and this is the substitution effect.

Consider the income effect now. A standard model would assume that consumption and leisure are both normal goods, that is, as your wealth increases, you have more of both. At the current optimum point, as the wage has increased, your wealth has increased, and thus you undertake more leisure and also consume more. This is the income effect.

So, what I think you were asking was more about the total effect on working hours, that is, the cumulative result of both the substitution and income effects on the number of hours worked. Because in this example, the substitution effect pushes leisure up and the income effect pushes leisure down, the total effect is indeed ambiguous.

However, just in case you were actually asking about the income effect per se, the ambiguity may be due to letting go of the normality assumption. If leisure isn't normal, but is inferior instead, then the income effect pushes leisure downwards. Here, the total effect isn't ambiguous however, as the total number of hours worked unambiguously increases.


What is the iff condition for a preference with linear Engel curves (all Engal curves are linear)?

Are negative incomes (perhaps even negative prices) also allowed? Unless they are, non-horizontal or non-vertical lines are impossible as Engel curves because they have to intersect the negative quadrant of the coordinate system. I will come back to this later.

Without additional restrictions (perhaps a lot of additional restrictions) on the preference relations, for any non-negative sloped line I can fabricate a utility function such that the line will be its Engel curve.

Let the line $L$ consist of the points $((x,y)\in\mathbb{R}^2|c = a \cdot x + b \cdot y)$ where $a,b,c \in \mathbb{R}$. I will denote this set of points by $L$, the line. It is assumed that we are talking about an actual line, so $a,b$ are not simultaneously equal to $0$.

Let
$$
U(x,y) = \left\{\begin{array}{lc}
\arctan (b \cdot x + a \cdot y) & \text{ if } (x,y) \in L \\
-\pi & \text{ if } (x,y) \notin L.
\end{array}
\right.
$$

Claim The utility function will always be maximized by a point on $L$.

This is easy, because $\arctan$ maps to $(-\frac{\pi}{2},\frac{\pi}{2})$, thus its points always yield greater utility than other points. We specified that the slope of $L$ is non-negative, so for any positive price pair the budget line will cross it, enabling us to chose a point on $L$.

We now come back to the question of whether income $I$ can be negative. If they can be, than every point $L$ will be a solution of the consumers utility maximization problem for a certain $I,p_x,p_y$. The reason for this is that $b \cdot x + a \cdot y$ creates an ordering of the points of $L$. (Lines perpendicular to $L$ have a slope of $-b/a$, these are the level curves of $b \cdot x + a \cdot y$.) Select a point $(x_0,y_0) \in L$. By setting $I$ to
$$
I = p_x \cdot x_0 + p_y \cdot y_0,
$$
the consumer has barely enough money to purchase this basket. If $L$ is non-negative sloped and prices are positive then no points with higher utility are attainable.

The construction I gave is not unique. One could also use the continuous utility function
$$
\hat{U}(x,y) = \min\left(a \cdot \left(x - \frac{c}{a} \right); - b \cdot y\right)
$$
to achieve the same result. (If $a = 0$ one may use
$$
\hat{U}(x,y) = \min\left(a \cdot x; - b \cdot \left(y - \frac{c}{b} \right)\right)
$$
instead.)


My understanding is that the literature has so far used three kinds of empirical analysis of the directed technical change.

Illustration based on the relationship between the relative supply of skills and the college premium as in Acemoglu (2002).
Quantitative Analysis to shed light on the theory as in Acemoglu, Gancia, and Zilibotti (2015). They match key moments and simulate their model.
Estimate elasticities of substitution between more and less educated workers. Behar (2009) use four types of data to estimate those elasticities, namely (i) cross sections, (ii) a long range panel of 40 observations,(iii) time series for Brazil and (iv) an annual panel for Latin America.


What's the combined wealth of the US banks?

The total assets of all US commercial banks are about \$17.2 trillion.

You'll have to be clearer about what you mean by "their own money". The total equity capital of US commercial banks is about \$1.9 trillion.


microeconmics: can a monopolistic firm be profitable if its marginal revenue is negative?

As a counterexample to the statement, consider the following:

Let demand be $p=10-q$, so that $MR=10-2q$. Suppose there’s no cost, so that profit equals revenue. If the monopolist produces at $q=6$, $MR=-2<0$, but $TR=(10-6)6=24>0$, thus contradicting the statement.


A couple of suggestions:

Acosta, P.A., Lartey, E.K.K. & Mandelman, F.S. (2009): Remittances and the Dutch Disease
Bresser-Pereita, L.C. (2008): The Dutch Disease and its neutralization: a Ricardian approach
Ismail, K. (2010): The Structural Manifestation of the 'Dutch Disease’: The Case of Oil Exporting Countries
Larsen, E.R. (2006): Escaping the Resource Curse and the Dutch Disease?
When and Why Norway Caught Up with and Forged Ahead of Its Neighbors
Matsen, E. & Torvik, R. (2005): Optimal Dutch disease
Roemer, M. (2015): Dutch Disease in developing countries: swallowing bitter medicine in The Primary Sector in Economic Development (Routledge Revivals): Proceedings of the Seventh Arne Ryde Symposium


What would mortgage payments be if monthly payments were constant in real terms?

Note most of the argumentation adapts off the standard derivation for constant nominal payments from Wikipedia.

Let us assume $c$ is a yearly payment for the sake of simplicity. Consider how much is left to be repaid after each year. The principal remaining after the first year is $P_1=(1+r)P-c$. That is, the initial amount plus interest less the payment. After the second year $P_2=(1+r) P_1 - c$ is left, so $P_2=(1+r)((1+r)P-c)-c(1+\pi)$. Note we need to pay $c(1+\pi)$ instead of $c$ so that we are still paying $c$ in real terms for the second year. If the whole loan was repaid after two years, $P_2=0$, so $P = \frac{c}{1+r}+\frac{c(1+\pi)}{(1+r)^2}$.

This equation generalizes for a term of $n$ years, $$P = \frac{c}{1+r} \sum_{k=0}^{n-1} \frac{(1+\pi)^k}{(1+r)^k}$$. This is a geometric series which has the sum
$P=\frac{c}{1+r}\left(\frac{1-\frac{1+\pi}{1+r}^n}{1-\frac{1+\pi}{1+r}}\right)$
which can be rearranged to give $$c=\frac{P(1+r)}{\left(\frac{1-\frac{1+\pi}{1+r}^n}{1-\frac{1+\pi}{1+r}}\right)}$$. The nominal payment for year $n$ will be $c(1+\pi)^{(n-1)}$.


The budget set is always defined given a price vector $p=(p_i)_{i\leq l}$ (it seems like $l$ is the number of goods in your problem) and an income $w$. We usually implicitly assume that prices are strictly positive and the income is finite.
Otherwise, as you correctly point out, we end up with infinite consumption being optimal and we don't even have an economic problem (loosely defining economics as the science of incentives regarding the allocation SCARCE goods).
A set is bounded if there exists an upper and a lower bound. We have the lower bound of $x_i\geq 0$ for all goods $i$ by construction of the problem. We can only have an upper bound on any quantity $x_i$ if $p_i>0$ and a finite $w$. Suppose some finite $\overline x= (\overline x_i)_{i \leq l}$ were an upper bound and $p_i=0$ for some $i$. Then you can see that this $\overline x_i$ cannot be the upper bound, because $\overline x_i+\epsilon$ is also in the budget set for any $\epsilon>0$.
EDIT: Sorry, I ignored the question on non-emptyness. Take any finite income and positive prices and note that $x$ is a continuous choice. Take the maximum price $\overline p= \max_i (p_i)_{i\leq l}$ and then consider a quantity vector $\widehat x= (\widehat x_i)_{i\leq l}$ with $\widehat x_i= \frac{w}{l \overline p}$ then $\sum \widehat x_i p_i \leq \sum \widehat x_i \overline p = l \frac{w}{l \overline p} \underline p=w$. So quantity vector $\widehat x$ is in the budget set such that it cannot be empty.


How can Finance influence economic activity?

This question, as mentioned, in incredibly broad. So, of course, this answer will not be complete. Also, I am unsure what you mean by "financial". I will try to answer this question, assuming that by "financial" you mean, specifically 'stock/bond' market.

Here is one link from NBER which should help you get started, kinda, since it only scrapes the surface.

There are two sections to consider:

  Theoretical: As far as traditional, not necessarily classical, economic theory is concerned, stock and bond markets are corollaries and ancillaries of several factors of economic activity, e.g. capital market, money market, investment, employment etc.. That is, the financial market should just be an indication and investing mechanism based on real economic factors.

and,

  Practical, Modern: Now, since so many different corporations and businesses invest and are invested in the stock/financial markets, volatility in the stock/financial market will affect them directly, or indirectly.

    Directly: If these companies invest and are invested in, their valuation and/or income will be affect by market swings. For example, many of the financial firms which went under in the crisis.

    Indirectly: The shifts in the market can affect the global position of investing. Attitude is perhaps the largest impact felt from the shifts in financial markets. If the general attitude on investing shifts, it makes it harder for new companies to gain investors, i.e. capital.


How does lending money contribute to society?

In this simplified hypothetical world, yes, person B could give 100 currency to person A to build the factory directly. But what if person B does not want to take on that risk? How does person B know that person A will be contracted to make 150 worth of goods properly? Person B could just wait to see if someone else finances person A and then buy from A if the factory is successful. A single individual lending, or even small businesses lending out money can be a relatively large amount of money for that entity (so, high risk), whereas banks are able to diversify risks, by making many different loans and spending time evaluating how risky different people are.

Banks take on risk that other people/firms can't afford to or don't want to. Interest rate is very roughly, the price needed to be willing to take on these risks, or more precisely, the rental rate of money.


How do calculate whether selling a product leads to profit?

The process for solving this type of problem is very general--it's not a set of rules.  You don't need to attribute fixed costs to one good or the other (if there is only one fixed cost).

Here's the procedure:

Write out the whole profit function
Find the maximum of the profit function

If the maximum is less than the scrap value, then scrap it.

If your profit function is continuous, then you can maximize it by differentiating with respect to the quantity produced of the two inputs, setting the first order conditions to 0, and solving.  Just be sure to check that you have found the maximums and not the minimums.  If it is not continuous you will need to use a different method to find the maximums.

How to write the profit function?

$$
\pi= TR(q_1,q_2) -TC(q_1,q_2)
$$

$TR$ is usually easy.  It's

$$
TR(q_1,q_2) = P_1(q_1,q_2)\cdot q_1 + P_2(q_1,q_2)\cdot q_2
$$

Insert your price functions (which will be constants if you are a price-taker) and your cost function so that $\pi$ is a function only of $q_1$ and $q_2$.  Then find its max.  The $q_1$ and $q_2$ that maximize $\pi$ are what you want.

This type of problem seems hard if you get caught up letting your right brain trying to figure out how to solve it heuristically.  Don't do that.  Write out the profit function and maximize it.


When a firm announces a dividend, it pledges to pay a given amount at a given date ($t_p$) to its stock holders. As traded shares may change hands frequently, the firm defines who has the right to claim the dividend as the shares owners at a given date ($t_e$), that usually is some date between the announcement date and the payment date. During the $t_e$ to $t_p$ period, the stock can trade, but its buyers won't have the right to receive the dividend, even though it hasn't been paid yet. This is why we say ex-dividend, because the share has no longer attached the right to claim the next dividend.


Why would a central bank buy government bonds?

Because this is how the "government issues new money" in the era of (quasi) independent central banks: instead of directly issuing new money, the government in the narrow sense borrows from the central bank (which has been awarded the "printing privilege") so this new money is recorded as debt burdening the government (plus interest), which then is accountable for this debt burden and how it affects the balance of public finances.


Prove that $h(p,u) = \nabla_p e(p,u)$ is implied by Roy's identity

a) The convention, when differentiating a real valued function $v$ wrt to a column vector $p$ of dimension $(G \times 1)$, is that:
$$
\nabla _{p}v\left( p,w\right) \equiv \frac{\partial v}{\partial p}\left(
p,w\right) =\left(
\begin{array}{c}
\frac{\partial v}{\partial p_{1}}\left( p,w\right)  \\
\vdots  \\
\frac{\partial v}{\partial p_{G}}\left( p,w\right)
\end{array}%
\right).
$$

b) When expenditures $w$ are not constant, the column vector $p$ with respect to which we take the derivative occurs twice in $v(p, e(p,\overline{u}))$, and the chain rule has to be applied in order to obtain the total effect of the price change on the indirect utility: the first partial effect on utility is due to the price changes, and the second effect is through the adjustment in expenditures implied by the price change. Hence the expression:
\begin{eqnarray*}
&&\frac{\partial v}{\partial p}\left( p,e\left( p,u\right) \right) +\frac{%
\partial v}{\partial w}\left( p,e\left( p,u\right) \right) \frac{\partial e}{%
\partial p}\left( p,u\right)  \\
&=&\left(
\begin{array}{c}
\frac{\partial v}{\partial p_{1}}\left( p,e(p,u)\right)  \\
\vdots  \\
\frac{\partial v}{\partial p_{G}}\left( p,e(p,u)\right)
\end{array}%
\right) +\frac{\partial v}{\partial w}\left( p,e(p,u)\right) \left(
\begin{array}{c}
\frac{\partial e}{\partial p_{1}}\left( p,u\right)  \\
\vdots  \\
\frac{\partial e}{\partial p_{G}}\left( p,u\right)
\end{array}%
\right).
\end{eqnarray*}


How to interpret increases in percentage with different sample groups

If you ask me your question is better suited to cross validated (stats.stackexchange.com).

Anyway you're probably confused because the percentages refer to different base lines. Note that the 60% of disease at B in 2017 refers to 40% of the population, whereas the 30% in 1917 refers to 30% of the population, so they are hard to compare.

Try working it out in absolute numbers to get a clearer picture:

In 1917 if there are 100 people, 70 of ethnicity A, 30 of B. Of the 70 people of A 7 (10% of 70) get sick, of B 9 get sick (30% of 30)

In 2017 there are 100 people, 60 of ethnicity A, 40 of B. Of the 60 people of A now 18 catch the disease (30% of 60), and of B 24 catch the disease (60% of 40).

Observations: overall sickness has increased (16 vs 42). The absolute increase in sick B people (15) is larger than that in A people (11), but this in part due to the fact that there are more B people now


Why are vacancy rate and unemployment rate negatively correlated?

Adopting your notation, the vacancy rate at any given time is defined as $A/Q$. There is no mechanical relationship between the unemployment rate $U/Q$ and vacancy rate (A/Q).

They are negatively related because when vacancy rate is high, it means there are many unfilled job openings, and they are unfilled probably because most people already have a job, thus lower unemployment rate. And vice versa.


What is the prime sign that a country is transitioning from a developing to a developed economy?

I don't know what the 'first sign that a country is transitioning from a developing to a developed economy', but I would look at the rates of output growth and study its acceleration.

In a layman's words, a transition from low to high income regime requires convergence to a high income teady state. This is only possible if the growth rate of aggregate output, underpinned by improvements in all aspects of the economy, has been consistently higher than that observed in high income (which you call developed) economies. Transition is complete (or the country achieves a 'developed' status) when the economy's growth rate reaches its long run steady state rate (typically low (<2% p.a.) positive growth rate).


How are payoff values in game theories calculated / estimated?

Though I'd ask for a bit more clarification which papers you are referring to for a more specific answer, overall the answer depends a bit on what the purpose of the paper/game and payoffs being proposed is.

Papers like Selten's 1978 Chain Store Paradox use payoffs that are important in their relative magnitude (not their absolute value) as a means of creating a perceived disconnect between Industrial Organization and Game Theory. Similarly, in the classic Prisoner's Dilemma game, the actual values for "fink/tattle" and "mum/don't" aren't as important as the fact that the dominant strategy is for both players to tattle. That payoff can be (0,0) or (-10,-10), so long as that payoff makes tattling the dominant strategy, it works for the game.

One of the larger reasons why "exact" values are given instead of confidence intervals is because of how strategies are calculated. Indifference is the key to defining mixed strategies, which is a much simpler feat if exact values are given.

That being said, there is a variety of games of imperfect information and, perhaps most important for your question, games with imperfect monitoring which do exactly as you suggest- allow payoffs to be dependent on some random variable not observed by one (or more) of the players.

I'm sorry if that wasn't very direct, or didn't really fully answer your question- please let me know if I can improve the answer. Confidence intervals can be difficult to use because they are so dependent on the statistical results as opposed to the theoretical model. However ranges of values are a relatively common phenomena in certain types of game theory models (for example, see auction theory). Usually it is assumed that players know the relative likelihoods of different payoffs, but even that can be relaxed in certain circumstances.


Definition In economics, stagflation, a portmanteau of stagnation and inflation, is a situation where the inflation rate is high, the economic growth rate slows down, and unemployment remains steadily high.

Stagflation hence requires a high unemployment rate. The unemployment rate can be defined as

$$ u = \frac{U}{POP}\\
u =\frac{U}{LF}
$$

Direct increase of unemployment?

that is, either the number of unemployed over population (more common) or over the labor force. Removing women and teens from the labor force does not affect anything in the first definition, and decreases the unemployment rate by the second definition.

Indirect increase of unemployment?

As stagflation requires a high unemployment rate, reducing the labor force size cannot directly create stagflation. Your only argument could then be that a reduced labor force somehow leads to a higher unemployment rate. Most likely, this is orthogonal. Equilibrium unemployment is a composite of frictional unemployment and voluntary unemployment. I cannot think of a reasonable argument why a reduction of the labor force should increase the relative share of either of these (unless of course, voluntary unemployment is higher among men than women, which is false).

Level effects and growth rate effects

Even more importantly, there is an important distinction between shocks to the level versus shocks to the growth rate. To the extent that - after women and teens have left - the growth rate of the labor force is the same as before, long term effects are negligible.

In a world with exponential growth, a shock to the level is - in the long run - negligible, as we will catch up quite quickly. A shock to the growth rate however, is permanent. Of course, "in the long run" is relative, but there is some idea of persistence behind stagflation. An exit of women from the labor force would, in my opinion, cause a sudden drop in output, but would not affect growth rates.


Understanding the arithmetic in this 1690 text

It seems that the gentlemen is speaking in the english dialect of England.  The Years purchase phrase is still used today.

https://en.wiktionary.org/wiki/year%27s_purchase

Years Purchase: The amount yielded by the annual income of property; used in expressing the value of a thing in the number of years required for its income to yield its purchase price, in computing the amount to be paid for annuities, etc.

An example of its use is here: https://investmentproperty.co.uk/property-investment-calculators/years-purchase-single-rate/


Is it possible for a firm does not have shutdown point?

$TC(q)=10+3q+0.5q^2$ is a quadratic cost function and has a shutdown point at $P=3$.


List of EU banks that received state aid

Pro Publica keeps an ongoing list of bailout recipients in the United States.

Here is a report on the UK bailouts during the crisis:
The Comptroller and Auditor General’s Report on Accounts to the House of Commons: The financial stability interventions

The SoFFin Wikipedia page has a list of participating German institutions.

FACTBOX - Bank bailouts around the world covers a few additional countries's bailouts between September 2008 and February 2009. A further update came in April of 2009 but it doesn't seem exhaustive.

Page 28 of Stability Programme of the Netherlands April 2011 Update contains information on their biggest bailouts but I am unsure if it is exhaustive.

The Spanish Fund for Orderly Bank Restructuring appears to be the key Spanish bailout program. The FROB FAQ lists entities receiving FROB support but again, this may not be exhaustive.


Price discrimination is when a firm charges different prices for a homogeneous product to different consumers.

In this example we see no such discrimination among consumers. This is just a case where there is competition between two firms.


Equivalence of definitions of continuity

HINT: it is easier if you go through the intermediate step that the set $\{\alpha \in[0,1]: \alpha x+(1-\alpha) z \succsim y\} $ is closed for every $x,y,z$


Is there an economic term for decreased demand due to expected decrease in price due to a product's value lasting for a finite amount of time

You are describing a market where a "non-storable" (or "perishable") good is traded. This is the (in terms of historical precedent) model of a market, since in the old days, most goods were agricultural, and many amongst them were "non-storable".

The argument behind "market clearing", i.e. that prices will adjust so that all quantity available will be sold, owes a lot to the goods being non-storable. You can observe this today in the so-called "farmer's markets", where if you monitor a day of trading, you will observe that prices go down as the day nears to an end, as suppliers attempt to sell all their quantity.

Why some consumers nevertheless buy early on and so at higher prices, even though they know that later on prices will go down, has to do with

a) constraints imposed on the consumer schedule (say, he has to buy early because he has to prepare lunch), and/or

b) quality considerations: early on the suppliers may give you more room to "pick and choose" item per item, or  they themselves will offer the better quality, to justify the higher price. It may be the case that some consumers have a lexicographic preference over a certain threshold of quality, which they expect it won't be around at the end of the day when prices will fall.

On the other hand, some consumers will hold on, given their constraints/preferences. As you can see, the phenomenon of all consumers waiting till "the last minute" presuposes a situation were the consumers are perfectly "flexible", and the quality of the good does not deteriorate with time, but the good looses all its "good-properties", instantly, at the end of its "useful life". In reality quality erodes gradually, in almost all cases.


Giskard is right lexicographic preferences would make the job being dicsontinuous, but the problem is to find relations with closed contour sets. Maybe an idea would be to start from constructing a preference relation  for which the only possible upper and lower contour sets are $X$ and $\emptyset$. In that case, you take the indiscrete topology, and all nets (or generalized sequences) converge to all the points in $X$. Anyway, I have not been able to construct such a relation yet.


The recommended books are decent. From these two I'd go with Bailey first and if you're comfortable with that, then LeRoy & Werner. The latter requires some background in linear algebra and optimization theory.

If you want to study some econometric applications for financial economics, you might try:

Cuthbertson & Nitzsche: Quantitative Financial Economics: Stocks, Bonds and Foreign Exchange
Campbell, Lo & MacKinlay: The Econometrics of Financial Markets


Removal of tax excise on producers, will increase supply?

A producer will always maximize profit by setting marginal cost equal to marginal benefit. Removing an excise tax will reduce the cost of every gallon of gas, so it would indeed effectively lower the marginal cost of production. Assuming the production function isn't falling into a diseconomy of scale (where the more effort/resources you put it, the less efficiently you produce), then yes, production of gasoline should increase as a result.

Hope this helped.


What determines the exchange rate of two fiat currencies if the supply of each is known?

There isn't any good way to rehabilitate the quantity theory when there are other currencies that are perfect substitutes for dollars - so in that sense, there isn't any answer to Landsburg's question. Indeed, the irrelevance of the quantity theory under perfect substitutability - which has always been theoretically clear - has become a practical reality recently, as reserves become a perfect substitute for other short-term nominal assets once the nominal interest rate hits zero.

That said, I'd make two points.

1. Anything less than perfect substitutability rescues the quantity theory.

Suppose we replace the left side of Landsburg's quantity theory equation with the more general form
$$F(M,XB) = PD$$ where $F$ is a homogenous production function that produces aggregate "monetary services" using money $M$ and bitcoins (with value stated in terms of money) $XB$.

Landsburg's equation corresponds to the case of perfect substitutes, $F(M,XB) = M+XB$. In fact, there's another assumption that's implicit in Landsburg's formulation, which is that either $X$ is constant over time or, if $X$ varies, the risk-adjusted expected return on bitcoins relative to money is zero: otherwise, you'd strictly prefer to hold one or the other, whichever gives the highest return, given that they have equal transactional value. I'm going to continue assuming that this assumption holds for simplicity - but keep in mind that in a full-fledged dynamic model, allowing returns to differ and individuals to substitute on this basis might endogenously produce the extra equation that Landsburg seeks.

If money and bitcoins have the same returns, then anyone holding them will want to equate the marginal transactional value of the two, setting $F_M=F_{XB}$. This holds for any relative quantities of $M$ and $XB$ in Landsburg's perfect substitutes formulation, which is why he's struggling, but for general (homogenous) $F$ it will only hold for one ratio $M/XB$ of the two. This will pin down relative demand.

For instance, if $F$ is Cobb-Douglas, with $F(M,XB) = M^\alpha (XB)^{1-\alpha}$, then $F_M = \alpha F/M$ and $F_{XB} = (1-\alpha)F/XB$, and equating the two gives us $M/XB = \alpha/(1-\alpha)$. Suppose $\alpha=1/3$. Then we have $XB = 2M$, and it is trivial to solve for $P$ from $M$ and $D$:
$$F(M,XB)=PD\Longleftrightarrow M^{1/3} (2M)^{2/3} = PD\Longleftrightarrow P = 2^{2/3}\frac{M}{D}$$
Cobb-Douglas is just one parameterization I'm using for illustrative purposes, but we'll similarly be able to solve as long as $F$ has a declining marginal rate of substitution between $M$ and $XB$ - which would be true, for instance, if $M$ and $XB$ were almost perfect substitutes, but not quite. Landsburg's case of perfect substitutes is very much non-generic in this sense: it's probably not true that fiat currency and bitcoins will ever be perfect substitutes in absolutely every application.

By the way, the idea that two forms of currency combine in an imperfectly substitutable way to provide overall monetary services isn't just something I made up - you can see assumptions like this in the literature in a number of places, like equation (3) in Ireland (2011).

2. The central bank can pin down the price level in other ways, even without the quantity theory.

The modern view on monetary policy is that what really matters is the central bank's ability to set the short-term interest rate. Traditionally, this has been done by changing the supply of money through open-market operations, but that doesn't need to be the case. Indeed, Woodford's canonical text shows how it is possible to implement monetary policy even in a "cashless" world where there is no demand for money: the central bank simply pays interest on money. (By the way, this result is hard to escape when you try to microfound the "quantity theory" equation by writing down a dynamic, internally consistent model: you realize that the quantity theory operates in general equilibrium via the response of interest rates to money, and that by manipulating interest rates directly we can get the same outcome.)

Indeed, we're moving closer to Woodford's hypothetical world all the time: for instance, one of the options for the Fed when it decides to raise interest rates in the coming months will be to push up the rate of interest on reserves, while keeping its expanded balance sheet intact.

From this viewpoint, Landsburg's observation just isn't very relevant. The central bank is dedicated to price stability, and it'll enforce this by adjusting interest rates in response to deviations of inflation from trend. If it can adjust interest rates through the traditional method of adjusting $M$ via open-market operations, great. But if it can't do this (because we live in Landsburg's world of perfect substitutability), then the central bank will just adjust nominal interest rates by changing the interest it pays on reserves, and ultimately accomplish exactly the same thing.


Why does an increase in jobs in the USA devalue the the Australian dollar, oil, iron, gas, gold, and copper?

When you look at many things in economics it is worth noting that everything is relative. $1 is only worth what it can purchase. The decisions you make are made in the contexts of your next best option (opportunity cost). In this case, the Australian dollar isn't worth any less domestically. The US has just gotten stronger. With the US outlook being quite positive at the moment, its assets and investments are looking more attractive, thus the appreciation of the USD wrt to the AUD (and many other currencies).


Compound interest ($r$ in terms of $\frac{dA}{dt}$)

You are not doing anything wrong. Since
$$
\lim_{r \to 0} \frac{\ln(1+r)}{r} = 1
$$
(to see why: use l'Hôpital's rule or Taylor approximation) holds, for small numbers $r$
$$
\ln(1+r) \approx r.
$$


Why is a fire service considered to be a non-rival service, while a highway is considered to be a rival good?

One common definition of a non-rival good (see here and (A)) is a good for which the consumption of additional units involves zero marginal social costs of production. On this definition the service provided by a fire service would always be rival, because (even with plenty of capacity in terms of fire engines and staff) each visit to attend a fire involves some extra cost (eg vehicle fuel).

More loosely, a good may be said to be rival if consumption by one person prevents simultaneous consumption by others (this can also be found here), implying that a good is non-rival where that is not the case.  In this looser sense, infrastructure such as highways and services such as the fire service might be held to be non-rival in circumstances where capacity is large enough that consumption by some rarely prevents or interferes with consumption by others.

For those who adopt this looser definition, the difference between a highway network and a fire service, which may lead to the former but not the latter being characterised as a rival good, relates I think to the likely level of supply or capacity.  Given high levels of vehicle use with peaks at certain times (rush hours), and given the costs of road-building and other demands on land use, it isn't likely to be practicable to provide enough highways in the right places to make traffic jams a rarity.  On the other hand, given that a fire at any one location is a rare event, and given normally a high degree of independence between fires at different locations, it should be possible to provide sufficient capacity in a fire service that the scenario of fewer fire engines than fires will rarely occur.

Reference

(A) Nicholson W (9th edn 2005)  Microeconomic Theory: Basic principles & Extensions p 596.


Nested/Recursive Dynastic Utility Functions

Here are two nice papers on recursive utility functions, a generalization of additive utility functions and compatible with "utility of the dynastic head to be partly a function of the utility of his children and grandchildren's utility, but where his children's utility is again partly a function of his grandchildren's and great-grandchildren's utility, and so on":
Blackorby, C., Nissen, D., Primont, D., & Russell, R. (1973). Consistent Intertemporal Decision Making. The Review of Economic Studies, 40(2), 239-248.
Blackorby, C., Nissen, D., Primont, D., & Russell, R. (1974). Recursively Decentralized Decision Making. Econometrica, 42(3), 487-496.


The "wealth" of a company is not a well-defined term. The worth of a company, its market value, is whatever someone is willing to pay for it. The market's best guess of that value, for listed companies, is equal to the share price multiplied by the number of shares in circulation. In theory, for both listed and unlisted companies, the worth of a company is equal to the sum of of all discounted future cashflows arising from the company.

As such, it is a stock variable, not a flow variable.

Conversely, the GDP of the country is a flow variable, not a stock variable. It's the total value of transactions in the economy in a year; as such, it's somewhat analagous to a company's turnover.


Isn't Bull Put Spread's Maximum Loss = Spread Credit + Loss from sold call + Profit from bought call?

You have your minus signs mixed up for buying and selling the underlying.  When your short call is exercised, you sell the stock for 44 (credit).  When your long call is exercised, you buy the stock for 48 (debit). So in your terminology, the maximum loss  is:

(-P + 44) + (P - 48) + spread credit = 44 - 48 + 3 = -1

A much easier way to remember this is that with vertical spreads:

If it's a credit spread, the maximum gain is the credit received and the maximum loss is the difference in strikes less the credit received.

If it's a debit spread, the maximum gain is the difference in strikes less the debit paid and the maximum risk is the debit.


Perfect multicollinearity when estimating a gravity model

My problem is: because Brazil is the origin for the exports to all other countries, GDP_O, POP_O and any other variables representing Brazilian data will be equal for all observations and as such there will be perfect multicollinearity.

This isn't perfect multicollinearity. It's a lack of time/space variation.  Multicollinearity occurs between features, not observations.  You would of course not have space variation in any given year, for the reason you note - but in a comment you said you ran a panel regression, so you should have that time variation.

  I want to regress exports as a function of, among other things, Brazil's GDP. As such, I want to see the effect of GDP on Brazil's exports and not the effect of GDP on exports to each country

Then, why are you using a gravity model for trade flows? Doing so spreads the effect size across a bunch of variables you seem to be uninterested in.  Total exports is just a number that relates to GDP through the accounting identity.  Since that identity is additive, a log-log form for your regression does not make sense.


How do tax payers have to pay for the tax deductions someone else receives due to a charitable donation?

Well you actually answered this question yourself. It is because of this:

While I do understand, that the government loses money in tax incomes, I don't understand how this directly increases the tax burden on other tax payers (except for, in the long term, the government might raise taxes to make up for lost tax revenues).

To be more precise an alternative is also for government to spend less but I think that the articles are arguing that ceteris paribus keeping government spending fixed government will have to somehow eventually collect the resources from economy in one way or other. In a long run government has to respect its intertemporal budget constraint. If someone pays less, taking government spending as fixed, someone will have to pay for that shortfall either directly or indirectly eventually at some point.


Does the UK have a trade agreement with Australia?

No, the UK does not have a trade agreement with Australia.

Since it is still a member of both the European Union's Single Market and Customs Union (EU SM&CU), it is covered by all of the EU's trade deals. However, the full EU-Australia trade deal is, at the time of writing, still being negotiated.

Should the UK decide to go ahead and become the first country in history to impose economic sanctions on itself, withdrawing from EU SM&CU, then its own home-baked trade agreements will come into force. The embarrassingly unimpressive list of trade agreements that the UK has negotiated is listed here.


the difference between $\hat{e}^2_i$ and $\sigma^2_i$

Certainly the true variance is not equal to the squared residual.

In the case or robust standard errors, it has been proven that if we use $\hat{e}^2_i$ we obtain a consistent estimator of the variance-covariance matrix as a whole, even though $\hat{e}^2_i$ is not equal to $\sigma^2_i$ and even though  each single $\hat{e}^2_i$ does not converge in probability to $\sigma^2_i$ (it converges to the true squared error, which is a random variable).

The fundamental reason for this result is that

$$E(\hat{e}^2_i) \to_{n \to \infty} E(e^2_i) =\sigma^2_i$$

So the OP's phrase

  When we calculate the robust robust standard error, we simply repalce
  $\sigma^2_i$ with $\hat{e}^2_i$. Thus, theses two are considered to be equal.

is simply wrong.


Envelope Theorem in Keen and Slemrod (2017)

At first glance, it appears the answer should be $\frac{dW}{dt}= -(z + tz_t) + v'*(z + tz_t)$, as $l$ and $e$ are chosen optimally and the envelope theorem cancels these terms out. Furthermore, one could expect both $z_t$ terms in the final derivative to drop out due to the envelope theorem. However, one and only one $z_t$ term remains in the authors' calculations.

The envelope theorem is just an application of first order conditions. In this case, it is better to simply use the first order conditions we know to derive the results, instead of using the theorem directly. I believe the authors invoke the term "envelope property" to concisely convey that they are using the previously found first order conditions.

Taking the total derivative we arrive at:

$\frac{dW}{dt}= wl_t -z - tz_t -c_ee' - \psi'l_t + v'*(z + tz_t)$

Inserting the expression for $z_t$ and grouping some terms we have:

$\frac{dW}{dt}= l_t[w(1-t)-\psi'] + e'[t-c_e] - z + v'*(z + tz_t)$

Note that the terms in square brackets are the same as the first order conditions for $l$ and $e$ and hence equal to zero. This leaves us with:

$\frac{dW}{dt}= - z + v'*(z + tz_t)$


Microeconomics: first fundamental theorem of welfare

The most common version of Pareto dominance says that an allocation is Pareto-dominated if there is another feasible allocation in which at least one agent is better off and everyone else is at least as well off. In particular, the latter allocation can include consumers that are just as well off as before.
It follows from the very definition of a competitive equilibrium that (1) holds. That a consumer chooses the best bundle they can afford means that every better bundle must be unaffordable and, therefore, more expensive.
Now, the sum in (3) is not just over those agents that are strictly better off, it is also over those who are equally well off. For those, one only gets the weak inequality (2) using local-nonsation.
Here is an example where local-nonsation is violated and in which a competitive equilibrium is not Pareto-efficient, showing the importance of the assumption of local non-satiation for the first welfare theorem:

There are two goods and two consumers. Both consumers have the initial
endowment $\omega_1=\omega_2=(1,1)$. For consumer 1, both goods are
perfect substitutes; consumer 1 wants the sum of the amounts of both
commodities they consume as large as possible. Consumer 2, does not
care what they get, they are indifferent among all consumption
bundles. You can check that there is a competitive equilibrium in
which both consumers simply receive their endowment and both goods
have the same price, $p_1=p_2>0$. But the unique Pareto-efficient
allocation gives everything to consumer 1, they receive the bundle
$(2,2)$ and consumer 2 the bundle $(0,0)$. Every allocation that
Pareto-dominates the proposed equilibrium allocation has consumer 1
being better off and therefore receiving a more expensive bundle (for
the equilibrium prices), but consumer 2 receiving a cheaper bundle
that is still as good as before.

There is also a weak notion of Pareto-efficiency: An allocation is weakly Pareto-efficient if there is no feasible allocation in which everyone is strictly better off. One can prove that every competitive equilibrium is weakly Preto-efficient even without local non-satiation: If everyone is better off, (1) implies that everyone receives a more expensive bundle and then we can directly go to (3).


Why isn't the Fed funds rate greater than interest rate on loans to the pubic

I think you are missing the credit risk attached to any loan. You, as an individual borrower, are far more likely not to reimburse your loan than a bank. Thus the loan is far less risky.

Central bank money could only be qualified as "better" (even that does not mean much) because it is issued by the Central bank, deposits are issued by commercial bank, so you bear the risk that your bank fail, but it happens that in most country individual deposits are insured by some state agency. So I cannot really see why central bank money would be better than deposits.

Nowadays, the amount of reserves that a bank must hold does not actually play a big role in the economy (as opposed to what most of economic textbooks and economic professors are teaching in undergrad schools). The most important part is the interest rate that is paid on those reserve, which is fixed by the central bank and which act as an opportunity cost for the bank when it decides to lend to an individual rather than holding reserves.

However I see your point, but the fact is that this way of thinking is misleading. Banks create (in theory) as many loans as they want, because there are no physical constraints, if you want, that prevent them to do so. The only constraint is a regulatory constraint that states that in order to remain solvent (in order to be able to repay the depositors) banks should invest x% of the assets in reserves. Those reserves would be held on a central bank account in central bank money, and the return on those reserves will be the fed fund rate decided by the central bank (the same way a commercial bank decide how much interest it will pay on your savings account). So it is true that one more unit of reserves gives the right to issue more than one unit of loans but this does not mean that one money is better or worse than another. It's is just a matter of regulation.

It may be hard to get, but the point is really that central bank money is safer so cheaper than deposits, no matter what regulatory constraints apply to banks.

You may want to read this paper which may be a bit complicated but you should get the intuition. This two docs may be simpler (with nice pictures) : doc 1, doc 2.


Risk with unknown probability distribution of the outcomes

A sure outcome is just a special case of a risky alternative with a degenerate probability distribution.
Under a set of appropriate conditions, the choice of a sure outcome over an ambiguous alternative can be attributed ambiguity aversion. But since different authors define ambiguity aversion somewhat differently, the "appropriate conditions" also differ. On the other hand, choosing a sure \$100 over an uncertain distribution over \$1 and \$2 can hardly be called ambiguity averse.


What are the identification problems in measuring peer effects?

Well, imagine you don't know how students are grouped. Maybe better students are deliberately grouped together, for example. Then your estimation is clearly biased. You'd basically reach the conclusion that classes with better students are better.

Peer effects are hard to identify with observational data because similarities in the behavior of peers could be due to peer effects or to selection into peer groups based on common unobserved characteristics, or to factors that affect all peers in a peer group.

To do a proper estimation we need a quasi-experiment, preferably a true experiment. Because in such context you indeed know that student allocation is random, so there's a clear causal link and possibility to separate the treatment effect from homophily and confounding factors.

Angrist and Pischke's Mostly Harmless Econometrics is a must-read for anyone interesting in working in this topic. Angrist (2014) discusses this matter in more depth and this NBER digest may help you with the intuition.


Why is Algeria the #2 milk powder importer in the world?

Four-fifths of Algeria is just desert --which leaves no room for agriculture.  The Algerian economy is mostly powered just by the oil industry, which contribute about ~70% of government revenues. Oil and oil related products constitute more than 80% of Algeria's exports.

For example, Turkey has twice the population of Algeria but it has more than 7.5 times the cattle Algeria has. (14.2 in Turkey vs 1.6 in Algeria). So the short answer is that Algeria isn't just suited for Agriculture.


Why CPI differs so much from consumption deflator?

The discrepancy between these two standard measures of inflation stems mostly from the differences between the basket of goods used by each index.

The CPI is based on survey data taken from some representative sample of consumers. This data represents a basket of "typical" household consumption. The PCE index is also based on survey data, but taken from firms instead. PCE index data represents a basket of sales derived from firm production. It is not difficult to imagine that this discrepancy in baskets can be magnified by certain prices, corresponding to goods in the difference set, receiving a much higher weight than others, thus driving a wedge between the indices. You allude to this in your question.

Additionally, and by construction, the CPI covers a smaller set of expenditures than those covered by the PCE index. The PCE index includes expenditures which are not paid for directly by the agent, such as medical care expenses from employer-provided health insurance, employee retirement payments, etc. The CPI does not take these sorts of transactions into account and instead covers strictly out-of-pocket expenses.

Lastly, the evolution of baskets characterized by the two indices over time is quite different. The CPI has a "stickier" basket that is much more resistant to quality-change and price substitution over time than the PCE index, which directly incorporates price substitution between goods in forming its representative basket.


Binding constraints at second-best optimum

I think its not a complex issue:

a) You need to keep the high type agent from pretending to be the ow type agent, so you give him an extra compensation from showing his high type. But this nice deal pays him more than necessary to keep him in the game so his IR is more than satisfied, i.e. not binding.

b) Similarly, you will make it painful to show that you are a low type, so that the high type doesn't go there, but you can;t make it so painful that the low type doesn't want to play so you pay him the minimum amount that he will take, i.e. his IR is binding, but  he would never choose to show himself as the high type anyway because with his low type, the incentives don't work in his favor, so the IC is not binding.