Outputting Regressions as Table in Python (similar to outreg in stata)?

You can use code like the following (making use of the as_latex function) to output a regression result to a tex file but it doesn't stack them neatly in tabular form the way that outreg2 does:

import pandas as pd
import statsmodels.formula.api as smf
x = [1, 3, 5, 6, 8, 3, 4, 5, 1, 3, 5, 6, 8, 3, 4, 5, 0, 1, 0, 1, 1, 4, 5, 7]
y = [0, 1, 0, 1, 1, 4, 5, 7,0, 1, 0, 1, 1, 4, 5, 7,0, 1, 0, 1, 1, 4, 5, 7]
d = { "x": pd.Series(x), "y": pd.Series(y)}
df = pd.DataFrame(d)
mod = smf.ols('y ~ x', data=df)
res = mod.fit()
print(res.summary())

beginningtex = """\\documentclass{report}
\\usepackage{booktabs}
\\begin{document}"""
endtex = "\end{document}"

f = open('myreg.tex', 'w')
f.write(beginningtex)
f.write(res.summary().as_latex())
f.write(endtex)
f.close()

The as_latex function makes a valid latex table but not a valid latex document, so I added some additional code above so that it would compile. The result is something like this for the print function:

                        OLS Regression Results
==============================================================================
Dep. Variable:                      y   R-squared:                       0.129
Model:                            OLS   Adj. R-squared:                  0.089
Method:                 Least Squares   F-statistic:                     3.257
Date:                Fri, 29 Apr 2016   Prob (F-statistic):             0.0848
Time:                        20:12:12   Log-Likelihood:                -53.868
No. Observations:                  24   AIC:                             111.7
Df Residuals:                      22   BIC:                             114.1
Df Model:                           1
Covariance Type:            nonrobust
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
Intercept      0.9909      0.908      1.091      0.287        -0.893     2.875
x              0.3732      0.207      1.805      0.085        -0.056     0.802
==============================================================================
Omnibus:                        3.957   Durbin-Watson:                   0.999
Prob(Omnibus):                  0.138   Jarque-Bera (JB):                1.902
Skew:                           0.380   Prob(JB):                        0.386
Kurtosis:                       1.849   Cond. No.                         8.50
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

and like this for the latex:

Update:
Not as full featured as outreg but the summary_col function does what you ask.

import pandas as pd
import statsmodels.formula.api as smf
from statsmodels.iolib.summary2 import summary_col
x = [1, 3, 5, 6, 8, 3, 4, 5, 1, 3, 5, 6, 8, 3, 4, 5, 0, 1, 0, 1, 1, 4, 5, 7]
y = [0, 1, 0, 1, 1, 4, 5, 7,0, 1, 0, 1, 1, 4, 5, 7,0, 1, 0, 1, 1, 4, 5, 7]
d = { "x": pd.Series(x), "y": pd.Series(y)}
df = pd.DataFrame(d)
df['xsqr'] = df['x']**2
mod = smf.ols('y ~ x', data=df)
res = mod.fit()
print(res.summary())
df['xcube'] = df['x']**3

mod2= smf.ols('y ~ x + xsqr', data=df)
res2 = mod2.fit()
print(res2.summary())

mod3= smf.ols('y ~ x + xsqr + xcube', data=df)
res3 = mod3.fit()
print(res2.summary())

dfoutput = summary_col([res,res2,res3],stars=True)
print(dfoutput)

Which has the following output:

=====================================
            y I       y II    y III
-------------------------------------
Intercept 0.9909   -0.6576   -0.2904
          (0.9083) (1.0816)  (1.3643)
x         0.3732*  1.7776*** 1.0700
          (0.2068) (0.6236)  (1.6736)
xcube                        -0.0184
                             (0.0402)
xsqr               -0.1845** 0.0409
                   (0.0781)  (0.4995)
=====================================
Standard errors in parentheses.
* p<.1, ** p<.05, ***p<.01

As before, you can use the dfoutput.as_latex() to export this to latex.


Textbook covering calculus and probability theory

In a course similiar to yours at the Radboud University Nijmegen, the Netherlands, they use two different lecture notes (in Dutch) and each set of notes recommends a book.

For probability theory, they recommend the following book:

Murray R. Spiegel, John J. Schiller, R. A. Srinivasan: (Schaum’s Outline of Theory and Problems of) Probability and Statistics. McGraw-Hill
Companies, 2000, 408 p., ISBN: 0071350047.


Prove that budget constraint is Lower Hemi Continuos (LHC)

I don't believe it is lower semicontinous.
Let $w = (0,\dots,0)$, $p \in \mathbb{R}^n_+$ be any vector such that $p_1 = 0$ (the first coordinate being 0).
The allocation $x=(1,0,\dots,0) \in B(p,w)$.
Define the sequence $p_n = p  + (\frac{1}{n},0,\dots,0)$ and $w_n = (\frac{1}{n},0,\dots,0)$. $w_n \rightarrow w$ and $p_n \rightarrow p$.
For any $x^n \in B(p_n,w_n)$, $p_n x^n_1 \leq w_np_n$, so $x_1^n \leq \frac{1}{n}$.
Hence for any sequence such that $x^n \in B(p_n,w_n)$, $x^n \not \rightarrow x$.


What is the relation between probability of default and country's interest rates?

The answer depends on the situation for the country in question. Since the answer is correspondingly very large if we covered all cases, I am only attempting to offer an overview of the cases. Feel free to ask follow up questions on more specific questions.

I will cover three important sinplified cases.

“Gold Standard”/“hard currency borrowing”.
Free-floating fiat currency.
Euro area.

(1) The “gold standard” example is not currently of interest, but if you want to look at historical governmental defaults, they were associated with the gold standard, or the countries borrowing in a foreign currency (“hard currency”). (Countries in currency pegs are also in a similar situation.)

In a gold standard, the government’s money in some sense can be converted into gold. Ultimately, it needs to have enough tax revenue in its local currency to force the private sector to turn in enough currency so that the consolidated government (fiscal arm/central bank) can honour the conversion parity. If that level of taxes is not politically sustainable, the government will be forced to default.

The usual operating procedure in a gold standard is for the central bank to raise interest rates to attract gold inflows. However, this raises the interest expense of the government, and so it makes it harder to sustain the gold parity price. So in this case, higher interest rates can raise default rates. However, the causality is ambiguous - the central bank is often forced to raise interest rates to defend the currency value, snd the currency is falling in value because of an expected default. Therefore, one could argue that high interest rates in a currency peg country are the result of an elevated default risk.

For countries with a currency peg or borrow in a foreign currency, they are in a similar boat, as they need to impose taxes in order to be able to obtain the external instrument.

(2) A floating currency sovereign (e.g. the United States, the United Kingdom, Japan, Canada...) does not attempt to fix the value of its currency versus external instruments. It can create arbitrary amounts of its currency unit without using any real resources. The only real concern is the purchasing power of the currency unit (inflation).

Since there are no external constraints (lack of gold), default is essentially a political decision.

There are operating procedures that might entail default (e.g. the debt ceiling in the United States). It is a political decision to not override those procedures.
Interest spending may be such a high part of the budget that politicians would rather default than pay the interest.
Government debt may be viewed as illegitimate. For example, debts incurred by a radical government that was later overthrown.

High interest rates might help push a country towards default, but that was not the case in the 1970s. The number of defaults of countries in this situation is so low and usually associated with other problems that there are almost no data to draw conclusions from.

3) The euro area is a special case: the currency itself is a fiat currency, but it is under the control of a central bank that is a supranational entity. The euro area nations are not sovereign in the same sense that the federal Government of Canada is, rather they are in a situation similar to a Canadian province.


Let me answer here putting together all the hints in the comments.

You've figured out that the equilibrium bid for a type $v$ bidder is $b(v) = \frac{2}{3}v$.

The bids are strictly increasing in $v$, so bidder $i$ wins whenever $b(v_i) \geq b(v_j)$ or $v_i \geq v_j$.

The expected payoff for a type $v$ bidder is thus
$$
(v -b(v))P(b_1 \geq b_2)
$$
or
$$
(v - \frac{2}{3}v)P(v \geq v_2) = \frac{1}{3}P(v \geq v_2)
$$
So what is the probability that $v \geq v_2$?

By definition of CDF $P(x \leq v) =  F(v) = \int_0^v f(x) dx$

So $P(v_2 \leq v) = F(v)$.

Thus expected utility for a type $v$ bidder is:
$$
u(v) = \frac{v}{3} F(v)
$$


How is Turkish economy handling the pressure of so much defense purchase?

Turkey had an estimated military spending of $14.8 billion in 2016.

A 30% increase would be a $4.5 billion increase. That's 0.5% of GDP. Turkey's government budget deficit was 1.1% in 2016. All else equals, the budget deficit would rise to 1.6% with that kind of increase in spending. Either that or they cut spending elsewhere.


Start here
$$0.75/p_xx= 0.25/p_yy$$

Multiply both sides by $p_xx p_yy$, you get:
$$0.75p_yy=0.25p_xx$$

Add $0.75p_xx$ to both sides, you get:

$$0.75(p_xx+p_yy)=p_xx$$

So:
$$\frac{p_xx}{p_xx+p_yy}=0.75=\frac{3}{4}$$

And you spend the rest of your budget, so $1/4$ of it, on $y$.


Did a similar property boom take place in another country or throughout history like we have seen in London? If so how did it effect the economy?

According to the ONS the house price-to-earnings ratio was 12.8 in London and 7.8 in England as of 31/12/2019.
According to Schroders the house price-to-earnings ratio in England was consistently above 8 until the late 19th century (see graph below). In 1845 it was 13 in England, so it was arguably higher than that in London at the time.
What I don't find very clear is if Schroders are referring to housing units when they speak of houses. They say that "houses" where bigger in the 19th century and hence more expensive. Obviously the average earner did not live in a bigger house but would have shared it with other average earners.
A more sensible statistic would hence be something like a "minimum housing unit price-to-earnings ratio". Comparable prices would still likely be higher in the past. But it shows that creating comparable data would require in-depth study not just of the economics of housing but also how these houses are used, i.e. what can realistically be considered a minimum housing unit in the 19th century given the housing conditions at the time (Engels published The Housing Question in 1872).

Source: Schroders (30/01/2020)


Given a Utility function, U(x,y), why is multiplying U(x,y) * x not a monotonic transformation?

Because $x$ is not a constant, and therefore multiplying $U(x,y)$ by $x$ does not necessarily preserve the ordering between bundles.

For instance, consider the function $U(x,y)=x+y$, and the bundles $(x_0,y_0)=(2,0)$ and $(x_1,y_1)=(1,2)$. We have
\begin{equation*}
U(x_0,y_0)=2 < U(x_1,y_1)=3
\end{equation*}
but
\begin{equation*}
x_0 U(x_0,y_0) = 4 > x_1 U(x_1,y_1) = 3
\end{equation*}
Hence multiplying $U$ by $x$ changes the preferences, therefore is not a monotonic transformation.

As a side note, multiplying the utility function by a constant provides a monotonic transformation if the constant is positive, and not only nonnegative as you assumed (if the constant is zero the orderings are obviously not preserved).


There’s a huge literature on this topic. I would start by looking at the IMF Fiscal Monitor, and see what is interesting. Their research papers will refer to other papers that can be pursued. IMF Fiscal Monitor page.

Otherwise, you can search for papers on “fiscal sustainability.” As a disclaimer, I am not a fan of most the literature. This paper by Scott Fulwiler discusses interest rates and sustainability, and it critiques some of the technical issues in the literature - link to paper.

In any event, there is no “magic tipping point” for default. It is up to the government involved to decide at what point interest costs are excessive, and whether they will default rather than keep going. There have been almost no default of floating currency sovereigns, so there is no data set to work with. (Most sovereign defauits have occurred when they borrow in another currency; the euro area is discussed below.)

The euro area countries face a particular problem that is not really an issue for other developed countries: they do not have control of their domestic interest rates. Their debt may be issued at high spreads versus other euro interest rates, and so a default becomes a self-fulfilling prophecy (as seen in Greece). Most of the “fiscal sustainability” literature assumed that a country controls the interest rate it borrows at, so the situation is much less risky.


An isoquant is a contour line drawn through the set of points at which the same quantity of output is produced while changing the quantities of two or more inputs.

Having production function: $f(x_1, x_2) = min \{x_1, x_2 \} + x_2$ for $x_i \in [1, 10]$

We can produce isoquants for varying number of inputs. Below you will find 10 isoquants for quantities ranging from 1 to 10 (keep in mind that $Q = 1$ and $Q = 2$ are overlapping). I hope it helps to visualize the problem.


Banks are allowed to print money out of thin air. Black magic or what?

Very good question.

Let's break it down:

  A bank give a Merry loan of 100,000$

Concretely, the bank records an asset of 100,000 (the loan) and a liability of 100,000$ (the amount recorded in Merry's deposit account).

Note that at this point, the liability is what appears on your internet banking statements as 'money' and those liabilities to you from banks are today what is traded in exchange for goods and services. In this sense the liability is 'money',  and it is why people consider this money to have been synthesized commercially ex nihilo. This is essentially correct. Banks are licensed to create this money from nothing.

  Merry buys a house with this money.

Analyse it in detail: Merry finds a supplier, who implicitly trusts the bank, who probably has an account at another bank (though not necessarily), and agrees to accept payment in the form of a bank liability. Merry initiates a bank transfer from MerryBank to SupplierBank. MerryBank decreases its liability to Merry by 100,000, keeping only the asset, but adds a new liability to SupplierBank. SupplierBank records an asset of 100,000 (MerryBank's liability to SupplierBank) and a liability to Supplier of $100,000.

  For a period of 1 year, Merry has paid back 10,000 to the bank.

Yes, in 99% of cases in the form of bank transfers from other newly created debt from other banks. These become assets in the books of MerryBank (liabilities from SomeOtherBank) and the loan asset is reduced. So while MerryBank gains a 10k asset from SomeOtherBank it actually cancels 10k of loan asset, so its position has not changed - only its risk position.

  Afterward Merry's finance condition worsen and she no longer can pay
  her debt. So in theory, if all the bank borrowers end up in the same
  situation as Merry did, the bank is going to bankrupt.

No. The bank still has an asset of 90,000$ and will use debt collectors to take Merry's property. The bank will then cancel the 90,000 loan asset and record a property asset at market value.

  But if the bank never had the loan money in the first place,

Yes. Fractional reserve requirements for loans was/is often 0%. Also, the asset (for example) at SupplierBank is then used as reserves to issue further debt. Two banks in a fractional reserve system act as feedback-loop amplifiers, so if the reserve requirement was 10%, that does not mean that the money multiplier would be 10 times, but depending on flow rates and dynamic systems calculations you can end up with hundreds of thousands times the base reserve. Incidentally, this was the keystone to the financial crisis.

  every penny received from borrowers is a pure profit. Even if Merry returned
  only 10% of her loan, the Bank still profits 10,000 because the given
  100,000$ to Merry were not bank's money in the first place.

No. See above. The asset position is unchanged.

  This leads me to the question: Why do banks put interest on top of
  this fraud? And what prevents banks from putting huge amount of money
  to some people's account (for example to the Bank's CEO )? This black
  magic is not quite clear to me.

Interest rate is primarily determined by interbank lending rates to cover reserve and capital requirements.

I don't know if it applies to the rest of the world, but the main source of income for banks where I live is actually transaction fees.

As for CEO payments, auditors. But in general CEOs reward themselves by issuing debt (bonds) to cover stock buybacks, to boost stock price, to which their bonuses are tied. Calling this kind of activity fraud is to some extent true, as outside the US a lot of this kind of thing is regulated.

Banks could be pure digital banks that have a employees in the hundreds. Typically they are dinosaur institutions that employ in the 10s or 100s of thousands of people. The banks use fees and interest to cover the cost of employees mainly. Incidentally, this is one reason why governments chose to assist banks in 2008. Directly and indirectly they employ a big chunk of the economy, a consequence of Western deindustrialization and financialization.


Strongly and strictly increasing utility functions

The term strongly increasing function is non-standard in economics (and I believe also in math) and should have been clearly defined in the main text.

The following definitions are given only in their Appendix 1, p. 529 (2011, 3rd edition):

As the following text states:

  a strictly increasing function need not be strongly increasing, but every strongly increasing function is strictly increasing


What's the relation between economic growth and buying power?

Probably no, not in the same year. But maybe in the year after.

If you're talking about the average citizen, the intensity of this expansion of buying power depends on wealth distribution, the importance of those industries which had grown and whether those industries are more labor intensive or capital intensive.

Also, this expansion doesn't happen instantly; don't expect that the whole nation will realize this growth in the same year that it happened. It takes time for the money to flow from the main industries to industries of the same network and than to other firms not related.

That process is even slower through the employee chain. One of the reasons is that frequently employee and employer negotiate salary for long periods (there's some "stickiness"). When a type of workforce is unionized, it takes even more time for them to negotiate wages as a group.

The pathway you're asking about is like this: More production means the industries needs to increase their capital, inputs and workforce. They buy more capital, more inputs and hire more workers. This increase in demand means those other industries and workers are able to negotiate better salaries than in the year before. To be able to satisfy this new demand, the industries that produce capital and input need to increase their production as well, by hiring more workers, buying capital and inputs. And this goes on and on till it,theoretically, affects the lower paying jobs. Of course it all depends on how much the links in the chain pass the wage increase to the next link.


Intuition behind Real Money Demand and Real Money Supply

They're in monetary units! Let's say an economy has a money supply of $100$ $mu$ and a price level of $1$. Another economy has the same money supply but a price level of $1.5$. Which economy has the bigger real money supply? The first, obviously. In the first hypothetical economy, you can think of the average transaction as one involving smaller amounts of currency. Real money demand will of course depend on nominal income $Y$. An increase in nominal income makes people demand more money because there's more transactions to be made.


When shall I use bootstrap standard error?

Normally, you use bootstrapping when you cannot derive a formula for the variance-covariance matrix of your estimator. Since you do not have this formula, you cannot obtain the standard errors of parameters, and so your econometric analysis is very limited. In these cases, using bootstrapping is the only option available. This is common in complex, non-linear models.

Similarly, you do not need them for those cases where you can derive robust, close-form formulas for the standard errors (as in OLS, or linear panel data models, or most of models found in most of textbooks). In this cases, you already have a formula which is asymptotically efficient, and bootstrapping would be redundant, and probably counterproductive if done incorrectly.

Notice that bootstrapping is not only useful for obtaining the standard errors of parameters. You can also bootstrap full tests. For example, you can bootstrap the Hausman test that compares RE versus FE. To go this way would imply a longer answer.

For a quick and hands-on introduction, check Cameron and Trivedi (2010), Chapter 13.


The Key to BNE is that players that know something (about the state of the world or their type) can condition their strategies with their information. That is, for example, in question 3, type A could choose strategy U, while type B could choose strategy D. Therefore from the second player's perspective, there are four possible pure strategies of his opponent. Namely UU, UD, DU, and DD. Which represents the actions of each type of player 1.

What you want to do is construct another table where player 1 has these four strategies and player 2 has strategies L and R whose payoffs are the expected payoff. For instance, in the cell (UU, L) the payoff to player 2 is $0.5(4)+0.5(2)=3$, and for (UU, R) it has a value of $0$. This table is only useful to asses player 2's payoffs since player 1 knows his type, so no need to fill a number for player 1. After constructing the table you realize that player 2 has a weakly dominant strategy (L). You have to go back and forth between the two tables (for type A and B) and the third table for player 2 to find all the fixed points.

For this game, there are several equilibria; so maybe that is throwing you off. For example, suppose player 2 plays her weakly dominant strategy, L,  in that case, both types are indifferent between U or D, so they both could choose U. Given the strategy UU, player 2's unique best response is L, so this is an equilibrium. I.e. (UU, L) is an equilibrium.

Try to convince yourself that (DD,R), (DD,L) are the other equilibria, and there are no more equilibria.

As for question 4, it is actually simpler. Since no player knows the state of the world, they cannot condition their actions on the state of the world. so the pure actions for player 1 are U or D and for player 2 are L and R.

You will have to construct another table where players average out the payoffs of each of the four possible combinations of actions. For example for the cell (U,L), the payoff for player 1 is 1 and for player 2 is zero. Similarly, the payoffs on cell (U,R) is $(0,0.5)$. After constructing the new table you can find the BNE by the usual algorithm to find standard simultaneous-move NE.

Try to convince yourself that there is a continuum of equilibria for this game. In all of them, player 2 chooses $R$ since it is a dominant strategy, and player 1 is indifferent between both actions so chooses any mixed or pure strategy (which technically speaking is a  type of mixed strategy too).


Are many of the traditionally oil dependent Gulf states really successfully diversifying their economies?

Several points need to be considered.
1) Oil dependance
First of all, it is not correct to assert that 30%-50% of the GDP being concentrated within a single industry is "diversified". This is actually a high focus. And I can only guess that a lot of other activities are strongly linked to the oil and gas sector: for example, banks might have a lot of lending and trading activity in the sector, even if they aren't directly counted as being part of it.
Moreover, an interesting thing to look at is how much the government relies on this sector. For Saudi Arabia, which budget was discussed recently, it is obvious that oil plays a major role, as it makes up 90% of the revenue. FOr the UAE the answer is more tricky, as the different emirates have significantly different profiles: Abu Dhabi relies strongly on the Oil industry, whereas the other provinces such as Dubai rely only slightely on oil - though they receive high subsidies from Abu Dhabi.
As for the progress made, there are indeed some. These countries are aware that they will not always be able to pump up oil, and that they will need, in the long run, to have other strong industries. That is why, pretty much the same as in Norway, huge sovereign funds (check nb 4 & 6) are setted up in these countries.
2) Development

I'm rather surprised by this, as I've always assumed that countries
such as Portugal and Mexico would be well ahead of the UAE/Saudi
Arabia in terms of development/education/technology etc.

GDP per capita is not a index of development. If you are looking for this, the HDI (Human development indicator) is a much more appropriate index. This index is composed by several measures of education, wealth (GDP), life expectancy and other indicators which relate to development and not only to economics.
The UN compiles this indicator on an annual basis, here is for 2015. It shows the following for the countries you mentioned. The HDI ranges from 0 (worse) to 1 (perfect score):

#6 Ireland: 0.916
#36 Poland: 0.843
#39 Saudi Arabia: 0.837
#41 United Arab Emirates: 0.835
#43 Portugal: 0.830
#72 Turkey: 0.751
#74 Mexico: 0.756

You can see from this numbers that the gap is thin between Portugal and the two Oil countries. You can also guess that the contribution of their GDP to the index is much higher than for any other countries mentionned above. Thus, the pure non-economic indicators are probably better for the other countries than Saudi Arabia and the UAE.
Furthermore, the Gini index can also be used to have some idea of the level of development of a country. It shows how big inequality are (regarding income or net wealth for example). Sadly this index is not computed for Saudi Arabia nor for the UAE. This is probably intentional, as these countries are well known to have very high inequality, in particular between immigrants (the huge majority of UAE's inhabitants) and natives.
Finaly, your intuition is correct: it is indeed quite 'naive' to multiply the GDP per capita by the proportion of GDP of a sector. Indeed, different sectors display significantly different productivity and efficiency of the labor factor. Therefore, it can be that the productivity of the Oil industry in those countries largely outweigh the productivity of the other sectors. Therefore it is hard to compare this clumsy calculation with 'full' GDPs of other countries.


You have the government's flow budget constraint (re-written in real terms):

$b_{t} + m_{t} + \tau_{t} = g + \frac{m_{t-1}}{\pi_{t}} + R_{t-1}\frac{b_{t-1}}{\pi_{t}}$ (1)

Now all you need to do is substitute (2) and the policy rules (also, I don't think Leeper's utility function had a $\delta$ but that's not important) and linearise. I.e linearise:

$b_{t} +δc[\frac{R_{t}}{R_{t}-1}] + (\gamma_{0} + \gamma_{1}b_{t-1} + \psi_{t}) = g +\frac{δc[\frac{R_{t-1}}{R_{t-1}-1}]}{\pi_{t}} + (\alpha_{0} + \alpha_{1} \pi_{t-1} + \beta \theta_{t-1}) \frac{b_{t-1}}{\pi_{t}}$ (2)

Note: I left some $R_{t}$ and $R_{t-1}$ in there but you will have to substitute the monetary policy rule there as well. Then all you do is a Taylor approximation around the deterministic steady state. Just a warning, the algebra will be very hairy, it took me several tries to get it right.

OP requested some more reading on this topic so I'll add a list here:

[1] Loyo (1999), Tight Money Paradox on the Loose: A Fiscalist Hyperinflation

[2] Woodford (1998), Public Debt and the Price Level

[3] Woodford (2000), Fiscal Requirements for Price Stability

[4] Leeper & Leith (2016), Understanding Inflation as a Joint
Monetary-Fiscal Phenomenon

I have many more but start with these.


No, this is not how wealth tax works. Wealth taxes do not transfer company ownership to the government.
In order to understand how wealth taxes work consider following example. Let us assume that there is individual owning 10 shares each worth \$10,000 dollars in company A. The individual has no debts or nothing else directly affecting the wealth of individual.
Given the numbers assumed above the wealth of that individual would be 10x\$10,000 = \$100,000.
Next let us assume no deductible so the wealth tax is applied to whole \$100,000. Lastly, lets set the wealth tax to 3% as in your question.
In the case above the wealth tax will be:
$$ (0.03) {\\\$} 100,000 = {\\\$}3000$$
per year. However, government does not get any of the shares in the company. The individual is free to get the \$3000 to pay the wealth tax from their labor income or from their dividend income, profits from some venture or rent they get etc (neither of which are wealth). So even with 3% wealth tax they get to keep their company and shares in their company forever.
The individual could pay for the wealth tax by selling some of their shares in the company, for example in year 1 they could sell 1 share for \$10,000 to someone else pay \$3000 tax to the government and consume rest or keep the rest on bank account for future. But the individual does not need to sell it to the government.
If government would want to buy the shares it could presumably do that on the same capital market where individual will sell their \$10000 share, but the government would have to compete with other buyers there or the individual could also just decide to sell it someone directly where government would have no access to actually buy the share (although I don't know why would government want to do that in the first place).
Consequently, even with 3% wealth tax it might take anything between 1 year (in case if the person just decides to directly sell all shares on market) or infinity number of years (in case that individual (and their descendants) will never decide to sell their shares, or never sell them to the government) for government to get more than 50% ownership in a firm owned by an individual that is being taxed.


Is there any specific significance of inverse demand?

Inverse demand maps from quantity to value. It is useful to identify how much some units of the good are worth to the consumers. The central welfare concepts of consumer's and producer's surpluses rely on inverse functions in their definitions.

In addition there are special cases (e.g. perfectly inelastic demand) where defining inverse demand is just easier.


Why does increasing the money supply decrease the interest rate in layman's terms?

On most fundamental level it is because interest rate is price for money. In the same way as price of milk goes down when supply of milk increases (ceteris paribus) price of money goes down when supply of money increases. For example, consider the following diagram from Blanchard et al. Macroeconomics below. You have supply of money (by central bank) and then you have demand for money by people. Interest rate ensures that demand for money = supply of money. If supply increases (shift to the right) interest rate has to decrease otherwise people would not be willing to get and hold that additional money.


How to calculate constant GDP in international $ from nominal GDP in local currency units

You need to use PPP rates. These convert basket of goods in one country (in LCU) to that of (normally) the US, in US dollars. You can find these rates here or here. With these rates you can transform nominal GDP in Yuan into nominal GDP in USD. Then, you can use a US CPI or GDP deflator index to transform these nominal international dollars into constant international dollars. The base year is given by the price index you use.

You can rear more about PPP and how to use it here.

Finally, in case you do not want to bother to calculate this yourself, or that you want to confirm your calculations, here is the actual series of China's GDP in 2011 international dollars.


There is no need to complicate things here - just try to find annual exchange rate used by IMF. Worldbank's data are here. LCU stands for Local Currency UnitsYou may see a drop of Rub/$ rate from 31.84 to 38.38. (Drop of Ruble, of course.)
National statistics provides input figures for international accounts. According to Rosstat, Russian nominal GDP grew in the Y2014:

Y2013 - 66,190 bil. Rub,
Y2014 - 71,406 bil. Rub.

Using  NGDP dollar evaluation from your post, annual exchange rate for Y2013 is 31.5 Rub/$; in Y2014 rate is 34.7 Rub/$, which makes some sense. But indeed Rub/$ rate was somewhat closer to 38.5 (see here). So, IMF figure for the Y2014 from your wiki reference seems to be more correct: bil. $1,857,461 for Russian NGDP. (do not forget about Crimea correction to appreciate the difference).
So, in fact Russian nominal GDP did drop significantly in current dollar terms in the Y2014.
Imho, PPP is not applicable for your question as Russian GDP by PPP is of different magnitude.


How to measure expected price growth rate for oil?

There is a futures market for crude oil: for example http://www.cmegroup.com/trading/energy/crude-oil/light-sweet-crude.html gives a price now for May 2018 of US$\$49.54$ compared with a cash price now of US$\$47.82$

The futures price being higher than the cash price is described as contango and is why there are excess stockpiles of crude oil.  In the past the comparison has sometimes been reversed, described as backwardation and associated with low stockpiles.  For a graph of this over time, see http://marketrealist.com/2017/02/return-backwardation-mean-crude-oil-prices/ though the relationship is more complicated than described there

For historical oil prices the Federal Reserve Bank of St. Louis has several sets of numbers, such as https://fred.stlouisfed.org/series/WTISPLC going back to 1946


USD/CAD fx: What's the difference between Buying USD and Selling CAD?

Should be no difference.

You are either (1) buying USD by selling CAD or (2) selling CAD to buy USD.


market for lemons with signals - equilibriums vs constraints

The two questions you ask are related but distinct. First recall what a separating equilibrium is: its an equilibrium where distinct type takes distinct actions, supported by a system of consistent beliefs.
Now lets come to your questions one by one:

In the Lemons Market, a separating equilibrium would involve the seller setting different prices for the lemons and peaches. Can such an equilibrium exist? Think about what the buyer's beliefs would be and if the lemon can mimic the peaches?

I guess you mean Incentive Compatibility constraints instead of Participation Constraints. You must have encountered them in Principal Agent models. In these models, the agents play a (bayesian) game and the Principal can commit to design the rules of the game. The Participation Constraint ensure that the agents are willing to take part in the game designed by the Principal. The Incentive Compatibility Constraint ensures that the ensuing equilibrium of the game is a separating equilibrium - i.e. different types of agent will be choosing different actions in the game. The specific rules of the game designed by the Principal depends on the context.


In economics Too Big To Fail (TBTF) can have slightly different meaning depending on what research you are looking at but generally speaking literature seems to agree that what matters is how interconnected or systemically important firm is (e.g. see Bernanke 2010;  Zhou, 2009).
Systemically important firms are such that other firms critically depend on them. In fact in recent literature you will often see people arguing that TBTF is a misnomer and better terms would be “too complex to fail” or  “too interconnected to fail” (see Kaufman, 2014), because really TBTF has much less to do with actual size than interconnectedness of a firm.
This is the reason why you usually hear the term TBTF being applied to banks or financial institutions. Financial, institutions are an economic equivalent of infrastructure as they help to allocate capital to its most efficient uses and many other firms depend on them (see Mishkin & Eakins Financial Markets and Institutions).
Consequently, whether General Motors or Snapchat are TBTF depends on how interconnected are with the rest of an economy. If there would be significantly many  'influencers' and 'content creators' that they would become significant part of an economy snapchat might be TBTF. This being said although I was not able to find specific papers on examining how interconnected GM or Snapchat are I am going on a limb to say that probably neither of them are TBTF. As argued by Strahan (2013) TBTF in most cases really applies mainly to financial sector due to reasons already mention above.
In fact Strahan specifically argues in that paper that non-financial firms, using General Motors as an example of non-financial firm that would not be TBTF, can only be TBTF in rare cases. Although the paper makes just analytical argument for this and draws only on general literature without actually examining networks of General Motors specifically (importantly note just because firm is bailed out that does not mean it was TBTF - every company has incentive to argue its TBTF to get bailout and policymakers often cannot examine networks in real time so they have to make an educated guesses about whether firm is actually TBTF or not - if a non-financial firm claims to be TBTF its good to be skeptical given literature but I would not dismiss it out of hand either).


Reference request on "post-scarcity" economics?

A related question is in asking when (if ever) will we be post scarcity. The paper Are We Approaching an Economic Singularity? Information Technology and the Future of Economic Growth by William D. Nordhaus should be of use.

Abstract:

  What are the prospects for long-run economic growth?, the present
  study looks at a more recently launched hypothesis, which I label
  Singularity. The idea here is that rapid growth in computation and
  artificial intelligence will cross some boundary or Singularity after
  which economic growth will accelerate sharply as an ever-accelerating
  pace of improvements cascade through the economy. The paper develops a
  growth model that features Singularity and presents several tests of
  whether we are rapidly approaching Singularity. The key question for
  Singularity is the substitutability between information and
  conventional inputs. The tests suggest that the Singularity is not
  near.

(emphasis mine)

Another possibility is to consider those sectors, like digital media, where we are already post-scarcity, in that marginal costs are zero or very near to it. I recommend Carl Shapiro and Hal Varian's Information Rules for an accessible and interesting discussion of the economic issues that arise in this setting.


Advertising is hard to understand if you take classroom economics models too literally.  I highly recommend diving into the literature on behavioural economics if you are truly interested. But for the purposes of your question, Pepsi runs advertising because it operates in a market in which consumer choice is primarily driven by two sources of cognitive bias: the anchoring and the availability heuristic.

On the one hand, new customers are born every day, and if Pepsi doesn't run advertising then those new customers will see a Coke ad first.  On the other, since existing customers don't actually make purchasing decisions based on a careful weighting of product merit, they need to be reminded that Pepsi exists and that there are warm, fuzzy feelings associated with the brand. This way, when they find themselves craving something to drink, Pepsi might be the brand that comes most readily to mind, because it's the last soft drink ad they saw.

This is just my opinion (built on some of the theory, but by no means do I claim it is the Complete Truth), but the anchoring heuristic in particular seems to explain brand loyalty in the cola market.  I grew up in the era of the "blind taste test" and my takeaway from those social experiments is that when the most significant product differentiation is along branding lines, consumers will tend to just stick with whatever product they encounter first, because absent labeling and the social utility that comes from a trite form of tribalism, the consumer is indifferent to the choice.


Relationship between strong and weak axioms of revealed preference

The only journal that published it is behind a paywall:
http://www.jstor.org/stable/2296210?seq=1#page_scan_tab_contents but check your school library access for it.

The proof is pretty complicated and is based on an induction argument. When I've tried to link SARP and WARP, I've only ever found references to his paper.


Applying Economic Principles to Other Studies

This paper seems to be about practical policy. It seeks to understand what drives the killing of elephants. While what you write makes sense in theory, there might be some hidden factor that we do not perceive that has a much bigger effect (for example the actors of the market may not be rational) so looking at the data is always welcome.

I would also say that your economic reasoning is not the only possible one. I could also provide a line of reasoning reverses causality

More elephants are killed (this may be random, or it may be a result of unemployment in other sectors).
Because demand is unchanged, prices fall.
Since authorities always seize half of the ivory contraband, more contraband is seized.

If this is true then the seizures and the price change is a result, implying a different market dynamic. Unfortunately the paper only tested correlation, not causality, but this highlights that there can be more than one explanation to a phenomenon. In fact it is even possible to argue that the price of ivory and the number of elephants killed is negatively correlated:

More elephants are killed (this may be random, or it may be a result of unemployment in other sectors).
The authorities, who until now looked the other way, notice this and start seizing ivory, leaving almost none on the market.
The price of ivory rises.

Which of these explanations you find logical is somewhat subjective. Because there can more than one, a logical explanation may not be the right one. That is why econometric papers (could) provide useful insights.


I will be using the following definition of continuity for binary relations.

Definition 1: A binary relation $\mathcal{R}$ on $\mathbb{R}^2_+$ is continuous if for every $x$, the following sets are closed.
$$
USC_x = \{y\in \mathbb{R}^2_+|y\mathcal{R}x\}
$$
$$
LSC_x = \{y\in \mathbb{R}^2_+|x\mathcal{R}y\}
$$

Definition 2: A set $Z$ is closed if $z_n$ is a sequence in $Z$ and $z_n\rightarrow z$, then $z\in Z$.

Now, let $\mathcal{R}$ be defined as in your question, i.e.
$$
x\mathcal{R} y\iff f_\alpha(x) \geq f_\alpha(y)\text{ and }f_\beta(x) \geq f_\beta(y)
$$
Let $x\in \mathbb{R}^2_+$. Let $x_n$ be a sequence in $USC_x$ converging to $x^\ast$. Then, for each $n$,
$$
f_\alpha(x_n) \geq f_\alpha(x)\text{ and }f_\beta(x_n) \geq f_\beta(x).
$$
Since $f_\alpha$ and $f_\beta$ are continuous, we have
$$
f_\alpha(x^\ast) \geq f_\alpha(x)\text{ and }f_\beta(x^\ast) \geq f_\beta(x).
$$
implying that $x^\ast\in USC_x$; hence, $USC_x$ is closed. We can show also that $LCS_x$ is closed by using identical arguments. This concludes that $\mathcal{R}$ is continuous.

Remark 1: The arguments above provide a general strategy of analyzing statements regarding $\mathcal{R}$. Transitivity of $\mathcal{R}$ can be proven in a couple of simple steps. Your statement regarding this property is a bit vague, and should be more rigorous imho.

Remark 2: Your explanation about completeness needs a clearly written argument as well. While it is true that $\alpha=\beta$ implies completeness of $\mathcal{R}$, the reasoning that makes it true is not exactly what you are proposing. It is also necessary to show that $\alpha\neq \beta$ implies that $\mathcal{R}$ wouldn't be complete (I believe that this should be the case). This would require a bit more work on your part. Let me know if you run into troubles in this.

Bottomline: In my sincerely humble opinion, you should exercise a lot with proving things. Maybe grab a book on introduction to abstract mathematics and such.


How to interpret this regression coefficient?

Assuming that from the beginning you made the regression with prarrest with transformation *100 (say you have for example [0.012,0.093] and you transform it into -> [1.2,9.3], I remark this cause its critical to the interpretation).

Ceteris paribus, one percentage point increase in prarrest causes the number of crimes committed per 1000 county residents to go down by 67.

It is correct, you could se this with this approach (assuming its linear regression), taking into account the your hypothesis must have this from:
$crrate=\hat{\beta_0}+\hat{\beta_1}(prarrest)\;\;\;\;\;$where $\hat{\beta_1}=-0.067$
$\frac{\Delta(crrate)}{\Delta(prarrest)}=-0.067 \implies \Delta(crrate)=-0.067\Delta(prarrerst)$
Now, if $\Delta(prarrest)=1$ then $\Delta(crrate)=-0.067$.
Since prarrest is in the terms clarified from the beginning, this $\Delta(prarrest)=1$ means a whole percentage point increase (e.g. from 34(%) to 35(%)), then looking at the terms or the $crrate$ variable (#crimes/population, so for one individual of the population how many crimes are committed), a -0.067 change means a decrease in the order of 0.067 per each person, so multiplying this by 1000, would give us decrease of 67 per 1000 people. Yes, your try is correct.


Is charging of interest inherently rent seeking?

rent is about uncompetitive markets. In very competitive banking sector, interest will still exist. So it's not rent seeking. If laws forbide charging real interests, no private bank would be on business.

If you look at Investopedia for "Economic rent", you find:

  "Economic rent" is a term that defines an amount of money earned that exceeds that which is economically or socially necessary. This can occur, for example, when a buyer working to attain a good or service that is considered exclusive makes an offer prior to hearing what a seller considers an acceptable price. Market imperfections thus lead to the rise of economic rents; it would not exist if markets were perfect since competitive pressures would drive down prices. Economic rents should not be confused with normal profits or surpluses that arise in the course of competitive capitalist production. This term also differs from the traditional use of the word "rent," which applies to payments received in exchange for temporary use of a particular good or property, such as land or housing.

Notice: "Market imperfections thus lead to the rise of economic rents; it would not exist if markets were perfect since competitive pressures would drive down prices. "

Bill's definition is the "old version", which is not what you mean by "rent".

https://www.investopedia.com/terms/e/economicrent.asp


The influence of restricted type space on incentive compatible mechanism design?

The type space is the support of the belief about the types. It seems a bit weird to allow a mechanism designer to arbitrarily restrict their belief support. However, you are right: if the designer can find out more about the private types by acquiring information, this would intuitively allow them to reduce information rent, because it makes the type "less private." Of course it would be best to fully unveil types such that no incentive for revelation has to be provided and full surplus extraction becomes possible. In your constructed example it may not be helpful to exclude very high types $>M$ as more profitable deviations are often found in closer vicinity to the real type. That depends on the problem at hand of course.
I can't see clearly where you want to go with this. So let me suggest something else in a similar spirit: mechanism design with evidence. Here, agents can make their private information hard by providing evidence that their true type belongs to some subset of the type space. See, e.g., the work by Ben-Porath, Dekel & Lipman (AER 2014, Ecta 2019 and more), Bull and Watson (GEB 2007), Hagenbach, Koessler & Perez-Richet (Ecta 2014) and others.


Why is the derivative of a monotonic transformation of a utility function assumed to always be greater than 0?

Let's look at the use of monotonic transformations of utility functions (which I guess is the most frequent occurrence of this concept in econ).
Let $u: \mathbb{R}^n_+ \to \mathbb{R}$ be a utility function. We say that $g: \mathbb{R}^n_+ \to \mathbb{R}$ is a monotonic transformation of $u$ if for all $x, y \in \mathbb{R}^n_+$:
$$
u(x) \ge u(y) \iff g(x) \ge g(y).
$$
Let $D \subseteq \mathbb{R}$. Then $h: D \to \mathbb{R}$ is called strictly increasing if for all $x, y \in D$:
$$
x \ge y \iff h(x) \ge h(y).
$$
There's the following result:
Th Let $D$ be the range of $u$, i.e. $D = u(\mathbb{R}^n_+)$. Then $g$ is a monotonic  transformation of $u$ iff there exists a strictly increasing function $h: D \to \mathbb{R}$ such that $g(x) = h(u(x))$.
proof: ($\leftarrow$) If $g(x) = h(u(x))$ and $u(x) \ge (>) u(y)$ then $h(u(x)) \ge (>) h(u(y))$ so $g$ is indeed a monotone transformation of $u$.
($\rightarrow$) For the reverse, define
$$
h(z) = g(x) \text{ whenever } u(x) = z.
$$
First we need to check that $h$ is indeed a function. As such, let $x, y$ be such that $u(x) = u(y) = z$. We need to show that $g(x) = g(y)$. Indeed as $g$ is a monotone transformation, we have that $g(x) \ge g(y)$ and $g(y) \ge g(x)$ so $g(x) = g(y)$. To see that $h$ is strictly increasing, let $z \ge (>) w$ and let $x$ and $y$ be such that $u(x) = z \ge (>) w = u(y)$. Then as $g$ is a monotone transformation $h(z) = g(x) \ge (>) g(y) = h(w)$. $\blacksquare$
To see the connection with the derivatives we have the following:
Th if $g: D \to \mathbb{R}$ is differentiable, $D$ is convex and if $g'(x) > 0$ for all $x$ in $D$, then $g$ is strictly monotone.
proof: Let $x \ge (>) y$ then by the mean value theorem there is a $c \in [x,y]$ such that:
$$
g(y) - g(x) = g'(c)(y - x) \ge (>) 0.
$$
Remark: as @Michael Greinecker said, the reverse is not true, there are strictly increasing functions whose derivative is zero at some points, like $g(x) = x^3$.

But how is that true for a negative monotonic transformation such as g(x)=−x where g′(x)=−1?

In principle we could define the reverse notions. Say that $\tilde g$ is a "negative" monotonic transformation of $u$ if for all $x, y \in \mathbb{R}^n_+$:
$$
u(x) \ge u(y) \iff \tilde g(x) \le \tilde g(y).
$$
Notice that $\tilde g$ reverses the ranking given by $u$.
Next, we can look at functions $\tilde h: D \to \mathbb{R}$ that are strictly decreasing:
$$
x \ge y \iff \tilde h(x) \le \tilde h(y).
$$
We have the following:
Th The function $\tilde g$ is a negative monotone transformation of $u$ iff there exists a strictly decreasing function $\tilde h: D \to \mathbb{R}$ such that $\tilde g(x) = \tilde h(u(x))$.
The proof is similar to the proof for the monotone/strictly increasing case.
Remark: In economics, "negative" monotonic transformations are not really used. We would like the functions $u$ and $g$ to give the same ranking over all bundles $x \in \mathbb{R}^n_+$ then they should be monotone transformations of each other. If you use a negative monotonic transformation, you are reversing the order.


How to estimate parameters in a utility function?

as @ts_highbury mentioned above you can take the natural logarithm on both sides of Cobb-Douglas equation $$\ln(U) =a\ln(X)+b\ln(Y)$$ after that "obviously" you can notice the equation became linear in parameters (i.e linear equation), so you can use a various types of estimation methods but most famous also easy one is the Least squares method.

P.S: be careful in interpreting the results after taking the natural logarithm or you can just take the exponentiation on both side of the linear Cobb-Douglas equation (after taking natural logarithm) and you will have the original form of Cobb-Douglas equation.


Why so many suppliers of inexpensive ordinary products

There can be several reasons or combination of reasons that could be responsible. This is too broad topic to go into every minutiae detail but main well known reasons from industrial organization literature would be (see Peitz and Belleflamme: Industrial Organization: Markets and Strategies):

Small businesses might have cost advantage over large businesses. Not every industry has production function with increasing returns to scale. If returns to scale are decreasing there might be an optimal sweet spot size for a firm beyond which it does not pay to grow output more.
If there are no significant barriers to entry, markets with homogenous products will typically attract a lot of competition.
Government policy might be responsible for that as well. Many countries support small and medium size businesses with various policies. If government positively discriminates towards such businesses with lower corporate taxes, subsidies etc, then this might discourage expansion of these businesses since loosing subsidy is equivalent of facing tax on a margin and thus it would discourage businesses from expanding unless there would be sufficiently large economies of scale that would outweigh this.

Furthermore, note you are likely getting causality in a reverse. It is well established that competition among many suppliers of similar/same product leads to lower prices in most models (see overview of industrial organization literature in the above mentioned Peitz and Belleflamme). Consequently, the reason why common items are cheap is that there are many suppliers in the first place.


What would a negative externality diagram look like if the initial price is equal to 0 and then increased with tax?

Prior to the tax, the marginal private cost is zero (so the MPC curve is a horizontal line along the x axis).

The individual chooses to consume where marginal private benefit is equal to marginal private cost (i.e., where the MPB curve crosses the x axis).

A tax of 0.05 implies that the marginal private cost (i.e. the cost per unit consumed) is now 0.05. This means that we get a new MPC curve that is a horizontal line at p=0.05. The new privately optimal consumption occurs where this new MPC curve intersects the MPB curve, which results in a lower quantity consumed.


Consumer's problem in default scenario:

\begin{eqnarray*} \max_{x_1, x_2} & \ \frac{x_2}{(1+x_1)^2} \\ \text{s.t.} & \ x_1 + x_2 \leq 10 \\& \ 0 \leq x_2 \leq x_1  \end{eqnarray*}

We can easily show that the solution will satisfy $x_2 = x_1$, and therefore we can rewrite the above problem as:

\begin{eqnarray*} \max_{x_1} & \ \frac{x_1}{(1+x_1)^2} \\ \text{s.t.} & \ 0 \leq x_1 \leq 5   \end{eqnarray*}

Solving this problem we get $x_1= 1$. Consequently, $x_2 = 1$ and the utility of the consumer in optimum is $\dfrac{1}{4}$.

  Consumer's problem with a possibility of freely disposing good 1 after paying the fee:

\begin{eqnarray*} \max_{x_1, x_2} & \ x_2 \\ \text{s.t.} & \ x_1 + x_2 \leq m \\& \ 0 \leq x_2 \leq x_1  \end{eqnarray*}
where $m$ is the net income of the consumer after paying the lump-sum fee.

Solution to this problem is $\left(\dfrac{m}{2}, \dfrac{m}{2}\right)$, and the utility of the consumer in optimum is $\dfrac{m}{2}$. Comparing this with the default scenario yields that consumer is indifferent between the scheme and the default when $m = \frac{1}{2}$. Therefore, the maximum willingness to pay to dispose the good at $0$ price is $10 - \frac{1}{2} = 9.5$.


How to prove that a function is a positive montone transformation

Other people have already hinted at the correct answer. Here is a complete justification:

Note that $V(x,y) = g(U(x,y))$ where $g$ is the function defined by $$g(u) = \dfrac{1}{\beta} \cdot \log(u).$$ Assuming that only positve levels of consumption are allowed (otherwise $V$ is not well defined), $g$ is continuously differentiable on the image of $U$ and satisfies
$$
g'(u) = \dfrac{1}{\beta u} > 0
$$
Therefore, $g$ is a monotone transformation. $\blacksquare$


The U.S. Bureau of Labor Statistics (BLS rather than BoLS) publishes the results of its Job Openings and Labor Turnover Survey (JOLTS)

For example, its summary news release for May 2018, published on July 10, started with:

  The number of job openings edged down to 6.6 million on the last business day of May, the U.S. Bureau
  of Labor Statistics reported today. Over the month, hires and separations were little changed at 5.8
  million and 5.5 million, respectively. Within separations, the quits rate and the layoffs and discharges
  rate were little changed at 2.4 percent and 1.1 percent, respectively. This release includes estimates of
  the number and rate of job openings, hires, and separations for the nonfarm sector by industry and by
  four geographic regions.


Why are cost functions often assumed to be convex in microeconomics?

There are several reasons:

Didactic Reasons: Other users seem to have missed it but in your question you specify you are talking about "(introductory) microeconomics" [emphasis mine].
Well the most prosaic answer is simply that it is much easier to solve
cost minimization, or various other models when costs are assumed to
be convex.
This in itself is sufficient reason to construct problems with convex cost functions in introductory microeconomic courses. Demand and supply are not linear, yet in most textbooks and introductory problem they will be assumed to be linear. In addition, in real life demand can be sometimes even upward sloping if a good is a Giffen good, and supply can actually be downward sloping (e.g. some labor supply in some special cases depending on people's preference between consumption and leisure). Yet introductory textbooks typically show downward sloping demand and upward sloping supply (e.g. see Mankiw Principles of Economics that discusses these concepts but only briefly, or more narrowly micro introductory books such as Frank Microeconomics & Behavior).
This is to a great degree for didactic reasons. It is much better for students to first master basics with simple models and when it comes to learning about costs having nicely behaved convex cost functions with single minimum makes learning easier than having to teach cost minimization with concave cost curves. Hence, even if empirically most cost curves would be concave not convex it would be very bad teaching practice to start with concave functions (or just go for full blown realism where cost functions might be piecewise, have different concavity/convexity at different points, be ill defined somewhere etc).

Because of Decreasing Returns to Scale - This was covered in great detail by Bayesian, but let me add more arguments and also rebuff some of your arguments in the question.
First, it is not unreasonable to assume that costs are convex in a long-run. In a world of scarcity firm cannot forever increase its demand for factors of production without affecting costs of these factors or inputs as well, their prices will rise eventually (ceteris paribus). We have crystal clear evidence that wages rise in tight labor markets, or that generally speaking shift in demand to the right (ceteris paribus) rises prices.
You argue that in perfect competition models firms are assumed to be small, but that is not a good argument in this case. This is because firms are assumed to be too small in terms of their output being able to affect market price of their output so price of output can be taken as given (See Frank Microeconomics and Behavior pp 337). Perfect competition does not require price of inputs to be taken as given. In fact, firm might operate on perfectly competitive market while facing just monopolistically competitive factor market (where the firm is consumer not producer).
Next, you argue that thanks to fixed costs one firms could just continuously invest in a new factories, but this argument should be false. A fix cost by definition cannot vary with output. If firm increases output by building new factory, the cost of factory ceases to be fixed costs. In fact fixed costs primarily exist in short-run as in a long-run most costs are variable (see Mankiw Principles of economics pp 260). In a long-run as you try to build more and more factories you run into the same problems of scarcity of land, capital and labor and thus bid up their prices. In fact this is nicely visualized and explained in the Mankiw textbook with the picture below:

Empirically, we observe that many industries have decreasing returns to scale (although constant returns to scale are common as well), and increasing returns to scale are rare (although not completely uncommon). See for example: Basu & Fernald, 1997; Gao & Kehrig 2017.
Introductory texts by their nature will not deal with specific cases but more general ones. Most introductory textbooks again do not spend too much time on Giffen goods not just because modeling them would be difficult for 101 students but also because they are not very often seen (although, I am not claiming non-convex cost functions are as rare as Giffen goods).

On the Aesthetics: I think Giskard raises a valid point that there are probably many economists who assume convex costs just for mathematical elegance. However:

I think Giskard slightly exaggerates the problem and is bit too cynical about it. For sure there are economists who value mathematical elegance uber ales, but there is increasing trend in share of empirical papers (see Angrist et al 2017), even in microeconomics, and I think that a reasonable non-cynical explanation for the small share of micro empirical papers is that until very recently there was always lack of good micro data (in addition this is also due to breakdown, you can see the share of industrial organization empirical papers (that also heavily use cost functions) is quite high).

Empirically, most industries do not exhibit increasing returns to scale. While non-convex functions are definitely real (especially along some points of cost curve), empirical evidence does show that decreasing returns to scale (although constant returns to scale as well) are quite common (e.g. see Basu & Fernald, 1997; Gao & Kehrig 2017), but I think Giskard has definitely valid point that some modelers will ignore empirics for sake of mathematical elegance.

Lastly, but not least, I think mathematical elegance can explain why such assumption features heavily in some published theoretical work, I don't think it can explain why it is featured in introductory micro texts. Is really quadratic cost function $c=q^2$ mathematically elegant? I don't think so but that is probably the most commonly used cost function you will ever see in intro texts.

Regarding the Varian quote. Varian on page 67 states that he will first cover situation with fixed factor costs and later move to variable factor costs. Hence, unless I am misreading Varian I think the statement on the page 68 is made under assumption of constant factor prices. However, the explanation above by Mankiw does not assume that.


Does a fall in the wage rate always reduce hours worked?

It is more common to consider an increase in the wage rate: in that case, assuming leisure is a normal good, the income effect should increase leisure (more money available to pay for it, making it more affordable) and the substitution effect should reduce leisure (each hour worked brings greater benefit than before the increase, encouraging work), with a net effect which could go either way

So with a reduction is the wage rate, you can simply reverse this: assuming leisure is a normal good, the income effect should reduce leisure (less money available to pay for it, making it less affordable) and the substitution effect should increase leisure (each hour worked brings less benefit than before the reduction, discouraging work), with a net effect which could go either way

In the improbable situation that leisure were an inferior good, the income and substitution effects would work in the same directions, so a wage rate increase would lead to less leisure and a wage rate reduction would lead to more leisure


Bank of England decided to lower reserve interest rates to 0.25%.

This means the central bank wants to reduce the interest paid on reserved stash, discourage bankers parking too much money in central bank.

With smaller return from central bank interest rate, this is just a policies that central bank hope the bank will use the money to invest or buy bonds.

In short, this has nothing to do with borrowing rates.

Long story
Most country central bank enact rules that require banker to deposit some reserved, which the stash can be used when one of the banker face a sudden surge of withdrawal,etc. It is just a measurement to prevent banker gone rogue, loan too much money to borrower than a safety threshold.  But money in this reserved are "unproductive", thus most central bank will give some interest return.

However, in high risk condition or common rent seeking situation , bankers might do the opposite, they park more money in central bank to earn some dime, when the bond or other money making tools just too expensive or risky to them.

The speculation follow by the question are mostly flaw, perhaps IMHO, those are fallacy impost by media.

In real life.
 1. i. Moving money is costly  ii. it is difficult to find a safe haven that is cheap.
 2. In fact, the move is just pragmatic (Beware that pragmatic doesn't mean it is logical or good decision)

And to OP question title : What stops a government in partnership with its central bank from lowering interest rates and then borrowing at this lower rate to fund investments?

Interest rates DOESN'T SOLVE the problem of money supplies.
Central bank only control money supplies, it doesn't do the "borrowing"
Government borrow money by printing bonds. HOWEVER, if too much bonds are printed and too much money are used to buy the bonds, it will affect liquadation of money. So there is always some "money printing" (AKA inflation) action follows.


Supply/Demand Curve for Excise tax that's being passed 100% on to consumers

The reason you have trouble with that normal graph is beacause it contradicts your premise. Therefore, you can't use it.

I assume you mean the real burden (incidence) is passed on 100% to the consumers.

In that case you need a graph that allows for that. Your graph implies that both share the tax to some degree. To analyze a tax passed on completely to the consumers you need adequate supply and demand functions that allow for it.

As you mentioned, one possibility is a perfectly elastic supply function and a standard demand function.

Another possibility is a perfectly inelastic demand function and a normal supply function.

The reason both of these work is because the tax burden is carried by the less elastic side. For one side to carry the full burden we need something to be perfectly (in)elastic.

You have suppliers completely avoiding the tax burden. So you need suppliers to be very elastic compared to demanders. This can be done by making suppliers themselves perfectly elastic. Alternatively you can just making demanders perfectly inelastic and then by contrast the suppliers will be the much more elastic side and not carry any of the tax burden.

After you do that your tax burden area should be only one color.


From the aggregate supply equation to the definition of inflation rate

This is because for small values $x$, $$\ln x_{t+1} - \ln x_{t} \approx \frac{x_{t+1}-x_{t}}{x_{t}}.$$
This holds since the growth rate $g$ can be expressed as follows:
$$g= \frac{x_{t+1}-x_{t}}{x_{t}} \implies x_{t+1} = (1+g)x_{t}$$
taking logs we get that:
$$ \ln x_{t+1} = \ln (1+g)+ \ln x_{t} \implies \\ \ln x_{t+1} -\ln x_{t} = \ln (1+g) $$
Finally, for small values of $g$ we get that $\ln (1+g) \approx g$ thus we get:
$$\ln x_{t+1} -\ln x_{t} \approx g $$
By the same token for small values of inflation (and inflation typically will take values $\pi<0.1$) it is completely reasonable to just define inflation as $\pi = \ln P_t - \ln P_{t-1}$ (although it would be more appropriate to use $\approx$).


Who is the first one to equate "rational" with "complete and transitive preference"?

As pointed in the comments this was done by  Ragnar  Frisch.  At least Barten and  Böhm. (1982) as well as Johansen (1969) attribute these axioms to one of these two publications:

Frisch, Ragnar (1926). "Sur un problème d'économie pure [On a problem in pure economics]". Norsk Matematisk Forenings Skrifter, Oslo. 1 (16): 1–40
Frisch,(1926). "Kvantitativ formulering av den teoretiske økonomikks lover [Quantitative formulation of the laws of economic theory]". Statsøkonomisk Tidsskrift. 40: 299–334.

Also I think this question has a misconception.
Von Neumann and Morgenstern (1944) indeed in their Theory of Games and Economic Behavior equate maximization of expected utility $E(u)$, with rational behavior.
However, note that the $E(u)$ of vNM is already based on axiom of transitivity and completeness. The issue here is that vNM were not just deriving utility of rational person. They were deriving cardinal utility of rational person.
The assumption of transitivity and completeness is only condition for preferences to be rational in general (see MWG pp 6.).


The efficient frontier in mean variance criterion

Using $\mathbb E$ for the expected value symbol, $E_R$ and $V_R$ for the mean and the avriance of returns $R$, for a utility function of the form
$$U(R) = \ln(1+R)$$
the second-order Taylor expansion gives (ignoring the remainder)
$$U(R) \approx \ln(1+E_R) + \frac{1}{1+E_R} (R- E_R) -\frac{1}{2(1+E_R)^2}(R-E_R)^2.$$
Then
$$\mathbb E[U(R)] \approx E_R + 0 - \frac{V_R}{2(1+E_R)^2}. $$
We then have
$$\frac{\partial \mathbb E[U(R)] }{\partial V_R} = -\frac{1}{2(1+E_R)^2} < 0$$
so decreasing the variance always increases expected utility, while
$$\frac{\partial \mathbb E[U(R)] }{\partial E_R} = 1 + \frac{V_R}{(1+E_R)^3} <0$$.
The reason why this derivative is always positive, is that even with negative returns, by construction "returns" cannot fall below $-100\% = -1$.
So increasing mean returns always increases expected utility.
I don't see where the problem is. What am I missing?


Yes, this is consumption.

There are 3 ways to calculate GDP, income effect, value added effect, and value of final products.

The simplest to understand is probably the income effect where the value of all wages and profits at every stage of production are added together to find final GDP. Since the household has paid rent and paid the bank for a service both payments count towards the income of both the landlord and the bank.

If the household in your example borrowed $1000 to spend on rent at 2% interest and then paid back that $1020 next year it would be a $1020 net gain for GDP with $1000 going to the landlord's income and $20 going to the bank's income for their service of lending out money. If the household defaulted on its debt then the bank would report a loss of $1000 on their income which would cancel out the $1000 gain on the part of the landlord.


When the base year changes, does it affect data at current prices as well?

Yes,

If you initially had GDP data at some base year $Year\;x$ for some $Year\;y$

$$I_{\frac{Year\;y}{Year\;x}} = \frac{GDP_{Year\;y}}{GDP_{Year\;x}} \times 100$$

If your base year has now changed to some base year $Year\;a$ for some $Year\;b$
, $a > x$, then you should convert the GDP data from year $b$ onwards to reflect the initial base year, $x$.

$$I_{\frac{Year\;b}{Year\;x}} = I_{\frac{Year\;b}{Year\;a}} \times I_{\frac{Year\;a}{Year\;x}} $$

Further proof,
$$\frac{GDP_{Year\;b}}{GDP_{Year\;x}} = \frac{GDP_{Year\;b}}{GDP_{Year\;a}} \times \frac{GDP_{Year\;a}}{GDP_{Year\;x}} $$

This way you will be able to get a continuous and consistent series based on some base year, $x$. If there were multiple base years, the data would be meaningless since it would reflect on changes with regard to different periods of time - ie, you would not be able to draw a definite conclusion.

An index compares the value of the current year against the value in the base year.
In this case, you might be looking for the current GDP value for every year from your data-set. As an example, if $$I_{\frac{2001}{2000}} = 110, \; and \; GDP_{2000} = 200, \; then \; GDP_{2001} =220.$$

When comparing data with changing base years, for eg,
$$I_{\frac{2001}{2000}} = 110, \;I_{\frac{2002}{2001}} = 105,$$
You can see that this does not mean the GDP in 2002 is $10\%+5\%$ higher than 2000. Instead, the actual GDP =

$$I_{\frac{2002}{2000}} \times GDP_{2000} = I_{\frac{2002}{2001}}\times I_{\frac{2001}{2000}} \times GDP_{2000} = 115.5 \times GDP_{2000} = 231$$
or you can break this down to
$$GDP_{2000} = 200$$ $$GDP_{2001} = GDP_{2000} \times I_{\frac{2001}{2000}} = 220$$ $$GDP_{2002} = GDP_{2001} \times I_{\frac{2002}{2001}} = 231 $$
 Thus this is why you have to change your indexes to reflect on the same base year, since your calculated values of actual prices will differ.


How can anyone possibly predict how the price of anything (stocks, Bitcoin, fiat, etc.) will change without "insider knowledge"?

How can anyone possibly predict how the price of anything (stocks, Bitcoin, fiat, etc.) will change without “insider knowledge”?

You can do that by forecasting based on data.  Exact forecasting model will depend on exactly what you want to forecast but most economic variables can be forecasted with a degree of accuracy that is generally higher than just 50/50 random chance (see Elliott & Timmermann 2013 Handbook of Economic Forecasting).
Some series are more difficult to forecast,  for example it is very difficult to beat random walk when forecasting forex exchange rates (Kilian & Taylor 2003), but even in those cases it is not considered impossible only exceedingly difficult.
When it comes to bitcoin forecasting specifically you might want to inquire more about exact models used there at either Quantitative Finance or Bitcoin stack exchange, but just casually search of google scholar reveals plethora of models for forecasting bitcoin returns or trends in Bitcoin prices. For example, Xiaolei et al (2020) report relatively accurate forecasting of Bitcoin value trends with Light Gradient Boosting Machine.
In addition, you would likely want to have not just once forecasting model but multiple ones and analyze their forecasts together.


What would the determinant of the input-output matrix represent? What is the intuition?

Your conjecture seems unlikely. For example the unit matrix would show an economy where no new goods can be produced (it takes 1 unit of something to make 1 unit of that same thing). Yet the determinant of this matrix is 1. If we were to multiply the $3 \times 3$ unit matrix by say 10, the new determinant would be a 1000. But the economy did not get better, if anything it got worse.


What is the necessity of the FDIC in the US banking system?

You provide a generally correct impression of history (i.e., there were lots of runs on banks by depositors in the US until federal deposit insurance was established— a good history is provided by Gorton (2012)), and then ask three questions: why isn't deposit insurance provided through the private sector, is it because the free market equilibrium premiums would be too high, and if so, does this mean that the FDIC premiums are set too low?

To answer, we should first back up to the reason for deposit insurance generally, which is twofold:

Monitoring costs
Bank runs are a stable equilibrium

Then, we can discuss the reasons for government provision of deposit insurance specifically, which are:

Externalities
Monitoring costs (still)
Moral hazard
Systemic crises

The monitoring costs issue is a real one, and has been discussed in the literature by Diamond (1984), among others. It's derived from a prinicpal-agent issue: foremost, depositors want bank management to make safe investments that ensure that their deposits can be returned with certainty, while bank management is encouraged by its equity holders to take on as much risk as possible (for a general discussion of this principal-agent problem in the theory of the firm, see Hart (2001)). In short, management isn't looking out for depositors, and it's really expensive to ask all depositors to be aware at all times about the state of a bank's investments. This is additionally difficult as a practical matter, as bank balance sheets are not paragons of transparency.

Further, as Diamond and Dybvig (1983) showed, even if one ignores monitoring costs, bank runs are one stable equilibrium of a system without deposit insurance. The logic for this is described in your question: if an institution that holds illiquid assets is forced to liquidate them prior to maturity and within a short time period, it may face solvency issues as a result of that forced liquidation. This is known as the "solvency-liquidity nexus" and is discussed in a context including wholesale funding, as opposed to only deposit funding, in Pierret (2015). As a result, if depositors believe that a bank may face a run, it's rational for depositors to run on that bank even if it is fundamentally solvent. In fact, if one considers that a depositor's only recourse (in the event that monitoring reveals that management is taking on too much risk) is to withdraw his or her deposits, it should be clear that zero-cost monitoring of banks does not eliminate the possibility of runs.

So, why government-provided deposit insurance?

Perhaps the simplest reason, but the one that is least discussed in the literature, is that bank failures tend to occur together, as widespread episodes with significant externalities. For this reason alone, it's easy to understand why society (and on behalf of society, government) would have an interest in ensuring that they don't occur. But that doesn't really answer your question, which is, why doesn't the private sector handle the problem of bank runs through private insurance (and, contra another answer to this question, the private sector has never handled this issue well) so that the problem never becomes one that the government must address?

Well, a second reason would be that depositors would still face monitoring costs, as the value of the liquidity guarantee provided by a private insurer would only be as good as that entity's capacity to make good on its guarantees, which would require monitoring in turn. So that problem would still exist.

A third reason, as Diamond and Dybvig (1983) discussed in their conclusion, is that an insurer must be able to regulate the bank it's guaranteeing, because otherwise, the bank managers again have an incentive to make riskier choices:

  The riskless technology used in the model isolates the rationale for
  deposit insurance, but in addition it abstracts from the choice of bank
  loan portfolio risk. If the risk of bank portfolios could be selected by a
  bank manager, unobserved by outsiders (to some extent), then a
  moral hazard problem would exist. In this case there is a trade-off
  between optimal risk sharing and proper incentives for portfolio
  choice, and introducing deposit insurance can influence the portfolio
  choice. [...]
  Introducing risky assets and moral hazard would be an interesting
  extension of our model. It appears likely that some form of government
  deposit insurance could again be desirable but that it would be
  accompanied by some sort of bank regulation. Such bank regulation
  would serve a function similar to restrictive covenants in bond indentures.
  Interesting but hard to model are questions of regulator "discretion"
  which then arise.

Given this need for a regulator where deposit insurance exists, it makes quite a bit of sense to build a legal framework around a single regulator-insurer, which we've done in the US in the form of the FDIC and related entities (including Federal Reserve supervision and the OCC). This is largely because market-based regulation of banks by private deposit insurers (such as by allowing insurers to cancel coverage if banks breach certain risk limits) would be difficult (at best) to structure without the broad legal authority the government has to seize control of an institution that breaches risk limits and make necessary changes. If private deposit insurers can't rescind coverage, their ability to discipline banks is limited; if they can, then banks face a likely run if their coverage is rescinded, which solves nothing. It is also, of course, difficult as a practical matter because of the aforementioned opacity of bank balance sheets.

Fourth and finally, as discussed by Caballero (2009a, 2009b), bank failures don't occur in isolation; institutions tend to be impaired at exactly the same moment that other institutions and markets are impaired. While you correctly note that a form of partial insurance is in use (in the form of credit-default swaps) today, it's worth considering what happened during the crisis, when outstanding CDS contracts caused the failure of the insurer AIG.

This leads us to a reasonable question: what would a private deposit insurer look like? We know at least two things. One, it would have to be very well-capitalized, for if it were not, the Diamond-Dybvig "run" equilibrium would still exist. Two, it would have to hold its significant reserves in the form of information-insensitive "safe" assets, most likely government-guaranteed securities, or else it may not be able to liquidate those assets under the conditions that lead to bank failures (which can be accompanied by fire sales and other phenomena that impair asset values).

This leads us to an answer to your second question, which is yes, this would be relatively expensive, as you'd have to insure the whole stack of deposits, rather than allowing the government to provide a tail risk guarantee as it does with the Deposit Insurance Fund (if the FDIC's DIF is run to zero during a crisis, the US government will make up the difference). That tail risk guarantee provided by the government is a conditional put— it costs nothing until the DIF needs a bailout, at which point the government can fund it by issuing debt at the very low rates that prevail (for safe assets) during times of financial distress.

To answer your third question, the DIF has never required a bailout, and its rates are adjusted to prevent it from holding reserves that are too great, so— the FDIC is probably charging an actuarially fair rate over a long time horizon, though the rate may be too high or too low at certain points in time.


Calculating the impact of a per-unit Subsidy given to Consumers?

This is a great question which economists have a pretty good answer to. Let's first suppose the market is in an equilibrium with no intervention at price $P^*$.

If there is a tax of value $t$ we have a new equilibrium with a consumer price $P^*_c$ and producer price $P^*_s$. These prices surely do not equal each other any more, and the difference of the two is not necessarily equal to the value of the tax.

You can think of a subsidy as a negative tax.

The change in consumer price due to a marginal tax or subsidy is called the pass through rate, $\rho$. For a sufficiently small tax, the relationship is $P^*_c=P^*+\rho\cdot t$.

Clearly, if the pass-through rate is one the consumer completely pays for the tax. If it is zero, the consumer does not at all pay for the tax. It is not impossible for $\rho$ to be negative or greater than one.

In perfectly competitive markets the pass through rate depends on the elasticity of demand, $\varepsilon_D$, and the elasticity of supply, $\varepsilon_S$. $$\rho=\frac{1}{1+\frac{\varepsilon_D}{\varepsilon_S}}$$

From this we can see that the pass through rate is larger when demand is more inelastic, and smaller when supply is more inelastic. This is where we get the adage "the more inelastic side of the market bares the burden of a tax."

For pass through to be equal to one, we either need (a) perfectly inelastic demand or (b) perfectly elastic supply. A lot of the time (b) can be satisfied by assuming constant marginal cost of production.

For imperfectly competitive situations, the pass through rate depends on firm conduct as well as the curvature of demand. I suggest Weyl and Fabinger "Pass through as an economic tool" 2013, Journal of Political Economy, publication for the details.


Do corporations outside of the Eurozone issue bonds in Euro?

Answer by @Wecon in the comments:

  With respect to your first question: they certainly do. The BIS quarterly review of sept. 2015 (p.9) even shows that "firms in the United States increasingly issued euro-denominated debt to benefit from the low borrowing costs." This may already be a set-up for answering your second question: its (relative) importance may be highly dependent on economic circumstances


What trick can be used to calculate mixed-equilibria?

The "trick" is to identify an equilibrium action where you know (or can pin-down) the payoff. You can then impose indifference between this action's known payoff and the payoffs from other actions (which will depend on the mixing probabilities) to determine those equilibrium mixing probabilities.
A nice example can be found in a Varian's "A Model of Sales". Let me describe a simple version of that model before discussing what it teaches us about finding continuous mixed equilibria.

Set up
Two firms each set a price, $p_i$, for a homogeneous product they product at zero cost. There is a continuum of consumers, each willing to pay up to $v$ for the product.
A fraction $a$ of the consumers (call them "shoppers") will buy from whichever firm sets the lowest price (breaking ties randomly). A fraction $\frac{1-a}{2}$ of the consumers (call them "1's captives") always buy from firm $1$ (as long as $p_1<v$). The final  $\frac{1-a}{2}$ of the consumers (call them "2's captives") always buy from firm $2$ (as long as $p_2<v$).
You should be able to quickly check that this game has no pure strategy equilibrium (think about why we can't have an equilibrium with $0<p_1<p_2$, $0<p_1=p_2$, or $0=p_1=p_2$).
Thus, we look for a mixed strategy equilibrium where prices are drawn from the distribution $F$. Let the support of $F$ (i.e., the range of prices over which firms mix) be $[\underline{p},\overline{p}]$.
No firm will ever choose $p_i>v$ because it then serves no consumers, but could earn a profit of $v(1-a)/2$ by serving its captives at a price of $v$. Thus, $\overline{p}\leq v$.
A firm's profit if it sets price $p$ is
$$\frac{1-a}{2}p+ap[1-F(p)].$$
The first term is the profit from captives (who the firm serves for sure). The second term is the profit from shoppers, who are served only if the rival firm sets a higher prices (which happens with probability $1-F(p)$).
Finding the mixed-strategy equilibrium
We know that if the firm is to mix over prices $p$ and $\overline{p}$ then it must be indifferent:
$$\frac{1-a}{2}p+ap[1-F(p)]=\frac{1-a}{2}\overline{p}+a\overline{p}[1-F(\overline{p})].$$
Notice that if we could eliminate the $\overline{p}$ terms from the right-hand side then we would have an equation that could be solved for the equilibrium strategy, $F$.

The next part is the key to being able to find the equilibrium, so I'll put it in a box:
Let's think about what happens when a firm sets $p_i=\overline{p}$. Because $\overline{p}$ is the highest price ever charged in equilibrium, we know that the firm never serves the shoppers at that price (i.e., $F(\overline{p})=1$). Moreover, since the firm serves only its captives who will buy at any price below $v$, it must be the case that $\overline{p}=v$.

We therefore know that:

$\overline{p}$ is in the support of $F$, meaning firms must be indifferent between $\overline{p}$ and any other price
$\overline{p}$ results in profit $\frac{1-a}{2}\overline{p}$
$\overline{p}=v.$

Putting these facts together with the indifference condition, we have
$$\frac{1-a}{2}p+ap[1-F(p)]=\frac{1-a}{2}v,$$
which implies
$$F(p)=1-\frac{\frac{1-a}{2}(v-p)}{ap},$$
which is our mixed strategy equilibrium. The only part of the equilibrium we haven't solved is the value of $\underline{p}$. But this is easily found by solving $F(\underline{p})=0$ for  $\underline{p}$ now that we know $F(\cdot)$.

Final remarks
The trick used in the above example is pretty useful more generally.
Unless the equilibrium strategy, $F$, has an atom, we know by definition that $F(\underline{p})=0$ and $F(\overline{p})=1$. Thus, if we can use economic reasoning to determine the value of either $\overline{p}$ or $\underline{p}$ (as in the box above) then we can eliminate all of the terms that depend on $\overline{p}$ (or $\underline{p}$) on the right hand side of our indifference equation and then solve for $F$.


Question about Gilboa and Schmeidler "A Maxmin Expected Utility With Non-unique Prior" 1989

Notice, $z$ is in $Y$, and so it is a lottery. Lemma 3.1 uses the standard vNM techniques to identify $u: Y \to \mathbb{R}$. Now, $J$ is defined as a function from $L_0$ to $\mathbb{R}$, so the statement $J(z) = 0$ is via the natural identification between $Y$ and constant acts, $L_c$.

Condition (ii) of Lemma 3.2 says that $J(y) = u(y)$ for all $y \in L_c \cong Y$. So, $u(z) = J(z) = 0$, or, in the other notation, $u \circ z = 0$.


What, in the most literal sense, are eurodollars?

They actually literary are a deposit accounts of US dollars outside US. Also no those foreign banks will not generally have accounts at Fed. In fact one of the reasons why eurodollar deposits pay higher interest than a dollar deposit in US as they are more risky since they are neither subject to FDIC insurance and those banks will not get liquidity from Fed if there is run on them.

After all, to a commercial bank, electronic deposits are just a local accounting fiction, the movement of which gets accounted for at the end of the day by a net transfer of central bank reserves.

But there is no 1:1 correspondence between amount of dollars in circulation and Fed reserves. You can have a look at Fred data for volume of reserves and money stock $M2$ and you can see that at any point in time money stock is several orders of magnitude larger than volume of reserves.
If you want to know how eurodollars work on conceptual level then have a look at Niehans, & Hewson (1976).
Actually a good primer was also provided by Milton Friedman (1971). Friedman, also includes concrete example.


Set $y \equiv c+\varepsilon$. So $y$ represents changed consumption around and "close" to a given level $c$. Take a 2nd-order Taylor expansion of the function $E[u(y)\mid c]$  around $c$, which is treated as fixed since we condition on it :

$$E[u(y)\mid c] \approx E[u(c)\mid c] + E[u'(c)(y-c)\mid c] + E[\frac12u''(c)(y-c)^2\mid c]$$

But $y-c = \varepsilon$
so

$$E[u(y)\mid c] \approx E[u(c)\mid c] + E[u'(c)\varepsilon\mid c] + E[\frac12u''(c)\varepsilon^2\mid c]$$

Due to the conditioning, and the independence of $\varepsilon$ from $c$ the expected value distributes and applies only to $\varepsilon$:

$$E[u(y)] \approx u(c) + u'(c)E[\varepsilon] + \frac12u''(c)E[\varepsilon^2]$$

Since we assume

$$E[\varepsilon] = 0 \Rightarrow {\rm Var}(\varepsilon)=E[\varepsilon^2] =1$$
we obtain

$$E[u(c+\varepsilon)] \approx u(c) + \frac12u''(c)  \tag{1}$$

Now consider $u(c-\frac 12 ARA)$, with $ARA \equiv -\frac{u''(c)}{u'(c)}$, and take a 1st-order Taylor expansion in this case, again around $c$:

$$u\left(c-\frac 12 ARA\right) = u(c) + u'(c)\cdot (c- \frac 12 ARA - c) = u(c) - \frac 12 u'(c)\cdot ARA $$

Using the definition of $ARA$ to replace it we obtain

$$u\left(c-\frac 12 ARA\right) \approx u(c) - \frac 12 u'(c)\cdot\left (-\frac{u''(c)}{u'(c)}\right)$$

$$\Rightarrow u\left(c-\frac 12 ARA\right) \approx u(c) + \frac 12 \cdot u''(c) \tag{2}$$

The right-hand sides of equations $(1)$ and $(2)$ are equal therefore, approximately, so  are their left-hand sides or,

$$E[u(c+\varepsilon) \mid c] \approx u\left(c-\frac 12 ARA\right)$$

which is valid as long as $ARA$ is defined the way it is. QED.

If the variance of the gamble is not unity but $\sigma^2 \neq 1$, then the more general equation is

$$E[u(c+\varepsilon) \mid c] \approx u\left(c-\frac {\sigma^2}2 ARA\right)$$


Help with Income Elasticity Exercise in Becker's Economic Theory

The precise question the OP asked was what does $K_j N_j$ represent. As Alecos' response says, the statement  $\sum_j K_j N_j = 1$, i.e. the weighted average of the income elasticities equals one, means that if total income goes up by one percent then total expenditures/consumption also increases by one percent. The total market expenditure for some goods (with income elasticity of demand > 1) will go up more than one percent, for some goods (with income elasticity of demand between 0 and 1) will go up but less than one percent, and for some goods (with income elasticity <0) will go down. But taking all goods together, expenditures will go up one percent when total income goes up one percent.
(Becker makes this point in the context of an individual rather than the complete market on p. 16 and footnote 2 of the text.)

Now I provide the solution to the exercise statement:

Start with the notion that when individual $i$'s income $y_i$ increases by one percent, his demand for good $j$ increases by $\eta_{ij}$ percent. What effect does this have on total market demand for good $j$? It depends on what fraction $w_{ij}$ of the total demand for good $j$ individual $i$ accounts for. Total market elasticity for good $j$, $N_j$ is the weighted sum of the increases in demand of each individual: $N_j = \sum_i w_{ij} \eta_{ij}$

Algebra can get this in terms of $k_{ij}$ and $\eta_{ij}$ (and $y_i$) as the exercise statement requests.

First find an expression for $w_{ij}$, the fraction of total demand (or expenditure) of person $i$ for good $j$: Letting $q_{ij}$ denote $i$'s demand for $j$, this fraction is
$$w_{ij}=\frac{q_{ij}}{\sum_i q_{ij}} = \frac{k_{ij} y_i}{\sum_I k_{Ij} y_I}$$
 where the second equality comes from the definition $k_{ij} \equiv \frac{p_j q_{ij}}{y_i}$.

Then $N_j = \sum_i f_{ij} \eta_{ij} = \sum_i \frac{k_{ij} y_i}{\sum_I{k_{Ij} y_I}} \eta_{ij}$. The weight/fraction $w_{ij}$ is increasing in $y_i$: for a given $k_{ij}$ a higher income person's elasticity is weight more. (Actually, what matters for how heavily and individual's elasticity is weight is not just income but actually $k_{ij} y_i$, their expenditures on good $j$, relative to the total market expenditures on good $j$. This makes intuitive sense: the elasticity of a heavy consumer of a good will have a bigger effect on that good's market elasticity than that of a light consumer, regardless of income.)

The fraction of total market expenditures spent on good $j$ is $K_j = \frac{\sum_i k_{ij} y_i}{\sum_i y_i}$.

So with some simple algebra we have $K_j N_j = \frac{\sum_i k_{ij} y_i \eta_{ij}}{\sum_i{y_i}}$.

With more algebra and using the identity $\sum_j k_{ij}\eta_{ij} = 1$ (see p. 16) you can confirm that $\sum K_j N_j = \frac{\sum_i (y_i \sum_j k_{ij}\eta_{ij})} {\sum_i{y_i}} = 1$.