Deriving the Euler equation from a Continuous Time Dynamic Programming Problem (HJB)

As already commented, the equation you probably meant is
$$
\rho V(k)= \sup_c \{\, u(c) + V'(k) ( f(k) -\delta k -c ) \,\}.
$$
I have never seen this equation called the HJB equation (probably missing a basic reference on my part). I'll call it "dynamic programming PDE".
What you're really asking is the connection between two approaches to solve control problems---Calculus of Variations/Optimal Control vs. Dynamic Programming.
Optimal Control (Pontryagin's Maximum Principle) is a first-order perturbation argument and Dynamic Programming Princple is a backward induction argument.
"Euler equation" arises from a first-order perturbation argument.
(In continuous-time, it's a classical Calculus of Variation equation. In discrete time, I've only heard it used in economics, describing intertemporal consumption smoothing, but it's a perturbation argument
just the same.)
In continuous-time, first-order perturbation of optimal path means a perturbation along the entire path.
In contrast, in discrete time, it suffices to perturb a single period--therefore one can obtain the Euler equation by simply differentiating the Bellman equation (under e.g. Benveniste-Scheinkman assumptions that ensure differentiability of value function).
In continuous-time, I don't believe the answer is as trivial.
There is, however, a classical connection between Optimal Control and Dynamic Programming via the method of characteristics.
Part of the connection is that, if the value function is sufficiently smooth, then the characteristic equations of the dynamic programming PDE gives
Pontryagin's Maximum Principle.
In your growth example, the maximizer of RHS is given by $u'(c^*) = V'$. Substituting then gives an implicit
first order ODE
$$
F(k, V, V') = \rho V  - u(c^*( V')) - V'\cdot( f(k) -\delta k) + V' \cdot c^*( V') = 0.
$$
Following the method of characteristics heuristicly, one of the characteristic equations would be
\begin{align*}
\frac{d}{dt} (V') &= -\lambda (F_k + F_V V') \\
                  &= -\lambda (- V' f'(k) + V' \rho),
\end{align*}
for some $\lambda \geq 0$.
Since $u'(c^*) = V'$,
\begin{align*}
\frac{d}{dt} \log (u'(c^*)) &= -\lambda (F_k + F_V V') \\
                  &= \lambda ( f'(k) - \rho),
\end{align*}
which is an Euler equation.
(Incidentally, the ODE $F(k, V, V')$ does not seem amenable to guess-and-verify, even when $u(c) = \log c$.)


What he's saying is that once the burger is already made, the cost of making the burger is a sunk cost, and thus the marginal cost of the burger is just the cost of the tiny labor involved in picking it up and selling it to the customer.

That's sort of an odd position to take.

Obviously once the burger is cooked you can't recuperate the costs, but in the aggregate, the decision to cook some number of burgers in such and such a time period is made before actually cooking them. So the cost of making a burger (including labor) isn't a sunk cost; it's a prospective cost.

Additionally, just because the monetary cost of making the burger is the same for each burger doesn't make it a sunk cost, just fixed (perhaps). But more to the point, there's a time cost which varies the more burgers you cook. We pay for convenience. Part of the marginal cost of cooking the burger is the opportunity cost of that time that could be spent not running a nice little fast food restaurant.


theoretical macroeconomic topic for bachelor project?

I would be careful about confusing topics and tools. Theoretical mathematics and data analysis are ways that you can approach a specific topic. No matter what topic you choose, you can apply your mathematical tools. I would recommend looking for something that interests you at a conceptual level and then seeing how you can apply your given set of tools to answer it.


In general, no, it can't impose stricter rules on imports than it does on domestic production. Or more accurately, it could, but the countries exporting fur to Norway could challenge this. The guiding rule is that domestic industry cannot receive preferential treatment over importers.

Norway can do it the other way around - that is to say, WTO rules would allow it to continue to import furs after closing down its domestic industry.

But there are always way around this. For example, by setting very specific standards (which are not explicitly to do with country of origin) that apply to everyone, but that only domestic industry complies with. If the standards are crafted well enough, they'll pass WTO's scrutiny.


What kind of tax diagram can show the full consumer tax burden?

I would recommend that you read what Harvey S. Rosen and Ted Gayer have to say about taxation and tax burdens on their book, Public Finance.
Your hypotheses for perfectly inelastic demand might be flawed, to the extent that (1) the house market is competing with Airbnb-type companies and (2) -for inelastic supply- there is an affluence in housing available (that is, there are vacant houses -which is most always true in most first-world countries).


Is there an index, parameter, ecuation, or anything to compare the tax pressure exercised by a country in relation to others countries?

I am not sure how you calculated the tax revenue in relation to GDP, but one measure would be tax revenue as % of GDP. This is quite common measure of how heavy the taxation is because it tells you what percentage of total output produced by country is collected by government. OECD provides data on this measure here.

However, it can always happen that even a country with high tax rate collects less taxes than country with lower tax rate. First, you have classic Laffer curve - if you set taxes to 100% you collect zero tax revenue because nobody will work for fee so you collect the same tax as at 0 tax rate - hence having higher tax rates does not necessarily mean higher tax revenue. This being said empirically most real life countries don’t have taxes so high as to be over the peak of the Laffer curve so this will probably not apply to your case. Another, issue is tax avoidance. In some countries it’s child’s play to avoid VAT or many people can work in informal sector to avoid income tax. If government does not have good institutions that prevent this then again even government with 50% tax rate can collect less tax revenue than government with 10% tax rate if in second country tax authorities are efficient in preventing tax avoidance but in first country not.

Another possible measure could be looking at average or marginal tax rate across the economy and compare these between two countries. However, this is just crude comparison of how high taxes are.

If you would care about comparing the burdens the tax systems in the two countries create then a good measure is the marginal cost of public funds or marginal excess burden of taxation. This is a measure of by how much government must reduce economic activity (by its distortionary taxes) to extract 1 euro (or local currency). For example marginal marginal excess burden of  0.5 would imply that to raise 1e in tax revenue output on margin contracts by additional 0.5e so in total government has to take “out” of the economy 1.5e to get 1e. This gives you some idea of distortionary effect of taxes (in case of no distortions the marginal excess burden should be 0e).

You can also calculate average excess burden which is the same measure but it gives you information about average burden instead of burden at the margin (i.e. the last euro collected). I don’t know of any dataset that just gives you this measure though because it has to be estimated and based on different estimation techniques or assumptions behind the model used you can get little bit different values, but you can try to search for it in google scholar. For US there are many papers trying to estimate the excess burden of almost any tax you can think of, but if you come from small country research might be not available on this.


Why would foreign asset purchases by PBOC temper yuan appreciation?

Quote...

"Suppose the PBOC buys a bunch of dollar-denominated bonds, using its
dollar reserves. All else equal, what effect should this have on the
yuan-USD exchange rate?"

Quote...

" I am having trouble seeing whether or why it might cause the yuan to
depreciate".

All else is not equal. To return to the original quantity of USD money held it will buy some USD from commercial banks, issuing renminbi in order to do so. Issuing renminbi will have the tendency to cause it to depreciate.


Assumption of sufficient wealth in quasi-linear preferences

This depends on the exact preferences, but usually the utility function
$$
U(x,y) = v(x) + y
$$
is such that
$$
\lim_{x \to 0} |MRS(x,y)| = \lim_{x \to 0} \frac{\text{d}v(x)}{\text{d} x} = \infty.
$$
In this case it is the consumption of an additional marginal unit of the nonlinear good $x$ that is infinitely useful compared to the consumption of an additional marginal unit of the linear good $y$. Hence the consumer will consume some units of $x$ before buying any of $y$. If the consumer is not 'wealthy enough', the corner solution will be such that $x > 0$, $y = 0$.


is univerisity's scholarships a price discrimination?

Scholarships certainly count as price discrimination. Charging different prices to different groups of consumers for the same good is third degree price discrimination, which is what occurs with school sponsored scholarships. First degree price discrimination occurs when a firm charges a different price for every unit consumed, and while theoretically possible for a school to do, is not done in practice and would require a much more complex pricing scheme than that offered by scholarships in the usual sense of the word.


What does it mean when someone says “2 trillion wiped off markets”?

That's connected to the way currency or prices fluctuate, particularly in the stock exchange. It comes from the approach that value (e.g. of a firm) equals the sum of all the stocks at current price. That's often called the "Market Cap(italization)".

Think of a firm A with 10.000 stocks. If from one day to the other the stock of company A goes from 100€ to 50€, then its market capitalization/market value goes from 1M€ (=10.000 stocks * 100€) to 500k€. In this conditions you could say that half a milion was wiped off the market or from this company.

That answers directly to your question, but keep in mind that there are many other ways of calculating the value of a firm.


max {a, b, c} means the highest of these elements.

For example, if "b" is higher than a and c, then max {a, b, c} = b


When was fractional reserve banking introduced?

Fractional reserves banking is at least as old as the 1400's:

  A balance sheet from the London branch of the Medici Bank, dated
  November 12, 1477, shows that a significant number of the bank’s debts
  corresponded to demand deposits. Raymond de Roover himself estimates
  that at one point, the bank’s primary reserves were down to 50 percent
  of total demand liabilities. 60 If we apply the standard criterion
  used by A.P. Usher, this implies a credit expansion ratio of twice the
  demand deposits received by the bank. There is evidence, however, that
  this ratio gradually worsened over the bank’s life-span, especially
  after 1464, a year that marked the beginning of growing difficulties
  for the bank. The roots of the general economic and bank crisis that
  ruined the Medici Bank resemble those Carlo M. Cipolla identifies in
  his study of fourteenth-century Florence. As a matter of fact, credit
  expansion resulting from bankers’ misappropriation of demand deposits
  gave rise to an artificial boom fed by the increase in the money
  supply and its seemingly “beneficial” short-term effects.
  Nevertheless, since this process sprang from an increase in the money
  supply, namely credit unbacked by growth in real savings...

Page 73-74 of Money, Bank Credit, and Economic Cycles Jesus Huerta de Soto

The Wikipedia page on Scali Bank claims that the Austrian school or at least Soto believes that fractional reserve banking did in Scali bank in the 1300's but I couldn't find evidence for the claim in the citation.

Middle eastern banks were offering transaction accounts and loans in the  7th–12th centuries and this makes it likely that fractional reserve banking was occurring there as well.


In order to cover the supply and demand imbalance following methods can be used in addition to the ones mentioned by you:
TO DEAL WITH THE IMBALANCE WHEN DEMAND > SUPPLY:

In the long-run period, Innovation of substitute goods
First come, first-served basis
Innovation of better production techniques which would increase the supply
Commodities are sold only to the frequent costumers of the producers
Better transportation and communication systems on the producers part to increase the supply and to meet excess demand.

TO DEAL WITH THE IMBALANCE WHEN DEMAND < SUPPLY

Decrease in the prices of complementary goods.
Better credit facilities (ex: lowering the bank rates)


Can setting bank's reserve requirement ratio in 100% simulate gold standard?

Examine this statement:

  If you want a gold standard, it's easier to set bank's reserve requirement to 100%.

The only sense in which this is true is that it eliminates fractional reserve lending.

However, a key defining characteristic of the gold standard is that government-issued money can be converted into gold upon demand. This creates a real constraint upon government fiscal policy. Having a 100% reserve requirement does not imply such a constraint on government policy, the currency can be a fiat currency.


Profits of a price taking firm take the form
$$ p q - c,$$
where $p$ is the price, $q$ is the output and $c$ is the cost. Of course there is a relation between the output $q$ and the cost $c$ in the sense that higher $q$ will correspond to higher $c$. As such, these two quantities cannot be chosen independently.

There are two possibilities to make this relation explicit: (i) you express everything in terms of output or (ii) you express everything in terms of inputs.

In case (i), one usually defines a cost function $c = c(q)$ which gives the cost necessary to produce an amount $q$. In this case, the profit maximization problem takes the form:
$$\max_q  p q - c(q).$$
In case (ii) everything is expressed in terms of inputs.  If we call $x$ the amount of inputs then we can write $q = f(x)$ as the total output that can be produced using inputs $x$ and $c = w x$ as the total cost of using an amount $x$ of inputs. Here, $w$ is the unit cost per input (e.g. the wage rate if $x$ is labour). In this case the profit maximization problem is given by:
$$\max_x p f(x) - w x.$$

The various functions are related by the identity:
$$w x \equiv c(f(x)).$$


Can fiscal consolidation be expansionary?

There has been some papers and articles put forth from the IMF over the last few years suggesting that Austerity has caused many European economies to drag and not recovery from the great recession as well as they otherwise could have.  I pulled the first article I found referencing the IMF research (from Washington post): https://www.washingtonpost.com/news/wonk/wp/2012/10/12/imf-austerity-is-much-worse-for-the-economy-than-we-thought/

I think the largest concern with respect to expansionary fiscal policy in today's economic climate is the crowd-out effect that huge public debt can have on private investment.  Perhaps someone can link a solid paper about that effect.


M0 are notes (e.g. dollar bills in the US) and coins Held by the public. It seems likely that almost all the notes and coins come from the public withdrawing them from banks.

Banks get the notes and coins via exchanging settlement balances at the Federal Reserve (“reserves”) for them, which are put in vaults for distribution.

The only alternative is for the Federal government to directly pay out notes and coins, which is possible, but this does not appear to be common practice.

If we ignore that alternative creation mechanism, we can answer your questions.

Firstly, an increase of M0 is reliant on the public withdrawing notes and coins from a bank. This is only possible if the bank has notes and coins in its vault, so this process is limited. In order to be sustained, the vaults need to be replenished.

Yes, banks need to exchange reserves for notes and coins.
A bank could borrow reserves from another bank or at the discount window to get vault cash, but those are two transactions.
Settlement balances are destroyed when exchanged for notes and couns.


The trick is to apply the total derivative.
Going from $Y = C(Y-T(Y)) + I(r) + G + NX(Y)$, we have:
$$
\operatorname{dY} = \underbrace{\frac{\partial C(\cdot)}{\partial Y}}_{\equiv C_Y > 0} \cdot \operatorname{dY} + \underbrace{\frac{\partial I(\cdot)}{\partial r}}_{\equiv I_r < 0} \cdot \operatorname{dr} \therefore \operatorname{dY} - C_Y \cdot \operatorname{dY} = I_r \cdot \operatorname{dr} \Leftrightarrow \boxed{\boxed{\frac{\operatorname{dr}}{\operatorname{dY}}\bigg\vert_{IS} = \frac{1-C_Y}{I_r} < 0}}
$$
Now, with the new equation, we have:
$$
\operatorname{dY} = \underbrace{\frac{\partial C(\cdot)}{\partial Y}}_{\equiv C_Y > 0} \cdot \operatorname{dY} + \underbrace{\frac{\partial C(\cdot)}{\partial r}}_{\equiv C_r < 0} \cdot \operatorname{dr} + \underbrace{\frac{\partial I(\cdot)}{\partial r}}_{\equiv I_r < 0} \cdot \operatorname{dr} \therefore \operatorname{dY} - C_Y \cdot \operatorname{dY} = (C_r + I_r) \cdot \operatorname{dr} \Leftrightarrow \boxed{\boxed{\frac{\operatorname{dr}}{\operatorname{dY}}\bigg\vert_{IS'} = \frac{1-C_Y}{(C_r + I_r)} < 0}}
$$
Since $|C_r+I_r| > |I_r|$, the IS curve is now flatter and thus the income responds more to changes in the interest rate than before.


Condorcet's paradox: Is the majority rule transitive?

As you stated, transitivity is that overall $x \succeq y$ and $y \succeq z$ implies $x \succeq z$. I will show an example where majority rule isn't transitive and hopefully it will answer your question.

Imagine that we live in a world with three people: Person 1, Person 2, and Person 3. Each of these people have preferences over three outcomes $x$, $y$, and $z$. Every decision in this world is made according to majority rules, i.e. The outcome chosen is the one preferred by at least two of the individuals.

Now imagine that the three people have preferences according to:

Person 1: $x \succeq y \succeq z$
Person 2: $y \succeq z \succeq x$
Person 3: $z \succeq x \succeq y$

The majority rule decision making implies that we rank the pairs according to the following:

$ x \succeq y$ because Person 1 and Person 3 prefer $x$ to $y$
$ y \succeq z$ because Person 2 and Person 3 prefer $y$ to $z$
$ z \succeq x$ because Person 2 and Person 3 prefer $z$ to $x$

Notice that this is exactly a violation of transitivity.


Why did the Greek GDP grow so fast between 2002 and 2004?

You are using USD exchange rate GDPs, which are very volatile. The main effect is probably the fact that the Euro began to circulate in 2002, replacing the cheap domestic currency.

Here are some more general reasons why the peripheral countries did well before the recession:

There was abundant oil supply and the developed countries were splurging on FDI and debt in the periphery
Greece had just joined the Euro and was getting tons of transfer payments
The Euro was still fairly high and hadn't fallen yet.
Greece may have fudged its numbers a bit to get into the Euro.

Eur/usd since 2000:


Interest rate - Price of Credit vs Price of Money

Interest rates are not to be directly considered as the price of credit. Instead, they are unequivocally linked to it. This is simply due to how bonds are priced.

$$
P = \sum_{n=1}^N \frac{C + M[{I_{n=N}}]}{(1+i)^n}
$$

Where $i$ is the contractual interest rate, $C$ is the coupon payment, $N$ is the number of payments, $I_{n=N}$ is an indicator function taking the value of $1$ when $n=N$ is $true$ or $0$ otherwise, $P$ is the market price of the bond, $M$ is the value of the bond at maturity.

In a second time, there is a link between money and credit, but indeed, transitively saying that interest rates are "the price of money" is a bit simplistic since it omits a great part of institutional and financial mechanisms.


Is it possible to give, in economics, an example of a relation ( set of ordered pairs) that is not a function?

I feel that your question might be a little broad, but there are certainly many areas of economics where non-functional relations are used. Two simple examples (there are many others):

One of the most fundamental models of behavior in economics relies on the idea that choices can be represented by "preferences" which mathematically are binary relations that do not have to be functions (they very often are not, e.g., $A \succ B \succ C$ is not a function, since $\succ = \{ (A,B), (A,C), (B,C)\}$.
Economics also relies on correspondences, or ``multivalued" functions (which, depending on the definition, can be just another way to view binary relations). Correspondences are used in a lot of subfields of economics but perhaps most notably in consumer theory, where the choice set of consumer with preference $\succeq$ and budget set $B$ can be a whole subset of the consumption space $C(\succ,B) \subseteq X$ (where $X$ denotes the consumption space).


What is some intermediate or advanced macroeconomics textbook use general approach and focus on general topics?

Have you had a look at Dornbusch and Fischer's Macroeconomics? I would say that is a great book for both Introductory and Intermediate Macroeconomics. It emphasises on basic concepts (I used a really old edition but it still worked for me!) does a review after each section, the formulas are crystal-clear (I may be speaking only of myself though and I apologize in advance), brings up mathematical verification only occasionally, and there is nothing that the diagrams do not explain.

In addition, I used Greg Mankiw's Macroeconomics (still do) to run through topics- it covers most concepts and does not intimidate the reader at any point, but it is not so heavy on the math.

I do not have much idea as to what topics are covered under advanced Macroeconomics, but do take a look at these.


You could have completeness and give up unrestricted domain. A non social choice example:

The relation "larger than" or $\leq$ is not complete over the complex plane $\mathbb{Z}$, but it is complete if it is restricted to a subset of it, real numbers.

Similarly you could have unrestricted domain and give up completeness.

E.g. majority voting is not complete if you allow unrestricted domain.

Therefore these are separate properties. You can argue you need both, but perhaps you are willing to part with one or the other. Personally I agree that giving up completeness rather than unrestricted domain seems strange.


First we need to discuss what wealth is. Is it just the amassment of valubles? In the 16th-18th century, rulers thought so. This notion is called mercantilism. This idea was rejected by Adam Smith in book IV of The Wealth of Nations.

The current meaning of wealth is not the amount you own, but rather how much you can consume (sustainably). Formally, the definition is Total of all assets of an economic unit that generate current income or have the potential to generate future income. This is of course the same amount as the amount that you can sustainably consume. This means that we can measure wealth either as production or as consumption.

Thus, there is not just how much riches you own, but also how much you can create. Also, remember that you cannot consume money. You can only consume goods and services. Amassing riches has little value in itself. It is mainly a way (for an individual or group) to postpone consumption to some point in the future.

Thus, wealth is measured not just by the current riches you own, but also by the ability to create more riches in the future. A man with 100,000 and no income is poorer than a man with no money but with an income of 50,000 a year.

  These things [...] don't generate new wealth.

But they do.

Labor

Let's look at how labor increases wealth by looking at a couple of examples:

Example 1. I work at a farm. The result of my work is some farm produce worth X. For that I am paid Y. The farmer that hired me will receive X-Y.
The net wealth gain for the society is Y + (X-Y), or X.

Example 2. I work as a barber. I charge X for a haircut. The customer gets a haircut worth X. Net wealth gain for the society is X + (X-X), or X. Since the haircut is consumed immediately, the amassed riches don't increase. Note that the customer didn't lose out - to him a haircut was worth at least X or he wouldn't have gotten one. In macroeconomic terms the GDP increases by X and net savings is unaffected.

Trade

Trade increases wealth by transferring a good or a service to someone that values it higher.

Example 1: I buy a used car from a neighbour for X amount of money. I get a car that is worth Y to me (Y>=X, or I wouldn't have bought it). My neighbour gets X for a car that was worth Z to him (Z<=X, or he would have kept the car). The wealth produced is (Y-X) + (X-Z), or Y-Z, which is >= 0 because Z<=X<=Y. In GDP terms the wealth produced is X - X, or 0, because GDP only measures how much is paid.

Example 2: An electronics company imports batteries from China for 10 cents each. They sell the batteries to customers for \$3 each. The wealth increase in GDP terms is \$2:90 per battery. The wealth produced is shared by the electronics company, the shipping company, the store salesman and any others that provide services to the electronics company.

Innovation

Innovation is both the invention of new and the improvement of the existing. Actually, most innovation is small improvements in existing technology or design.

Example: I improve a machine that makes nails so that it can make twice the number of nails in the same time. This means that more nails can be produced (and thus consumed) using less labor.


There are many ressources available online

Synth is a statistical software that implements synthetic control methods for causal inference in comparative case studies with aggregate data as described in Abadie and Gardeazabal (2003), Abadie, Diamond, and Hainmueller (2010), and Abadie, Diamond, and Hainmueller (2013). The Synth package for R is also described in detail in Abadie, Diamond, and Hainmueller (2011).
A lecture on Synthetic Control Methods from Yiqing Xu.
A video lecture by Alberto Abadie himself on  Synthetic Controls for Policy Evaluation!


what are Investments (as defined in macroeconomics) that Households make?

The terms investment and capital are highly confusing because they are different in everyday language and in economics.

Definitions in economics:

Investment is the purchase of any (new) capital goods.
A capital good is any good that is used to produce other goods.
A good is anything that satisfies human wants.

(a) By convention, a house counts as a capital good, the idea (or fiction) being that it generates a stream of housing services over time. Hence, the purchase of a new house counts as investment.

(b) A share in a company does not count as a capital good, because it cannot be used to produce other goods and services.

(c) A taxi driver buying a NEW taxi and a shop-owner buying a NEW shop both count, because as in (a), the idea is that these goods in turn generate a stream of taxi/retail services over time.

The use of fertilizer counts as intermediate consumption (i.e. the use of intermediate goods in the production of other goods and services).

Generally, major improvements to existing buildings also count as investment. So a shop-owner renovating his shop, building a parking lot, a new apartment building all count.

However, ordinary maintenance and repairs do not count as investment (they instead count as intermediate consumption). As you can probably guess, in practice this gets tricky because it can be difficult to distinguish between a "major improvement" and an "ordinary repair". (See UN SNA 1.56.)

Paying for improvements in human capital does not count as investment. (See UN SNA 1.54 and 1.55, which also discusses how like you, many argue that this should count as investment, but right now, by convention, it does not. Schooling counts as consumption, while training by companies counts as intermediate consumption. Perhaps in future editions of the SNA it will be counted as investment.)


Marginal Utility with Supply and Demand Curves

Suppose there is only one good and I fix a price, $p$. The demand curve implies a corresponding quantity $q$.

Now, reduce the price by a tiny amount to the new level $p'$. The demand curve tells us that this reduction in price will cause an increase in quantity to $q'$.

Now, let's think about what happens here. Our consumer was willing to buy the some extra units at the price of $p'$, but not willing to buy those units at the price of $p$. We can infer that the consumer's willingness to pay for these units satisfies $p'<\text{WTP}<p$.

Remember that I said the change in $p$ was tiny? If this change is almost zero then the number of extra units demanded will also be small and we can get an exact value for the consumer's willingness to pay for these units: $p\approx \text{WTP}\approx p'$.

So the willingness to pay for the marginal unit is just the price. But the price is also the height of the demand curve!$^†$ Note that if I had started with a different price and performed the same exercise then I could have calculated the willingness to pay for a different unit of the good. So the height of the demand curve at a given point measures the consumer's willingness to pay for the unit of the good below that point.

Lastly, how much should the consumer be willing to pay for a unit? The answer is that he should be willing to pay as much utility as that unit gives him (a.k.a. the marginal utility). Thus, the demand curve's height measures willingness to pay, which is just another way of saying it measures marginal utility.

$^†$ Note that the distance $PQ$ in your figure is just the height of the demand curve too.

Appendix

To see how this works in maths and with an arbitrary number of goods: note that the optimal solution to the consumer choice problem occurs where the marginal rate of substitution between each pair of demanded goods is equal to their relative price
$$\frac{\text{MU}_x}{\text{MU}_y}=\frac{p_x}{p_y}.$$
Rearranging:
$$\frac{\text{MU}_x}{p_x}=\frac{\text{MU}_y}{p_y}=\text{some number}.$$
Rearranging
$$\frac{1}{\text{some number}}\text{MU}_x=p_x$$
so the price is just some constant multiplied by the marginal utility. By appropriate normalisation of either the units in which money is measured or the utility function (which is invariant to monotone transformations) we can make $\text{some number}=1$ so that $\text{MU}_x=p_x$.


Why did the Fed raise interest rates between 2004 and 2006?

Good question!

It seems that one of the reasons was precisely that inflation did not respond. They kept raising interest rates,  hoping it would respond at some point. In fact, nominal interest rates have to increase more than expected inflation for the real interest rates to go up, so it made sense.

The other reason is also related to the question. It seems that they were aware that a relatively long period of low interest rates was potentially fueling "asset price bubbles",  such as in housing, but also in asset backed securities, corporate bonds and equities. These also did not seem to respond quickly to the rate changes.

Of course, apparently, they did not realize that it was dangerous moment to increase rates quickly. It would have potentially been a good idea to raise the rates very very slowly, so that leveraged agents had a chance to de-lever as the prices of their assets slowly came down... Alternatively they could have slowly tightened the regulation on lending instead of raising the rates at all. This last point seems to be the key to avoiding a new crisis: having a good too to increase and decrease economy wide leverage distinct from interest rates.


Why do Reserve Banks only move in 0.25 basis point 'leaps'

The Fed normally only change rates during their regular monetary policy meetings, of which there are 8 per year (roughly six weeks between meetings). Therefore, they cannot change rates every month.

The amount 1/4% comes from interest rate quote conventions; historically they also used 1/8% (12.5 basis points). If they were to go to smaller increments, it would probably be 1/16%.

The first reason why they do not announce finer increments is that the Fed Funds rate that they target is a market interbank rate; they do not have complete control over it. The Fed publishes the "effective fed funds rate" in the H.15 report, which is their estimate of where the market is trading; it does not exactly match the target. Therefore, they cannot credibly deliver such small incremental changes to the overnight rate that matters.

The second reason is that such a small increment would probably make the Fed look silly. How do they have enough confidence in their ability to fine tune the economy so that moving the policy rate by 1/16% is now the right level?

Realistically, the current level of interest rates does not matter, so much as they expected path. At 25 bps per meeting, the total change over one year is 2%. That should be enough to match changes in economic conditions. An increment smaller than 1/8% would not be enough to keep up with conditions.

As for the current situation, it is unusual. Interest rates are very close to 0%, and it is difficult for them to go negative (the "zero lower bound"). Everyone knows that interest rates can only go one way (assuming no negative rates), so it is just a question of timing. This creates a different mood for interest rate watchers than when interest rates can go either way. Fed Reserve officials have been cautious about raising rates, since they have no leeway to cut rates if the economy rolls over. For this reason, they would not have considered any rate hikes until they were certain that rates needed to be well above 0%. (Even when the rate hikes occurred, they have been quite tentative.) Furthermore, if you want by news stories, "everyone" knew that the policy rate was "unsustainably low" every year after 2010. In other words, we need to be careful about what "everyone" knows.


The function $\tau$ is the 'line of best fit', the growth trend. The 'residual' $y-\tau$ is the output gap. So, $\tau$ detrends $y$ over time, to give the detrended business cycle $y-\tau$. $\quad y-\tau$ fluctuates around 0—it is positive when the output gap is positive and the economy is in a boom, and negative when the economy is in recession.

The first term makes the line of best fit follow the points in a more curvy way than just a straight line; the second term makes the line of best fit straighter (the higher $\lambda$ is).


If a utility function is quasi-concave, can we say that the IC curve associated with it is convex?

In your example the IC curve is not convex in the usual meaning of the word convex when applied to sets. What you probably mean is that the IC curve implicitly defines a convex function $f$ where $f(x) = y$.

If this is indeed what you mean then as long as you also assume that $U$ represents monotone preferences, then the quasi-concavity of $U$ will imply the 'convexity' of the IC curve and vice-versa. In this case the upper contour set of any level of $U$ will be a convex set, the lower border of which is the 'convex' IC you mention.


Does land value tax create an incentive to merge land ownerships?

Some people strongly disagree that LVT is not distortionary:

  George was right that other taxes may have stronger disincentives, but
  some economists now recognize that the single land tax is not
  innocent, either. Site values are created, not intrinsic. Why else
  would land in Tokyo be worth so much more than land in Mississippi? A
  tax on the value of a site is really a tax on productive potential,
  which is a result of improvements to land in the area. Henry George’s
  proposed tax on one piece of land is, in effect, based on the
  improvements made to the neighboring land.

  And what if you are your “neighbor”? What if you buy a large expanse
  of land and raise the value of one portion of it by improving the
  surrounding land. Then you are taxed based on your improvements. This
  is not far-fetched. It is precisely what the Disney Corporation did in
  Florida. Disney bought up large amounts of land around the area where
  it planned to build Disney World, and then made this surrounding land
  more valuable by building Disney World. Had George’s single tax on
  land been in existence, Disney might never have made the investment.
  So even a tax on unimproved land can reduce incentives.
  ... Zachary Gochenour and Bryan Caplan have pointed out that while the
  surface value of land is more apparent, especially for farming
  purposes, many lands have hidden natural resources, such as gold,
  water, and oil. These resources require investment on the part of
  owners to discover and produce. “Information about the land can be
  considered an improvement in its own right.” To tax the entire, or
  even a large, value of the mineral resources would create enormous
  disincentives for exploration and production

The Concise Encyclopedia of Economics: Henry George (1839-1897 )

To answer this more definitively we'd need to take a stand on how we think the tax assessor would measure unimproved value. We could then ask if combined plots would distort this measure. I read the quote above as saying that the assessor looks at the sale prices of undeveloped lots in the area. But the value of these lots is necessarily higher due to the improvements of others (or even your own). Combining plots, to the extent that it makes big, valuable projects possible, distorts this measure but in the wrong direction. Some other method that tried to strip out the indirect value of your own improvements on the unimproved value of your own land but left in the increase in value of the improvements of others would bias towards combining properties.


Any databases with CEO letters and shareholder letters?

Perhaps you could try something like the Bloomberg Terminal; I hear they have a lot of information.

If you were looking for annual reports from American companies, I'd tell you to go to the SEC website. However, I've noticed that http://www.luse.co.zm/ has some annual reports from companies on the London Stock Exchange, such as British American Tobacco, here.


Private Investment Impact on Bond Yields

I think you've answered your own question. Not only would lower inflation be a factor in lower bond yields but also consider that a lower level of demand for loanable funds for private real investment means lower corporate bond yields. That view is cited in the following Larry Summers quote taken from this economics.stackexchange.com question on secular stagnation...

  Think about what the world was like when General Motors or AT&T or
  Exxon or IBM were iconic companies. They were issuing debt. They were
  investing on a massive scale to expand capacity and to build networks.
  Now think about iconic companies of today like Apple and Google. They
  have more cash flow, fundamentally, than they know what to do with.
  And the result of that, of course, is an excess supply of savings.
  Another way to approach the question is to say, what's happened to the
  relative price of durable equipment, either producer equipment or
  consumer equipment, appliances? And the answer is those prices have
  gone way down. Well, when those prices go way down it means a given
  unit of savings goes much further.

  All of that, it seems to me, operates to reduce real interest rates


Model (1) is a regular linear probability model. Your results say that $x_1$ is significantly correlated with $y$.

Model (2) is strange. It means that the probability is quadratic in $x_1$ and the turning point is exactly $x_1=0$. You will have difficulty justifying the restriction.

The coefficients in Model (3) being insignificant may be a symptom of multicollinearity, i.e., $x_1$ and $x_1^2$ are strongly correlated. To see if that's the case, you can test $H_0: \beta_1 = \beta_2 = 0$. I am quite confident they are jointly significant considering the significant results in (1) and (2). If they are jointly significant but individually insignificant, it's a typical symptom of multicollinearity.

Plotting the fitted probabilities for (1) and (3) often helps. Also see if the range of $x_1$ contains the turning point ($-\frac{1}{2} \beta_1/\beta_2$) in Model (3). You usually have insignificant quadratic terms if the turning point is not in data range, because $y$ is already explained well by the linear model, and then the linear term also becomes insignificant due to multicollinearity.


LS book (Recursive Macro Theory). Chapter8: Complete Market

Since this is a Markov chain

\begin{eqnarray}
\pi_t(0,0,\cdots,1,1) &=&  \color{blue}{\pi(s_t = 0 | s_{t - 1} = 0)} \cdots \color{magenta}{\pi(s_2 = 0 | s_{1} = 1)} \color{red}{\pi(s_1 = 1 | s_{0} = 1)}\color{orange}{\pi(s_0 = 1)} \\
&=& \color{blue}{1} \times \cdots\times \color{magenta}{0.5}\times\color{red}{1}\times \color{orange}{1} = 0.5
\end{eqnarray}

But you are right if $t > 2$ then $\pi(s_{t + 1}  = 0 | s_{t} = 0) = 1$, so if $s_t = 0$ then $s_{t + 1} = 0$, but at $t = 2$ there's a 50/50 chance that the state changes from $1$ to $0$, so the state may remain $1$ as in the first history, or switch to $0$ as the second one


Help trying to find Economic Profit for a Monopoly

Profit is revenue minus costs. So we have $q = 12500$ and $p = 35$. (Note that I'm assuming you've correctly figured this out.) Then we have revenue:

$$R = \int_{0}^{12500} MR dq = \int_{0}^{12500} (60 - 0.004q)dq$$

And costs:

$$C = 300000 + \int_{0}^{12500} 10dq = 300000 + 125000 = 425000$$

The profit is $\Pi = R - C$.

The average cost is $C/12500$.

I leave you to crunch the integrals.

Edit: In looking at your problem, $p(q) = 60 - 0.002q$ is definitely an inverse demand function. The revenue $R = qp(q) = (60q - 0.002q^{2})$ (quantity times prices). This yields the profit maximization problem (maximizing with respect to quantity produced):

$$\max_{q} qp(q) - c(q) = (60q - 0.004q^{2}) - 10q - 300000$$

The First Order Conditions are:

$$60 - 0.004q - 10 = 0 \implies 60 -0.004q = 10$$

You did the work from there to solve for $q$.


Replicate Blundell and Bond (2000) results using R

3. How do we recover parameters from production function estimates (INCOMPLETE ANSWER - will be updated with how to do this in R once I have time to figure it out, or if somebody else knows...)
Blundell and Bond aren't estimating the parameters of a Cobb-Douglas production function. They're estimates the parameters of a "dynamic (common factor) representation" of a production function, specifically:
$(1) \ \ y_{it} = \beta_n n_{i,t} - \rho \beta_n n_{i,t-1} + \beta_k k_{i,t} - \rho \beta_k k_{i,t} + \rho y_{i,t-1} + (\gamma_t - \rho \gamma_{t-1}) + (\eta_i(1-\rho) + e_{it} + m_{it} - \rho m_{i,t-1})$
...which they get by quasi-differencing their production function and doing something weird with the $v_{it}$ term (not shown above). Changing the symbols, they re-write this as:
$(2) \ \ y_{it} = \pi_1n_{it} + \pi_2 n_{i,t-1} + \pi_3 k_{it} + \pi_4 k_{i,t-1} + \pi_5 y_{i,t-1} + \gamma^*_t + (\eta^*_i +w_{it})$
Where the * presumably indicates steady-state values, but again, I wish I had access to a teacher to explain precisely why they write that. Anyway, what their SYS-GMM estimator returns is a vector of parameter values $\boldsymbol{\pi} = (\pi_1,...,\pi_5)'$. We now have to use this to get the vector of parameters $\boldsymbol{\theta} = (\beta_n,\beta_k,\rho)$.
The only place I can find an explanation of how one does that is this paper applying SYS-GMM to Cuba. They refer to Woolridge 2002 (I don't have it), and assert that one can consider the problem as defined by the relationship $\boldsymbol{\pi} = h(\boldsymbol{\theta})$. We have $\boldsymbol{\pi}$, we need $\boldsymbol{\theta}$, and we will have to employ some function h() to make the link. It sounds like h() is taken to be an instance of a general class of minimum distance estimators, and we can (according to Woolridge 2002, apparently) solve the problem by solving:
$min (\boldsymbol{\theta}) = (\hat{\pi}-h(\theta))'\Omega(\hat{\pi}-h(\theta)) $
where $\Omega$ is some weighting function. And again, apparently $\Omega$ should be chosen as the variance of the matrix of parameters, obtained by "the delta method".
...but now it gets confusing, because although it's easy to find info on the delta method online, the matrix of parameters includes the unknowns $\boldsymbol{\theta}$. I guess the way to tackle this is to multiply it all out and plug into optim, using some arbitrary choice on starting values (factor shares of output, if $ valued?). I will update if I get a chance to try that...
In going from equation (1) to (2) it is pretty obvious that:
$$h(\beta_n,\beta_k,\rho) = \begin{pmatrix} \beta_n \\ - \rho \beta_n \\ \beta_k \\ - \rho \beta_k \\ \rho \end{pmatrix}= \begin{pmatrix} \pi_1 \\ \pi_2 \\ \pi_3  \\ \pi_4 \\ \pi_5 \end{pmatrix},$$
and $$\Omega = var(\pi)$$


How to calculate Debt to GDP ratio quarterly?

Debt is a stock variable, measured in dollars. (I will use “dollars” for the local currency for simplicity.)
GDP is a flow, with units of dollars per accounting period.
These two concepts are different, and so any ratio is a convention, without any “correct” answer.
Let’s assume that we want to follow the usual convention, and divide the stock of debt by the GDP generated during time it takes for the Earth to go around the sun. There are a number of standard ways of doing this.

Take the debt level at the end of the year, divide by the total of GDP for the same year (sum of four quarters). This gives us an annual series.
For any quarter, take the debt at the end of the quarter, and divide by the rolling 4-quarter sum of GDP.
For any quarter, take the debt at the end of the quarter, and divide by the annualized quarterly GDP.

One could imagine even more alternatives, such as comparing the average level of debt in the year to annual GDP.
You just choose one, and explain your choice.


That was tricky. The idea is as follows: First, under standard assumptions, demand is continuous. If you change prices a little bit, demand will not change a lot. In particular, if your excess demand for the first good was initially strictly positive, it will still be strictly positive for small changes in price. Now, the new bundle after the price was slightly raised was affordable at the old price with some money left to spare on more stuff. So the old bundle must be at least as good as the new bundle with some additional stuff on top and, therefore, be better than the new bundle.
Let's make this formal: So let $p=(p_1,p_2,\ldots,p_n)$ and $p'=(p_1+\epsilon,p_2,\ldots,p_n)$ with $\epsilon>0$
and we assume that $x_1^*(p,pw)>w_1$ and $x_1^*(p',p'w)>w_1$.
Let $x=(x_1,\ldots,x_n)=x^*(p',p'w)$. We first show that
$px<pw.$ Indeed,
$$p'x=(p_1+\epsilon)x_1+ p_2 x_2+\cdots+p_nx_n\leq(p_1+\epsilon)w_1+p_2 w_2+\cdots+p_nw_n.$$
$$(p_1+\epsilon)(x_1-w_1)+ p_2 x_2+\cdots+p_nx_n\leq p_2w_2+\cdots+p_nw_n.$$
$$p_1(x_1-w_1)+\epsilon(x_1-w_1)+ p_2 x_2+\cdots+p_nx_n\leq p_2w_2+\cdots+p_nw_n.$$
Therefore,
$$px=p_1x_1+ p_2 x_2+\cdots+p_nx_n\leq p_1 w_1+p_2w_2+\cdots+p_nw_n-\epsilon~\underbrace{(x_1-w_1)}_{>0}<pw.$$
Now, we need an assumption that guarantees that a consumer is better off if they spend all their income. Then we know that the consumer is better off under $x^*(p,pw)$ in which they would spend their whole income of $pw$ instead of choosing the bundle $x$, which costs only $pw-\epsilon(x_1-w_1)$. An assumption that guarantees it is optimal to spend the whole income is exactly what guarantees Walras' law.


What you are seeing in the data provided in the google search is an annual estimate  of GDP.  In the FRED data download is GDP reported on a quarterly basis.

Does last quarter value represents the GDP of the year? I thought we generally add all 4 quarters to get the annual number in finance?

Add up the all the quarters nominal estimates and divide the total by four to get the annualized numbers. (but when it reals you do not add them up, as real terms are not additive)

Is GDP culimnative? So the GDP created Q2 1950 is actually ~10 billion, and we just add that to the value of the previous quarter (to get ~290 billion)?

Nominal GDP estimate is the market value of the goods and services produced by labor and property located in the United States for that particular time period (quarter).  The estimate is not a running total from previous periods.

Now if you are asking if the nation's economy grow or contract compared to the previous period measured?   Then the GDP growth rate tells you how fast a county's economy is growing. It compares real GDP from one quarter to the next.

new: 2. Does GDP reset to 0 every year? So Jan 1st 12:00am, value starts at 0?

It depends on the country.  In the US, the source data for GDP is collected every month and the GDP is estimated at every quarter  once all final estimates are collected for each quarter in the year then the annual estimate is published (note there are revisions to the data and on a quarterly basis revisions are published 3 times).

https://www.bea.gov/resources/learning-center/what-to-know-gdp

If the GDP value is culminative, and you only add to the value of last quarter, the only reason why you can have a lower overall GDP is export is higher? The GDP equation is (GDP = C + I + G + (X – M)). The only subtraction I see in the equation is export. Or can consumer spending, business investment, and goverment spending be negative???

It would be really bad if GDP in nominal or real terms are negative.  However, GDP growth (percentage change)  can be negative but the interpretation is different as the growth rate tells you how fast a county's economy is growing.
The (X – M) term is defined as “net exports” .  So in the case where imports M is greater than exports X then net exports would be negative.      The other components would typically not have a negative values unless something catastrophic happens.


Without additional information, it is hard to answer your question but I would say that a moderation model is like any econometric model (and in fact, it is an econometric model).

Therefore, I would tend to agree with @Student, adding control variables is a good idea but it must represent your theoretical considerations (is there an omitted variable bias, a confounding variable, etc ?). So you should only add relevant variables to your model in order to increase its precision.

Just adding random/useless variables will unnecessarily increase the complexity of your model; and as my econometrics professor used to say: "why use a bazooka to kill a fly when you can use a fly swatter ?"

Edit: I (scholar) googled "moderation effect economics" and I randomly chose a paper published in a good journal, you can see that they also use relevant control variables (p. 512).


Supply curve when the marginal cost is zero

Took me a bit, but I have figured its out:

The single firm's supply curve is defined as it marginal cost, as long as MC is above it average cost
A firm would not sell below its average cost, since it would loae money, Mathmatically we can define, for the single firm
S = MAX(AC, MC)

Our basic assumption,for this discussion, is that MC = 0

Under the basic assumption and that costs are positive, obviously, then, the supply curve will always equal AC:
S = MAX(AC, 0), AC > 0
S = AC

The other conculsion from the basic assumption is that the production function, under the costs limit, is completely unaffected by quantity
Intuitively: its costs the same to produce 100 products and 1000 products.

If the Total costs function, as a function of quantity, is fixed, then the average function is:
AC = TC/x

In that case, we'll get a supply curve, though a very special one:
It'll decrease price as a result of increasing quantity, and prices will aspire to zero. always

Update:
Allowing myself to add a link to an article I wrote on the topic:

Start-Up economics: the disrupt model


Why is perfect price inelasticity of demand not considered an exception to the law of demand?

Because although some goods have perfect inelasticity over some interval, they are not perfectly inelastic over the whole demand curve - over the rest of the curve, quantity and demand will be strictly inversely related. And the law of demand doesn't have a "strictly" in there. Price and quantity are inversely related, but not strictly so, which (in the mathematical sense) means that parts of the demand curve are allowed to be perfectly inelastic.


What are the constituents of a country's inflation

The components of aggregate demand are government consumption, consumption, investment, net exports.

While people typically assume there is a relationship between demand shocks and inflation, this does not mean that inflation will necessarily go up whenever consumption goes up, because consumption depends not only on demand shocks but also many other things that will not necessarily affect inflation in the same way. For example, if consumption goes up because labor productivity and incomes increase, inflation may even go down. On the other hand, if the central bank decreases the interest rate, demand will increase and this will cause an increase in inflation.


Trouble understanding hamilton 1996 specification testing in MS

In my question, I rewrite the joint probability as the product of the conditionals probabilities but I wrongly implemented the markov assumption of this model. The markov assumption is:
$$
p(s_{t}=j|s_{t-1}=i,\Omega_{t-1})=p(s_{t}=j|s_{t-1}=i)=p_{ij}
$$
But, when conditioned to information available at date t
$$
p(s_{t}=j|s_{t-1}=i,\Omega_{t})\neq p(s_{t}=j|s_{t-1}=i)=p_{ij}
$$
Meaning that

$$
p(s_{t}=j,s_{t-1}=i|\Omega_{t})\neq p(s_{t}=j|s_{t-1}=i)p(s_{t-1}=i|\Omega_{t})
$$


Ratios to determine company's current purchasing power

You could probably get a very good idea of a company's ability to spend money - that is, make capital investments, develop new ventures etc. - by investing its balance sheet and develop both liquidity and leverage ratios. High liquidity with low leverage means that a company is probably not investing enough. Low liquidity with high leverage ratios would indicate a company has already invested heavily and is not in a position to undertake new ventures etc.

Here is a good introduction to leverage ratios
 Here is a good introduction to liquidity ratios


Currency triplets - arbitrage opportunity?

Think currency rate as simply the price, on the currencies market, of one currency in terms of another. But this is not a perfect market, so to answer your question with some concrete cases:

even if you want to purchase at time T(price) a certain amount of currency at price P, it's not guaranteed that at the time T(transaction) the price is still P; market is open 24/7 with several transactions every minute that modify the price with rapid changes in supply/demand;
currency market is not like a grocery store where there are two actors, a price, a transaction, and calculations can be done on the back of the envelope. To sell / buy currencies you need to involve external agents like banks and third-party agencies, all contributing to the final real price;
risk is not 0: as per first bullet, events can cause currencies fall several points, and the losses should be considered in the real-rates model.


Economic Theories for the creation of a new or separate Industry?

Have a look at monopolistic competition. It basically details how many industries can find their own niche in a differentiated product market. Some prominent papers in the subject:

Dixit and Stigliz: Monopolistic Competition and Optimum Product Diversity
Krugman: Increasing returns, monopolistic competition, and international trade
Salop: Monopolistic Competition with Outside Goods


Industrial capitalism, characterized by its use of heavy machinery and a much more pronounced division of labor.  It is a system  whereby industry and resources are owned by few for profit. In this capitalist system means of production were privately owned.  Its is marked by the rise of monopolies and trusts dominating industry and other aspects of society. Often used to describe the economy of the late 19th and early 20th century.

Agrarian capitalism refers to the description of agricultural economy. It is centered upon the production, consumption, trade, and sale of agricultural commodities, including plants and livestock, is rural rather than urban-based.

mercantilism was based on the idea that a nation's wealth and power were best served by increasing exports and limit imports.

The Industrial Revolution began in the 18th century, when agricultural societies became more industrialized and urban.  During this period society moved form primary Agrarian capitalism  to Industrial capitalism.

An agrarian society Wealth comes from the land. Industrial society refers to a society driven by the use of technology to enable mass production, supporting a large population with a high capacity for division of labor. Industrial society is characterized by the use of external energy sources, such as fossil fuels, to increase the rate and scale of production.

Mercantilism developing during 16th century came about  as governments created incentives to only buy its own countries goods especially among colonies.   However, these views changed with the introduction of capitalistic beliefs and value to the late 1700s.    The industrial revolution came about because industrialization required significant work and investment from individuals and not necessarily the government.

Keynesianism - the theory made an argument for active government intervention during recessions, leveraging tools of monetary and fiscal policy to restore confidence, boost output, and smooth the ups-and-downs of the business cycle.

To answer your question agrarian capitalism does include the use of plantations.

During the industrial revolution women entered the workforce in textile mills and coal mines in large numbers.  Women and children often worked in the factories and mines in order to help pay for the families cost of living.   Women were not valued the same as men in the workplace, and were often paid much less than men.  However, during this time the early feminist movements started as women began to organize and protest for more equality in society.

https://www.historycrunch.com/role-of-women-in-the-industrial-revolution.html#/


If the consumer is not a price taker, does she still set marginal rate of substitution equal to the price ratio?

We have the utility function  $U(x,y) = Ax^ay^a$ and we want to solve

$$\max U = Ax^ay^a,\;\;\; s.t. xp_x + yp_y(y) = I$$

The Lagrangean is

$$\Lambda = Ax^ay^a + \lambda [I-xp_x - yp_y(y)]$$

and the first order conditions are

$$\frac {aU}{x} = \lambda p_x$$
$$\frac {aU}{y} =\lambda p_y + \lambda y\frac{\partial p_y}{\partial y}$$

It follows that at the optimum

$$xp_x = yp_y + y^2\frac{\partial p_y}{\partial y}$$

or that

$$\frac{x}{y} = \frac{p_y}{p_x} + y \frac{\partial p_y/\partial y}{p_x}$$

But the previous expression is more interesting because it can be viewed as a quadratic polynomial in $y$, and be solved for its roots


Open/modern question in labor economics?

If you want a recent state of the art in Labor Economics, you can have a look at the last Handbook of Labor Economics (2011, edited by Card and Ashenfelter). Though it is getting old, it provides an overview of specific approaches or questions in Labor. It is especially relevant if you want to define a research project for a PhD.

To be more specific, here is a (non-exhaustive) list of trendy research questions (you can use these keywords to find research articles and reviews):

Polarization of the labor market. There is evidence of a decline in demand for middle-skilled workers (less employement and reduce in wages). The two main explications are globalization and skill-biased techonological change. Researchers try to weight the role of each explanation from data. In particular, a recent approach interests in the type of tasks in each occupation. This is related to the chapter by Autor and Acemoglu in the Handbook. Jobs with routine tasks are likely to be affected by technology as they can be more easily automated. Looking at the type of tasks, whether it uses cognitive or non-cognitive skills, brings new insigths in other interesting questions (returns to education, gender/racial wage gaps, local labor markets). You can look at some work by David Autor or David Dorn for instance.
Productivity puzzle. In many advanced countries, productivity has not found its before-crisis trend, and we do not know why. Research focused in country-specific situations (the UK productivity puzzle, the French productivity puzzle, etc) to find the reasons of this productivity slowdown in the national labor markets. It is also a puzzle that so many countries are affected despite their differences in the functionning of their economy.
Labor market, household decisions and marriage market. A rich literature studied the labor supply choices within the household as a joint decision. A new research area relates labor market outcomes to the formation of households. The novelty is to consider that individuals are not matched in a pure randomly way on the marriage market, and decisions to marry may account for individual abilities and skills. In particular, it has been shown that high-skilled female workers marry high-skilled male workers more often than in the past, and reciprocally. This has consequences in terms of inequalities and labor supply decisions. There are recent works conducted by Pierre-André Chiappori among others.


Yeah, that's pretty much it for any horizontal or vertical line:

P = p0


How to distinguish Full-time (FT) employment and Part-time (PT) employment when calculating employment rate?

They must be counting full time/part time equivalents rather than individuals. They are basically comparing the hours worked versus the hours people report desiring to work. Thats the only explanation.


Variance in the Context of a AR (1) Model

$$Var[y_t] = E[(u_t+\theta_1u_{t-1}+\dots+\theta_1^{t-1}u_1)^2] = E[u_t^2]+\theta_1^2E[u_{t-1}^2]+\theta_1^4 E[u_{t-2}^2] + \dots +\theta_1^{2t-2}E[u_1^2]$$
the latter equality follows from the assumption that $u_t$ is not serially correlated (i.e. $E[u_{i}u_{j}] = 0\ \forall\ i \neq j$). Then, since $E[u_{t}^2] = \sigma^2\ \forall\ t$ it follows that
$$Var[y_t] = \sigma^2(1+\theta_1^2 + \theta_1^4+\dots+\theta_1^{2t-2})$$
it remains to find $(1+\theta_1^2 + \theta_1^4+\dots+\theta_1^{2t-2})$ which is equal to $\frac{1-\theta_1^{2t}}{1-\theta_1^2}$ according to geometric series sum formula.
Thus, $$Var[y_t] = \sigma^2\frac{1-\theta_1^{2t}}{1-\theta_1^2} $$


Is it possible to peg a cryptocurrency to a stock or bond index?

There are already cryptocurrencies that are backed 1:1 by dollar, see Tether and USDC, or even backed with gold, see Paxos Gold, and yes all of that are redeemable. there is no problem with it so far.
There may be some trust problem, are the issuer really back the cryptocurrency with real asset? (this can solved easily by external auditing though) At 2018 there are some issues that tether aren't pegged 1:1 with USD, the result is Tether value are decreasing to around 0.9 USD. Weirdly enough, even without an external audit(they never got an audit since that scandal), people still believe in tether and the price of tether jumped out to 1 USD Again.


Is the labor force participation rate related more to labour supply or to labour demand? Or both?

Traditionally, labor force participation is best thought of as a supply-side phenomena. That is, a certain subset of our nation's population is jobless but is jobless by choice. For example, consider a stay-at-home parent. So, we have agents making a decision to not supply labor.

However, there is a hint of the demand side at play here. For example, a very weak demand side might discourage workers, thereby compelling the marginal participant to exit the labor force.

So, to be explicit: LFP is a supply-side concept. However, there is a mechanism through which demand side forces can factor into LFP.


Mixed Strategy Nash Equilibrium for this particular 3x3 matrix

You generally have the right approach with expected payoffs correctly, but there are some errors in your logic.
First, take a look at your reduced matrix. You're missing a way to reduce it further and simplify the math a bit! You've left an extra strategy in place for Player 2. Player 2 knows that Player 1 will never play $M$, so Player 2 only needs to consider best responses to $L$ and $R$.
Second, note that you're using Player 1's payoffs in your expected payoff calculations for Player 2. For example, you have
$$E[A] = \frac{1}{2}q + 1(1-q)$$
The payoffs of $\frac{1}{2}$ and $1$ are to Player 1, not to Player 2. You want to use $\frac{1}{2}$ and $0$ in your calculation.
Third, the way you find a mixed strategy Nash equilibrium is by setting players' expected payoffs to be equal. Player 1 and Player 2 each need to be indifferent between their strategies, and that occurs when they play their strategies with probabilities so that the other player's payoffs are equal.
So, for Player 2's equilibrium strategy, you'll take Player 1's expected payoffs and set them equal: $E[L] = E[R]$. If they aren't equal, then one's greater than the other. Say $E[L] > E[R]$. Then, Player 1 will play $L$, and Player 2 will want to play $C$, then Player 1 will want to play $R$, and you end up in a cycle (and not an equilibrium). When the payoffs are equal, you have a stable solution, and neither player has an incentive to deviate from that strategy.
The same rationale applies to Player 1's equilibrium strategy - it's the set of probabilities that result in all of Player 2's expected payoffs being equal, and your Nash equilibrium is the set of all the strategies and their corresponding probabilities.


Why could China, much poorer than US, lend so much money to US?

You don't have to be richer than someone to lend them money. You only need to have money to lend. China has a lot of money to lend because it has a high savings rate. In the past, it preferred to lend the money to the US, because that would allow the US to buy more goods from China. However, the situation is now changing.


Dealing with Missing Data when Testing the CAPM

Easiest fix: if you're worried about it you should value weight your results. This is suggest by, for instance, Kothari, Shanken and Sloan (1995). Firms that are delisted tend to have extremely small market cap, so value weighting gives them very little impact on summary statistics. Delisted returns should also be used, although I'm not sure how much impact they'll have. I've seen the delisted return stuck into the month after a stock ceases to be traded.

In finding $\beta$'s, I tend to see the regression used only on those dates for which the stock return is observed. The correction really comes in value weighting summary statistics afterwards. Whether all this is "correct" or just the practice I've seen is not something I'm sure about.

Edit: here's a different perspective.


Delay in the transmission of knowledge to poorer countries

By the notation used, labor quantity appears to be constant. Then the only source of growth of output is the growth in $A_N(t)$, and they will have the same growth rate. Take logarithms and then time derivatives on the output per capita equation to see this.
Then remember introductory differential equations on how to express $x(t)$ as a function of $x(0)$ and a constant growth rate. And don't forget to explicitly solve for the 10-factor difference required, at the end.


Impact of working capital constraints onto output

I can think of at least one paper where binding accounting rules are shown to have negative consequences on real investment:

Can Tight Accounting Oversight Distort Investment? Evidence from Mortgage Restructurings after the JOBS Act

  Abstract: I study the effect of accounting oversight on a bank’s level
  of troubled mortgage restructurings. If banks manage their accounting
  numbers (e.g. regulatory capital or earnings) in the wake of the
  financial crisis, then they may be hesitant to restructure troubled
  loans because such restructurings require the bank to mark the value
  of the loans down. I show that giving a bank more accounting
  discretion leads the bank to restructure more troubled mortgages. My
  identification strategy uses a provision of the Jumpstart Our Business
  Startups (JOBS) Act of 2012 that allows a subset of banks to
  deregister from the SEC. The first stage of the analysis shows that
  accounting oversight is of first-order importance in the
  deregistration decision; in particular, there is a sharp increase in
  the probability of deregistration from 7% to 30% for banks that can
  escape accounting oversight. By instrumenting for the decision to
  deregister with the bank’s newfound eligibility to deregister after
  the JOBS Act, I show that deregistration leads banks to slash their
  accounting and audit fees and make fewer provisions for loan losses. I
  next show that deregistration induces banks to double their investment
  in mortgage restructurings, which entail large upfront accounting
  expenses; this effect is driven by banks with low capital ratios, and
  estimates for non-mortgage troubled debt restructurings react
  similarly. These results suggest that banks manage their capital using
  real activities, and this is one reason for the low level of mortgage
  restructurings since the recession.


What's the term for the debt you have to pay today?


Homothetic production function and Profit Function

I have figured this as the answer to this question;
As we know profit maximization problem is given as,
$$
\pi(p,w) = \mathop{max}_{\textbf{y}}\quad p.y - C(\overrightarrow{w},y)
$$
When $f(\overrightarrow{x})$ is homothetic,

$$
C(\overrightarrow{w},y)=h(y).C(\overrightarrow{w},{1})
$$

Substituting in the profit function gives;

$$
\pi(p,w) = \mathop{max}_{\textbf{y}}\quad p.y - h(y). C(\overrightarrow{w},1)
$$

First order condition gives us;
$$
p=h'(y)C(\overrightarrow{w},1)
$$

Which can be written as;
$$
h'(y)=\frac{p}{C(\overrightarrow{w},1)}
$$

or

$$
y=(h')^{-1}\frac{p}{C(\overrightarrow{w},1)}
$$
$$
\Rightarrow y= \gamma(p).\beta(w)
$$

Therefore $y(p,w)$ is separable.
Since
$$
\pi(p,w)=\int{y(p,w)}dp
$$

$$
\pi(p,w)=\int{\gamma(p).\beta(w)}dp
$$

$$
\pi(p,w)=\beta(w)\int{\gamma(p)}dp
$$

$$
\pi(p,w)=\beta(w)\alpha(p)
$$

Thus $\pi(p,w)$ is also separable in factor prices and output prices.


Why marginal benefit is decreasing as more and more pizzas are being produced?

You can think of it as the marginal benefit per person. The first piece of pizza you eat is amazing, it has a very high marginal benefit. But say you eat five whole pizzas in one sitting and then are immediately  offered another slice. That next slice doesn’t seem so appealing, does it? You’re already full and starting to get sick of it. That slice has a low marginal benefit. Certainly lower than that first slice. You can also think of this as a continuous process (i.e the second slice is a little less satisfying than the first, and the third is a bit less satisfying than the second and so on).
So, as the total consumption of pizza increases, the marginal benefit (i.e. the benefit of the next unit) gets smaller.


What are the pros and cons of a "long stock, short call" position compared to a "short put"?

The primary consideration in practice is that the initial investment is larger if you are long the stock, and thus consumes more balance sheet.
The next consideration is that if you don’t hold to maturity, the implied volatility exposure is opposite (and the associated greeks) for the positions.
After that, there are only voting rights, jurisdiction-dependent tax issues, and the reality that one instrument might be mis-priced.


Solvency and liquidity. How to understand them? Should Central Banks only lend to solvent but illiquid banks?

Yes, your thinking about solvency and liquidity is in the right direction. An easier way to think about them is: Liquidity means that a bank has enough resources to meet its obligations and Solvency means the bank owns more than it owes.

Lending to solvent but illiquid banks is a less risky thing to do for the central bank because the bank needs more liquidity at the moment but it owns enough long-term assets to repay the debt. Bagehot was the first to publicise this insight. It is called Bagehot's rule.

Lending to insolvent banks (banks that owe more than they own) may be risky because it's long-term assets are significantly limited for the bank to repay this debt. This has the potential to cause a moral hazard because the bank is likely to come back to borrow more from the central bank in the future.


Do banks pay each other from their reserves?

The details of payments systems are complex, and vary by country. I am slightly familiar with how it works in Canada, and I believe that the principles would not be radically different in other developed countries. Here’s a primer on the Canadian payments systems: link.

The advantage of Canada is that there are no reserve requirements, so it’s easier to explain. If you look at the balance sheet data published by the Bank of Canada, the Canadian banks normally have a balance of \$0 at the central bank at the end of the day. In fact, that is what is expected. (In practice, there are small misses of the zero balance; a lot of money sloshes through the payments system.) In a country with reserve requirements (like the U.S.), banks would like to have a target balance equal to their required reserves. (At present, the Fed has forced excess reserves into the system.)

The amount of interbank transactions are very large. They send amounts back and forth on the payments system based on various transactions - their own, and client transactions. The job of a Canadian bank Treasurer is to make sure that the net balance of all payments ends up at \$0 at the end of the day, so that the net position at the Bank of Canada is unchanged. (In a country with reserve requirements, they will want to hit at least the required reserves level.) They will achieve this by any number of methods:

issue new securities;
borrow/lend in the interbank market;
buy or sell securities (mainly government bonds);
borrow/lend through repurchase agreements (repos);
if all else fails, run to the central bank and borrow against their assets. This option is greatly frowned upon, and will prompt an investigation from regulators as to why the bank cannot borrow in the markets. (This is termed a “lender-of-last resort” operation, and should only be necessary in a crisis).

(Since you asked about bank lending in particular, it is achieved by Bank A wiring money to Bank B through the payments system, in exchange for a contractual obligation to make a reverse future operation with interest.)

The multiplicity of options means that we cannot trace exactly what happens when customers transfer funds to someone at another bank; all we know is that at the end of the day, all of the bank’s transactions with the payments system should net to zero.


Do the three ways of measuring producer’s surplus give the same result?

Ask your teacher what he meant, because there is either a misunderstanding or he is mistaken.

Producer's surplus will come down to $y \cdot p - VC(y)$.

In the first graph, by definition of $AVC$ we have
$$
y \cdot (p - AVC(y)) = y \cdot p - VC(y).
$$
In the second graph, using $\int MC = VC$ and $VC(0) = 0$, we have
$$
\int_0^y p - MC(x) \text{d} x =
\left[x \cdot p - VC(x)\right]_0^y = y \cdot p - VC(y).
$$
The area in the third graph is
$$
\begin{align*}
\int_0^z p - AVC(z) \text{d} x +
\int_z^y p - MC(x) \text{d} x.
\end{align*}
$$
Let us calculate it by parts:
$$
\begin{align*}
\int_0^z p - AVC(z) \text{d} x & = z \cdot p - z \cdot AVC(z) = z \cdot p -  VC(z) \\
 \\
\int_z^y p - MC(x) \text{d} x & = (y - z) \cdot p - VC(y) + VC(z).
\end{align*}
$$
Summing up the two, we get $y \cdot p - VC(y)$.


Simply shifting the demand curve down will typically reduce the combined surplus, especially the producer surplus.  But you could devise two non-parallel demand curves, one below the other, where consumer surplus rose, for example like this:

If the demand curve fell from the higher to the lower red curve, than the combined surplus would fall by the orange and brown amount while the producer surplus would fall by the green and brown amount.  But the consumer surplus could rise, providing that the green amount is larger than the orange amount.  You could achieve this without such kinky curves, but it might not be so obvious.

If consumers' tastes (or utilities) change then with a constant budget constraint, demand could increase for other products, raising the combined  surplus for those.


Is there a per country estimation of the GDP drop due to COVID?

There are some cross-country studies but not that many of them yet. For example, Maliszewska, M., Mattoo, A., & Van Der Mensbrugghe, D. (2020) from world bank provide some cross-country estimates of the cumulative effect of covid-19 for various countries and regions across the world. Their estimated cumulative effect of covid on GDP (measured as deviation from their benchmark) is shown below.

However, I would advise to take these results with a pinch of salt. These are preliminary estimates that were published in April and better ones will be fore sure forthcoming in near future.


Why imports increase if higher relative inflation?

At first, U.S. exports increase because US exports become cheaper due to the low inflation rate. Inflation itself only influences the exchange rate indirectly through its effect on supply and demand for currency. That effect goes through trade. For that to happen, trade must first take place and so US exports must first increase.

So at first the exchange rate doesn't adjust at all (no difference in supply/demand of the currencies just yet). US goods become cheaper so they're exported more. As US exports increase the demand for dollars by South Korea increases. This will later on lead to the south korean currency to depreciate and for US exporte to go down again (if the Marshall Lerner condition is fulfilled).


Show that $x(p,w)=w\cdot x(p,1)$ with homothetic preferences

An indirect proof. Suppose
$$
x(p,w) = w\cdot x(p,1)
$$
does not hold. This is equivalent with stating
$$
U(x(p,w)) \neq U(w\cdot x(p,1)).
$$
(To be precise: $x(p,w)$ and $x(p,1)$ may be set valued. In this case we are talking about two elements at least one of which is not included in both sets.)

Case 1.
$$
U(x(p,w)) > U(w\cdot x(p,1))
$$
As $U$ is homothetic
$$
U(x(p,w)) =  U(w \cdot \frac{1}{w}\cdot x(p,w)) = w \cdot U(\frac{1}{w}\cdot x(p,w)).
$$
Using this we have
$$
w \cdot U(\frac{1}{w}\cdot x(p,w)) = U(x(p,w)) > U(w\cdot x(p,1)) = w\cdot U(x(p,1))
$$
and thus
$$
U(\frac{1}{w}\cdot x(p,w)) > U(x(p,1))
$$
However as $\frac{1}{w} \cdot x(p,w)$ is clearly an element of $B(p,1)$ this is impossible as $x(p,1)$ gives maximal utility in that budget set.

Case 2.
$$
U(x(p,w)) < U(w\cdot x(p,1))
$$
As $w \cdot x(p,1)$ is clearly an element of $B(p,w)$ this is impossible as $x(p,w)$ gives maximal utility in that budget set.

Thus we have proven that
$$
U(x(p,w)) = U(w\cdot x(p,1))
$$
which is equivalent with
$$
x(p,w) = w\cdot x(p,1).
$$


inflation rate vs interest rate on calculating present value of future money

The question has some ambiguity. If you want to calculate something, you need to define what you are interested in. If you want to think in terms of purchasing power, you can use the inflation rate.
However, using inflation would be the wrong answer in the context of finance or project analysis. In the financial context, you are comparing money now versus money in the future. The basis of comparison is an instrument that bridges money from the present to the future, which are debt instruments.
For example, if you are borrowing to finance a project, your profitability depends on the return of the project in nominal terms versus the cost of financing. What the value of the currency does versus a basket of goods during that time does not matter.
The idea of inflation trading off future returns is a beloved concept of conventional economists, but its usefulness is greatly overrated. Since you cannot purchase the CPI basket and store it indefinitely, there is no obvious necessary relationship between inflation rates and the return on investable assets. (People might invoke model relationships, but that puts a lot of faith in the models.)


Most governments (outside of typically oil-producing nations that have sovereign wealth funds) are net debtors, so public deficits show up as increases in government debt outstanding, and public surpluses (public saving) show up as decreases in debt outstanding.

In the model used in Mankiw’s book, the assumption is that government spending never constitutes investment (i.e, it doesn’t create productive capital). In that model, it’s also assumed that households hold government debt: government debt just soaks up some of the funds that would otherwise be available as “loanable funds” to private firms to produce “productive capital,” shifting the supply curve to the left.

So, in that model, the quantity axis represents something like “private investment,” and government saving shows up as an outward shift in the supply curve, which results in lower interest rates and more private investment.

There’s another way of thinking about it, which doesn’t require that one assume that government spending never forms capital like roads, bridges, airports, etc., nor that government deficits are paid for directly out of private saving. In that model, government deficits just show up as increases in demand for loanable funds, which increases interest rates. Implicitly, this does still mean that less private sector investment will occur. The opposite is of course true for government surpluses. The interpretation of the quantity axis then changes to something like “amount of financial intermediation.”

Either model of loanable funds is, of course, a radical simplification of how financial intermediation works, and like anything in an intro course, it’s worth remembering that tons of caveats apply.


What am I doing wrong in the derivation of Bass diffusion model?

You are missing an integration constant

$$
\log\left(\frac{p + qF(t)}{1 - F(t)}\right) = (p + q)t + \color{red}{\tilde{C}}
$$

This constant you can name it whatever you want, I'm going to name it as

$$
\color{red}{\tilde{C}} = \color{blue}{C}(p + q) + \ln q
$$

where $C$ is just another constant. So I basically changed one constant for  another one (completely allowed). Now the problem becomes

\begin{eqnarray}
\ln\left(\frac{p + qF(t)}{1 - F(t)}\right) &=& (p + q)t + \color{blue}{C}(p + q) + \ln q = (p + q)(t + C) + \ln q\\
p + qF(t) &=& e^{(p + q)(t + C) + \ln q}(1 - F(t)) = qe^{(p + q)(t + C)}(1 - F(t)) \\
[q + qe^{(p + q)(t + C)}]F(t)&=& qe^{(p + q)(t + C)} - p \\
F(t) &=& \frac{1}{q} \frac{qe^{(p + q)(t + C)} - p}{1 + e^{(p + q)(t + C)}}
\end{eqnarray}

Rearranging a bit the terms

$$\bbox[5px,border:2px solid blue]
{
F(t) = \frac{1}{q}\frac{q - pe^{-(p + q)(t + C)}}{1 + e^{-(p + q)(t + C)}}
}
$$


The approach is correct, but it is not true that firms have the same equilibrium production quantities because we are dealing with Cournot competition - it is because the firms have the same profit functions. You can see for yourself with an example that if the two firms had different cost functions, they would not necessarily produce the same outputs as each other.

The step of "substituting one into the other" implicitly uses the concept of Nash equilibrium, i.e. each firm's production quantity is a best response to the other's. Given that $A$ produces some amount $q_A$ and $B$ knows it, $B$ would make the amount $q_B^*$ is profit-maximizing for $B$. Then we fix $q_B^*$ and find what $A$ would make as a profit-maximizing response - call it $q_A^*$. If I then fix $q_A^*$ and find $B$'s profit-maximizing response is the same $q_B^*$, $(q_A^*,q_B^*)$ is a Nash equilibrium: each firm's action is optimal, given the other's. This line of reasoning can be implemented by a solving a system of two equations, given by the profit-maximizing condition $MR=MC$ for each firm that you mentioned.


How can the HKMA peg the Hong Kong dollar at 7.8 HKD/USD for so long?

To maintain the currency peg, the Hong Kong Monetary Authority (HKMA) has to buy HKD and sell USD when the currency comes under depreciation pressure and sell HKD and buy USD when the currency comes under appreciation pressure.

Depreciation pressure: The HKMA operates a currency board, which means that more than 100% of outstanding HKD is backed by USD reserves. It cannot run out of reserves and the peg could be maintained indefinitely in the face of depreciation pressure, if this was the only goal the HKMA wished to pursue.

Appreciation pressure: The HKMA can print more HKD and sell it. There is no limit to this in theory (although the Swiss National Bank chose to abandon its peg in 2015). The peg could be maintained indefinitely in the face of appreciation pressure as well.

However, as long as the HKMA is committed to maintaining the peg, it can't run an independent monetary policy suitable for Hong Kong. (The HKD is sold/bought to maintain the peg instead of being sold/bought to stimulate or rein in economy.)

If the Hong Kong economy slowed, the HKMA would maybe prefer to simulate the economy by injecting more HKD into the system (printing more money). It might not be able to do so though if the HKD was under depreciation pressure. It would actually have to BUY the HKD to maintain the peg. This is the exact opposite of what would help the economy.

At the moment, Hong Kong is importing US monetary policy by pegging its currency to the USD. This has worked as Hong Kong is a small open economy in sync with the global economic cycle and the US is a main driver of that very cycle.

Nonetheless, some speculate that the peg will eventually be abandoned in favour of a peg to the Chinese renminbi as Hong Kong is more and more linked to the Chinese economy and US monetary policy is becoming less suitable for Hong Kong.


What does the seasonal adjustment curve for major macroeconomic metrics look like?

Quarterly GDP seasonally adjusted

source:  https://aneconomicsense.org/category/econ-data/gdp-productivity/
AnEconomicSense.org by Frank Lysy

Monthly unemployment rate seasonally adjusted

Source: https://www.researchgate.net/figure/Seasonal-adjustment-of-US-unemployment-using-the-defaults-of-seasonal-The-result-is_fig2_329822239
by  Christoph Sax and Dirk Eddelbuettel

These are just some examples.

To manually do it in excel this link  from Catherine Hood shows an example

http://www.catherinechhood.net/SeasAdjShortTimeSeriesExcel.pdf

simplest case- – A Quarterly Series with a Flat Trend

Steps to calculate the seasonal adjustment:

Calculate the average for the series. This will be use as the trend.
Calculate the difference between the original series and the trend. Label this as "residual"
Calculate the seasonal factors (SF), which are the average of the residuals for a given quarter.

For example, the seasonal factors for all the Quarter 1 values will be (111+87+145) / 3 = 114.3333 and so on.

Subtract the seasonal factor from the original series to get the seasonally adjusted series. For example, for Quarter 1, 1991, we have 864 – 114.333 = 749.6667.


First, remember that a Cobb-Douglas production function is a special case of a CES production function (see e.g. here).

Secondly, there are critical assumptions that have to be met when one tries to determine TFP (or the "Solow residual" as it is often called). The most important one is perfect competition.
Furthermore constant returns to scale. Without those the factor shares (capital, labor over total output) and output elasticities are not identical.
Thus the standard "growth account exercise" within the Solow-Swan framework is not possible with a general CES function.


Cobb-Douglas and Logarithm Utility Functions

Utility functions are invariant with respect to positive monotonic transformations (PMT).
Take $U(x,y)=x^\alpha y^{1-\alpha}$, and let $V(x,y)=\log(U(x,y))$ be a PMT of $U$.
Thus $V$ and $U$ both represent the same preference, and thus demand functions for $x$ and $y$ are the same.


Is it common to see hedge funds go bankrupt?

Have any hedge funds gone bankrupt as a consequence of the "GameStop scandal"?

This is very difficult, almost virtually impossible to say as events unfold. In order to understand why consider the Greylock Capital Associates that filed for bankruptcy on 31 Jan. this year.
It would be tempting to say that this bankruptcy occurred due to GME stock shorting, but if you actually read the news around the bankruptcy you will discover that the company was already suffering heavy losses for about 3 years.
This example shows you that hedge funds and generally most companies can operate even when in red for multiple years before they need to file for bankruptcy. As such, most of the time, it is really difficult to tie one single event to firm bankruptcy.  Generally speaking, from economic theory we know that in the short-run even operating at loss might be more optimal than firm exit.
Even if some hedge fund representative would openly say that they went bankrupt because of GME shorts during this GME bubble fueled by reddit speculators, you can't take that statement at face value. The manager might simply say that as an excuse for years of poor management and underperformance.
What you need to answer this sort of question is some research that can show that there is a relationship between shorting the GME bubble and probability of bankruptcy, which would show that probability of hedge fund going bankrupt was significantly affected by this event. Otherwise all you have are just anecdotes.
Since this case was so prominent I expect that eventually someone will try research it but I doubt you will find any empirical peer-reviewed research on this in next 2-3 years even more. This is not just because research takes time but also because you need to gather more data than possibly can be avaiable in real time.

And more generally is it common to see hedge funds make bankrupt.

Depends on how broadly you define bankrupt (e.g. are you only interested in bankruptcy in legal sense or more broadly in English language sense when people often call any firm closing bankruptcy?). For example, according to Liang & Park (2010) between 1995-2004 hedge funds had attrition rate of about 8% on yearly basis. Although, not all of that is bankruptcy as some firms might leave before filing for bankruptcy. There is reason to believe this rate accelerated over time. For example, in 2019 more hedge funds closed than new opened.
Hedge funds have relatively high rate of attrition because:

They have very few limits on how to operate so hedge funds industry is very dynamic. In a dynamic industry you will often see old firms 'dying' while new innovative firms coming in (e.g. see the NBER working paper of  Getmansky et al (2015)  on this topic). This is explanation why hedge funds close quite often in general but probably cannot explain why they close so fast in recent few years.
It is widely believed that hedge funds perform poorly relative to their high fees. This is often quoted reason for why many hedge funds loose customers and are forced to close or go bankrupt recently. See the Investopedia article on 'the great exodus out of hedge funds'.  This is topic on which research is not settled. On one hand raw data clearly show hedge funds underperform. On the other hand this might be due to biases since hedge funds are not obliged to disclose their data to anyone but their customers and regulators. This can create biases in the data as the best firms might not disclose publicly their data to protect their trading strategies that are not patentable (e.g. see Fung et al (2000) or above cited Getmansky et al (2015) again). Depending on how particular research tries to correct for these biases results can change and some are more favorable to hedge funds but there is a lot of uncertainty about the results here. This being said customers would not be leaving the hedge funds if they were not believing that they are having better opportunities elsewhere.


Cost Minimization of $f(x) = min(x_1,x_2) + x_3$

While not said explicitly in the question I am guessing from the Langrangian function you set up that the problem you intend to solve is
$$\min_{x_1,x_2,x_3} p_1x_1 + p_2x_2 + p_3x_3 \\[8pt]
h(x_1,x_2,x_3) = \min\{x_1,x_2\} + x_3\geq z$$
This assignment combines perfect complements in production $f(x_1,x_2) = \min \{x_1,x_2\}$ with perfect substitues $g(y_1,y_2) = y_1 + y_2$.
One approach is sometimes to solve problems in two stages. Intuitively, with $y_1:=\min\{x_1,x_2\}$ firm will produce $y_1$ at the unit cost $p_{y_1} = p_1+p_2$ for whatever level of $y_1$ is produced it can never be cost minimizing to use more of $x_1$ than $x_2$ or other way around. Hence $x_1=x_2 = \min\{x_1,x_2\} = y_1$ and the seperate cost for each unit of $y_1$ is therefore $p_1+p_2$.
Next, solve cost minimizing with production
$$g(y_1,y_2) = y_1 + y_2 = y_1 + x_3 = z$$
with input prices $p_{y_1}$ and $p_{3}$. Since factors $y_1$ and $x_3$ are equally productive (constant equal marginal productivity) production will happen with cheapest factor so cost must be $C(z,p) =\min\{p_{y_1},p_3\}z = \min\{p_1+p_2,p_3\}z$.
I avoid using Lagrangian multiplier method because the production function is non-differentiable.


Why Nobel prize winners studied economics?

The Nobel Prize website is a good source of information. On the economics prize page they have a link to a page for each laureate. From there you can find, in particular, a biography for each prize winner where the laureates often talk about the genesis of their interest in economics.
For example, Eric Maskin writes

I became a math major at Harvard College, where I studied algebra with Pierre Samuel and Richard Brauer and analysis with George Mackey and Lars Ahlfors − all of them inspiring, some of them amusingly eccentric. Almost by accident, I wandered at one point into a course on "information economics" taught by Kenneth Arrow, later my Ph.D. advisor. The course was a hodgepodge of topics from the frontier of economic theory, but a good part of it was devoted to Leonid Hurwicz's work in the nascent field of mechanism design. This work was a revelation to me: it had the precision, rigor, and sometimes the beauty of pure mathematics and also addressed problems of real social importance − an irresistible combination.
In fact, I ended up essentially doing an economics Ph.D. The degree was nominally in applied mathematics. But the applied math program at Harvard in those days was remarkably flexible, and allowed students to study whatever they wanted, as long as they wrote a thesis with "significant mathematical content." I took quite a few economics courses (although none, I regretted later, in macroeconomics or economic history), including Truman Bewley's general equilibrium course, where I first got to work with my classmate and later co-Laureate Roger Myerson (we sometimes tackled the rather demanding problem sets together) and Jerry Green's analytic seminar, whose student participants included Elhanan Helpman, Bob Cooter, and Jean-Jacques Laffont.
As an advisor, Ken Arrow was amazingly generous with his time; and I learned an immense amount from our many one-on-one discussions in his office.


who is responsible as international court to solve economic conflicts between countries and is it obligatory?

As far as I am aware, there is no "general" international court for economic affairs.
The ICJ (Interntaional Court of Justice) is the judicial organ of the UN, and it in fact decided few cases with economic background, but usually in the context of other questions of international law.
The closest to what you are looking for is probably WTO dispute settlement system, which is described here: https://en.wikipedia.org/wiki/Dispute_settlement_in_the_WTO
In theory, it is for WTO members only, but in practice, 96 % of global trade is covered under WTO.

Other possible court is the International Court of Arbitration: https://en.wikipedia.org/wiki/International_Court_of_Arbitration
It has no clearly defined jurisdiction, instead, the parties must agree to bring the case to it, but it is becoming increasingly popular.

Last, but not least, an ad hoc panel can be established specially for given dispute.


Prove that the set $X = \{x \in R^L_+| u(x) \geq \bar u\}$ is closed

The statement does not seem to be true.

Define $u$ as $u(x) = -1$ if $x \leq 0$, $u(x) = 0$ if $x \in (0,1)$ and $u(x) = 1$ if $x \geq 1$. The set of points $x$ for which $u(x) \geq 0$ is $(0,\infty)$, which is not closed.