Actually they're saying that, when cash is available, people need not to deposit their money into the bank, and hence is "guaranteed" (not taking into account risks associated with holding on to cash, like fire or getting stolen) a 0% interest rate.

Having cash, therefore, means that if the deposit rate goes into the negative territory (you're charged to deposit your money), then people could simply stock cash at home. As a result, the interest rate could not go much into the negative territory.


What are the assumptions behind calculating the market equilibrium from supply and demand curves?

Public information about prices--not necessarily about costs or willingness to pay--is indeed the key here.

If S5 is willing to sell to B5 at $1, she must also be willing to sell to any other buyer at this price. Given the low price, there would be an over-demand for the unit sold by S5, and a resulting upward pressure on the price asked by S5.

Similarly, if B1 is willing to pay S1 $5 for a unit, he must also be willing pay the same amount to the other sellers for one unit. Then Sellers with lower costs will rush to compete for B1's business, thereby creating a downward pressure on the amount bid by B1.

The process settles when the market reaches equilibrium described by your second scenario, where three units are traded at $3/unit.

In Vernon Smith's canonical market experiment (and countless replications thereafter) wherein buyers and sellers with varying values/costs trade in a double auction,  the equilibrium outcome predicted by theory, rather than the assortative matching outcome described in your situation 1, is observed.

This is also related to the Law of One Price.


Optimal allocation refers to whom the seller awards the item, in order to maximize revenue. In a Vickrey auction, it is a Nash equilibrium for everyone to bid their valuations. This is stronger than a Bayesian Nash equilibrium, which is the solution concept in Bayesian games (which include auctions). The Bayesian Nash equilibrium is the solution concept in which every player seeks to maximize their expected utilities, subject to their respective beliefs about the world. In the Vickrey auction, agents' beliefs about the world do not really matter.

The goal of the Vickrey auction is to illicit truthful responses from the agents. That is, they bid their precise valuations. The first-price auction is revenue-equivalent to the Vickrey auction; however, agents do not truthfully bid their valuations.

In particular, if we have a symmetric $n$-player auction where players' valuations are drawn from a probability distribution $F([0, \omega])$ with density $f$, we have the expected revenue for the seller as:

$$n \cdot \int_{0}^{\omega} (n-1) \cdot y \cdot f(y) \cdot F^{n-2}(y) \cdot (1-F(y)) dy$$

Combinatorially, we select the highest two bidders in $n(n-1)$ ways, since order matters. Suppose the second highest bid is $y$. This occurs with probability $f(y)$. The remaining $n-2$ players bid less than $y$ with probability $F^{n-2}(y)$. The highest bidder has valuation greater than $y$ with probability $1-F(y)$. Each of these selections are independent; so by rule of product, we multiply. As this is an expected value, we include $y$ in each product. By rule of sum, we add up over each possible value of $y$ (hence, the integral).


Yes.

Example. In 2017, a country's GDP was \$100 and its population was 100.

In 2018, its GDP was \$110, its population was 105, and the price level rose by 3%. Then:

2017 GDP per capita was $\$100 \div \ 100 = \$1$.
2017 Real GDP per capita (in 2017 prices) was also $\$1.$
2018 GDP per capita was $\$110 \div \ 105 \approx \$1.048$.
2018 Real GDP per capita (in 2017 prices) was $\$1.048 \div 1.03 \approx \$1.017$.


Markets that are linked to options prices

what are the characteristics of markets that could be predicted with options prices of the present?

There aren't any. Options might be descriptive of the present, today's expectations, and of today's uncertainty; but they cannot be predictive of the future.

Putting aside the intrinsic value of in-the-money options, option prices reflect traders' expectations about the future price of the underlying asset. But expectation does not mean determination or prediction of the future. In fact, traders try to factor volatility in their valuation of options, which further weakens the notion that option prices are predictors. Volatility represents uncertainty.

Therefore, it is false that option prices predict what the future price of the underlying asset will be.

The lecturer is wrong in the distinctions he stated. Had he substantiated his rationale for the distinctions, we would be able to either follow his point or identify where exactly his argument goes astray.

What could happen is that an option or its underlying asset are under- or overvalued, thereby creating arbitrage opportunities while price adjustments take place. However, that has nothing to do with options prices predicting any future prices.

  While I understand the second example, I was to surprised with the
  other two because no reason was given. If anything, shares are
  controlled by the society, and therefore, should have been easier to
  control compared to say food, which is susceptible to natural
  calamities.

Whether an asset is controlled by either nature or society has no bearing on this matter because nature and society are not even disjoint factors, and they are not mutually exclusive factors.

Trade and production of rice are subject not only to nature calamities, but also to acts of man: strikes, increase of technology, changes in work culture, sudden news that rice cures a chronic disease, and so forth.

Conversely, the stock price of an oil company might go down as a result of nature calamities and thus be unrelated to the acts of man.


Interchangeability between knowing an event obtains with probability 1 and knowing an event obtains with absolute certainty?

If you take the standard Aumann model, but allow for states occuring with probability zero, you can taje some nonempty event $N$ such that for the agents prior $p$, one has $p(N)=0$. Let $K\cap N=\emptyset$ and $p(K)>0$. If the agent knows $K\cup N$ but not $K$, then she assigns probability $1$ to $K$, even though she does not know $K$.


What were "scissors prices" (China, 1950s)?

Under Mao the Chinese government was a planned economy, prices were set by the party, not by the free market. Procurement prices of agricultural produce, what the government payed to farmers, were lower than consumer prices of the same produce, what the government got from selling the produce to the consumers. The difference in prices becomes state revenue. The widening of the gap between the two prices is likened to the opening of a pair of scissors, hence the name.


Can you recursively forecast one variable using two variables?

If it is possible to have actual values of the other variables for example let’s say variable $y$ is always provided by statistical bureau with 5 year lag but for second variable $x$ the values are always up to date you can use them.

However, if you have data for both $y$ and $x$ up to date and you want to make forecasts for next year and further in the future you will have to do auxiliary forecasts for $x$ as well.

When you are doing some out of sample evaluation and you want to make a thought experiment of how the model forecasts $y$ conditional on knowing what was the $x$ then you can do that, but such evaluation won’t be indicative of actual performance when in real life both future $y$ and $x$ is unknown. This is the biggest downside of multivariate forecasting.

However, you can always make an auxiliary forecast for the $x$. Let’s say with an auxiliary AR model for the $x$ variable.


So I figured I'd post the proof for the third condition now that I know it.

  We want to show that if $U$ and $V$ are linear representations of $\succeq$ on  $\mathscr{L}$, then $\exists \ \beta > 0, \gamma \in \mathbb{R}$ such that $V = \beta U + \gamma$

Case 1: All the lotteries are all equally preferred.

If $\forall \ L_1, L_2 \in \mathscr{L}$ we also have $L_1 \sim L_2$, then $U$ is constant.

$V = \beta U$ holds $\forall \beta > 0, \gamma  > 0$

Case 2: All the lotteries are not equally preferred.

$\forall L \in \mathscr{L}$ where $L \neq L_b, L_w$
$$L_b \succeq L \succeq L_w$$
$$L_b \succ L_w$$

For a lottery $L \in \mathscr{L}$,

$$\lambda_L \equiv \frac{U(L) - U(L_w)}{U(L_b) - U(L_w)}$$

for $\lambda \in [0,1]$

$$1 - \lambda_L \equiv \frac{U(L_b) - U(L)}{U(L_b) - U(L_w)}$$

So now we can write $U(L), \ \forall L \in \mathscr{L}$ as

$$U(L) = \lambda_L U(L_b) + (1-\lambda_L)U(L_w)$$
$$\implies L \sim \lambda_L L_b + (1-\lambda_L)L_w$$

For $V$ to also represent $\succeq$,

$$V(L) = \lambda_L V(L_b) + (1-\lambda_L)V(L_w)$$

$$V(L) = \bigg[\frac{U(L) - U(L_w)}{U(L_b) - U(L_w)}\bigg] V(L_b) + \bigg[\frac{U(L_b) - U(L)}{U(L_b) - U(L_w)}\bigg] V(L_w)$$

$$V(L) = \frac{U(L)V(L_b) - U(L_w)V(L_b)}{U(L_b) - U(L_w)} + \frac{U(L_b)V(L_w) - U(L)V(L_w)}{U(L_b) - U(L_w)}$$

$$V(L) = \frac{V(L_b) - V(L_w)}{U(L_b) - U(L_w)} \cdot U(L) + \frac{U(L_b)V(L_w) - U(L)V(L_b)}{U(L_b) - U(L_w)}$$

Define $\frac{V(L_b) - V(L_w)}{U(L_b) - U(L_w)} \equiv \beta$ and $\frac{U(L_b)V(L_w) - U(L)V(L_b)}{U(L_b) - U(L_w)} \equiv \gamma$

Thus $V(L) = \beta U(L) + \gamma$, $\forall L$

$\square$


What does it mean to have the marginal cost equal to 7?

It means that, at the current size of your production capabilities, if you want to produce one more unit of your product, you will have to spend 7 Euro.

This, MC value is a quantitative description of the cost you pay if you want to adjust your initial production plan to a larger output and it is determined by your production capacities.

Here is an example:

You have a bus company and your service is to offer a one way trip from point A to point B, every day at a given hour. Now, your buses have 50 seats and your total cost for the trip is 100 Euro (fuel, road taxes, etc). In addition to that your licence says that  you have to provide additional bus if the passengers are more than 50. In this case, if you have 49 passengers and in the last minute there is a reservation for one more seat, your marginal cost is effectively zero, however, if there are two reservations in the last minute and the total number of passengers is now 51, the 51th passenger's marginal cost is 100 Euro, because you need a second bus only for him.


When a country adopts the Euro, what happens to its debt?

Yes, upon the introduction of the euro on January 1, 1999, all debt (indeed, all nominal contracts) in participating countries was converted from national currency to euros at a legally defined conversion rate. See this press release from December 31, 1998, which states:

  In accordance with Article 109l (4) of the Treaty establishing the
  European Community, the irrevocable conversion rates for the euro were
  today adopted by the EU Council, upon a proposal from the Commission
  of the European Communities and after consultation of the European
  Central Bank (ECB) for effect at 0.00 on 1 January 1999 (local time).
  In compliance with the legal framework for the use of the euro, the
  irrevocable conversion rate for the euro for each participating
  currency is the only rate to be used for conversion either way between
  the euro and the national currency unit and also for conversions
  between national currency units.

  The euro conversion rates are the following...

For the particular case of Greece, which joined the euro on January 1, 2001 instead, the conversion rate with the drachma was apparently set at 1 euro = 340.750 drachma by this regulation in June 2000.


The government is issuing massive debt for coronavirus relief - why aren't interest rates rising?

One standard way of determining the fair value of a bond yield (for a country that controls the currency it borrows in) is to say that it is the sum of two factors.

The expected (Geometric) average of the overnight rate (that is set by the central bank) over the life of the bond. This equals the expected return of holding money market instruments instead of the bond.
A term premium, which is compensation for taking the returns risk associated with a bond instead of investing in money markets.

Under this view, the lack of a rise in bond yields is easily understood - almost no market participants expect rate hikes.

It should be noted that there are a variety of academic disagreements with the previous explanation. However, if you browse the FRED database, you will see that during recessions, bond yields fall and deficits rise. This is because the rate expectations view is much closer to reality than supply/demand arguments based on deficits being large.

There should be other related questions on this website, but I did not spot any that were directly useful.

FRED 10-year yield
FRED Federal deficit


Two-good economy, can we tell if the demand of good 1 rises or falls when the price of good 2 rises (price of 1 is constant)?

Check out the demand functions for three cases:

CES Utility
Leontief Utility
Cobb-Douglas Utility

You should be able to see that the first case gives you that demand for good 1 increases, the second case demand for good 1 decreases, and in case 3 it is unchanged.


Book recommendations: Introduction to economics

It's great that you're developing an interest in economics. I would suggest Mankiw's Principles of Economics to start with. I believe it meets both your requirements and covers the two major areas of economics, microeconomics and macroeconomics, so you would get a decent overview of this field of study. Good luck!


Why does central banks lowering interest rate incentivize more borrowing?

Banks lend because it is profitable for them to lend out money.  Banks do become less profitable in low interest environment, but they are still empirically able to maintain profitability, although there is evidence that goes hand in hand with some increase in risk taking (e.g. see Bikker & Vervliet 2018). Consequently, the direct answer to your question is that banks expand their lending because it is still profitable for them to do so.
Additionally, if central banks would be really determined they could expand money supply by open market operations (e.g. purchase of government bonds) and various other channels, so if a determined central bank would want to expand money supply it can do that whatever the private banks are doing.
Furthermore, inflation does not necessary means that the bank will become worse off. If there is 2% inflation in an economy but bank's profits increase by 5% during the same period then a bank is still better off in real terms.
Lastly, there is not necessarily 1:1 correspondence between increase in money supply and inflation.  For example, even in a simple money market equilibrium model (which determines price level - increase of which is inflation) given by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY$$
Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ output.
Solving for price level and log-linearizing we get:
$$\ln P = \ln M + \ln V - \ln Y $$
Consequently, ceteris paribus, if $M$ increases by $1\%$ and $Y$ increases also by $1\%$ then the net effect on price level will be zero (and consequently inflation will also be zero). Even in this simple model you will only get inflation if $ \ln M + \ln V - \ln Y > 0$ so when bank increase money supply it won't necessarily result in an equivalent inflation. Furthermore, in more complex models inflation depends on other factors such as people expectations meaning the relationship is not necessarily even proportional, but discussing this in depth would be beyond scope of this answer.


Did capitalism exist before the 18th century?

According to some yes. In this (historically oriented) view, whatever succeeded feudalism is called capitalism.

Capitalism, broadly accepted as a phase of history, is dominating the last half-a-millennium. It has also become conventional that capitalism itself had various phases and stages. Its antecedents go back to ancient history. Barter and exchange of one thing for another, as Adam Smith, the main advocate and ideologue of market capitalism argued, are a part of human nature. The pockets of private-market economy and its institutions, especially in urban settlements, flourished in medieval Europe. Moneylenders and urban merchants were its main representatives.
Its continuous development, however, began mostly from the sixteenth century. Historiography produced a vast literature on the transition from feudalism to capitalism. These writings maintain that declining productivity, a demographic crisis, and the scarcity of peasant labor generated the lowering of rents, a decrease of labor services, and an increase in peasant mobility and freedom in Western Europe. In most of these countries, consequently, serfdom was gradually loosened and disappeared by the sixteenth century. Commercialization of agriculture played a central role in the development of capitalism. The flourishing textile industry in Britain with an excess production over consumption led to the enlargement of productive capacities in an unheard scale and made England the ‘workshop of the world.’
This stage between the sixteenth and eighteenth centuries is often called ‘commercial capitalism’ in history. The system was based on private ownership of warehouses and ships and buying and selling goods all over the world. The merchants also lent out money at interest and established connections with production. They bought and distributed raw materials for peasants who worked at home in the traditional way (putting-out system), and gradually subordinated the rural cottage industry. It led to a proto-industrialization, the development of a decentralized putting-out system coordinated by merchants, and later the establishment of (nonmechanized) factories, which introduced division of labor in the production procedure.

Quoted from I.T. Berend, "Capitalism" in International Encyclopedia of the Social & Behavioral Sciences, 2001
Alternatively, what he calls "commercial capitalism" is called "merchant capitalism" by others. And as the Wikipedia article linked shows, others (Ellen Meiksins Wood in particular) disagree on whether that was really capitalism.

She argues that commerce is not equivalent to capitalism and does not produce it because 'the dominant principle of trade everywhere was not surplus value derived from production but "profit on alienation", "buying cheap and selling dear" ': that is, although trade involves the seeking of profits, it does not in itself affect how goods are produced. Wood argues that polities which flourished from trade such as medieval Florence and the early-modern Dutch Republic that did not become capitalist are not examples of a 'failed transition', but rather examples of how trade and urbanisation has flourished widely in human history without capitalism developing. She likewise argues that various past societies, such as early modern France, had peasantries with ample opportunity to enter into commercial production, which, however, did not take this opportunity. Thus Wood concludes that capitalism must have arisen from changing imperatives in fundamental relations of production.

Both Berend and Woods have serious Marxist leanings, so even in this relatively narrow camp there's disagreement on when capitalism bagan...
In yet other terms, this is the divergence of views between the "commercialization" vs the "agrarian" model of the emergence of capitalism. It seems to me that the latter is mainly favored by Marxists, while the former has a broader appeal (so Berend is a bit of a heretic among Marxists in this respect.)


I am not sure this will answer your question, but let's me try with a concrete example using Stata and its 1978 Automobile Data in order to regress car's price on mileage (mpg).

sysuse auto, clear
reg price mpg, robust

This gives a price estimate of -238.9, with a standard error (s.e) of 57.5. Then, if I run the same regression after duplicating the observations three times.

expand 3
reg price mpg, robust

I get as expected a much lower s.e. of 32.9. Finally, clustering the s.e. by make (a Car Id)

reg price mpg, cluster(make)

gives a s.e. of 57.2 and makes the trick.

However, if you expand twice some observations and thrice others (as in your example). Then, you will get a larger s.e but also a different estimate. But, again, clustering by make reduces the s.e. Here is the code:

sysuse auto, clear
expand 3 if price>4000 & price<=6000
expand 2 if price>6000
reg price mpg, robust
reg price mpg, cluster(make)

Note that the price and mpg variables have been expanded without any change for each car. In your example, if the ones who have taken their temperature multiple times have each time a different temperature, then you may consider introduction an individual fixed effect.


The Principle of Optimality and the Bellman Equation

Recall that the Principle of Optimality states that the solution to Our Bellman Functional Equation is the same as the solution to the sequential problem if:
Assumption 1: $\Gamma(x)$ (our set of feasible values) is non-empty for all $x\in X$.
Assumption 2:$\lim_{t\rightarrow\infty}\sum_{t=0}^\infty \beta^t F(x_t,x_{t+1}) $ exists for all $\tilde{x}\in \Pi(x_0)$ ($\Pi(x_0)$ being the correspondence of $x$ considering an initial $x_0$)
Assumption 3:$|V^*(x)<\infty|$ for all $x \in X$.
Assumption 4: For any $x_0$, there exists a plan $\tilde{x}\in \Pi(x_0)$ such that $u(\tilde{x})=V^*(x_0)$.
In your story here since $\beta<0$ we know that this is bounded all we are doing is adding an additional constraint. This is equivalent to writing the legrangian:
$$\mathcal{L}=\sum_{t=0}^\infty\{\beta^tu(c_t)+\sum_i\lambda_{t,i}(F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}\}$$
Again I may not be iterating the whole story because there are two sectors here but this is how I'd go about thinking about if the principle of optimality is satisfied.
Hope this helps
Edit: After looking at your addendum im guessing the proceedure would be the same
$$V_t(k_t)=max_{c_t}\{u(c_t)+\lambda [F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]\}+\max_{c_{t+1},...,c_{T}}\{\sum_{t+1} \ [\beta^{t+1}u(c_{t+1})+\sum_i\lambda_{it}[F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]]\} $$
or
$$V_t(k_t)=max_{c_t}\{u(c_t)+\lambda [F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]+V(k_{t+1})\}$$


MMT is propounding to be a theory. A theory can't be sustainable or unsustainable.  However, I think you meant to ask if the policy proposals of MMT-ers would be sustainable so I will answer assuming that is your question.
Now MMT is not a proper scientific economic theory* as it is not rigorously defined (Mankiw 2020), so it is not actually clear what MMT policy implications are but many MMTers generally claim that central bank should monetarily finance government spending, and that generally any amount of real spending can be funded by monetary expansion. Moreover they generally claim that countries that borrow in their own currency should not worry about government deficits because they can always create money to finance their debt.
These claims of MMT are rejected by virtually all professional economists. For example, recently IGM forum made poll among top US policy economists to gauge how many economists believe in the above mentioned propositions of MMT,  and the results shows that none of the top economists polled (all economists in the poll are a representative sample of Ivy league economists) did agree with any of the stated MMT propositions (see here). But then even MMT proponents generally do not agree on what MMT is. I seen people claiming it is a monetary theory and others claiming it is whole new economic paradigm. Moreover, the MMT is generally criticized for its policy prescriptions and even generally as a theory itself in the academic literature (e.g. see Mankiw 2020 or Palley 2014)).
Furthermore, you should note that I do not know of any central banker at Fed or elsewhere that would be supporter/adherent to MMT (I suppose some might exist but nobody who is publicly known).

Additionally, will the low-rate environment remain permanent?

This is a good question without a good answer. It might be. Summers (see Summers 2014 in  Teulings and
Baldwin  (2014) or Summers 2015) argues that we will live in low rate environment due to secular stagnation and overabundance of savings (indeed Fed is forced to keep interest rates low due to enormous supply of savings). The secular stagnation hypothesis has its proponents but it is far from being completely accepted by profession.  To an extent we really live in secular stagnation and there is permanent over-supply of saving due to aging populations, slow technological growth and so on it might be permanent. However, it might as well be just transitory phenomenon caused by the sheer magnitude of Great Recession (indeed before pandemic hit Fed and other central banks were already planning to hike the interest rates but then pandemic hit - see this Dec. 2018 news article).

I can't imagine the Fed, for example, raising rates substantially as 1. It would tank most assets and 2. It would make the already incredibly unwieldy sovereign debt that much harder to manage. It would seem we are locked into an endless money printing path that essentially can't be stopped without immense pain. Am I right in this understanding?

Fed can do it in small steps in order to avoid popping potential bubbles. In fact that was the plan before covid-19 hit, to slowly rise interest rates instead of dramatically hiking them. See this NY times article on what Fed officials were claiming they planned to do, from before covid hit,   here.

 * from the mainstream perspective on demarcation of science where theories have to be rigorously defined and testable to be considered scientific, of course there might be some alternative interpretations of what science is see discussions in Reiss Philosophy of Economics


Why does equity have historically poor performance after rate cuts?

If this trend holds for a longer series, my guess is that this is due to the correlation between the federal funds rate and general economic conditions. The federal funds rate is usually cut in anticipation of a weakening economy or during a period of low growth.
It is well known that the expected returns on financial assets, including stocks and many other kinds of assets, vary with the business cycle. For example, see Fama and French (1989), "Business Conditions and Expected Returns on Stocks and Bonds." The most common explanation for this is that effective risk aversion is higher in bad times than in good times (e.g., as measured by GDP or aggregate consumption). This phenomenon is discussed more generally in the paper, "Discount Rates", by John Cochrane (2011). (Ungated access here.) For example, on p.1052, we read

  There is a strong common element and a strong business cycle
  association to all these forecasts. Low prices and high expected
  returns hold in “bad times,” when consumption, output, and investment
  are low, unemployment is high, and businesses are failing, and vice
  versa.


How long can a population grow once it has reached the replacement fertility rate?

Population size isn't just about birth rate.

It's also about death rate and net migration.

If the health of India's population continues to improve, then life expectancy will increase, and population size will continue to grow until life expectancy stops increasing, assuming zero net migration.

Net inward migration would also increase the population size.


How does an import quota affect the demand / supply of currency?

The goal of import quotas is to get people to buy domestic goods, which can only be bought for local currency. Therefore import quotas increase demand for local currency. But...

Setting import quotas is a very undesirable economic policy. It is basically a state intrusion into a market economy. Quotas for imports are usually implemented as a desperate measure when there is a continuously negative trade balance, local currency rapidly loses value, and every other measure (increasing taxes, customs, subsidizing exporters, etc.) failed. Distribution of quotas usually involves plenty of corruption. On top of that, World Trade Organization does not normally allow import quotas.

The more common use of import quotas is in agriculture, to support local farmers or production of a certain crop. But such limited use of import quotas should have very limited effect on the currency, unless it is a 'banana republic' where agriculture is the main economic activity.


Where can I find statistics for numbers of 16-20 year olds in the world?

That's not really an economics question, so it's probably off-topic. However, the US Census Bureau seems to track this here over time by sex.


Shopping example in Kőszegi / Rabin (2006)

I think it is just a matter of definition. The disutility from spending on the shoes is not the utility obtained by purchasing the shoes: it is the difference between the utility obtained when she does not buy the shoes and the utility she gets when she buys them.

Suppose for instance that she expects to buy the shoes at price $p$. Her "money" utility of buying the shoes equals
\begin{equation*}
U(\text{Buying}) = -p
\end{equation*}
since there is no gain-loss utility in that case, and $-p$ is her final wealth (normalizing initial wealth to zero).

However, if she finally decides not to buy the shoes, her money utility equals
\begin{equation*}
U(\text{Not Buying}) = \eta p
\end{equation*}
since her final wealth equals $0$, and in addition she experiences a gain-utility of $p$ with respect to the reference point (spending $p$).

Therefore the disutility from buying is
\begin{equation*}
U(\text{Not Buying}) - U(\text{Buying}) = (1+\eta) p
\end{equation*}

I hope it helps.


Target2 Balances at the ECB-will it lead to the collapse of the Euro?

Although possible it is not necessary for several reasons.

Optimal currency area (OCA) can in principle exist even without fiscal transfers/excessive borrowing if there is perfect factor mobility within the OCA that can absorb the asymmetric shocks that countries in the monetary union face (see De Grauwe Economics of Monetary Union).

In the European Monetary Union (EMU), at a present, the factor mobility is not perfect because while there are no direct legal restrictions on movement of capital and labor it is not easy for people to move due to language barrier and poor integration of social security at EU level. However, as time progresses we see English becoming world-wide lingua franca. For example, in the Netherlands (where I live) a reasonable EU citizen can live and work never learning to speak Dutch in virtually any industry that does not require direct contact with consumers. All government forms and interactions are in both Dutch and English, and the same holds for doctors, and  virtually all retail/service businesses. This significantly lowers barriers to labor mobility. In addition all the social security systems are getting more and more integrated within EU. For example, when it comes to healthcare the system in EU is already integrated in a way that switching is not a hassle but same does not hold for pensions.
Since we are talking about some unspecified future it is possible that at that time the labor mobility will be sufficiently perfect that any shocks can be absorbed via  labor mobility and mobility of other factors as well. Indeed, if in past there would be no implicit restrictions on people moving from one EMU country to another, instead of fiscal transfers you would see large flow of people that would eventually correct the imbalances.

Germany presently sends fiscal transfers to the Southern Europe because it was fortunate enough to experience positive productivity shocks while southern Europe experienced negative ones. I am not trying to minimize the role of good governance and policy in German robust productivity and also economic growth in recent decades, but no matter what policy  this is to a significant extent also driven by random technological (where I am using the word in broad economics sense) or other macroeconomic shocks. Even the best govern country can't do much about that. In addition, random demand shocks or even monetary policy (e.g. low interest rates for too long) is also one of the issues that can contribute to macroeconomic imbalances that necessitate fiscal transfers (Pierluigi,  Sondermann 2018; Blanchard and Giavazzi 2002; Lane and Pels 2012;  Chen et al 2012).

Since  we are discussing the future it is entirely possible that in 10 or 20 years Germany will be the country experiencing negative productivity/demand etc shocks while southern countries will be experiencing positive ones, as often macroeconomic shocks can be asymmetric.  If the Germany starts insisting that South redressing its debt/fiscal transfers imbalances to Northern countries at that point in time - when it is from macroeconomic perspective optimal for South to actually do so, it won't cause any problem for the Euro, if anything it would be the optimal response.

Euro, is not just economic but also political project. In principle, even if Germany does not want to send fiscal transfers or lend to southern countries, if ECB conducts monetary policy in a way that addresses only macroeconomic shocks in countries experiencing negative shocks (which in case of asymmetric shocks would be at an expense of countries not experiencing negative shocks), or if EU leaders decide they are fine with living with financially instable EMU it is still possible to keep the euro.


How can someone claim that this return is guaranteed, since we need to use $\mathbb{V}ar(\widetilde{W})$ to measure the utility gain of an individual?

Maybe I misunderstand your point, but though $\widetilde{W}$ is a random variable, $\mathbb{E}(\widetilde{W})-\frac{\rho}{2}\mathbb{V}ar(\widetilde{W})$ and hence $CE(\widetilde{W})$ is a real number. So $CE(\widetilde{W})$ is a measure of a certain income which to the consumer has the same utility as $\widetilde{W}$.


Can't Derive Expectation in Second Price Auction

There are $N+1$ bidders. So we only care about the second highest bid. That's the same as finding the highest bid amongst the remaining $N$ bidders. The expected second highest bid is given by:

$$b \cdot Pr[b] \cdot Pr[b \geq b_{-i}] = b \cdot f(b) \cdot F(b)^{N-1} = b \cdot dF(b)^{N}$$

As expected value is an average, we divide out by the probability that $b \leq \beta$, which is $F(\beta)$.

Thus, we have:

$$\mathbb{E}[b^{[2]} | b^{[2]} \leq \beta] = \dfrac{1}{F(\beta)}\int_{0}^{\beta} bdF(b)^{N}$$


Why is the prime rate typically 3% higher than the fed funds rate?

The Prime Rate in the US is supposed to represent the rate at which banks make loans to customers.  For many years is has been set at Fed Funds+300bp, whereas the rate at which banks actually make loans varies a lot depending on the creditworthiness of the borrower.


In general the demand for a certain good (say from a consumer) can be written as a function of the prices of all available goods and the total amount of money that the consumer has available.
Take the setting of two goods, $q_1$ and $q_2$ with prices $p_1$ and $p_2$ and total income $y$. Then the demands can be written as:
$$
q_1 = d_1(p_1, p_2, y)\\
q_2 = d_2(p_1, p_2, y).
$$
If you look at Engel curves, you consider prices as fixed, so the only thing that is allowed to vary are the income levels. In this case, you can simply write:
$$
q_1 = e_1(y),\\
q_2 = e_2(y).
$$
If you want to estimate the full demand system (i.e. the functions $d_1, d_2$), you have to make sure that you have variation both in income levels ($y$) and (relative) prices ($p_1$ and $p_2$), so either time series data for a given individual or panel data for a group of individuals.
If you do not have price variation (e.g. because you are using a cross sectional dataset and everyone faces the same price, or if prices do not change over time) then you can only estimate the Engel curves $e_1$ and $e_2$.


Here is a counterexample (i.e. an example showing that the given statement is false). Assumptions:

The cost of producing each unit of the good is \$1.
All consumers are exactly identical, with each consumer willing to pay up to \$10 for one unit of the good. (Also, each consumer is willing to pay at most \$0 for a second unit of the good.)
Market 1 has 200 consumers, while Market 2 has 100. (And so, it is true that "quantity demanded at any positive price is always higher in market 1 than in market 2."†)

Since all the consumers are exactly identical, the profit-maximizing price in both markets is simply the maximum each consumer is willing to pay, namely \$10 (and the producer will produce 300 units of the good). Thus, it is false that the monopolist "will certainly charge a higher price in market 1 than in market 2".

†This statement is actually wrong. At a sufficiently-high price, the quantity demanded of a good in any two markets will be equal to zero. To correct this statement, one would either change "always higher" to "no lower" or append to the start of this statement "if quantity demanded is non-zero, then ..."


Please help a layman distinguish among sorts of normal goods

Would graphics cards be different sorts of goods in these three
scenarios?

No. What we have is different sorts of consumers, with different utility functions. The good is the same, it is the demand function that changes.
In a dynamic setting of course, the preferences of consumers (and which are the predominant ones in terms of demand share) will possibly drive how graphics cards evolve as products.
As for the second question, I am pretty sure that somewhere in the vast and centuries old economics literature someone has defined yet one more subcategory of a good.


What is the difference between Herfindahl Index and the Concentration Ratio

From the Industrial Organization by Belleflame and Petiz (Page 34/35, Chapter 2):

  While the m-firm concentration ratio adds market shares of a small number of firms in the market, the so-called Herfindahl index (also known as Herfindahl–Hirschman index) considers the full distribution of market shares.

We can conclude that the mathematical approach in HHI is more adequate for a full market setup, rather than observing a particular concentration ratio in one firm.
Also from the same book

  the Herfindahl index provides a better measure of concentration as it captures both the number of firms and the dispersion of the market shares.

Hence the squared market shares.

For your information, there is another concentration measure called the Lerner Index, although the Lerner index is a snapshot of the intensity of competition. You can calculate L by finding the difference between price and marginal costs as a percentage of the price. More formally:
$L=\frac{p-C'}{p}$

But it is easily observed that L ignores some dynamics of the markets. Lower prices do not necessarily mean high levels of competition.


How can I obtain Leontief and Cobb-Douglas production function from CES function?

The proofs I will present are based on techniques relevant to the fact that the CES production function has the form of a generalized weighted mean.
This was used in the original paper where the CES function was introduced, Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capital-labor substitution and economic efficiency. The Review of Economics and Statistics, 225-250.
The authors there referred their readers to the book Hardy, G. H., Littlewood, J. E., & Pólya, G. (1952). Inequalities , chapter $2 $.

We consider the general case
$$Q_k=\gamma[a K^{-\rho} +(1-a) L^{-\rho}  ]^{-\frac{k}{\rho}},\;\; k>0$$

$$\Rightarrow \gamma^{-1}Q_k = \frac 1{[a (1/K^{\rho}) +(1-a) (1/L^{\rho})  ]^{\frac{k}{\rho}}}$$

1) Limit when $\rho \rightarrow \infty$
Since we are interested in the limit when $\rho\rightarrow \infty$ we can ignore the interval for which $\rho \leq0$, and treat $\rho$ as strictly positive.

Without loss of generality, assume $K\geq L \Rightarrow (1/K^{\rho})\leq (1/L^{\rho})$. We also have $K, L >0$. Then we verify that the following inequality holds:

$$(1-a)^{k/\rho}(1/L^{k})\leq  \gamma Q_k^{-1} \leq (1/L^{k}) $$

$$\implies (1-a)^{k/\rho}(1/L^{k})\leq  [a (1/K^{\rho}) +(1-a) (1/L^{\rho})  ]^{\frac{k}{\rho}} \leq (1/L^{k}) \tag{1}$$

by raising throughout to the $\rho/k$ power to get

$$(1-a)(1/L^{\rho}) \leq a (1/K^{\rho}) +(1-a) (1/L^{\rho})  \leq (1/L^{\rho}) \tag {2}$$
which indeed holds, obviously, given the assumptions. Then go back to the first element of $(1)$ and

$$\lim_{\rho\rightarrow \infty} (1-a)^{k/\rho}(1/L^{k}) =(1/L^{k})$$

which sandwiches the middle term in $(1)$ to $(1/L^{k})$ , so

$$\lim_{\rho\rightarrow \infty}Q_k = \frac {\gamma }{1/L^k} = \gamma L^k = {\gamma }\big[\min\{K,L\}\big]^{k} \tag{3}$$

So for $k=1$ we obtain the basic Leontief production function.

2) Limit when $\rho \rightarrow 0$
Write the function using exponential as

$$\gamma^{-1}Q_k=\exp\left\{-\frac k{\rho}\cdot \ln\big[a (K^{\rho})^{-1} +(1-a) (L^{\rho})^{-1}\big]\right\} \tag {4}$$

Consider the first-order Maclaurin expansion (Taylor expansion centered at zero) of the term inside the logarithm, with respect to $\rho$:

$$a (K^{\rho})^{-1} +(1-a) (L^{\rho})^{-1} \\= a (K^{0})^{-1} +(1-a) (L^{0})^{-1} -a (K^{0})^{-2}K^{0}\rho\ln K- (1-a) (L^{0})^{-2}L^{0}\rho\ln L + O(\rho^2) \\$$

$$=1 - \rho a\ln K - \rho(1-a)\ln L+ O(\rho^2) = 1 +\rho \big[\ln K^{-a}L^{-(1-a)}\big]+ O(\rho^2)$$

Insert this back into $(4)$ and get rid of the outer exponential,

$$\gamma^{-1}Q_k = \left(1 +\rho \big[\ln K^{-a}L^{-(1-a)}\big]+ O(\rho^{2})\right)^{-k/\rho}$$

In case it is opaque, define $r\equiv 1/\rho$ and re-write

$$\gamma^{-1}Q_k = \left(1 +\frac{\big[\ln K^{-a}L^{-(1-a)}\big]}{r}+ O(r^{-2})\right)^{-kr}$$

Now it does look like an expression whose limit at infinity will give us something exponential:

$$\lim_{\rho\rightarrow 0}\gamma^{-1}Q_k = \lim_{r\rightarrow \infty}\gamma^{-1}Q_k = \left(\exp\left\{ \ln K^{-a}L^{-(1-a)}\right\} \right)^{-k}$$

$$\Rightarrow \lim_{\rho\rightarrow 0}Q_k =\gamma\left(K^{a}L^{1-a}\right)^k$$

The degree of homogeneity $k$ of the function is preserved, and if $k=1$ we obtain the Cobb-Douglas function.

It was this last result that made Arrow and Co to call $a$ the "distribution" parameter of the CES function.


Do people really care about higher moments?

Perhaps there is some evidence toward your claim, but I would argue that in most situations, people do not use point estimates (although some "smoothing" likely occurs). In particular, there is no doubt that people care about the second movement (variance). The commonly employed, and robustly empirically documented, notion of risk aversion captures exactly this. Given two risky prospects with the same expectation (say different assets) it seems that most people prefer the one with lower variance (i.e., the less risky prospect). Think: which would you chose when given the option between \$50 for sure or a 50/50 toss up between \$0 and \$100? The difference between these two options is entirely in their second moment.

The evidence for this is so vast and varied it is hard to even give an encompassing reference; but since you ask about the stock market, it is worth pointing out that one of the key explanations for why stocks have a higher average return than bonds is that they are more risky. I imagine a google scholar search of 'risk aversion in the stock market" will be fruitful.

It is worth noting that there is also empirical evidence that people care about the third moment (skewness). This working paper by Yusufcan Masatlioglu, A Yesim Orhun, and Collin Raymond is an experiment in which subjects

  reveal a strong preference for positive skew over negative skew; in other words for ruling out more uncertainty about the desired outcome (and tolerating uncertainty about the undesired outcome) compared to ruling out more uncertainty about the undesired outcome (and tolerating uncertainty about the desired outcome).

As for even higher moments, the jury is out, but I would imagine their importance diminishes rather quickly. The complexity of the distributions needed to generate differential higher moments increases and I would think heuristics would start to kick in.


Does the envelope theorem hold at a corner solution?

First, you made a sign error in the computations. After correcting for your error, a crucial hypothesis you miss is that $X$, the choice set, does not depend on the variable $t$ in the theorem (with the notations of the theorem). To apply the theorem properly, the interval $[0,L]$ should not depend on $L$.

A) The sign error

$$\frac{\partial H}{\partial L_K}=-\alpha (L-L_K+1)^{-1}H+(1-\alpha)(L_K+K)^{-1}H=0$$
We define $L_K^0= (1-\alpha)(L+1)-\alpha K$.

B) Why we could think the envelop theorem's result may fail

Presuming that $0<\alpha<1$, there are four possible cases.

(1) $L_K^0\in [0,L]$. One can check the objective function is concave, thus $L_K^*=L_K^0$.

(2.i) $L_K^0\notin [0,L]$ and $H(L,0,K)< H(L,L,K)$. Then $L_K^*=0$.

(2.ii) $L_K^0\notin [0,L]$ and $H(L,0,K)> H(L,L,K)$. Then $L_K^*=L$.

(2.iii) (just to be exhaustive) $L_K^0\notin [0,L]$ and $H(L,0,K)= H(L,L,K)$. Then there are two solutions, $0$ and $L$.

In case (1), $$\frac{\partial F}{\partial L}(L,K)=\frac{\partial H}{\partial L}(L,L^*_K,K)+\frac{\partial L^*_K}{\partial L}.\frac{\partial H}{\partial L_K}(L,L^*_K,K).$$
The second term of the right-hand side is equal to zero thanks to the first-order condition. This is compatible with the envelop theorem's result for an interior solution.

In case (2.i), $F(L,K)=H(L,0,K)$ and so $$\frac{\partial F}{\partial L}(L,K)=\frac{\partial H}{\partial L}(L,0,K).$$
This is compatible with the envelop theorem's result for a corner solution here.

In case (2.ii), $F(L,K)=H(L,L,K)$ and so $$\frac{\partial F}{\partial L}(L,K)=\frac{\partial H}{\partial L}(L,L_K=L,K)+\frac{\partial H}{\partial L_K}(L,L_K=L,K).$$

We have to be cautious about the notations here, $\frac{\partial H}{\partial L}$ means the partial derivative corresponding to the first argument, and $\frac{\partial H}{\partial L_K}$ to the second one. The second term of the right-hand side is nonzero, which does not fit with the envelop theorem's result.

C) Why it actually does not fail

Write the problem as $F(L,K)=\max_{x\in [0,1]}H(x,L,K)$, with
$$H(x,L,K)=(L-x L+1)^\alpha(x L+K)^{1-\alpha}.$$
This problem is equivalent to the initial one.
The key difference is that the interval $[0,1]$ does not depend on $L$ or $K$. This is the reason why we can apply the envelop theorem, whereas it was wrong to apply it before.

We can check that the case (2.ii) is compatible with the envelop theorem, we have $F(L,K)=H(x=1,L,K)$ and so $$\frac{\partial F}{\partial L}(L,K)=\frac{\partial H}{\partial L}(x=1,L,K).$$


In 2013 researchers at the Federal Reserve Board published a paper {link} considering the costs and benefits of replacing \$1 bills with \$ coins.
Here are the takeaways from their executive summary: compared to notes, coins offer

Less payment system efficiency. Using \$1 coins instead of \$1 notes for transactions is inherently inefficient, requiring a replacement of one \$1 note with more than one \$1 coin to make up for the difference in the way coins and notes are used by the public.
Less cost-effective. Replacing \$1 notes with \$1 coins is also not cost-effective for the U.S. government and public more broadly, primarily because the higher cost to produce coins compared with notes is not offset by the longer life of the coin.
Higher costs under all projected scenarios. Circulating only \$1 coins costs more under every scenario we considered (in net present value terms) than would continuing to provide $1 notes to the public.
Increased costs to the private sector. Circulating only \$1 coins could result in increased costs to the private sector, perhaps in the hundreds of millions of dollars per year, and would more than offset any "seigniorage revenue" to the government reported in earlier GAO studies.

So the case for using coins is indeed weak. But reading the paper in more details offers some clues as to the historic reasons why coins are so widespread.
For example, low denominations tend to be used quite frequently in day to day transactions, meaning they wear out much faster. The UK, Canada, and Australia all replaced their equivalent of the \$1 bill with a coin in the 1980s. But this was at a time when electronic payment mechanisms (such as credit/debit cards) were not widely available, meaning many small transactions used the low denomination currency with high frequency. This lent an advantage to coins' high durability that is less important today.
Indeed, one of the historical advantages of coins has been their extra durability, which makes them cheaper to maintain in the long-run. But the FED report notes that the durability of notes has increased significantly. When the UK replaced its £1 note with a coin, the lifetime of a note was 10 months (many of the original coins are still in existence 30+ years later). But a 2012 \$1 bill has an expected lifetime of 70 months, tipping the economics of coins vs notes decisively in the latter's favour. Many countries are now adopting polymer (i.e., plastic) banknotes that have yet better durability.

In sum, notes appear to dominate coins both economically and practically (at least around the \$1 denomination value). Various countries have also eliminated lower denominations from their coinage altogether (see Wikipedia).
But there are costs involved in replacing coins with notes. The state must retool its currency production, provide public awareness campaigns, and administer the process of currency replacement; the private sector must modify vending machines, cash dispensers, and similar appliances, and incur other costs implied by the transition.
For example, The Guardian reports that it was cost up to £50m to replace just the nation's parking meters as the UK undergoes its switch to a new design of one pound coin. Another estimate is that it will cost £40-50 for each changing room locker that needs to be upgraded. A treasury estimate put the cost of introducing new 5p and 10p coins in the UK to the coin-operated industry at £80m.
These costs, along with general inertia and the fact the people increasingly avoid using cash altogether are probably the main reason why we still have coin currency.


MRS of Perfect Complements and MRS Equilibrium Condition

Your instructor seems to have a rather sloppy approach to the mathematics of microeconomics.

Let's begin with the easiest case, the bundle $(5,1)$. We have $\min\{3\cdot 5,1\}=1$. For $\delta<14$, we have $\min\{3\cdot 5, 1+\delta\}=1+\delta$ so the marginal utility of $y$ is $1$ at the bundle $(5,1)$. Moreover, for any $\delta>-14/3$, we have $\min\{3\cdot(5+\delta),1\}=1$, so the marginal utility of $x$ is $0$ at $(5,1)$.

A similar argument applies to the bundle $(2,8)$, but here the marginal utility of $y$ is $0$, and dividing by zero is not allowed. Maybe your instructor wants to hear that the MRS is infinite.

Now at the bundle $(3,9)$, increases in any commodity have zero effects, but decreases do. There is a kink and you cannot smoothly move along the curve. By any semi-reasonable standards, the MRS is undefined. But to answer the verbal question, the consumer is not willing to give up any amount of one commodity ( effect negative) to receive more of the other good (effect zero).

Now the idea that you get the optimal consumption bundle by setting the MRS equal to relative prices is complete nonsense. If prices $p_x$ and $p_y$ and income $m$ are all positive, we can still, however, still find the optimal bundle. First note that the optimal bundle $(x,y)$ wil satisfy $3x=y$. If $3x>y$, one could reduce the amount $x$ without decreasing the utility (the minimum is unchanged), but use the freed money to increase $y$, which does have a positive effect. In an optimal bundle, this is not possible. by a similar argument, we can rule out $3x<y$. So we must have $3x=y$, which means that at any optimal solution, the MRS is undefined!!! To solve for an optimal bundle, all that remains to do is plug the condition $3x=y$ into the budget constraint $p_x x+p_y y=m$. We can substitute $3x$ for $y$ to get
$$p_x x+p_y 3x=m$$ and solve for $x$, which gives us
$$x=\frac{m}{p_x+3p_y}.$$
Similarly, we get
$$y=\frac{m}{p_x/3+p_y}.$$


That’s what the source data goes up to at present: Fed H.6

The fact that it is weekly does not mean that it will be close to available in real time, they presumably need time to get the data for some components of the aggregate.


how is new money actually introduced in the United States?

The Treasury auction process is described here: Treasury Direct

Bidders can either enter competitive or non-competitive bids. Non-competitive bids are limited in size; the price is set by the competitive bids, which generally come from Primary Dealers (that are obliged to bid).

As a simplified example of the bidding process, assume that the government announces an auction for $2 million in 30-year bonds. (Note that this example ignores bidding size limits.)

There are three bids:

a bid for \$1 million at 4% from A
a bid for \$1 million at 4.25% from B
a bid for \$1 million at 4.50% from C.

The government looks at the bids (starting with the lowest yields) until the amounts cover the amount to be issued. The highest winning bid yield is used for all. In this case, that means that A and B get filled with a $1 million in bonds at 4.25%, and C gets nothing.

The connection to money is somewhat complicated. The issuance of the bonds reduces the money supply - the bidders have to pay for the bond. The coupon and principal payments will increase the money supply.


Is the convexity of production sets necessary for the welfare theorems?

Convexity of the production set is indeed not needed for the proof of the first welfare theorem but for the proof of the second welfare theorem. It is not a necessary condition though.

It is possible to interpret this as an existence issue. The first welfare theorem is about all competitive equilibria and holds trivially if there are none. The second welfare theorem, on the other hand, states that for a given Pareto efficient allocation, there is a price system and redistribution of endowments with respect to which it is a (quasi-)equilibrium.

The standard proof of the second welfare theorem uses a result of Minkowski on the separation of nonoverlapping convex sets by a hyperplane, but it is possible to prove a version of the second welfare theorem as a corollary to an existence result by a nice argument due to Maskin and Roberts. The argument is quite easy in the case of an exchange economy: Take a Pareto efficient allocation as the endowment distribution. If a competitive equilibrium exists from these endowments, everyone will end up with something at least as good as their endowment. Since the endowment distribution was Pareto efficient, nobody can end up with something better. So everyone must be indifferent between their demanded commodity bundle and their endowment, so they might as well just demand just their endowment. The argument generalizes to economies with production.


Good Site for Practicing Advanced Problems

There are lots of good courses (economics, math etc.) on MIT OpenCourseware or interestingly, there are also very very good lectures in YouTube.
But I think the web site for MIT opencourseware is a real treasure. (at least for me).


Could debt cause inflation instead of deflation?

Yes it can.

If debt originates from the banking system, then it potentially has a side effect of money creation. Whether or not money is actually created when a bank loan is made depends on the banking system's regulatory framework, and the lending policies of its banks, and these can vary widely.

However if there is net excess of bank lending over bank loan repayment and loan default, then money creation is occurring, and the additional money can and indeed usually does, result in inflation.


Measure of New Firms Born in Each Period

I don't know your model. Just some general input:

It seems that you have a model with a continuum of firms. That is, there are many firms and each one is strategically small. In mathematics, a measure of a set is a way to express the size of this set. Hence, the measure of firms alive represents "how many firms are active". The authors normalize this measure to 1. If it helps, think of 1 million individual small firms and think of "measure 1" as a normalization of this million.

In your model, each period a only a fraction $(1- \rho)$ of this measure 1 survives and additional measure $\rho$ of firms are "born". Thus, the total measure of firms remains 1.
Again, it may help to think of it as a normalization of a large finite number. For example, if $\rho=0.5$, only 500 000 out of 1 000 000 firms from last period survive and 500 000 additional firms are created such that there are still 1 000 000 firms.


Disclaimer: this answer comes from a microeconomic research perspective.  Time series / macroeconomic specialists will likely have other perspectives.

There is no general rule for what's too low across the entire field of economics.  Yes, microeconomic models (i.e., individual-level observations) will tend to give low R-squared values (often in single percentage point digits) because there are so many factors that can affect human outcomes, a lot of which just cannot be observed.  But more generally, R-squared will depend on your data and model, specifically the nature of your dependent variable, any transformations you may have applied to the variable, what explanatory variables you include and what variables you exclude.

In Economics research, R-squared is rarely of concern because predictive power is often not the main goal.  Instead, economists focus on finding reliable estimates of coefficients for variables of interest and deriving useful inferences from the structure of the model and the estimated values of its parameters.  And standard error of the regression is considered a better metric than R-squared because it comes in the units of the dependent variable, provides the absolute measure of the typical distance that the data points fall from the regression line, and it scales the width of all confidence intervals calculated from the model.

Source for some of the points included in this answer: https://people.duke.edu/~rnau/rsquared.htm


I don't buy the explanation of this being vagaries of advertisements, regardless of the job.

Instead, I see it as employers showing rational and predictable flexibility in response to a changing supply of labour.

At times of plentiful supply, they can reduce recruitment costs by being far more selective at the first stage - filtering by essential requirements in the job ad. Finding good staff can be an onerous process. All the filtering stages after writing the ad are labour-intensive, and the labour is often line-managers and upwards, so not cheap. If I know I can get a decent number of interview candidates even with very stringent essential requirements, I'll do that, and that will reduce the number of applications that I have to process. This reduces my costs without significantly impairing the outcome.

At times of restricted supply, it's more cost-effective to advertise for lower essential requirements. That way, you cast a wider net, get more applicants, and are more likely to find a suitable candidate, without overwhelming the staff who have to read through the applications and process them.

As an occasional recruiter, I've done all this myself.


exchange rate and domestic money supply part II

@Toby comments noted, I will attempt an answer.  First, the FX market is international so you cannot tell whether the foreign currency seller is domestic or foreign. However, if it is a foreign buyer, they will typically reinvest the dollars in either a commercial or financial asset such as a Treasury bond, which could result in the remittance of the dollars back to a domestic holder.   What you know for sure is that the dollars will be deposited in a bank somewhere and will add to the pile of excess reserves currently held by the US banking system.  Hence technically the money supply goes up.

Perhaps the greater question is whether raising the money supply in this way has any effect on the economy.  I would suggest not, since banks already have excess reserves.   Rather, the transmission mechanism for FX intervention is via changes in the exchange rate.   But that could be debatable.


What's a more accurate term than 'share price' that doesn't depend on (reverse) stock splits?

How about "value"? Market capitalization seems fine too.


On the Ms. Exponential vs Ms. Hyperbolic vignette

I believe the money pump works like this:

At $T=0$ borrow a hundred dollars from Henry the hyperbolic discounter for the full time period $t$. The interest rate should be zero because their discount rate over loans of that period is $1$. For simplicity, make it a zero coupon loan, with the payment due at maturity and assume you both agree their is no credit risk. Now step forward one period to $T=\epsilon$. Offer to sell the loan back to Henry. His discount rate is now $\frac{1}{1-\epsilon} < 1$ so the interest rate is lower than what he would charge on a new loan so the price must be above par. Call the price $100 + \psi_1$. Do the sale, tear up the loan, and pocket $\psi_1$. Ask for a new loan of a hundred dollars, again at zero coupon, now at interest rate $\frac{1}{1-\epsilon}$ per period. Step forward another period to $T = 2 \epsilon$. Again, $\frac{1}{1-2\epsilon} < \frac{1}{1-\epsilon}$ so you can sell this obligation too for more than par ($1 + \psi_2$). Do the sale , tear up the loan, and pocket $\psi_2$. Rinse, lather, and repeat until they don't have a hundred bucks to lend you.

Update:
Hopefully this is a simpler example. What you do is in each period buy T (T$\geq$2) period a bond from Henry the Hyperbolic discounter. You also sell back to Henry the bond you purchased in the previous period. You will be willing to do this as long as the time period is short enough that the profit on the transaction is high enough to make up for your own discount rate.

Consider a 2 year bond with the preferences above. The 2 year discount rate is $\frac{1}{3}$ which is less than the 1 year discount rate $\frac{1}{2}$.

Eddy the exponential discounter is willing to lend the cash to Henry because is profits on that loan are high enough to compensate him, even though his rate of time preference for both 1 and 2 year loans is higher than Henry's.


I don't like the question because money movement is a current flow and it is also capital flow.

Anyway, the current outflow (to pay for the goods from Mexico) will be matched by a capital inflow. Mexico will use the increase in earnings to buy goods and services and investments in Vietnam. If we assume proportions are constant before and after the increase in earnings, then some amount of the increase will go into investments in Vietnam. Investments are capital.


Why is the Free Rider Problem a problem?

According to the standard criterion used to evaluate welfare (Pareto efficiency), the problem with 'free riding' is that goods are not produced even though they cost less to produce than they are worth to consumers. That means that we are forfeiting an opportunity to help people without harming anyone else.

To illustrate using a standard example, suppose that 10 people live on a street, and that each would value an additional street light at £100 (i.e. each would pay up to £100 to bring a new street light into existence). The street light would cost £101 to produce, but if produced can be used by all for free: this is known as 'non-excludability'. Will it be produced? If people are self-interested, then the answer is no. Although any resident would benefit from the street light, building it would cost them £101, which exceeds the benefit (£100). That's a problem since the total benefit (10 x £100 = £1000) greatly exceeds the cost of the light. If the inhabitants could be compelled to pay for the street light, sharing the cost equally, then they would all be better off.

I should stress that, from the perspective of Pareto efficiency, the problem with 'free riding' has nothing to do with 'equity'. To use your example of parks, the problem is not that people get to use the park without paying. Rather, the problem is that, since people are expected to free ride, the park won't be created in the first place.


Directed search with privately informed buyers and capacities

I am aware of classic models such as Kreps & Scheinkman (1983). I am also aware of Burdett, Shi & Wright (2001), but note that their buyers are ex-post symmetric as both of them have value $v_i=v=1$.

I am also aware of the literature on competing auctioneers, e.g., Peters & Severinov (1997) and Virag (2010). The buyers' equilibrium selection strategy in such models is quite cool: If sellers post second-price auctions with reserve prices $r_1>r_2$, buyers of value $v<r_2$ abstain from buying, buyers of value $v \in [r_2,x]$ visit low-price seller 2 with certainty and buyers of value $v>x$ visit each seller with probability $1/2$. One can solve for this cutoff value $x>r_1$. Under a condition on the lower bound of $F$, there is indeed an equilibrium in which both sellers set the same reserve price (which is below the lowest possible buyer value).
Knowing this, one can write down an underlying wasteful costly rationing process that essentially replicates the buyers' selection equilibrium above. However, I find this a little far fetched.

Any other approaches? As I said, this is not my literature - I just find the problem too basic to be neglected.


How much real value has the stock market lost since Feb. 2020?

Increases in the money supply do not affect the real value of investments unless they lead to inflationary increases in the (quality-adjusted) prices of goods and services. Which this increase has not done. If you choose to measure the value of your investment as the proportion of the M1 money supply that it represents then sure, it has been devalued, but that’s a rather pointless and futile way to measure the value of investments.


Confusion about the effect of an increased expected inflation on the price level

From what I understand, everything here has been expressed in real terms. With the announcement, people now expect higher money growth and 'higher inflation'. True that the nominal interest rate rises, they earn greater return on savings, yet it is of no use to them if price level rises. Note that, here, only real rise in interest rate would induce people to save more. Thus now, according to the author, consumption tends to rise and how that might lead to rise in price level is clear.


Which economical consequences could happen to a country if it forbiddes importing cars?

1- Some basic macroeconomic fundamentals

I highly doubt a trade imbalance is causing high inflation. The trade deficit is likely happening because of economists call "Twin Deficits". I'll spare you the accounting that leads to this equation and present you a setorial balances identity. Let $S$ be private savings, $I$ be capital formation, $G$ be government spending, $T$ be government revenue (in taxes), $X$ be exports and $M$, imports. We have:

$$(S-I) + (T-G) = (X-M)$$

In intuitive terms, the private sector balance plus the public sector balance, which is basically the domestic balance, must equal the trade balance. This happens due to sheer accounting, it's not a "theory". This is the same in the US, Venezuela, China and in a tribe of herdsman.

When $(T-G)$ is negative, we have a budget deficit - your country's case. Assuming people are saving the same amount, investment must fall or imports must go up. The more realistic scenario is that both things happen.

And we also have the Long-Run Fiscal Equilibrium condition. Let $M^s$ be money supply, $B$ be government debt, $P$ the price level, $r$ be the real interest rate and $t$ be time, we have:

$$ \frac{M^s + B}{P} = \sum_{t=0}^{\infty} \frac{(T_t - G_t)}{(1+r)^t}$$

This basically means that real government passive must equal the liquid present value of expected government savings. If people expect the government to continue to have budget deficits, either interest rates or the price level must go up. That's what's causing inflation. This also means that if the government increases the money supply without credibly signalling it will save more in the future, it will also implicate an increase in the price level, inflation.

2- Some basic microeconomic fundamentals

If importing cars gets prohibited, the supply of cars will go down. The supply curve shifts to the left. This means less cars available and more expensive ones.

Higher car prices signal domestic producers to make more cars. Supply goes up again and prices tend to converge to a new equilibrium. The problem is making cars domesctically might no be very efficient. Your country employs more capital and labor in other activities because it is more productive to do so. If it weren't and there was an easy opportunity to make money by producing cars, people would just do it.

Restricting imports is basically making your country poorer by forcing it to have less goods and services available and allocating precious capital and labor to activities that aren't as productive. It's a very bad idea.


Thought about a very simple example: Suppose we have two distinct points $(X_{1},Y_{1})$ and $(X_{2},Y_{2})$. The best regression line for these two points would be the line that passes through both points. However, any line passing through $(\frac{X_{1}+X_{2}}{2},\frac{Y_{1}+Y_{2}}{2})$ would let $\hat u_{1}+\hat u_{2}=0$, just like the perfect regression line.


Can Autonomous Consumption be Greater than Income of individual?

Expressing consumption as

$$C = a + f(Y)$$

where $Y$ is income, there is nothing in the theory that forbids

$$a > f(Y)$$

even though the theoretical anticipation and the empirical regularity (i.e. for most people) is that $a < f(Y)$.


What does it mean when we say an iso curve has no interior?

From Wikipedia:

  In mathematics, specifically in topology, the interior of a subset S of points of a topological space X consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S. [...] If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is completely contained in S. (This is illustrated in the introductory section to this article.)


When demand increases why does the price decrease but equilibrium price increase?

On a demand curve when the demand increases the price will decrease.

You actually mean "along the demand curve, a decrease in price will increase quantity demanded, all else equal". This is the law of demand, and it holds for ordinary ("non-Giffen/Veblen") goods that have downward-sloping demand curves. There is no shift of the demand schedule in this scenario:

  However on a demand and supply graph, when the demand shifts to the right, the price will increase.

Yes, when the demand curve shifts to the right (all else equal), the equilibrium market price rises to account for the fact that consumers are now willing to pay more. The increase in price is the mechanism by which excess demand at the initial price is cleared:

There is no contradiction here in the supply-and-demand model. Note that movements along the demand curve can only occur when we are away from a market equilibrium. These price movements are traced out by shifts of the supply curve and they continue until the new market equilibrium is reached.

Based on your vocabulary, you are likely misunderstanding the demand curve with actual quantity demanded.


Why are averages preferred to extremes on the same indifference curve?

That is false. Whoever told you this is mistaken.

Or it could also be that you misheard/misread and are confusing this with the idea that the bundle of 100 apples + 100 oranges is (usually) preferred to the bundle of 200 apples or the bundle of 200 oranges.

  Doesn't everything along an indifference curve have the same preference?

Yes.


Anonymous/impartial SWFs focus only on the pattern of well-being,
  and not the identities of the people who end up at particular well-being levels.

Identities here simply means names. When applying an impartial SWF, one only looks at the profile of utilities, not who those utilities are associated with. Consider the following two cases:
\begin{array}{ccc}
&U_{\text{President}}&U_{\text{CEO}}&U_{\text{Farmer}} \\\hline
\text{Case 1}& 100 & 80 & 10\\
\text{Case 2}& 10 & 100 & 80
\end{array}
The patterns of the distribution of well-being/utility in the two cases are essentially the same, and yet they differ in who is better/worse off. An impartial SWF would generate the same outcome in both cases, since it only focuses on the pattern (i.e. the utility numbers) of the well-being distribution and not on who (i.e. the identity) has which level of utility. In particular, the fact that the President is the happiest person in Case 1 but the most miserable in Case 2 has no impact whatsoever on the outcome of the impartial SWF.


How come websites such as Khan Academy and Anatomy Zone do not charge for their services?

Khan Academy is a 501c non-profit. They do some fundraising, they have some regular backers, and some big time contributors like these.

I imagine it's a similar case for Anatomy Zone. They are affiliated with a bunch of other non-profits that you can see on the bottom of their front page.


As a comment noted, without a reference, it is difficult to give a definitive answer. However, I have often run into similar phrasing in financial market commentary, so I will offer an initial guess as to what you read. This answer is just giving an initial idea as to what topics you could search for to get more information.

This phrasing is often used by members of the Austrian school of thought in economics. (The name "Austrian" refers to the fact that many of the founding members of the school of thought came from Austria.)

The Austrian school is very free markets oriented, and highly critical of government interference in the economy. The Austrian Business Cycle Theory - which I am not an expert on - argues that interest rates are important for planning in the private sector. If the central bank sets the interest rate away from the "correct" level (I am unsure how they phrase this technically), this leads to distortions in private sector activity (away from where the private sector would have naturally been). This leads to imbalances and eventually a crisis/recession. Since the economic activity is not "natural", it is often referred to as "fake." (Since interest rates are determined by the price of bonds, this is the link to asset prices and QE.)

The Mises Institute link to website would be probably the best site to learn about Austrian economics.

However, the Austrian school is not well-represented in academia at present. Most other economic schools of thought currently do not take the Austrian criticisms very seriously. Austrian thinking has not changed much over the past decades, and most non-Austrians view them as having been debunked a long time ago. As a result, it would be difficult to find recent authoritative criticisms of Austrian business cycle theory. In other words, it would be probably be hard to find criticisms of the idea that QE has led to a "fake economy." For example, from the post-Keynesian perspective, it is unclear how we can even define "fake" in this context.

I believe that if you wanted to pursue this, you would need to look further into Austrian theory, and ask a narrow question about a particular topic.


Wouldn't an unconditional basic income lead to an inflation that negates its effect?

Theoretically basic income might have a tendency to increase inflation. But that largely depends on how much is the basic income provided. Usually, the idea of basic income itself means that it is 'basic' and is just sufficient for a subsistence. With such an income level, the problem of inflation does not crop up really.
However, basic income might have other serious problems with crowding out due to large government spending and the disincentive to work and hence loss of productivity.

Even if we suppose, there is a large basic money disbursement which can cause inflation, then I think it should not be unconditional. Usually, such plans are in the form of guaranteed employment programs, buying of ration from designated stores and many more measures. However, I still feel these measures are not for tackling inflation rather the disincentive to work.


While people are spending less, they are also earning less, especially the many millions who have lost their jobs. It is very possible that the earnings lost are much greater then the money saved on reduced spending, and so net savings will decrease. It will take time before we really know either way.

Here is a study which looks at the increase in savings at the start of the so-called Great Recession (2007-2008):

  As the crisis began, households cut spending and the saving rate shot
  up to about six percent within a year. During recessions or other
  periods of economic weakness, when any engine of growth is needed, an
  increase in the household saving rate could slow the pace of recovery
  because, by definition, when a household saves more it consumes less.

Increased savings could potentially be good for growth in the long run if it allows for greater investment in the near future, and of course saving is good from the perspective of individual households that are able to save. But for the overall good of the economy as a whole, we need spending right now much more than we need savings.


Is it accurate to state that an economist cannot assign a true numerical value for utility?

I think the Investopedia article does not use clear terminology which leads to confusion.
In Economics, utility can be split across two categories as ordinal and cardinal utility.
In the case of ordinal utility assigning numeric values is meaningless. For example, if A is proffered to B, assigning 200 utils to A and 100 utils to B is meaningless because that would imply that A is twice as good as B, but the ordinal utility can only measure preferences, not intensity.
In the case of cardinal utility, which measures both preference ordering and their intensity it is completely valid to assign some value like $u(A)=200$ utils or $u(B)=100$ utils. In this case, the numerical values can be assigned and they are, mathematically,  'true' numerical values.
However, from my reading of the Investopedia article, they seem to be referring to the measurability problem.  That is an old discussion in economic theory that revolves around whether utility can be actually measured in some way (i.e. can we somehow scan a person's brain to measure precisely utility).
Now from a purely practical perspective whether the utility is measurable or not we definitely cannot measure in any direct way presently. Right now we can only observe people's choices and their revealed preference which is only solid basis for ordinal utility, not cardinal utility. However, this does not mean that utility would be immeasurable. In fact, famously  Von Neumann and Morgenstern (1944) in their Theory of Games and Economic Behavior argued that utility even if we cannot measure it yet is measurable. The authors argued that utility can be viewed the same way as temperature in physics. Temperature also originally could not be measured beyond just saying it is getting colder or warmer but developments in thermodynamic theory later allowed us not just to cardinally measure temperature, but even allowed us to recognize that there is actually some absolute zero and absolute units temperature (presently even Celsius scale is actually a derived measure in terms of Kelvins).
Consequently, in the view of  Von Neumann and Morgenstern utility should also be measurable and their view is quite influential although not accepted by all. If utility is measurable then there is a 'true' underlying measurable utility value to any choice. Whether we actually have instruments to do the measurements or not is irrelevant for the problem.
However, from my reading of the Investopedia article I think that what Investopedia refers to is the present practical inability to directly measure utility. I cannot see any indication from the article that it is taking stance on the measurability problem itself. Economists definitely don't go around hooking jumper cables to peoples heads and directly measuring utility from their choices (yet). Consequently, it is an accurate statement from a present-day practical perspective but not necessarily valid statement from the perspective of measurability problem.
Response to second question after edit:
The issue here is that true numerical value does not necessarily depend on measurement. As explained in this article from Science Campus site:

The measurement value (which is sometimes referred to simply as the measurement) is the value given by a measuring instrument and the true value is the actual value of the property being measured.

Of course, there are no instruments to measure utility, however that does not mean that there are no true numerical values for it. In fact, as argued by  Von Neumann and Morgenstern (1944) in cardinal utility there will be some mapping of the underlaying preferences on the set of real numbers. Hence, cardinal utility actually does represent the 'true' values of utility (within the vNM framework at least).
However, Investopedia is correct in pointing out that economists do not measure it when constructing utility because (presently) there is no practically viable way that would in real life make it possible to create such mapping from the unobservable preferences onto real numbers. But just because we are not able to practically make such mapping does mean it cannot exist (again in past people were unable to map the true values of temperature onto real numbers - in fact for the  most part of the human history such mapping was impossible until technology and science advanced enough to make it possible).
Furthermore, of course there are economist who are pure ordinalist but from my reading of the Investopedia article they are not taking pure ordinalist position rather they mean that we are simply unable to measure utility in practical sense.


Is the forex market a form of bartering?

No by definition forex is not bartering. By definition (see here) bartering is

Barter is an act of trading goods or services between two or more parties without the use of money (or a monetary medium such as a credit card). In essence, bartering involves the provision of one good or service by one party in return for another good or service from another party.).

In forex market you are exchanging money for money. Even if you would want to consider the foreign currency a good on an account that it is not legal tender in your country of residence (which would be debatable), you are still exchanging money for that good - not another good. So this cannot be barter exchange by definition.


Is there any widely accepted downsides for Cuban system?

The Cuban system is widely criticized. Particularly by people who experienced it.

(Aside: In fact, this is consistent with the general trend that people who have actually lived under Communist, Socialist and Marxist systems tend to condemn those systems. And the only people who tend to praise such systems are people who never lived under those systems yet favor imposing those systems on others against their will.)

roughly 1 in 10 of all Cubans choose to live in the U.S. (despite the extreme hardship of making that migration)
roughly only 1 in 100,000 Americans choose to live in Cuba (despite the relative ease of making that migration and many of whom are FBI fugitives, avowed Communists and members of other radical leftist organizations)

Notwithstanding the above observations and statistics, let's study two of the question's premises.

  Cuban medical system is as far as I know well known for the quality of its formation and the medical workers and doctors it gave. Cuba has also a good educationnal system.

While it is true that the medical system has been praised. Many criticize the praise as being inaccurate. For example, this research paper from the University of Miami in 2007 cites:

  My own research, however, suggests that the unequivocally positive descriptions of the Cuban health care system in the social science literature are somewhat misleading. In the late 1990s, I conducted over nine months of qualitative ethnographic and archival research in Cuba. During that time I shadowed physicians in family health clinics, conducted formal and informal interviews with a number of health professionals, lived in local communities, and sought to participate in everyday life as much as possible. Throughout the course of this research, I found a number of discrepancies between the way the Cuban health care system has been described in the scholarly literature, and the way it appears to be described and experienced by Cubans themselves. [emphasis mine]

Further, the author writes:

  After just a few months of research, however, it became increasingly obvious that many Cubans did not appear to have a very positive view of the health care system themselves. A number of people complained to me informally that their doctors were unhelpful, that the best clinics and hospitals only served political elites and that scarce medical supplies were often stolen from hospitals and sold on the black market. Further criticisms were leveled at the politicization of medical care, the unreliability of health data and the overall atmosphere of secrecy surrounding the prevalence of certain infectious diseases such as HIV and tuberculosis.

We know that Socialist, Communist and Marxist regimes are not transparent with their information. So many consider the official reports of the quality of Cuba's health care system to be unreliable at best.

Perhaps the most objective indictment of the Cuban system, however, is to look at migration patterns into and out of the country.

In 2017, the Migration Policy Institute reported findings that:

  Cubans have been among the top ten immigrant populations in the United States since 1970, and in 2016 were the seventh largest group. Nearly 1.3 million Cubans lived in the United States in 2016,

Further, they report:

  Compared to the overall foreign- and U.S.-born populations, Cubans were less likely to be proficient in English, had lower educational attainment, and earned lower household incomes.

Researching migration patterns in the opposite direction: from the U.S. to Cuba, I found this Wikipedia article which estimates there were nearly 2,000 to 3,000 Americans living in Cuba in 1998. And that many of them were FBI fugitives, avowed Communists and members of other radical leftist organizations.

The fact that roughly 1 in 10 of all Cubans choose to live in the U.S. (despite the extreme hardship of making that migration) and roughly only 1 in 100,000 Americans choose to live in Cuba (despite the relative ease of making that migration) is a very objective and quantitative indictment against the Cuban system in addition to the more qualitative indictments provided from the above sources.


Will this policy advantage the financial sector over other parts of the economy?

I do not see how this would a priori advantage the financial sector over other parts of the economy. This policy is done to stimulate economic activity by putting pressure on lowering the long-term interest rates and to push pressure on inflation and lower exchange rate. Such expansionary monetary policy is conventionally considered to stimulate real economic activity in the short-run (see any undergraduate macro textbook for example Blanchard et al. Macroeconomics: a European Perspective). There is no reason to a priori think this specially advantages banks over other sectors, as this is macroeconomic policy with aim to stimulate the whole economy, although banks will likely benefit as well.

Why doesn't the reserve bank give this $100B to treasury directly, so that fewer bonds need be issued to support gov spending?

This is for several reasons. First, giving money directly to government would mean that money supply is permanently being increased by that new money (technically that money could be still destroyed in some ways but not unliterary just by central bank). When central bank buys government bonds the money supply is expanded only as long as the government bonds are not repaid. Central banks still have option of scrapping the bonds and hence making the money supply expansion permanent, but they also have other options. Central banks will usually want to expand money supply in recession but in expansions they might want to contract it. They could contract it also with other ways (like increasing reserve rations), but open market operations are usually one of the preferred tools of central banks.
However, this does not affect government budget negatively. Any interest that government pays to central bank is then returned to the government. So from the perspective of government this debt has no cost. In fact such extra demand on government bonds will likely make it cheaper for government to issue bonds on primary market so if anything it might indirectly help it.
Also, the reason why it is done on secondary market rather than primary market, is that central bank is supposed to be independent from government. Hence they try to always keep arms length relationship with it.

Alternatively, why doesn't the reserve bank distribute this $100B directly to ordinary households (e.g. \$4k per resident)?

This is what is called 'helicopter money'. Some economists are in favor of such policy, however arguments are being made that it threatens central bank independence. Helicopter money is very close to fiscal policy and if people learn they could get 'free money' from central bank they might put political pressures on central banks being involved in such transfers beyond what is warranted by monetary policy. It is argued that any fiscal policy should be done by elected representatives and not by unelected technocrats, and helicopter money is very fiscal-esque.   See more on this in Reichlin; Turner; Woodford (23 September 2019). Central bank independence is in economic profession revered in similar way as in law profession independent judiciary or in journalism independent press so anything that threatens central bank independence is hard sell. Of course, this raises additional question of why the central bank independence is so important - exploring that is beyond the scope of this answer but you can read more about it in this ECB explainer.    In addition in most countries this would not even be legal although laws can be changed.

Edit:
Theory above aside, even when we look empirically at aggregate measures of bank profitability we can see that in the post 2007 they are low, as you can see from the figure below which plots price-to-book ratio for banks, which is a measure of profitability and expected profitability that stock prices capture (see Carletti, Claessens, Fatás, Vives 18 June 2020, and sources cited therein).

QE was not common before 2007 and became quite commonplace after, due to the Great Recession. Hence, there is not much evidence to suspect that banks are somehow 'getting rich' out of QE. Furthermore, of course the above is just raw data so it has to be interpreted with caution. One could argue that maybe there are some other factors that depress  bank profitability inspite of all the QE.
However, when we look at some serious research it actually confirms that QE together with negative interest rates and other recent bank regulation erode bank profitability (e.g. see Florian 2018).  Hence, if anything there is research and evidence supporting the idea that QE, combined with other policies central banks nowadays commonly pursue, makes banks less profitable not more.


Does higher tax on luxury goods does more harm than benefit?

Luxury goods tend to be Veblen goods, so reducing taxes on them doesn't make them more attractive. They also tend to be goods whose value is mainly based on scarcity, so increasing demand doesn't increase the amount supply, it just increases the equilibrium price.  If diamonds were cheaper, would there be significantly more diamond jobs?

There's the further issue that taxes just move money around. Giving rich people a tax break means either raising taxes on poorer people, reducing government spending, and/or deficit spending (which reduces the amount of capital available for private investment).


What guarantees that endowed agents have non-zero prices in an Arrow-Debreu Economy

A pretty trivial assumption would do it:

If the consumers preferences are strongly increasing, i.e.
$$
x \succ_i y, \mbox{  if $x \geq y$ but $x \neq y$ },
$$
and unbounded for all $i$, then no $x$ can be an optimal bundle if $p \notin \mathbb{R}^C_{++}$ (assuming there are $C$ commodities).
So zero price cannot arise in an Arrow-Debreu equilibrium.


Can we have incomplete markets with a continuum of securities?

Thanks a lot for the reference. I think the result does hold. Here is what I found:
The validity of such a law of large numbers what subject to some debate in the 1980s.
See Judd (1985), Feldman and Gilles (1985) and Uhlig (1996) for representative papers.

Luckily Feldman and Gilles show that for a probability space $(Y; B(Y ); \Pi)$ there exists a continuum of random variables $y(i;.) : \Omega \rightarrow Y$ such that for all $i$ the random variable is distributed according to $\Pi$ and that for all $\omega \in \Omega$ and all $D \in b(Y)$
$$m(i \in I : \{y(i; \omega) \in D ) = \Pi(D)$$
that is, the population income distribution is nonstochastic and given by $\Pi$ The continuum of random variables cannot be pairwise independent.


Are there import taxes between US states?

No. Such interstate taxes are prohibited by the Commerce Clause as they would be "undue burdens on interstate commerce".

The Commerce Clause is an evolving doctrine that has been largely developed by the courts (in particular the US Supreme Court).

The most recent development was South Dakota v. Wayfair (2018). Section A gives a summary of this doctrine's development and states:

  Modern precedents rest upon two primary principles that mark the boundaries of a State’s authority to regulate interstate commerce. First, state regulations may not discriminate against interstate commerce; and second, States may not impose undue burdens on interstate commerce.


You are confusing your balance sheet with your income statement, and putting a few things in the wrong bins. The basic accounting identity is:

$$ Assets=Liabilities+Owners' Equity$$

Assets are generally valued not at cost (or as you put it, "the price I have paid them on the first place"), but at either their hold-to-maturity value (i.e., the present discounted value of all future cash flows from the asset), or their mark-to-market value (what you can sell the thing for).

Liabilities are money owed to others that is unpaid as of the date of the financial report— salaries do not generally appear on the balance sheet, except for a small amount that has accrued but hasn't been paid out yet. Salaries will primarily appear on the income statement. Lease accounting is complicated, but if you have capital (not operating) leases, those will show up on both sides of the balance sheet under GAAP.

Equity (including equity held by investors other than yourself) is simply the residual once the value of outstanding liabilities is subtracted from the value of assets. It is definitely not the "actual money [you] get from selling these things without subtracting costs."


Difference between CCAR and ICAAP stress tests

ICAAP stress tests are for internal capital or economic capital, which are requirements under Basel II pillar II.
CCAR stress testing is for regulatory capital, which is also under Basel II. US regulators provide specific scenarios and risk factor shocks.


Is there a typo in Sims(2002) paper on solving linear expectations model?

Yes the $z(t)$ column vector, as it appears in $(4)$ is written upside-down in the translation. Also the first equation in $(2)$ is translated lagged once, i.e. the translated system describes the $w(t-1)$ equation.

Finally there is a typo in the $\Gamma_1$ matrix, there is a void 6th column that should be ignored, and $\theta$ should be in the 4th column not in the 5th (because it multiplies $u(t-1)$.


There is some data like that, but not split by private/government entities holding it.

  Exhibit 5 compares foreign holdings of U.S. long-term securities with estimates of U.S.
  holdings of foreign long-term securities as of selected survey dates. At \$17,481 billion, foreign
  holdings of U.S. long-term securities remained considerably larger than the \$10,734 billion in
  foreign securities held by U.S. residents at end-June 2017.

Actually BEA has more up-to-date NIIP data. The latter includes much more than just long-term securities, e.g. derivatives too.

The US Fed does buy some foreign currencies as part of their SOMA (System
Open Market Account) operations, and the Treasury also does that through their Exchange Stabilization Fund (ESF). According to that source these

  are held in [...] in a variety of instruments that yield market rates of return in their respective currencies and have a high degree of liquidity and credit quality.

For example:

  As of June 30 [2018], the euro reserves held by both the SOMA and the ESF totaled \$25.4 billion [...] The amount of yen-denominated deposits and government securities held by the SOMA and the ESF fell to \$16.9 billion [...].

Note that these numbers are pretty insignificant relative to the private positions from the first graph I posted.

The same source mentions that the US does have reciprocal currency swaps with the ECB and the BoJ (about \$1 billion with each). It doesn't have any with China though and the latter's central bank holds large US reserves, trillions...