Why do banks take deposits if they do not need them to make loans?

I think there are a few separate issues here.
First, semantics: if an institute doesn't let you deposit money into your account, I think we'd be hard-pressed to call it a "bank".
This really doesn't matter to the fundamental aspect of your question, but I think it's of some use. Because a "bank" has to allow deposits (at least by a naive definition, if not a legal one), you're artificially limiting yourself to institutes that do so.
If we broaden the scope of the question to include more than just "banks", we start to get to the meat of the problem.
Second, the real question: why would an institute let you deposit money into their possession to begin with?
I'm sure a hundred super-smart, very educated experts on the subject will give you a hundred and five different answers that are all the absolute, only truth in all the universe.
But the gist of it is that people are willing to pay the bank to safeguard their money. Money in the bank is insured against theft and heavily guarded, so many people would rather have large sums of money in the bank than in their house.
In return, they might pay for the service directly or through microtransactions related to the account. Many banks charge fees for various things, such as cashier's checks, notary services, or financial advice, services you might not bother with if you aren't already in possession of a checking account.
Further, banks can take the money in your checking account and invest it elsewhere (an obvious form of investment being loans to other people). A single, smart person investing a thousand people's money can make enough profit to be worth their while without being directly paid a cent. Scale that up and you get large banking corporations.
Banks often make money off credit card transactions, even if you make payments before any direct interest is due. They also make money off overdraft fees (a form of lending with very high "interest" rates), interest on the credit card (more lending), and so forth.
And, of course, they hope that if you need to borrow money, you'll do it through them.
Third, the side question: do institutes need deposits to make money off loans?
No, they absolutely do not.
Payday loan establishments don't take deposits, instead relying on high-interest, low payback period lending to make their money.
Pawning an item is a form of payday loan with physical collateral: you leave the collateral with the pawn shop and get paid for it, then if you return on time you get most of your money back, with the difference being the shop's fee for the loan. Pawn shops make a lot of money through simply buying low and selling high, but they still make money off pawned items and don't take deposits.
That said, you can only lend money if you have it, so having a large amount of cash sitting in your vault from deposits allows you to lend more money than if you only lend from your own finances. There's more risk involved, but when you start getting thousands and millions of account holders, the aggregate risk goes way down and you can start treating random statistical values like static rates.
Because bank deposits allow the bank to loan more money, it raises the interest they earn on the loans, so it's a good incentive to allow people to deposit into bank accounts.
Fourth, the question not asked: what does the quoted article mean?
I'm not an expert here, but from my reading it's basically just a way of twisting around the semantics to describe how things can be treated in an opposite fashion to common convention, but doesn't actually prove the convention wrong.
The article claims that lending money "creates" that money. This is true to a degree, but is meaningless in the big picture. When a modern bank gives you a loan, they don't actually hand you any physical thing that's worth money. They just write a number on a (probably digital) ledger that says you "have money".
But they can only get away with that because they own assets that are worth real-world value equivalent to the amount they've "created", and because, in the long run, you will give them the amount of money they've lent you, balancing the sheet yourself.
A bank can only get away with "creating" money for a borrower if they can afford to transfer that money to whomever the borrower is trying to buy something from. It does no good to have fictional money if you can't actually spend it, and you can't spend it if the bank doesn't have a commodity the seller's bank wants.
In practice, the commodity transferred is often virtual money established by digital ledgers, but that commodity is very real, and very controlled, and therefore has actual, real-world value. As such, when a bank "creates" money out of thin air, it's doing so by spending a real-world commodity it only regains if you pay back your loan (or they sell your collateral after taking it back).
There's a particularly egregious bit of word-twisting the article makes. It claims:

Bank deposits are simply a record of how much the bank itself owes its customers. So they are a liability of the bank, not an asset that could be lent out. A related misconception is that banks can lend out their reserves. Reserves can only be lent between banks, since consumers do
not have access to reserves accounts at the Bank of England.

It's true that reserves themselves can't be lent. That's the definition of the required reserve. But it's also irrelevant. Only 10-12½ % of deposits are required reserves (Bank of England, relevant to the article; other banking systems will have other reserve ratios). The other 87½-90 % of deposits are free to be lent out as desired to make money for the bank. Deposits are absolutely an asset for the bank to lend out in the normal sense of the word. They're also liabilities in a strict sense, but they're liabilities that earn the bank a lot of money.
Imagine you loan me £1000 of tools. I owe you either the tools, in good shape, or £1000, or some equivalent combination. Additionally, I'm probably paying you a fee, say £100 a month, to borrow them. Financially, that's a "liability".
But I'm able to use those tools to earn £5000 a month doing something like construction or automotive repair. In normal speak, the tools are an enormous asset, not a liability. I just need to make sure I can return the tools and borrow other tools if you want yours back.
So while the text is "true", it's also completely misguided in rebuking common "misconceptions". In short, it's using smoke and mirrors to make you think you're wrong about something even though you really aren't.
Summary
Institutes don't require deposits to make money off lending, nor do they require lending to make money off deposits, but the two functions synergize very nicely with each other to create what we would typically think of as a bank.
The article you're quoting is mostly just using some hand-waving semantics to explain how modern banking uses a different mindset to approach the same thing we've been doing for millennia, with virtual currencies replacing physical bases for value.
But there's no epiphany behind the hand-waving; the fundamental nature of what's going on behind the layers of bureaucracy and pedantry hasn't changed since a caveman borrowed one sheep from his neighbor and gave back two sheep the next year.
Note
Most everything I've said is actually written in the quoted article. It talks about how a home loan that "creates" money is then balanced out by the bank paying the home seller's bank. It talks about gaining deposits so the bank can afford to lend more money. It talks about needing to pay interest to the central bank to maintain the assets upon which the commercial bank's finances are based.
The article isn't really bad or inaccurate. It just makes a few over-reaching, semantic claims that don't hold up under scrutiny in an attempt to make you think about banking a bit differently. And those are the claims that threw you off.
It also throws in the concept of growing the aggregate amount of money in the country (the "broad money"), a process typically called "inflation", to maintain a stable, thriving economy. So in that sense, banks do "create" money (though ultimately the central bank does the creation by effectively forgiving tiny amounts of debt here and there through carefully-controlled interest rates as well as simply handing out newly-printed notes), but that's really an entirely different topic that's muddying the issue.


Consider the CES production function over four goods defined as follows:
$$ x = (\alpha j^{\gamma} + k^{\gamma})^{1/\gamma}$$
$$ y = (\delta l^{\beta} + m^{\beta})^{1/\beta }$$
$$ z = (\zeta x^{\xi} + y^{\xi})^{1/\xi }$$
If $\gamma$ is near one then $j$ and $k$ are near perfect substitutes. If $\beta$ is near one then $l$ and $m$ are near perfect substitutes. If $\xi$ is a large negative value then the composite goods $x$ and $y$ are near perfect complements


There are several approaches to this problem, none of them accurate as the human happiness is internal subjective experience and thus not verifiable by people other than the afflicted (it is not intersubjectively verifiable).
Please see below several methodologies for pricing the non-economical values such as happiness developed by economists:

Willingness to pay

This problem can be viewed as ‘what is the market price of an item not sold on the market’. To solve it, economists can look whether happiness cannot be, at least in theory, bought on the market. For example, if a pharmaceutical company wants to test some drug that causes unhappines and has to pay USD 1000 for a person to take the test and willingly be unhappy at some level, then the price of this level of unhappiness is USD 1000.

Cost-base valuation

This can be applied when the lack of happiness is directly connected with measurable expenses, like therapy bills, antidepressant drugs, lost income, etc. In this methodology, if a decrease of a happiness is connected with cost of USD 1000 in some period, its value in this period can be measured at not less than USD 1000.

Comparability with damages won in courts

In this method you must find a statistical sample of court judgments, in which compensation was granted for non-pecuniary damages (e.g. for work related undue stress and dissatisfaction) under similar circumstances. As the circumstances can never be the same in different cases (happiness always is subjective and have different economical and social context for each individual), you rank the judgments (e.g. from the weakest to the strongest undue stress and dissatisfaction) and try to place your case on the X axis against court cases. This should result in a point or a range at which unhappiness at question should be valued.

The methods above are connected with many technical problems and are highly criticized. For example, such issues are usually tackled in economics by analyzing the relationship between the marginal cost and one unit of happiness. But what would be a unit of happiness? I recommend further reading of some legal papers on this issue as well as Steven Kelman criticizm in his paper ‘Cost-Benefit Analysis: An Ethical Critique’, where he argues against putting dolar values on non-market benefits and costs at all.

Hope this helps.


Is it possible to eliminate the U.S. national debt?

Yes, I imagine there's reason to contest that claim. Consider the GDP to debt ratio. While it has risen over time and with the recent recession, it was not too far away from Germany which is considered, I think, the absolute picture of fiscal health.

Consider also the critical point that the US government does not have a chance of death like a normal citizen. For example, if you take 30 year loan from JP Morgan, they consider the possibility that you might die. It is critical that they acquire this money before that happens- or they will lose their investment! The government does not have such concerns, and as a result face supernaturally low interest rates. Furthermore, they can pay back debt at a leisurely pace, extend their debt frequently, and if the interest rate is below the rate of inflation (Which it has been!), then they actually earn money by waiting to pay it back.

One of the main reasons for the increase in the debt-to-GDP ratio has been not an increase in spending, but a noticeable decline in government income.  The US tax rates are at historic lows. Even worse the US GDP has been smashed by the recession. The debt could be paid back eventually if it were politically feasible to raise taxes while keeping expenditures constant. The banks of the world have bet billions, trillions if you count derivative and second-stage investments, on this feasibility.

Finally, despite the fact that it is fiat money, it would be very unwise to act as if the debt didn't exist.  Note the panic over congress's threat to default. The consequences of even contemplating a default included: higher future interest rates, lower GDP, and  typically, a devalued currency. Conversely, if money is printed to cover the debt, the consequence is hyperinflation, leading to widespread drops in the standard of living. In my estimation,  if the United States did go into a spontaneous and total default, or hyperinflated to pay the debt- this would likely be the greatest economic disaster of all time, given the size of the US GDP currently. This is not hyperbole- I do not think there is a serious macroeconomist who would disagree.


What happens to a company whose share price goes down to very little over dividend loss?

What happens is completely dependent on the owners. They're the ones whose income has been taken anyway. They may be the only ones with the legal power to do anything (depending on the jurisdiction: there may be some countries where the State can intervene in such matters).

In some jurisdictions, the directors have a legal obligation to maximise returns to the shareholders over the long term; in other cases, they may have a contractual obligation to do so. And in most cases, that will be the expectation of the shareholders, anyway. They can intervene at an Annual General Meeting or an Extraordinary General Meeting: they can pass motions to change future dividend policy; or replace the directors - they could, of course just become the board of directors themselves. And if there's a legal or contractual obligation, then there may be recourse available through legal action too, to seek compensation for the directors' past transgressions.

If they decide to do nothing, then nothing in particular happens to the company's finances.


The profit of a firm $i$ is given by:
$$
\pi_i(p) = p q_i - C_i(q_i)
$$
where $p$ is the price, $q$_i is the output of firm $i$ and $C_i(.)$ is the cost function which differs across firms.
The first order condition gives:
$$
p = \frac{\partial C_i(q_i)}{\partial q_i} = MC_i(q_i^\ast)
$$
This shows how to obtain the optimal supply of firm $i$, i.e. where $MC(q_i^\ast)$ is equal to $p$ (see figures a,b and c in the question).
Total profits for the firm is then:
$$
\begin{align*}
\pi_i(p) &= pq_i^\ast - C_i(q_i^\ast),\\
&= \left(p - \frac{C_i(q_i^\ast)}{q_i^\ast}\right)q_i^\ast,\\
&= \left(MC(q_i^\ast) - AC(q_i^\ast)\right) q_i^\ast.
\end{align*}
$$
This corresponds to the shaded areas in figures a,b and c in the question: the areas equal the difference between $MC(q_i^\ast)$ and $AC(q_i^\ast)$ multiplied by $q_i^\ast$.
Now, consider the first order condition:
$$
p = MC_i(q_i^\ast).
$$
We can invert this function to obtain the supply curve of firm $i$:
$$
q_i(p) = q \text{ whenever } p = MC_i(q).
$$
The total supply in the market at price $p$ is determined by adding the supply curves of over all firms.
$$
Q(p) = \sum_i q_i(p).
$$
Notice that, as usual in economics, figure d in the question draws the supply and demand curves wrong, as they put $p$ on the vertical axis and $q$ on the horizontal. Mathematically, the two should be switched.
If we would do it correctly, it should look something like this:

If $p^\ast$ is the equilibrium price, then the producer surplus is given by:
$$
PS = \int_0^{p^\ast} Q(p) dp = \sum_i \int_0^{p^\ast} q_i(p) dp
$$
Now let us make a change of variables $p \to q$, where $p = MC_i(q)$. Then $dp = \frac{\partial MC_i(q)}{\partial q} dq$  so:
$$
\int_0^{p^\ast} q_i(p) dp = \int_0^{q_i^\ast} q \frac{\partial MC_i(q)}{\partial q} dq.
$$
Then use integration by parts to get:
$$
\begin{align*}
\int_0^{q_i^\ast} q \frac{\partial MC_i(q)}{\partial q} dq &= \left[q MC_i(q)\right]^{q_i^\ast}_0 - \int_0^{q_i^\ast} MC_i(q) dq,\\
&= q_i^\ast MC_i(q_i^\ast) - C_i(q_i^\ast),\\
&= p^\ast q_i^\ast - C_i(q_i^\ast) = \pi_i^\ast.
\end{align*}
$$
Here we assume $C_i(0) = 0$ and we used the first order condition to substitute $MC_i(q_i^\ast) = p^\ast$. We also used $\pi_i^\ast$ to denote the profit of firm $i$ at price $p^\ast$. From this:
$$
CS = \sum_i \int_0^{p^\ast} q_i(p) dp = \sum_i \pi_i^\ast,
$$
which shows that the consumer surplus indeed equals the sum of all profits in the industry.


Is there an externality if someone buys all stock of some merchandise?

From the Oxford dictionary: (emphasis added by me)

  externality (noun)

  A consequence of an industrial or commercial activity which affects other parties without this being reflected in market prices...

Another way to put this is that externalities are not the results of market rivalry. So if you and I are bidding on the same item on e-Bay, and you are buying it, I kind of suffer, but this is within the market context, this is not abnormal. Whereas if you buy a new stereo system on e-Bay and then keep me up all night by listening to loud music, this happens outside of the market context (unless I sold you the right to listen to loud music) and is an externality.

In case of a Cournot-oligopolies one firm's increased production would also harm other firms by lowering the price. But this is not an externality either, as it is the result of normal market rivalry for the consumers.

It seems like in the situation you are describing you are not facing an externality, as it everything happens in a market context. Perhaps there is information asymmetry, as the store was understocked. (This would require additional assumptions and is not the only possible explanation.)


Is a resource allocation problem in mechanism design a direct or indirect mechanism?

By definition, a direct mechanism is a mechanism that asks all agents for their types and then produces some outcome. Formally, it is a mechanism $\langle M,g\rangle$ in which $M_i$ is wlog equal to $i$'s type space, usually denoted by $\Theta_i$.

In your example, there is no such restriction. However, for any indirect mechanism there exists an equivalent direct mechanism. This result is called the revelation principle.


Methodology of Economics: interpretation and explanation

The standard reading on the methodology and the role of explanation in economics is Daniel Hausman's work.

Hausman, Daniel M. The inexact and separate science of economics. Cambridge University Press, 1992.

Nancy Cartwright has some work on the interpretation of causation in the social sciences.

Cartwright, Nancy. Hunting causes and using them: approaches in philosophy and economics. Cambridge University Press, 2007.

Another entry to this literature may be the Stanford Encyclopedia of Philosophy:

http://plato.stanford.edu/entries/economics/

However, I am not sure what you mean with a literature on interpretation in economics.


What is the justification for it being a problem if loans are not paid back?

It appears that the OP confuses  money with property rights

A) It writes "deposits owned by the bank" which is simply wrong, since deposits are liquid assets belonging to the persons that deposited them. The bank leases the funds from them and then sublets them to the debtors.
B) Looking also at an OP's  comment to another answer, indeed the bank may create the money, but it creates the money as a medium of exchange and representative of value -it does not keep the property rights on the money itself it creates.
 C) Finally we should not forget that the bank is a distinct legal entity from its own shareholders.

Stopping at the four explicit amounts the OP gave (after person $B$ pays person $B'$ and person $B'$ deposits the money in the bank), the bank's balance sheet is as follows:

\begin{array} {| r | r | r |}
  \hline
\hline
  \text {ASSETS} & \text {USD} & \text{LIABILITIES} & \text{USD}\\
  \hline
\text{Loans} & 1,700 & \text{Shareholders Capital} & 1,000\\
\text{Cash} & 1,000 & \text {Deposits} & 1,700\\
\hline
\text {TOTAL ASSETS} & 2,700 & \text{TOTAL LIABILITIES} & 2,700\\
  \hline
\end{array}

A Balance Sheet is just an inventory at a specific point in time, of Cash and Rights to Collect in the future ("Assets") and Obligations to Pay (also in the future) ("Liabilities").

It is important to understand the "Liabilities" (obligations to pay) side.
The initial USD $1,000$ with which the bank started, represent an "obligation to pay" for the bank towards its own shareholders. Second the bank is obliged to pay at any time USD $1,700$ total to persons $A'$ and $B'$ that deposited their wealth to the bank. The fact that the depositing transaction could happen because the bank in a previous instance has created the money, does not give the bank a property right on that money, just because it was its "creator".

That the bank has at the moment $1,000$ in Cash as when it started doing business, relates to the bank's "liquidity" situation. This is not unimportant, and in fact it is an index of financial health on its own right. But it has to do only with the short term horizon. For the long-term horizon though, it is property rights that matter. And so it is not the only financial criterion for the economic health of a bank.

So assume that person $A$ comes in and say "I cannot pay my loan of USD $900$". Certainly, in the short-term, this won't affect the liquidity of the bank -it will still have the USD $1,000$ that it now has. But if the bank writes-off the debt, it will do so against the Shareholders capital invested. So after the write-off the balance sheet will be

\begin{array} {| r | r | r |}
  \hline
\hline
  \text {ASSETS} & \text {USD} & \text{LIABILITIES} & \text{USD}\\
  \hline
\text{Loans} & 800 & \text{Shareholders Capital} & 100\\
\text{Cash} & 1,000 & \text {Deposits} & 1,700\\
\hline
\text {TOTAL ASSETS} & 1,800 & \text{TOTAL LIABILITIES} & 1,800\\
  \hline
\end{array}

Assume things with debtor $B$ go smoothly, the bank will eventually (in the long-term) collect his loan, and it will be able to give to persons $A'$ and $B'$ their deposits back... so where is the problem? Well, the problem is that it will give to its shareholders only USD $100$ back instead of the initial USD $1,000$ investment.

And who might care?

First, we live in societies built upon the principle of private property rights. The shareholders have a right to defend their property rights, and the bank acting as their representative attempts to defend them by not simply writing-off the debt (but demanding another asset -the car- in its place), even if such a write-off may affect the property rights only of the shareholders, and not of the people that keep their wealth to the bank as deposits.

But again, does this creates a problem to the economy now? In terms of actual existing liquidity the answer is no. But the Present is affected by what we project for the Future. After the write-off, the future of the shareholders and of the bank looks worse, because their current wealth is less. And this will affect negatively what the bank and the shareholders will and can do in terms of their involvement in the economic activity now.
So to the question "who cares?" the second answer is "the whole economy".


What is the latest thinking on the economic calculation problem?

Depends what exactly is the authors definition of “radical markets”. A predominant consensus in the economic field is that a mixed economy which relies predominantly on market form of organization but also has government stepping in correcting major market failures and some macroeconomic management is economic system that delivers the greatest amount of material welfare to greatest number of people.

I can’t see much challenge to this consensus within the profession but I would hesitate to make predictions about future - just few years before Einstein developed theory of relativity Lord Kelvin stated that: “There is nothing new to be discovered in physics now, All that remains is more and more precise measurement." Although, the consensus is quite strong and entrenched so I would also not say that the estimate of authors of at least few generations is wrong.

Wether that’s the best is matter of personal morals. Some radical ascetic may prefer a system where everyone lives on some agrarian subsistence level. If our objective would be to just bring  everyone down to the subsistence level then markets would of course not be the best form of organization. This last paragraph might look silly to you but I am just highlighting that there is no moral consensus on what the objective of society should be, and this objective is also what determines what’s the best system, and that this is ultimately question that transcends economics.

Edit: In response to your new question in your clarification about using computer technology, the following statement:

  Typically an alternative calculation method, more efficient than markets, is considered to be possible through technology: parallel processing and more recently artificial intelligence

is not considered correct in economics, in fact this whole statement puzzles me because it presents itself as a fact while it is not true that in the wider economic literature technology, as you defined it here as parallel processing or artificial intelligence, is considered an efficient alternative or even more efficient method of economic calculation.

The fundamental problems that markets help to solve are aggregating information and providing incentives. However, when we talk about information we often talk about information which is not easy or even possible to objectively verify. For example, important information that markets aggregate is the information on consumer preferences. No matter of computing power will help you with that if people dont reveal their real preferences through lets say questionnaires because they either might suffer from social desirability bias, people might have incentives to mislead or not even themselves fully understand the preferences they have.

Moreover, market mechanism through prices actually provides incentives for people to act in accordance with the information the markets aggregate. If the prices for apples increases due changes in consumer preferences then producers do not get just the information about that in an aggregate and simply understandable way in the form of price but they are also provided with an incentive to act accordingly and expand their output.

Parallel processing and machine learning does not help with solving the underlying  problem of economic calculation in any way. Parallel processing and machine learning algorithms can certainly help governments to conduct more efficient policy by helping to model and estimate its effect and getting more efficient feedback on the response of economy to government action. But they do not help with the fundamental problem of economic calculation at all, and frankly no amount of computing power probably will as the problem of economic calculation is not a problem of lack of computing power but a problem of eliminating information asymmetry between people and 'central planners'  and giving people incentives to provide accurate information and act on it.


A world only with two contries (net exporter)

So we have
$$S^A = I^A + NX^A $$
$$S^B = I^B + NX^B $$
where $S^A$ and $S^B$ are exogenous constants. And since there are only two countries we have
$$NX^A = -NX^B $$
which leads to
$$S^A - I^A = -( S^B - I^B ) $$
Use that last equation to solve for r (after plugging in the I formulas). Then solve the first equation and require that $NX^A$ be positive.


What mechanisms are at work when the market falls dramatically one day and then rises just as dramatically the next?

In short...human nature! (I'm only half-kidding). Of course, mechanically, what drives these abrupt changes is simply a large amount of buyers one day...and a large amount of sellers the next.

A great metaphor you might like is Mr. Market. Also this video of Warren Buffett talking about Mr Market. It's just a metaphor used to show how capricious, and seemingly irrational, the market can be.

I don't think there's some philosophically deep mechanism at play (other than human nature). Maybe you could make the argument that new information comes out that justifies a 5-10% swing...but that argument seems flimsy to me.


What explains why Puerto Rico has much lower GNP than GDP?

This is impossible to answer completely objectively given the data availability.
The GNI is given as:
$$GNI = C + I + G + NX + NFFI$$
where, C is consumption, I investment , G gov. spending, NX net exports,  and  NFFI is net foreign factor income. The first 4, terms are by definition GDP so the relationship between GNI and GDP is:
$$GNI=GDP+NFFI \text{ or } GNI-NFFI= GDP$$
Hence if $GNI<GDP$ we know that NFFI must be negative. NFFI is the net factor (e.g. not just wages but return on capital and other factor income) income that Puerto Rican residents.
However, note that since it is a net factor income separate observations like that more Puerto Ricans work abroad than foreigners there are not helpful. First, what matters is the value of wages so if Puerto Rican firm hires one foreign scientist whose yearly wages are 10x that of low skill Puerto Rican worker abroad than NFFI portion from wages will be still negative even if the ratio of foreign workers in Puerto Rico and Puerto Ricans working abroad will be less than 10.  Second, since it includes income from other factors it might easily be that many foreign own companies locate in Puerto Rico and so their incomes count toward GDP but not GNI.
Hence to answer your question definitively we would have to take apart the NFFI term and analyze its components. However, problem is that such data is likely not publicly avaiable. Most statistical offices do not provide the breakdown for NFFI. I checked the world bank statistics and couple of other main sources for publicly avaiable macro data but I was not able to find any source that would provide decomposition of NFFI for Puerto Rico.
Consequently, we can only conjecture why GDP is larger than GNP. I think likely reason is that Puerto Rico offers lower tax rate than US and does so for a long time (Boles, 1988; Grubert & Slemrod, 1998), which is attractive for US firms (especially since Puerto Rico is US territory with US citizenry). Subsequently, when these firms eventually get income and distribute it to their 'foreign' owners it counts negatively towards NFFI.
However, at the same time as found by Borjas (2007):

Puerto Rico attracts high-skill in-migrants and exports low-skill workers.

So the argument I mentioned above applies as well and likely every Puerto Rican working abroad contributes less to GNI than any high skill workers that come to Puerto Rico, meaning that even if large numbers of Puerto Ricans work abroad it might not be enough to offset the influence of relatively less high skill workers.
In my personal opinion, the former reason likely dominates but absent of any more detailed statistics you should take it with grain of salt.


Is there still allocative efficiency when a subsidy is placed on a good?

allocative efficiency occurs whenever total market surplus is maximized. This occurs whenever price is equal to the ‘free market’ equilibrium price. There is no surplus of goods at this equilibrium price. There is no DWL at this equilibrium. If there exists DWL, the market is not operating at the 'free market' (as phrased in a comment below) or the ‘unfettered’ equilibrium and the market is not allocatively efficient. To be clear, this market will still operate at some equilibrium. However, this equilibrium will not be efficient.


Not sure I understand your confusion.
Fruits provide external benefits worth $X$ to honeybees, honeybees provide external benefits worth $Y$ to fruits.
Knowing all this, farmers and beekeepers agreed that farmers should pay beekeepers amount $Z$, which probably comes close to making things even, meaning $X + Z = Y - Z$. Since $Z > 0$, $X < Y$.


First we must address the nature of savings. Savings are not bad or good per se. It depends on the context. It is true that in the short run, increased savings will lower consumption, thereby lowering GDP. However savings also increase investment, which in the long run increases economic growth. Therefore, the optimal savings rate is found by solving the trade-off between the short and long-term.

If you are interested only in the short-term (GDP right now and a few periods more) then you would see savings as bad. If you are interested only in the long-term you would see savings as good. If you are interested in both, then it is ambiguous.

The optimal savings rate is certainly not 0 for a standard economy. One way the optimal savings rate is characterized is through the so-called "golden rule" in a simple Solow growth model. You might want to look this up.

Empirically, we typically observe that most countries save too little, i.e. are below the golden rule savings rate. If you are below the optimal rate, whatever that may be, then more savings are good. If you are above it, then more savings are bad. Hence, it is not straightforward to make a case about the desirability of savings.

Your second question was about poverty and savings. Yes, they can and do coexist empirically. In fact, many economists are puzzled why savings rates are so low in poor countries, since more savings increase growth over the long-term.

To see why savings and poverty can coexist, you must think about the reason we have savings. Individuals save more today, so that they can consume more tomorrow. Hence, you cannot really detach the two completely and say that savings are bad. Imagine a poor country that is primarily agricultural, as is often the case. Farmers there make money a few times a year, maybe once or twice, when the crops are sold. However, they must live off this money for the whole year. The only way to make sure you can survive the winter is to save money from crop sales. Furthermore, the poor also need to make big purchases in the future and the only way to do so is to save. E.g. they are poor, but must save up for a wedding, etc. Or they need to buy fertilizer for the next season and the only way to do so is to save enough money. All these things are possible as long as death from starvation is not an imminent threat, which for the majority of poor people today it is not.

Savings do not exclusively occur when all other current consumption needs are totally met, as it seems to be implied in your question. In fact, they can never truly be met, as ressources are scarce. Savings are instead determined by the optimal decision of how much to consume today vs. tomorrow, or how many of my needs to leave unfulfilled today vs. how many unfulfilled tomorrow. Hence the fact that poor people cannot fulfill all their needs today does not imply that they should not save, since they also think about the future and how to fulfill those needs tomorrow. Think of surviving on an island with limited food. You would likely consume a bit everyday and be hungry in order to survive, than eat everything in one day, be full and then die in a few days.


Consider a set of consumption bundles $X$, which is convex and let $\succeq$ denote 'at least as good as', then convex preferences may be defined mathematically as follows.

Convex preferences. For any $x,y,z\in X$ and $\lambda\in[0,1]$ we have that $$x\succeq z\text{ and }y\succeq z\Rightarrow \lambda x+(1-\lambda)y\succeq z.$$

Informally we may say that convex preferences captures the concept of averages being at least as good as extremes.

Critique 1. Below is an excerpt from the direct link where the author points out the fact that under certain circumstances, the assumption of convexity may not be adequately assumed as a property or preferences.

  That is, [convex preferences says that] if an agent is indifferent between $x$ and $y$ and has convex preferences, then she will like $x/2+y/2$ at least as much as either $x$ or $y$. Of course, this doesn't always make sense. You might like a beer or wine, but not a mixture

However, it depends on the interpretation.

  Whether convexity makes sense often depends on the interpretation of the
  goods space. For example, if the components of $x$ are rates of consumption, then a half-half mixture of beer and wine might mean drinking beer half the time and wine half the time. Convexity of preferences seems more plausible in that interpretation than in the previous one. Similarly, some find convexity easier to rationalize if the "goods" are more highly aggregated – for instance, if the goods are "food" and "clothing," than if goods are highly specific.

Critique 2. The link directs you to McCloskey's critique of economics. She criticizes how economists focus too much on assumptions and mathematical proofs and too little on empirical facts. In this sense, if microeconomics or mathematical economics is to be about general properties of consumer behaviour, then the assumption of convex preferences is not always suitable from a methodological perspective, as non-convexity may hold in a given economy. Also, following McCloskey, one may say that the assumption of convex preferences, while it may guarantee that certain mathematical propositions are true (i.e., one may deduce them from the assumption of convex preferences and simpler axioms), does not by itself say something about economic behavior. It surely implies certain economic behavior $\mathcal B$, but we cannot from the observation of such economic behavior $\mathcal B$ draw the conclusion that economical agents' preferences are such that they are convex over economical entities like goods.

Critique 3. Instead of assuming convex preferences, one may study the wants and sentiments of individuals,  as Adam Smith does in his Theory of Moral Sentiments and later in his Wealth of Nations, or study cognitive biases, like Daniel Kahneman does in his best-selling Thinking, Fast and Slow. These two authors does not, to my knowledge, directly criticize the concept of convex preferences, but they show how economic analysis performed in another way may lead to other conclusions about the economy. If their conclusions do not agree with conclusions derived from axioms like that of convex preferences, then there is implicit criticism in the sense that there is a mismatch in theoretical conclusions.


Why life expectance is growing faster than retirement age? - source needed

Loss-aversion and political tactics.

A lot of the elderly are no longer physical broke down - which was the original reason for retirement. But, if you can retire in good health, it is completely rational for the individual to do this, and to vote for the politicians who will provide this for you.

Or, as a danish mayor once put it:

  "When you first throw something into the monky cage, it is hard to
  get it back again"

Aka, when you first give people major entitlements and life improvements, it is next to political impossible to take them away.


What happens when a subsidiary distributes its shares to its parent company?

The argument to have both share values converging is based on the fact that when the subsidiary redistributes the shares to the parent is similar to "unspining off".

The spin-off subsidiary is effectively being re-owned by the parent, becoming a part (a share) of the parent. As such, owning shares of the subsidiary becomes a proxy of owning shares of the parent.


How helpful is Hotelling's Rule in determining the price of oil?

It would be easy enough to subject the question to simple quantitative analysis - get the time-series of historical oil prices and away we go.

Or, we can take a quick qualitative approach and get the same answer:

The Rule just sets out what the most economically efficient path would be. That's nothing at all to do with how things happen in reality: it's just how things behave in textbooks for beginners. For example, it relies on an assumption of perfect foreknowledge; whereas, in the real world, we have previously-unanticipated innovation that radically changes both industry's cost structures, and the amount of known reserves - as your fracking example illustrates.


Mathematical derivation of the Production Possibility Frontier

The question is broad, but I believe there is plenty of literature that defines this concept in similarly broad terms.
The following is adapted from the Wikipedia on Pareto Efficiency,
which is the mathematical basis of the Production Possibilities Frontier.

There may be better definitions out there, but this one should probably work in a lot of cases:

The Production Possibilities Frontier, $P(Y)$, may be more formally described as follows. Consider a system with function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, where $X$ is a compact space of feasible decisions (including allocations of time and endowment goods) in the metric space $\mathbb{R}^n$, and $Y$ is the feasible set of criterion vectors (say, final goods and services) in $\mathbb{R}^m$, such that $Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}$.

We assume that the preferred directions of criteria values are known so that more of any good in $Y$ is better. A point $y^{\prime\prime} \in \mathbb{R}^m$ strictly dominates another point $y^{\prime} \in \mathbb{R}^m$, written as $y^{\prime\prime} > y^{\prime}$, means that for each element index $i$, $y''_i \geq y'_i$ and there is at least one element $j$ such that $y_j'' > y_j'$. The Pareto frontier is thus written as:

$P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} > y^\prime, y^{\prime\prime} \neq y^\prime \; \} = \emptyset \}. $


The law of supply and demand - How does it work?

You need to take into account the cost of producing a good as well. In most cases, the marginal cost of producing something is increasing. A profit-maximizing producer would want to produce until price is equal to marginal cost. That means that scenario 2 is responsible.

For the case of rising prices with no increase in demand, just think about the basic demand and supply model. What could cause an increase in equilibrium price without the demand's changing? (Hint: a shift in ____). What could cause that shift?


How can I prove $∇U(x).D_m x(p,m)= \text{shadow price}$?

Assuming certain regularity conditions, the first order conditions for
$$
\max_{x, \lambda} U(x) - \lambda (p \cdot x - m)
$$
are
\begin{align*}
&D_{x}U(x(p, m)) - \lambda p = 0
\\
\text{and} \quad
& p \cdot x(p, m) - m = 0.
\end{align*}
Moreover $x(p, m)$ will be differentiable with respect to $m$ at $(p, m)$, and this fact together with the second equation implies
$$
p \cdot D_{m}x(p, m) - 1 = 0.
$$
So, by the first equation,
$$
D_{x}U(x(p, m)) \cdot D_{m}x(p, m)
=
\lambda p \cdot D_{m}x(p, m)
=
\lambda.
$$
The derivative $D_{x}U(x(p, m))$ of the utility function at $x(p, m)$ corresponds to $\nabla U(x)$ in your notation.


Should I use current prices or volumes estimates?

If you are conducting empirical exercises, you should use real values i.e. volumes. If the data has time dimension, it is even more imperative to deflate the variables. The reason for this suggestion is following:

Take output data, over time output changes and usually around a rising trend. The trend growth of output could be due to rising prices instead of volume of output. The following image shows the effect of inflation on output:

The nominal GDP grows at a relatively higher rate than real GDP. The economy appears to have grown, in fact, much of this growth is due to rising price levels, not due to positive growth in volume of output. Using the data from the graph, you can see that

nominal GDP in 2012 is 1.3 (=16000/12500) times that in 2005, that's
a 28% increase in GDP.
real GDP grew by 8% only.
in the same period, prices rose by 18.5% (=1.28/1.08-1).

This shows that the volume of GDP, real output, did not grow as fast. The nominal GDP, on the other hand, gives us false impression that the economy is growing at a faster pace.

If you are using the data in statistical or econometric exercises, you should use volume data in constant prices.

And, if you are comparing performance across countries, you should convert the data to international dollars using purchasing power parity (PPP). I think OECD database offers PPP based macro-variables


Demand derived from Cobb-Douglas utility, interpretation, check

From (1) and (2) you get
$$\frac{x_j}{x_i}=\frac{a_j p_i}{a_i p_j},$$
or equivalently,
$$x_j =\frac{a_j p_i}{a_i p_j} x_i.$$
Substituting this into equation 3 for $j=2,...,n$ and $i=1$ (solving for the demand function for good 1) we get
$$M=p_1x_1 + \sum_{j=2}^n p_j  \frac{a_j p_1}{a_1 p_j} x_1$$
$$M=p_1x_1 + \sum_{j=2}^n  \frac{a_j p_1}{a_1} x_1$$
$$M=p_1x_1 + \frac{p_1}{a_1} x_1\sum_{j=2}^n a_j $$
$$M=p_1x_1(1 + \frac{1}{a_1}\sum_{j=2}^n a_j). $$
Finally, solving for $x_1$ we get
$$x_1^* = \frac{M}{p_1}(1+ \frac{\sum_{j=2}^n a_j}{a_1})^{-1}$$
$$x_1^* =\frac{a_1}{\sum_{j=1}^n a_j} \frac{M}{p_1}.$$

Analogously,
$$x_i^*=\frac{a_i}{\sum_{j=1}^n a_j} \frac{M}{p_i},$$
for $i=1,...,n$.

As is always the case for demand functions derived from a Cobb-Douglas utility function, the consumer spends a constant share of income on each good. To see this, rearrange the previous equation to get
$$p_ix_i^*=\frac{a_i}{\sum_{j=1}^n a_j} M,$$
for $i=1,...,n$.


Leontief preferences and 2nd welfare theorem

Hint:

  Second Welfare Theorem: Suppose that for every individual $i = 1,...,N$ the utility function $U^i(c_i)$ is continuous, locally non-satiated, and quasi-concave. If $\left\{c_1^P,...,c_N^P\right\}$ represents a Pareto optimal allocation in which $c_i^P >> 0$ for every $i = 1,...,N$, then there exists a set of prices, $p^E \geq 0$ and $p^E \neq 0$ and a set of transfer payments, $\left\{\tau_1,...,\tau_N\right\}$ such that:

  $\left\{c_1^P,...,c_N^P\right\}$ is feasible

  $$\sum^N_{i=1}c_i^P \leq \sum^N_{i=1}\omega_i^P$$

  For every individual $i = 1,...,N$, the consumption bundle, $c_i^P$, solves:

  $$\max_{c_i} U^i(c_i) \ s.t. \ p^E \cdot c_i = p^E \cdot \omega^i - \tau_i$$

  and $\sum^n_{i=1}\tau_i = 0$

Or in other words, for any efficient (Pareto) allocation, there exists a price vector such that the price vector and optimal bundle represent a quasi-equilibrium with transfers.

Also note that you need preferences to be convex and locally non-satiated and the production set must be convex. So now what happens when $U_i(c_i) = \min \left\{\frac{c_1^P}{\alpha_1},...,\frac{c_N^P}{\alpha_N}\right\}$? It is indeed continuous, locally non-satiated and quasi-concave.


Providing subsidies affects Marginal Private cost but not Marginal Social Cost?

The amount of external cost is determined by the level of production in this framework.

If the producers receive a subsidy the MPC shift to the right as you point out. That also means that for each level of cost production increases and therefore the external benefits are less, that is, they are internalizised. If the subsidy is enough the external benefits of production are zero and the MPC=MSC.


Determining whether preferences are rational given a utility function

As long as a preference can be represented by a continuous utility function, it is rational (and continuous). So from the very fact that you're writing down a utility function that is continuous in both $x$ and $y$ (on $(0,\infty)^2$), it follows that the function represents a rational preference (provided that both $x$ and $y$ are positive).

However, if you insist that $y=0$ is part of the feasible choice set, then the preference is not complete. The individual cannot compare a bundle $(1,1)$ with another bundle $(1,0)$ because the utility of the latter bundle is not defined.


Piketty's explanation of elasticity of substitution (from his book Capital in the 21st century)

The following is from Thomas Piketty and Gabriel Zucman (2015, From Handbook of Income Distribution, Volume 2, Chapter 15, Part 15.5.3 which is hard to link to directly but get it here):

Take a CES production function
$$Y=F(K,L)=(a⋅K^{\frac{\sigma-1}{\sigma}}+(1−a)⋅L^{\frac{\sigma-1}{\sigma}})^{\frac{\sigma}{1-\sigma}}$$

Me: $\sigma$ is the elasticity of substitution. They have a small typo here with the outer exponent of $\frac{\sigma-1}{\sigma}$ instead of $\frac{\sigma}{1-\sigma}$, but I'm pretty sure that is wrong so I corrected it here. It doesn't change the rest of what I'm doing.
Back to Piketty and Zucman:

The rate of return is given by $$r = F_{K} = a \cdot  \beta^{\frac{-1}{\sigma}}$$ with $\beta = K/Y$.
The capital share is given by $$ \alpha = r\cdot\beta = a\cdot\beta^{\frac{\sigma-1}{\sigma}}$$

Me: Now take the partial derivative of $\alpha$ with respect to to $\beta$
$$ \frac{\partial \alpha}{\partial \beta} = a \cdot (1-1/\sigma) \cdot \beta ^{-1/\sigma}$$
By assumption $a$ is always positive. Because $K$ and $Y$ are always positive so is $\beta ^{-1/\sigma}$. That means the sign of $\alpha$ is the sign of $1-1/\sigma$. For $0<\sigma<1$ you can see that $1-1/\sigma < 0$ and therefore $$\frac{\partial \alpha}{\partial \beta}  < 0$$.


Everybody seems to be worried about the raise in the consumption tax, and how it has thus far crippled the efforts to bring the Japanese economy out of stagnation. But Japan has 200% of DGP in public debt, and it cannot leave its public finances go south, not even temporarily.

My impression is that it was the choice to lower corporate tax and "in exchange" increase V.A.T. (perhaps under the impression that it was low, if they looked at European Union levels). Demand was supposed to strengthen through direct transfers, while lowering corporate tax is to boost business initiatives, towards investment but also give some added room to increase wages (to help demand).

We' ll see. Up to now, it doesn't go very well.

ADDENDUM

That the increase in the Sales Tax had as its target to take care of the public finance side of the equation is clearly stated in the IMF-report on the matter, in which we read

  "The consumption tax rate increase in April to 8 percent was a major achievement, but is only a first step towards fiscal
  sustainability...The second consumption tax rate increase in 2015 to
  10 percent with a uniform rate should be confirmed. Raising the tax
  rate further at a moderate pace would help establish fiscal policy
  credibility. Staff estimates that fiscal consolidation in 2015 would
  slow growth by ½ percent, leaving growth in 2015/16 still above
  potential under the baseline."

A quick journalistic take can be found here.


Why do we have to normalize the income of consumers when working with an Edgeworth Box in a simple trade model with Pareto optima?

If you by normalizing income mean normalizing one of the prices then yes you do not have to normalize it, but it makes everything easier since only the relative price matters.
All price vectors (2 dimensions) that solves the problem will have the same ratio. There are therefore usually infinitely many price vectors that solves the problem.

Regards


In general, if $c_1$ is endogenous, you need instruments for $c_1$ as well.

Example: Even when $x_1$ is exogenous, if $c_1$ is endogenous and $x_1$ and $c_1$ are correlated, then OLS (which is the IV estimator using $x_1$ as IV for $x_1$) is generally inconsistent.

Exceptions exist. For example, when exactly identified, if $(Ezc') (Ecc')^{-1} (Ecu)=0$, then your IV estimator is consistent, I guess. (Can derive this using standard technique involving the law of large numbers.)


Number of observations in different panel data regressions

It is conventional to report both the total number of observations (1000 in your case) and the number of groups (500 in your case). If you want or are allowed to report only one number as the number of observations, it is suitable to report 1000 for POLS because POLS doesn't mind the fact that it is a panel data set. For FD, it is slightly confusing to me to say that the number of observations is 500. I would probably ask again of the writer what 500 means.


Will high computing power substitute the certainty-equivalence assumption?

This is perhaps a good opportunity to point out that the "certainty equivalence" concept means one thing in microeconomics/choice under uncertainty theory, while it means something different in macroeconomics.

Microeconomics/choice under uncertainty
The Certainty Equivalent of a lottery/gamble, is the amount of wealth which, if given with certainty, provides the same utility as the lottery/gamble. See for example Jehle & Reny's "Advanced Microeconomic Theory" (2011, 3d ed.), p.113.

Macroeconomics
"Certainty Equivalence" is the situation in a stochastic model, where optimal decision rules prove to be identical to those that would have been derived in a deterministic framework (see for example Lungqvist & Sargent's "Recursive Macroeconomic Theory" 2004, 2nd ed, p. 113-115). Informally, this is sometimes described as "agents behave as if the stochastic processes are not stochastic", or "the decision rule is not affected by stochastic variability".

It is a funny coincidence that both references treat the subject in the same page number...

The concepts are evidently linked by the "as if there was no uncertainty" angle, but they are different in essential aspects: the micro-concept is a "buy-out" of uncertainty, providing a certain alternative towards which the agent that faces the gamble is indifferent in terms of utility, while the macro-concept emerges as a property of the solution (and only under restrictive model structures, or linear approximations thereof).


What is considered a "final product" for gross domestic product?

Since you're asking a question about the value-added approach to calculating GDP, the answer that follows talks about the value-added approach. It's worth noting, however, that GDP can (and is) also calculated through two other approaches: the expenditure approach and the income approach. In the US, the value-added approach is used primarily in the GDP-by-industry accounts.

The boundary between intermediate and final consumption seems blurry. How, and where is the line drawn?

Products and services provided to consumers are considered to be final products. Products and services provided to businesses are treated as intermediate inputs. See point 4 on page 3 in this NIPA primer.

When doing the accounting at an high level (city, state or national), how do the accountants know what part of the paint consumed take as intermediate consumption and what part to take as final consumption?

A simplified answer (ignoring issues like capital consumption and inventories) is that at the national level, business revenue is totaled, and then business expenses (other than labor costs) are subtracted.

So if a restaurant sells \$100 worth of dinners, pays a farmer \$50 for the food, and pays a designer \$10 for designing the restaurant, the totals look like this:

Restaurant: \$100-(\$50+\$10)=$40

Designer: \$10

Farmer: \$50

Total: \$40+\$10+\$50 = \$100

As you can see in this example, by subtracting business expenses from revenue, double-counting is eliminated.

At the state level in the US, GDP is actually allocated from the national level using a variety of techniques. Metro area GDP is allocated from the state level to counties based on income by industry and then aggregated. As a result, the value-added approach is simply not used to calculate state and local GDP in the US.

Is the service of cleaning a factory considered a final product or intermediate consumption needed to produce whatever the factory produces?

Intermediate consumption.

What about the plant's paint, electricity, training of workers and so on?

Intermediate consumption.

The consumption of decoration from a restaurant is a requirement (since people go to a restaurant in part, due to the style) and by that reasoning, it is intermediate consumption, but a luxury in one's house (therefore final consumption).

Intermediate consumption.

People need to eat in order to work and produce, and more so the more they work (hence food may be classified as intermediate consumption), and need medical care to live (and hence produce) just like car factories need steel. I have seen these things sometimes taken to be final consumption.

If you refer to the example above, you will see why they're treated as final consumption. The household sector is not treated as a business; if it were, the value-added approach would not work. Wages paid to workers would be subtracted as intermediate inputs, and workers' consumption of whatever they consume would also be subtracted as intermediate inputs, and everything would sum to zero.

If a restaurant sells \$100 worth of dinners, pays a farmer \$50 for the food, pays a designer \$10 for designing the restaurant, and pays workers \$20 for making the food, the totals look like this:

Restaurant: \$100-(\$50+\$10+\$20)=$20

Designer: \$10

Farmer: \$50

Households: \$20-\$100=\$-80

Total: \$20+\$10+\$50-\$80 = $0

What we've discovered above is called the circular flow of goods and services.

A simpler example can make this obvious: let's say that there exist two businesses in an economy. One grows cabbage, the other raises pork. The cabbage guy gets hungry for some pork, and buys \$10 worth of pork, while the pork guy decides he wants some cabbage and buys some of that for \$10. Under the value-added approach, \$20 of market output has been produced. If we said, "Hey, wait, but people need to eat food to live!" and did not count the food production, then we'd end up concluding that \$0 of market output had been produced. That would be incorrect.


Is it true that if marginal cost is constant, then average variable cost is also constant and equals marginal cost?

Your intuition is correct.
First, you're right that "marginal cost only depends on variable cost", since
\begin{equation}
MC(q)=\frac{\mathrm dTC(q)}{\mathrm dq}=\frac{\mathrm d(FC+VC(q))}{\mathrm dq}=\frac{\mathrm dVC(q)}{\mathrm dq}.
\end{equation}
Next, if marginal cost is some constant $k$, then variable cost must be $VC(q)=kq$, because we can integrate $MC$ to obtain $TC$, where the term that varies with $q$ is $kq$:
\begin{equation}
MC(q)=k
\quad\Rightarrow\quad
\int k\;\mathrm dq =\underbrace{kq}_{VC(q)} +\underbrace{K\vphantom{q}}_{FC}
\end{equation}
where $K$ is a constant resulting from the integration.
Lastly,
\begin{equation}
AVC(q)=\frac{VC(q)}{q}=\frac{kq}{q}=k,
\end{equation}
which is a constant, and same as $MC$.


Why is the Social Security tax split midway between employer and employee?

It is actually a little bit more complex because there are special cases where there might be no tax burden shifting (see discussion in Mankiw Principles of Economics ch 12) although, generally speaking the statement of D. Friedman would be correct.
As to why government does that, besides incompetence that should not be too casually dismissed, there might be completely rational reasons.
If people do not understand economics, and if people do not understand that they are still being taxed, they might be more likely to vote to candidates that they believe are not taxing them (regardless of whether they are actually being taxed or not). This is fully consistent with those voters being rational mind you since rationality only requires people to act consistently with their beliefs and  makes no assumptions on what peoples beliefs are (see more discussion of rationality in Tadelis Game Theory: An Introduction). Consequently it might be rational for politicians to do this to maximize their votes.
For example, consider analogy with food producers. Some food producers put on their food labels reading "no chemicals". This is prima facie absurd statement as anything made from ordinary matter even plain water is by definition made from chemicals. To have food without any chemicals the food would have to be made out of plasma  or dark matter. Even interpreted generously as meaning there are no synthetic chemicals, there is no real difference between synthetically made atoms and molecules and the same atoms and molecules being produced by nature.
Yet customers who are illiterate when it comes to physics and chemistry might still prefer such products 'without added chemicals' making it rational for profit seeking companies to produce such products.


Is there a journal for replications in experimental economics?

Experimental Economics states: "The journal will also consider
articles with a primary focus on methodology or replication of
controversial findings."
The Economics e-Journal takes replications from any field.

The Behavioral Economics Replication Project could be interesting for you. They presented their research at the 2016 American Economic Association Annual Meeting.


Declaring bankruptcy is a step taken by companies to get protection from creditors. (They cannot seize collateral unilaterally, etc.) The company can still operate, within the legal framework, and with the obvious limitation that nobody wants to be owed money by the firm. Although equity often ends up worthless after all creditors have their claims settled, this is not always the case. As such, equity shares may not be worthless.
The question of the value of the equity is distinct from being listed on an exchange. This question cannot be easily answered since each exchange has its own requirements to be listed.
For example, the NYSE criteria are here: link to standards.
Note the phrasing: “The NYSE reserves the right...” to delist under various criteria. As such, there are no hard rules on what must happen.
For each company of interest, one would need to go to the exchange(s) where the shares are traded to see the rules.


How's a levered hedge fund effectively selling a gigantic put option on its ability to finance its own positions?

This was a really good question, and a fascinating topic (one that was admittedly hard to wrap my head around). I found a great video on this topic here.
When people say you're selling a (cheap) put option, they generally imply you're taking a needlessly big risk for only marginal upside. For example: not having adequate cash in your portfolio is essentially selling a cheap put on your liquidity. You get the marginal gain of avoiding cash drag...but you get absolutely steam-rolled if you're hit by a margin call. Best case you enjoy extra 1% / year, worst case you get annihilated. Seems asymmetric, no? I think that's what they're implying.
So, in an ideal world, you take JUST the position of cash-synthetic basis mispricing. BUT, DE's point is you're taking that risk PLUS the risk of whether or not other leveraged hedge funds can finance their position. Ominously enough, you're likely part of that "leveraged hedge fund" group yourself. If a competitor loses their financing, they will have to liquidate their trade. If they liquidate, it makes the mispricing WORSE. As they liquidate (and your trade moves against you), you likely ALSO have to liquidate because your financing is also drying up. All of these things would happen at once, and would self-reinforce.
Put succinctly, you are taking two co-amplifying risks (financing options + basis mispricing) - but only really getting paid for one of them (basis mispricing).
Does that hopefully help?
Cash-Synthetic Mispricing
This is essentially when you can buy a bond yielding 3% / year, and then turn around & buy insurance on it for 2% / year. It is riskless arbitrage...and theoretically Capitalism should compete it away. The difference between my numbers above (the 2% and 3%) is the "Cash-Synthetic Basis" referenced in the DE Shaw article.
"Cash" is a misnomer that refers to the actual underlying bond. "Synthetic" means a CDS, or some sort of synthetic instrument.


It’s an indicator but I personally would never decide based on them.

$t$- statistics tests the null that individual coefficient $\beta_i$ is 0 against alternative that it is statistically different from 0.

$F$-test which is increasing function of $R^2$:

$$F=\frac{R^2/k}{(1-R^2)/(n-k-1)}$$

Which tests the hypothesis that all $\beta$ coefficients are 0 against alternative that some of them aren’t.

$F$-test is not affected by multicollinearity and it will be high if $R^2$ is high. Hence if all your $t$-tests say all variables are insignificant but $F$-test (based on high $R^2$) says some of them should be significant it’s an indicator for multicollinearity. However, it could be also due to other problems so it’s always best to do some direct test - calculate correlation between independent regressors or even better something like calculating variance inflation factor (VIF) or some more fancy model.

Generally the rule of thumb is to remove variable if it’s VIF is above 5 some sources say 10.


Efficient revealed preference check of whether two preferences over lotteries are the same?

Perhaps I'm way off base on this, but given the relatively strong requirements imposed in the question, I'm tempted to say that we can never truly be able to state with certainty that some individual's utility function $u_i$ is absolutely a linear transformation of some other function $u^{*}$ considered ex ante. While depending on the specific $u_i$, $u^{*}$ and initial questions asked, it could be possible to dismiss the two as not satisfying the transformation requirement, but I don't think one could ever actually confirm it.

The underlying reason for this thinking is the inclusion of vNM utilities and lotteries over outcomes. Since the former assumption imposes cardinality on our utility framework, there is always the possibility of discrepancies between $u_i$ and $u^{*}$. Even considering just one good (money, for example) any sequence of observed choices made by some individual following a utility function $u_i$ might only differ from how an individual following $u^{*}$ behaves in the "gaps." No matter how fine the probabilities and dollar amounts get, we can always come up with some alternative $u'$ which replicates the decisions up till that point. (We can always consider, for example, some $u^{'}$ that's a piecewise on some interval where $u^{*}$ is smooth.

Of course, as our ability to observe smaller and smaller intervals increases from repeated choice observational data, the possible deviations between $u^{*}$ and some alternative $u'$ trends towards zero. But again, finding out that all observations thusfar suggest the individual's utility $u_i$ is a linear transformation of $u^{*}$ doesn't guarantee that there isn't some interval of the function where they're different.

It should also be pointed out that this really only considers extreme cases- for all experimental purposes we only need a function that approximates an individual's utility function. But I do think it's tougher to truly prove one is a linear transformation of the particular $u^{*}$ in mind.


Directed technical change, Acemoglu (2002) framework - trying to understand how an equation is derived

The price $χ_L(j)$ is 1!

You only need to integrate a constant. That's why you have that equation.

(See p. 789, second paragraph, when he normalizes it to 1)


The Backward Bending Supply Curve, Asymmetric Information, and Monopoly

This definitely is not a complete answer. But I can imagine the case of increasing returns to scale. Or natural entry barrier. Increasing returns to scale is often discussed in international trade theory, but it can be also relevant here.

Example: think about commercial aircraft industry, dominated by Boeing and Airbus. Aircraft industry is seen as exhibiting increasing returns to scale, so monopolistic power those companies have (well, it is oligopoly, but that does not change the conclusion) may actually lead to cheaper quality aircraft.

Now there are people who really believe that all monopolistic powers are undesirable, but I believe those people are only minorities.


Quoting convention for Gold in the early 1900s

Have you considered googling the subject?
By doing so I have found:
https://en.wikipedia.org/wiki/%C2%A3sd

  £sd (occasionally written Lsd) is the popular name for the pre-decimal currencies once common throughout Europe, especially in the British Isles and hence in several countries of the British Empire and subsequently the Commonwealth. [...] In the classical Roman Empire, standard coinage was established to facilitate business transactions. 12 denarii were rated equal to 1 gold solidus – a 4th-century Roman coin that was rare but which still circulated; and, since 240 denarii were cut from one Roman libra of silver 240 denarii, therefore, equalled one pound (livre in France, peso in Spain, etc.). [...] Different monetary systems based on units in ratio 20:1 & 12:1 (L:S & S:D) were widely used in Europe in medieval times. The English name pound is a Germanic adaptation of the Latin phrase libra pondo 'a pound weight'. [...] In writing, there were several conventions for representing amounts of money in pounds, shillings and pence: £2.3s.6d. (two pounds, three shillings and sixpence)


In terms of attribution. Recall that Net Domestic Product is the total cost of goods produced minus capital depreciation. In terms of why farmers income is lower, than the NDP, you must subtract out all the costs the farmers have to pay to create that good. So fertalizer cost, reinvestment, wages, interest costs, supplies, insurance etc. etc. all need to be subtracted. Finally, the remainder is split between farmers (in terms of income) and shareholder of corporations that operate in agriculture who recieve a return on thier investment (retained earnings for corporation or dividend payments).

Finally, your farmer income dataset is not an accurate description of the rewards of thier labours. Many farmers are small business owner so, in addition to thier income, they are building equity by making the land more productive and creating a business.

In other words, you are almost looking at the gross margin of a farming business. The NDP is the Revenues (minus capital costs) to get to the farmers net income you have a a substnatial number of other costs to subtract (taxes, amortization, operating expense, interest expense etc.). Finally, the farmers net income is not nescairly the same as the FARM/ FARM BUSINESS' net income so the actual farmers income will be lower still.


Do governments really need to tax us, when they can create money at will?

It is true that government can print as much money as it wants but value of money is not constant. For example, under 101 econ quantity theory of money the price level is given by:
$$P = \frac{MV}{Y}$$
Where $P$ is the price level (with change in price level giving you inflation), $M$ money supply, $V$ velocity of money and $Y$ real output. Assuming $V$ constant government can increase money supply only by the growth rate of economy without incurring inflation. In more complex IS-LM AS-AD model the story is a bit more complex but the main message that government cant expand money supply freely without risking inflation holds (the equation above is actually an simplified equilibrium condition at money market in IS-LM).
Most modern governments run spending at a level of somewhere between $25\%-60\%$ GDP (see the OECD data) while real output grows at around $2\%$. There is no way that even the modest $25\%$ spending could be covered without government running two digits inflation in the best case scenario ignoring other issues this would create.

In fiat money system collecting taxes is what actually gives money value in the first place. See this thread that discuses this issue in greater detail (especially Holger's and Brian's +1 answers). If government will stop collecting taxes there is no guarantee that in the long term its currency will still be acceptable, especially since given the point 1 the currency would be rapidly loosing its value.

"Can't the government Quantitative Tightening?":
It can use it reduce the problem but it would not be any solution. Raising interest rates helps contract money supply itself but at the cost of  depressing economy and increasing unemployment. For example, prior to Volker disinflation the US inflation during late 70's and early 80's was even in the worst times below $12\%$ and to bring it down to low inflation it took interest rates that were almost as high as double the inflation rate with the funds rate reaching more than $19\%$ (see Goodfriend & King 2005).
Moreover, some point rising interest rates higher and higher would not help fighting inflation just due to the fact that $Y$, which varies negatively with interest rate $i$ (that is $\partial Y/ \partial i<0$ see for example Blanchard et al Macroeconomics an European perspective) would fall more than $M$. Furthermore, in the end raising interest rate can only reduce the broad money supply not actually the base money i.e. the actually printed coins and notes in circulation so if government will insist on continuing to print money to cover all its expenses raising interest rates would just slow the inevitable.

The view of those economists are not necessarily contradictory. One way how government could avoid some disastrous inflation is to slash its spending dramatically, borrow at rate that keeps debt to GDP stable due to growth and try to live of the seigniorage from some reasonable slow paced money supply expansion (assuming they would be still in demand due to issue covered in 2). According to Fed Dallas the US seigniorage revenue accounts for less than $2\%$ of real output so with some balanced borrowing my guesstimate is that US government could run spending on level $4-5\%$ of GDP in this scenario. That would be barely enough to cover US defense spending itself and not much else.  The second economist just assumes that government spending would not change and hence stresses the effect this would have on the value of money itself. There is no contradiction there both are valid options that government could pursue.


How would one take advantage of this arbitrage opportunity?

Hi: Don't think of a specific case. Just think of investment A and investment B and assume that the return on both investments is the same. Also, assume that investment A costs more to invest in than investment B. In that case, you would sell some amount, X, of investment A to Mr. Pushover and buy the same amount, X, of  investment B with the proceeds from Mr. Pushover ( and you'll have a little left over since A costs more than B. Call  what's left, L, for leftover ).

Then, when the investment horizon is complete, you pay Mr. Pushover the return with the return that you made from investing in B.

But,  now you still have L left so you made L with no risk. That's the arbitrage. I hope that helps. To summarize, if two investments return the same thing, they have to cost the same thing or else there's an arbitrage.


I wouldn't necessarily agree that inflation is universally preferable to deflation. Each has positive and negative consequences. Inflation hurts savers, deflation hurts current consumers. A little inflation (let's say, 2% per year) is useful -- if nothing else, it helps deal with wage stickiness. Deflation, on the other hand, would be a godsend for a country like Venezuela where prices have been growing incessantly.

This is all to say that there isn't a single rule for inflation vs deflation. The broader economic context is critical.


White standard errors correct for heteroscedasticity only, whereas Newey West standard errors are robust to both Heteorscedasticity and Autocorrelation (HAC) .


Consumption Externalities in partial equilibrium analysis

Without any constraint the social planer would set $x_B \rightarrow \infty$, and not solve the "first order condition" given by $x_B+1/x_B=0$ only valid for an inner solution. The centralized problem is not comparable with the budget constrained decentralized problems you solved. (By the way why should $m$ be the same for both consumers?)


Values of A,K,N,a in Cobb-Douglas function expressing GDP

K is just a stock of capital. This Fed data show that the stock of US capital is almost 70m at constant prices for the most recent year (2019). But there are other ways of measuring it. You can google papers that estimate multifactor productivity to see what measures for K and N people are using (not everyone uses labor hours too).
A is multifactor productivity. This OECD data show that using 2015 as index year the multifactor productivity in the USA in 2019 was 102. A cannot be measured directly. People estimate it using K and L and using time series/panel regressions. That is why this will be in datasets expressed just as a growth rate or index, we can't see what A is only how it changes.


Maximum-support Nash equilibria in zero-sum games

I am not sure about what exactly you mean by the maximum support NE but you may want to check the concept of Correlated Equilibrium. Any convex combination of NE payoff profiles is a Correlated Equilibrium (but not vice versa in general). So, what you wrote made me think about that. However, I don't think it makes much sense to use correlated equilibrium since, you know, there is no obvious advantage of correlation. Also, the NE payoff profile must be constant in your environment.

Another thing that comes to mind is restricting the attention to super-modular games. Then, you get a nice lattice of NE. So, I don't know if this makes sense either.

I would suggest playing around with some examples to understand what exactly it is that you want to characterize and asking a new one based on your examples.


Belo (2010) uses a production function
$$
Y_{j,t} = \epsilon_{j,t} F^j(K_{j,t})
$$
where a competitive producer $j$ is allowed to choose the state-contingent productivity level $\epsilon_{j,t}$, subject to the constraint set
defined by the following analytically tractable CES aggregator:
$$
E_t\left[ \left( \frac{\epsilon_{j,t+1}}{\Theta_{j, t+1}}\right)^\alpha \right]^{1/\alpha} \leq 1.
$$
In Cochrane's  2016 essay, "Macro-Finance", Cochrane mentions this specification as a potentially important ingredient for understanding the interaction of macroeconomics and financial markets:

  The second ingredient, I think, is to enrich the production technology
  so that consumer investors can shape the riskiness of the
  technological opportunities they face. Belo (2010) and Jermann (2013)
  have recently explored specifications of technology that allow such
  choices.


Comparing GDP contractions between Russia and US

In Rogoff's article (at least current version) the first sentence you quote links to IMF's World Economic Outlook Database.

My guess is that when he is referring to output he means Gross domestic product, constant prices. Looking up the numbers in the database we have 63038.4 billions of 2011 rubles in 2014 and 61097.5 in 2016 which translates to 3.1% decline (negative growth).

If I use per capita numbers than that also gives 3.1% decline which I guess is "comparable" to 4% decline.


Examples of non-differentiable problems in economics

Here are four I could think of:

Leontief and Lexicographic functions, used in preferences or
production functions are non-differentiable.
Labor models often employ a discrete labor supply (work or don't
work, sometimes alongside the decision of how much or how hard to
work).
Housing models often employ a non-convex adjustment cost to ensure
that there is a discontinuity in the cost function for the choice of
housing.
Kinked objective functions, where at one or more points the
left-hand and right-hand limits  differ, are also common in
economics (e.g. optimal option exercise at maturity)


How did Brexit wipe $2 trillion off world markets?

Think about it this way:

You own 100 houses on a private island in Italy. These houses could be sold for $1 million each.

All of a sudden, the Italian prime minister outlaws all boats. There is now no way to get onto the island apart from swimming.

Since you can no longer get onto the island by boat, and have to swim, no one wants to live in your houses, and thus no one wants to buy them for \$1 million. Your houses are still quite nice, though, so people instead will offer to buy them for $200,000.

Your houses were originally worth \$100 million in total (that's how much you would have got if you sold them) but are now only worth $20 million. You have 'lost' \$80 million.

In Brexit, instead of houses, we have stocks. These stocks were originally worth a certain amount of money. When Brexit occurred, people no longer wanted to buy or own British things, because they thought that they would be worth less. The total value of British things dropped by $2 trillion. It's important to note that the things are still there (the stocks didn't disappear!) but are simply valued less to investors.


The reference is World Bank (2006). You can see the detail of the methodology in Appendix 1, and their estimates of intangible wealth in Appendix 2.

There are other methodologies to measure intangible wealth (e.g. here and here). Alternatively, there are many ways to compute natural and physical capital, which will give different results to the one given by the WB, even if using the same formula to compute intangible capital. For example, see here. If you search in google or here things like "measuring capital stock", you get tons of hits too.

I think that because this is a relatively new research area, there is no "official" guideline about how to measure intangible wealth. Maybe in the future some sort of standard will be produces, pretty much like the standard for GDP came to be (which is still subject to criticism anyway).


I am the last person that should be answering continuous time questions like these, but if there's no one else I guess I'll give it a shot. (Any correction of my dimly remembered continuous-time finance is very welcome.)

My impression has always been that this is best interpreted as a consequence of the martingale representation theorem. First, though, I'll loosely establish some notation. Let the probability space be generated by the $n$ independent Wiener processes $(Z_t^1,\ldots,Z_t^n)$. Let there be $n+1$ assets, where the value of the $i$th asset at $t$ is given by $S_t^i$. Assume that asset $i=0$ is a riskfree bond $dS_t^0=r_tS_t^0dt$,  while assets $i=1,\ldots,n$ are each risky and are driven by the corresponding $Z_t^i$:
$$dS_t^i=\mu_t^idt+\sigma_t^idZ_t^i$$
Assume there is a strictly positive SDF process $m_t$ normalized to $m_0=1$, such that $m_tS_t^i$ is a martingale for each $i$ (basically the definition of SDF) and where
$$dm_t=\nu_t dt+\psi_t\cdot dZ_t$$
(I use $\cdot$ as the dot product, which will be convenient.)

Finally, let the $n+1$-dimensional vector $\theta_t$ be our portfolio at time $t$, such that net worth $A_t$ is given by $A_t=\theta_t\cdot S_t$. Assume that $A_0$ is fixed and that further we have
$$dA_t=\theta_t\cdot dS_t$$
Now I'll state the objective, which captures the essence of complete markets. Suppose that the world ends at time $T$, and that we want net worth $A_T$ to equal a certain stochastic $Y$, which can depend on the full history up until time $T$. Suppose that $A_0=E_0[m_TY]$, so that in a world with complete markets we could (at $t=0$) use our initial wealth $A_0$ to purchase the time $t=T$ payout $Y$. In the absence of these direct complete markets, the question is whether there is nevertheless some strategy for the portfolio $\theta_t$ that will allow us to obtain $A_T=Y$ in all states of the world. And the answer, in this setting, is yes.

First, one can calculate $d(m_tA_t)=\theta_t\cdot d(m_tS_t)$. Thus $m_tS_t$ being a martingale implies that $m_tA_t$ is a martingale. Thus we have $A_T=Y\Longleftrightarrow m_TA_T=m_TY$ iff
$$m_tA_t=E_t[m_TY]$$
for all $t\in[0,T]$. Note that this is true for $t=0$ by assumption; hence to get equality it is only necessary to prove that the increments are always equal on both sides.

Now the martingale representation theorem comes in. Since $E_t[m_TY]$ is a martingale, we can write
$$E_t[m_TY]=E_0[m_TY]+\int_0^t \phi_s\cdot dZ_s$$
for some predictable process $\phi_s$. So we need to be able to show $d(m_tA_t)=\phi_t\cdot dZ_t$. Writing
$$d(m_tA_t)=\sum_i (m_t\theta_t^i \sigma_t^i +A_t\psi_t^i)dZ_t^i$$
we see that we need $m_t\theta_t^i\sigma_t^i +A_t\psi_t^i=\phi_t^i$ for each risky asset $i=1,\ldots,n$, which we can invert to give the needed portfolio choice $\theta_t^i$:
$$\theta_t^i=\frac{\phi_t^i-A_t\psi_t^i}{m_t\sigma_t^i}$$
The riskless asset portfolio choice $\theta_t^0$ can then be backed out from $A_t=\theta_t\cdot S_t$.

The intuition here is simple: we need to always have $A_t$ adjust to maintain the equality $m_tA_t=E_t[m_TY]$, but both the expectation on the right and the SDF $m_t$ on the left are moving in response to the driving processes $dZ_t^i$. Hence we need to pick a portfolio $\theta_t$ such that $dA_t$ precisely offsets these movements and the equation continues to hold. And we can always do this as long as locally, our assets span all the risks $dZ_t^i$ -- which can happen more generally, even for $n$ correlated assets as long as their increments are locally linearly independent. (The case here of $n$ risky assets each drien by an independent Brownian motion is a special one.)


If I gain, then someone else loses. Correct?

I completely agree with denesp's answer, however I think you can make it even simpler.

  On a very small scale, it's certainly true that if I gain, somebody
  else might lose. If I take away my brother's chocolate, then he will
  lose it, and will most probably not get anything comparable.

OK, let's say I prefer chocolate to wine gums and my brother likes wine gums better than chocolate. Then taking his chocolate away and giving him my wine gums is good for both of us, so we both win and no one loses. So the answer is no.

You could even consider the extreme case in which your brother hates chocolate and you are doing him a favor by taking it. (Works not as well with chocolate, but you might think of recycling.)

In general these "trades" are called Pareto improvements.

But this is only one example, if you are interested in the subject, you might be interested in one of the following basic economic ideas:

Trade between two countries in which one of them is more efficient than the other: Ricardo's comparative advantage example
Your brother likes to give: altruism / warm glow (I really don't like the wiki page here, but was not able to find a decent non-scientific explanation of it.)
Or maybe other other-regarding preferences, for example fairness (your brother has a lot of chocolate and feels better if he gives you some): Theories of Fairness and Reciprocity (On page 3 is a brief "Non-technical summary" which might be interesting.)

As you can see there are many examples for a "win-win" situation and there are many many others, depending on the situation.


What Taylor approximation is used in this equation?

Even though your question does not allow a definite answer, I am pretty sure the author used a Taylor expansion around the logarithm of both sides of the equation. This process is called log-linearization, and is fairly common.

We can approximate (logs of) growth rates as $x \approx log(1+x)$ when $x$ is small. (and by the rules of logarithm:  $\log((1+x)^\sigma ) \approx\sigma x$

Rewriting your equation
$1+(r_t-\delta)=(1+\eta)^{\sigma}(1+\theta)$

where
$r_t=\alpha (k^{\# *})^{\alpha -1}$

and applying this rule gives

$r_t-\delta=\theta+\sigma \eta$

which gives the desired result.


NAICS seems to be what you're getting at (http://www.census.gov/eos/www/naics/). This is the industry classification mechanism in North America. If you are interested in groups of workers' tasks as opposed to groups of establishments, you might be interested in occupation, which is the Standard Occupation Classification (SOC) system (http://www.bls.gov/soc/).


ICAPM Factors

People have chosen different ways to pick factors. Chen, Roll and Ross are a classic example of attempts to find reasonable ICAPM factors. Fama-French factors are often explained as correlated with underlying ICAPM factors. Other researchers have chosen to look for factors without assuming outwardly observable exposures by analyzing returns using factor analysis or principal components.

Explanations of Fama-French

I don't know if there are "commonly accepted" explanations for the Fama French factors. This is not to say that there aren't explanations, there are a lot of them. Just no one seems to agree on which ones are "best". Candidates include both consumption based explanations AND production based models, each of which relates Fama-French factors to underlying economic variables. Cochrane has a nice summary of the performance of both in pricing Fama-French portfolios here.

Additional factors commonly used

Fama and French are now talking about a five-factor asset pricing model which also includes profitability and investment, similar to Novy-Marx's paper and work by Chen, Novy-Marx and Zhang which more directly relates to production asset pricing. Momentum is also a consistent candidate, although it's a real pain to put any kind of explanation to. There are tons of others. This paper lists many of them, as well as calling the significance of some of them into question.


Why does China spend so much on agricultural/farming subsidies?

The first thing that comes to mind is food security. China has a large population to feed and does not want to depend on foreign nations for its food.

A (speculative)  second reason may be prevention of domestic unrest. The largest share of China's population still lives in rural areas, and by supporting farmers they may silence protests.

A recent article analysing the drivers of China's agricultural policies by Lopez et al., including the fairly recent shift from taxes to subsidies can be found here:
https://www.sciencedirect.com/science/article/pii/S0305750X1630211X


Back to economy basic 101:
Inflation is due to excessive cash chasing limited goods.

So to relieve monetary inflation, an institution can only do either or all of this:

Slow down money printing
Find ways to produces goods/needs/services that absorb the money
Enact policies to make people stash the excess cash, e.g. high-interest long term bonds, tax incentive on saving.

Other claims just typical media noise.

So "Is falling inflation in India one of the reasons for the Nifty crossing the 10,000 points mark?"

The answer is No.

No government can uplift market speculation by reducing money. The good news is, India stock market cross the 10k mark is because more hot money(inflated money) is entering the stock market.  When excessive cash is snapped up in the stock market, there is little room for the cash to "chase" for the factors that cause goods price hikes, e.g. rents, raw material.

Thus, in a country with poor institutional control, stock rallies are extremely dangerous. Once the speculator cash out their profits from stock market, they may switch attention to other more profitable speculation, e.g. real estate.

Unless the country production sustains continuous growth (rule no 2) to absorb the overflow cash, otherwise you will see a repetition of 1997 Asia financial crisis.

Perhaps OP should ask, can Nifty dodge the fake of (East) Asia financial crisis and take the path of China (which thousands billions export surplus offset the inflation).

(update)
As in economics study, one must be very careful to distinguish between noise and facts. IMHO, all market analytical tools are noise, not facts.  That's why real investor like Warren Buffet simply ignore "market analytical tools" and went straight to costly and time-consuming facts digging.

To learn the bubble is outside the scope of the question. But just a hint, IMHO, index stocks are the best anchor to check against bubbles.


How can the VNM completeness axiom be derived from the transitivity and continuity axioms?

I think I just found a counterexample to your teacher's claim.

Suppose completeness is violated completely (sorry for the pun): $L\not\succsim L'$ and $L'\not\succsim L$ for any $L,L'\in\mathcal L$, i.e. no two lotteries are comparable.

Then I claim that transitivity and continuity are both satisfied vacuously.

Transitivity: $L\succsim L'$ and $L'\succsim L''$ imply $L\succsim L''$ for any $L,L',L''\in\mathcal L$. But since no two lotteries are comparable by $\succsim$, the antecedent is always false, and so transitivity is always satisfied.

Continuity: for any $L,L',L''\in\mathcal L$, the sets $\{\alpha\in[0,1]:\alpha L+(1-\alpha)L'\succsim L''\}$ and $\{\alpha\in[0,1]:L''\succsim\alpha L+(1-\alpha)L'\}$ are both closed. Again, since no two lotteries are comparable, both sets are empty and therefore closed. Hence, continuity is satisfied in a vacuous sense.


An overview of 4 books for an undergrad course in Mathematical Economics

All four of those books are widely used at the undergraduate level. They are written with the intention to be text-books of a class, so they should be good for self-study. I  recommend you to read 1 and 2 before reading 3 and 4.

Will give you a basic overview of the way we think in economics, clarify some basic concepts and explain work-horse models in the discipline like supply and demand, which can be used in a lot more domains than expected. Several similar books are also very good, a common one is Mankiw's book Principles of Economics.

In economics, we use a lot of math, but  disproportionally certain areas of math more than others. A course in Mathematical Economics will focus on these tools we use. For example, it will focus a lot on optimization, linear algebra, calculus, analysis, differential equations, etc. Optimization and "comparative statistics" is probably the most important thing in economics, so a possible approach would be for you to learn up to there from Chiang, and then go to the other books and come back to Chiang if and whenever you need it. There are a lot of good similar books,  for example, "Mathematics for Economists" by Simon and Blume.

I am not familiar with this book, but Macro has tons of good textbooks. It will not be  math-intensive actually, so you will not need all of the tools learned in Chiang to read this one. A good substitute for this one would be Barro's Macroeconomics.

This is a classic textbook for intermediate Micro. Almost everyone I know learned from it, and it is definitely advanced for an undergrad, but as friendly as possible. It is hard to find a good substitute for this one.

If you have access to a library and find a topic hard to understand, I would look at how other authors approach it. This community will also be a useful resource if you use it properly.


Opportunity profits vs. opportunity costs

Opportunity cost is simply the value not obtained of the highest value alternative. It can be positive or negative; meaning it doesn't really make sense to define the opposite as opportunity profit. There is only two ways to go about it rationally, either you are profiting from doing something, or you are not profiting by not doing something, which is defined as opportunity cost. Opportunity profit would just equate to opportunity cost because you are not doing whatever the opportunity is. (Also, I'm not the down voter).


Which market category does this sustainable product belong to?

I'm from Canada and I work on policies supporting "clean technology" development, but its a very nascent sector from a statistical standpoint.  Statistics Canada only recently formalized a (very imperfect) definition, and our federal export development branch politely puts your problem this way: "For most Canadian cleantech exporters, the ASEAN markets are remote and not well known. Companies who feel that these countries may provide export opportunities should consult with Export Development Canada and the Canadian Trade Commissioner Service."

If you want to do some tough research, I would use search terms like "clean technology", "environmental goods and services", "water management", "land reclamation", etc. But in short, you're coming up dry because there's very little out there.


An indifference curve can touch the axes, take for example the case of perfect substitutes, if the consumer his utility is given by
 U=x1+x2
 then obviously he is indifferent between bundle (10,0) and (0,10). So not touching the axes is not a characteristic of indifference curves as such, even though in the case of Cobb-Douglas functions it holds.

As for your other concern, whether an indifference curve can take as input only one good, with the other input being money: this is perfectly possible, it is even a more general case than the indifference curve for two goods. In the case of the inputs x1= a good and m=money held by the consumer we can formalize the utility maximization problem in the following way:
max U=u(x1,m)
Subject to the constraint: b=p1x1+m
As you see the standard form can be used with just one change: the price of a unit money held is of course 1.

And now the more general case, again let x1 be a good the consumtion of which we are interested in and now let y be the MONEY SPENT on all other goods. Now again we can use the modification of the budget constraint above. That is: price of y is 1.
In short: anything can be an input in an utility function, just think about your own preferences: you dont just enjoy goods, you enjoy holding money and giving to charity etc. but also spending time with friends/family, with some modifications all this can be put in the form of a utility maximization problem and thus an indifference curve can be constructed for any two inputs, holding other inputs fixed.